FUNDAMENTALS - 1

BASIC  PHYSICS  THEORY

This is a simplified discussion of the basic Physics which underlies our theory of Electricity and Magnetism.

LESSON 1

1.1   THE ELECTRIC CURRENT

     Classical theory holds that Matter can be divided into smaller and yet smaller particles until a point is reached where further division causes a change in the properties of that matter. This smallest unchanged particle is called a Molecule.   Most molecules can be sub-divided but often the resulting Matter has properties which are unlike those of the parent molecules;  these smaller particles of different form are known as Atoms.  Any attempt to sub-divide atoms results in their destruction and so atoms represent Nature's ultimate building-blocks. Atoms are known in more than 90 different kinds and, because they are the fundamental building-blocks, they are known as elemental atoms and the different kinds of Matter which they form are called Elements.  Matter in which the molecules are formed by bonding together different atoms is referred to as a Compound. (A mixture differs in that it contains different kinds of molecules which are not bonded together and so they can be separated from the mixture as Elements.)

      Elemental atoms bond with each other to give us the pure substances such as Copper, Oxygen and Iron (not Steel which is a compound of Iron). Compounds of different atoms are represented by such things as Water (Hydrogen and Oxygen), Salt (Sodium and Chlorine) and ultimately Life itself.

  ** Research into the nature of atoms gave rise to puzzling phenomena which, under mathematical analysis, has led to the conclusion that the atoms are themselves constructed from a large variety of very-small particles. For our purpose here it is necessary to consider only the first stage of that analysis which says that an atom contains a relatively-heavy Nucleus around which is gathered a number of extremely-small Electrons. The different Elemental Atoms owe their differing characteristics to the differing numbers of electrons which they possess.

  ** It is found by experiment that, when electrons are placed together, they actively repel each other; similarly when two nuclei are placed together they too actively repel each other.  However, when a nucleus and an electron come together, they exert a powerful mutual attraction.

  ** This behaviour is explained in terms of two differing Electric Charges which, by tradition, are labeled as positive (+) and negative (-). Also by tradition the positive charges are attributed to nuclei and the negative charges to electrons. The argument then put forward is that two like charges (either two nuclei or two electrons) mutually repel while two unlike charges (a nucleus and an electron) mutually attract.

  ** A complete atom does not show a tendency either to attract or to repel other atoms and so it must be concluded that overall they have zero electric charge;  this must indicate that there are an equal number of positive and negative charges. To cope with this the heavy nucleus is sub-divided into an appropriate number of Protons each of which carries a unit positive charge; there is one proton to match each electron in every atom.

      Some electrons, held only weekly by their parent atom, can be dislodged;  atoms which lose electrons in this manner are left with positively-charged holes which then exert a mutually-attracting force with any passing electron and so tend to re-fill themselves.  However, if the disturbing force is more powerful than the recovery force of the atom, the dislodged electrons can drift away and form a moving stream of  "free" electrons which is believed to constitute what we know as an Electric Current.

1.2   THE WATER ANALOGY

       Just before 1800 interest began in a new Science called Galvanism following the experiments of an Italian scientist named Galvani.  In an effort to make sense of the phenomena they were uncovering the learned men of the day drew an analogy between the electrical effects they were discovering and the water systems which they used and understood.

      They knew that a water pump, by raising a pressure of water, could drive a current of water through a system of pipes.  Similarly they imagined that their generators of electricity behaved as an electricity pump, that they raised an electrical pressure which then drove a current of electricity through a system of electrically-conducting paths.

      To a large extent that analogy is still accepted today. As described above a generator of electricity is considered to "pump" electrons continuously from a point of low-potential to a point of high-potential (it creates an electrical pressure);  the point from which negatively-charged electrons are removed is left with a residual positive charge while the point where the electrons are dumped acquires a surplus of negative charge.

      A circuit Designer provides a suitable conducting path external to the generator through which the surplus electrons, piled-up on the negative terminal stream back to the positive terminal where they are gathered up and pumped again and again to the negative terminal This is why there must always be a complete circuit before electricity will flow.

      This analogy with a water system is useful in explaining the ideas behind electrical theory and it is used often in the following text.

1.3   ELECTRIC SOURCES

      In practice it is not that easy to set electrons drifting and, as a general principle, they have to be pulled from one direction with a positive charge and pushed from the opposite direction with a negative charge. The prime-mover in such an operation is known as a Generator of which the two major types are the Chemical Cell and the Mechanical Generator.

1.3.1  The Chemical Cell

  ** There are many designs of Cell and they have many intended purposes but their general requirements are a mixture of chemicals in which are immersed two plates called Electrodes. These are usually made from different materials; see Fig.1. The different materials of the electrodes react differently with the chemical mix and this results in electrons being stripped from one electrode with an equal number deposited on the other.

  ** The basic construction of electric cells is shown in Fig. 1. The electrode which loses electrons :is left with a non-neutralised positive charge while the other, with too many electrons, acquires a negative charge.

      Experimenters quickly discover also that it is necessary to mechanically-restrain the electrodes because the unlike charges which they carry set up a mutually-attractive force which pulls the electrodes together.

**  As already discussed the holes on a positively-charged electrode will attract electrons while the surplus electrons on a negatively-charged electrode will repel electrons;  it follows therefore that, as the electrodes acquire their charges, so those charges try to force electrons back across the cell.   **  Ultimately a state of equilibrium is reached where the chemical action of the Cell is exactly balanced by the reverse action of the charged electrodes and the electro-chemical activity of the Cell is halted.   **  Between the two electrodes there now exists a distinct difference in their state of charge and this is referred to as a potential difference but is usually referred to simply as a p.d.  The essential character of a p.d. is that it is called into being by a transfer of electrons (a transfer of charge);  a p.d. can be generated across other electrical components where it can be used as a "secondary generator" to start another transfer of charge - another electric current.   ** A potential difference is an electric force which is equivalent to pressure in the water analogy and it is measured in units named volts in honour of the Italian scientist Volta.       An electric current, which flows under the influence of electric pressure (volts), is measured in terms of the number of electrons which drift past a given point in a period of one second; of course we cannot count them and current has to be assessed through the various effects which it produces. (The standard torch battery delivers approx. 1.2 to 1.5 volts:  this experimental cell delivers 0.6 volts approx.)

  ** The number of electrons involved is measured in coulombs (after Coulomb) which therefore are units of quantity (charge or no. of electrons); the flow-rate in coulombs-per-second (no.of electrons per second) indicates the strength of the current and this rate is expressed as amps. Properly the unit of current-flow is the ampère named after the French scientist Ampère but it is invariably shortened to the more convenient amp.

1.3.2    Work and Power

      To deliver water from a reservoir positioned at some height above the delivery point requires a pipe;  at the lower end of that pipe there is a pressure caused by the weight of the water;  this is measured in terms of the height of the water-surface above the delivery point and is known as the Head of Water.

      The pipe allows water to be lowered from the reservoir or, given a pump that can more than oppose the pressure, it allows water to be raised to the reservoir. We have all moved objects either horizontally over the ground or vertically and we complain that we have had to work; fuel/food consumed in doing that work manifests itself as heat and we have to take steps to cool down again or suffer heat-stroke. The faster the rate at which we work so the hotter we get.

      In such a mechanical world work is measured by multiplying the weight that is moved by the distance through which it is moved and the result can be expressed in units such as foot-pounds or centimetre-grams.

      In the water system referred to above there is a choice in that a small-diameter pipe can be used to deliver a given quantity of water over a long period or a larger diameter pipe can be used to deliver the same amount in an instant deluge. Should you be sprayed by the smaller pipe you may get wet but otherwise would be unharmed;  a similar acquaintance with the larger pipe however could have serious consequences. The ability of the large pipe to cause damage is described by saying that it delivers a powerful jet of water.

** Because the larger pipe works (delivers water) at the greater rate we say that it is a system of greater Power.   Power expresses the rate at which work is done and is measured here by multiplying the pressure by the amount of water delivered per second. (Note that a river delivers (at low pressure) - and therefore in innocuous fashion - much more water-per-second than a dangerous water hose.)

  ** To transfer this concept into the electrical world:

The amount of water delivered becomes the amount (the number)of electrons moved (or the amount of charge which has been transferred);   the amount is  measured in  coulombs.
The rate at which the water is delivered becomes the rate at which electrons drift and is measured in amps.
The head of water (the height through which the water falls) becomes the electrical pressure or p.d. and is measured in volts.
The power of the water jet becomes the electrical power dissipated as heat and is measured as before by multiplying pressure (volts) by rate (amps).

  ** The unit of power in an electrical circuit (volts x amps) is the watt named in honour of the Engineer James Watt.   Power appears as heat in the component through which the current flows and across which the voltage is developed and, of course, it causes a rise in the temperature of that component. This temperature rise continues until the rate at which heat is lost to the environment equals the rate at which heat is being generated. The heating effect is inescapable and so components must be designed so that they dissipate enough heat to keep the temperature within designed limits;  care must be exercised in placing hot components.

      When, as in Fig.2, an electrical path is connected between the terminals of a cell it is to be expected that an electric current will flow as electrons are propelled from the overcrowded negative terminal to the positive terminal.  The process deserves a closer look however because it conceals a very important principle.

      As soon as the external connection is made the electrons begin to slide, snakes-and-ladders fashion, down the potential gradient from the negative to the positive terminal.  In doing so they upset the equilibrium that had been established between the charged electrodes and the chemical mix and so the chemical generation of electric-charge re-commences. This causes the escaped electrons to be gathered up again and replaced on the negative terminal to go around the circuit again and again;  the cell is seen to be acting as a pump as envisaged when drawing the water analogy.

  ** The important action, which applies to every generator, is the destruction of the equilibrium;  while that remains intact the chemical action is halted and the call cannot maintain a current flow.  It is a fact of Life that, when any generator is called on to supply current, its terminal-voltage falls.  The greater the demand for current so the greater must be that fall;  the greater must be the failure to achieve that equilibrium.

      This natural phenomenon is an annoying inconvenience for the designer-engineer who, to calculate the current which will flow in the circuit he designs, needs to know the terminal voltage;  the problem is that he cannot know the terminal voltage until the value of the current is known!  There is a simple and elegant solution to this problem which involves thinking in terms of "perfect" theoretical generators which could not possibly have a practical existence;  the matter is dealt with under 1.15: Internal Resistance.

Fig.2 Cell with an External Path

** From the above description it is clear that current which is supplied by a chemical cell flows always in the same direction namely from the negative to the positive terminal. Confusion arises because, in early theorising, it was decided arbitrarily that current would be taken as flowing from positive to negative.  Enshrined in many of the classical "Laws", and still to be found in old textbooks, the convention survives and so it is important, where confusion can arise, to state which convention is being used. It is customary to refer to old-fashioned pos-to-neg flow as classical current and to the now-accepted neg-to-pos flow either as electron-flow or as electron current.       Current does not always flow in this one-way fashion and so, as described here, it is called direct current or simply dc.   Reference is often made to a "dc voltage";   this seldom causes confusion but it is best avoided by using the correct term direct voltage.   The word battery is commonly used wrongly also;  it is a contraction of "battery of cells" and implies a package which contains more than one cell.   A 1.4-volt torch battery should be referred to as a cell;   a (nominal) 9-volt battery contains six such cells.

1.3.4   Majority-carriers & Minority-carriers

      The discussion above proposes that an electric current is in fact a flow of "free" electrons.  When dealing with transistors and other semiconductor devices there is a great deal of talk about n-type and p-type semiconductors and about majority and minority carriers. The meaning of these terms is not particularly important for the RAE but, for those who wish to know, the following explains the basic theory.

      The basis of most semiconductor devices today is the Element Silicon.  However pure Silicon does not conduct an electric current and it has to be contaminated or doped.  Some doping materials increase the number of free electrons (carriers) which are available to create an electric current and the result is known as semiconducting material with majority carriers ;  others cause a decrease in the number of free electrons and these form semiconducting material with minority carriers.

      But how can a flow of electrons take place in a material which is deficient in free electrons ?  Imagine a pea-shooter which is filled with peas ;   if an extra pea is pushed into the left-hand end of the tube a pea is expelled from the other end as all the peas move along the tube from left to right.  This is likened to the action in a majority carrier.

      Now consider the same pea-shooter filled with peas but, instead of pushing-in one extra pea, a pea is sucked-out from the right-hand end so leaving an empty space or "hole".   Under the force of suction the next pea in line jumps from position 2 into the end position once occupied by the extracted pea so causing an impression that the hole has jumped to the left from position 1 to position 2. The pea in position 3 jumps into the hole vacated by pea no.2 thus causing the hole to move apparently one more step to the left.   In this manner the peas all move to the right as before but the hole appears to move in the opposite direction until a space is created at the left-hand end of the tube for the injection of another pea. This is the mechanism suggested for current-flow by minority carriers.

The two types of material produce transistor devices which have different properties.

END OF LESSON 1

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QUESTIONS

1. The very smallest particle of Matter is the Atom.   How is this related to the Molecule?

2. What are the constituent particles of the Atom as far as Radio Theory is concerned?   How do we allocate electric charges to them?

3. Although atoms bond together chemically to form large quantities of either Elements or Compounds they do not show any electrical attraction (or repulsion) to each other.     How do we account for this electrical neutrality?

4. What does Standard Theory propose as an explanation of the flow of electric current?  What is thought to be the action of a Cell by which it causes current to flow around an electrically-conducting path?

5. In constructing electric cells how do we ensure that the two electrodes offer different potentials?

6. Unless it is provided with a complete electrically-conducting path between the electrodes a cell ceases to produce current;  what sets-up the equilibrium that makes current-production impossible?

7. What are the Units that are used to express the magnitude of::

            (b)   electrical pressure or p.d.

            (c) power that is dissipated when current is driven through an electrically-conducting path?

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LESSON  2

1.3.5 Mechanical Generators

  **  These generators too come in many different forms and serve a variety of purposes but they have a common principle of operation.  Mostly they are rotary machines which combine a rotary mechanical force (a torque) with a magnetic force and, in some manner which is well documented but totally unexplained, cause a flow of electric current.

    The traditional argument is that an electric current flows under the influence of a p.d. and indeed this is the explanation usually accepted for mechanical generators whose action is thought to generate a voltage. However, as electrical theory is developed, problems can arise if this line of thought is taken too literally.  In as much as we are unable to explain the operation of these machines there is justification for accepting the reverse order namely that the machine causes current to flow and that this current transfers electrons from one of its terminals to the other. To accept that positive and negative potentials arise in this manner does not affect the above argument in the creation of equilibrium conditions and so does not invalidate the explanation of the fall in terminal voltage when current is permitted to flow.

  **  Whatever the truth may be it is usual to regard all generators as possessing an internal force that encourages the migration of electrons;  this motivating force is called the electromotive force of the generator and is invariably used in its contracted form emf. The emf represents the voltage that appears between the terminals of a generator when the current-flow is zero;  i.e. it is the equilibrium voltage referred to above. The lesser voltage which results when current is flowing is usually called the terminal voltage.

  **  The difference between emf and terminal-voltage is important particularly in practical calculations but the subject is dealt with in greater detail under 1.15: Internal Resistance.

  **  There are occasions when the concept of emf is abandoned and a generator is considered as delivering a current;  this matter too is dealt with under Internal Resistance.

  **  Because of the construction of mechanical generators and their mode of operation they do not produce direct current but one which changes cyclically in both strength and direction. The output from these machines is called alternating current but it is usually referred to by the initials ac; again it is not uncommon to see reference to an "ac voltage" but the correct term is alternating voltage.  Machines which produce ac are called alternators

    There is a rotary machine called a dynamo which allegedly produces dc;  in fact it is an alternator but has a built-in rotary switch (a commutator) that, literally, reverses the connections each time the current reverses direction. The "alternators" fitted to modern cars are in fact alternators but they have built-in semiconductor devices (see below) which imitate the action of the dynamo's rotary switch.

  ** In general it is easier and cheaper to generate in ac form than in dc and, where direct current is required, a means must be provided to convert between the two forms. This is dealt with fully in Part 5: Power Suppliers.   It is sufficient to note here that conversion from ac to dc is a process known as rectification and the devices used are called rectifiers.  Conversion from dc to ac is termed inversion and is achieved in equipment known as an inverter.

1.3.6 Polarity

  ** This term is used to describe the direction in which an emf or p.d. acts.  For example it might be said that the polarity is such that Terminal-A is more positive than Terminal-B.  At the output terminals of an alternator the polarity constantly reverses.   Perhaps from this has sprung the word pole which is used to indicate a point of concentration;    e.g. the terminals of a cell or a generator can be described as its positive and negative poles;  magnets, including the Earth, have North and South poles.

1.4   CONDUCTORS  AND  INSULATORS

  ** The discussion of electron-drift began with the words: "… Some electrons, held only weakly by their parent atom …" .  There are of course atoms which hold on very tightly to all their electrons and consequently it is difficult to cause electron drift.  Materials in which electrons are not easily dislodged and which therefore do not willingly conduct an electric current are called either non-conductors or insulators.

  ** Materials in which "free" electrons are easily dislodged are called conductors and, usually, they are also good conductors of heat.  Most good conductors are metals but a notable exception is the element Carbon;  this appears in many forms and is a most useful material in electrical engineering.

  ** None of the insulating materials are perfect in their action and they all allow passage to small leakage currents.  Leakages tend to remain small even when the applied voltage is increased considerably but, as the leakage current increases, so too does the amount of heat generated.  Because most insulators are poor conductors of heat excess leakage can cause a rapid local rise in temperature and so insulating materials tend to fail suddenly and self-destruct.

  ** There is a class of materials midway between conductors and insulators which are known as semiconductors;   these can be switched electrically between states of conduction and non-conduction without suffering damage and they form the basis of transistor and chip technology.

1.5   RESISTANCE  AND  RESISTORS

     The argument above has been that some kind of electromotive force is required to dislodge free electrons if an electric current is to be set in motion. Without that emf any current-flow in existence ceases and so the only logical conclusion is that the material fights back;   it does not approve of its electrons being harried and moved-on . In fact such action has been mooted already by pointing out that a dislodged electron leaves behind it a positively-charged hole with a distinct come-hither to any nearby electron.

  ** In short, even though a material may be classed as a conductor, it resists a flow of electric current. This action is referred to as the material's resistance;  it is measured in ohms in honour of the German physicist Ohm.

  ** For most materials resistance is a property which is determined only by the length and cross-section area of the conducting path. Considered as a standard unit, such as a 1-inch cube, the resistance does not change.  In terms of a standard unit however each material has its own specific resistance.  Metals such as Copper and Silver have very low resistances but glass offers a very high resistance.

  ** Some specially-made materials have resistances which vary according to some other parameter;  for example; their resistance may either rise or fall when they get hot.  See also under 1.12.1 Ohms Law.

  ** In one sense resistance is a nuisance to the electrical engineer and yet, without resistance, it would not be possible to control a flow of current.  The behaviour of electrical and electronic circuits depends on the values of resistance that are present in different parts of those circuits.  Components are manufactured therefore to provide known specified amounts of resistance and these are known as Resistors.

  ** Apart from possessing different values in ohms resistors are manufactured in different physical sizes;  large components provide a larger surface area through which to lose heat and so they can handle greater powers than small components.  Modern materials can operate at higher temperatures than the older carbon-type resistors and so the physical sizes have tended to decrease;   additionally in the low-power circuits of modern semiconductors power-ratings tend to be much lower than in valve circuits.  However it is necessary to be aware of the heat which resistors may generate and to place them accordingly in a piece of equipment.

  ** Resistors are manufactured also in forms which enable their value to be adjusted. One such has been mentioned above which is made from material which is temperature sensitive. Other types can be varied by mechanical adjustment, by varying the voltage applied to them, by illuminating them with light or some other radiation, by exerting on them a mechanical pressure.

  ** Valves and transistors in their various functions perform as variable resistors;  see Part 3: Electronic Amplifiers.

      In some branches of engineering resistance itself is the end product. There can be few homes which do not possess a physically-large resistor designed deliberately to get hot without self-destruction and known variously  as  an  electric  fire,  a kettle element, an immersion  heater,  a light  bulb … ?

1.6   ELECTROMAGNETISM

  ** Under 1.3.4  Mechanical Generators it was stated that an emf drives an electric current through a conducting material but there are also other effects of applying an emf.  Fig. 3shows a length of conductor connected across a cell via a switch.  It was probably discovered by accident but, when a small compass needle is placed beside the conductor, the needle is seen to move whenever the switch is closed and to return to its original position when the switch is opened once more.

  ** A compass needle possesses the property of magnetism which means that it exerts a magnetic force on certain materials the most common of which is the element Iron.  The exact nature and origin of the force are not known.

      A compass needle has at one end a North Pole and at the other end a South Pole;   the North Pole is named because it points toward the North Pole of the Earth and this indicates that the Earth too is a magnet.  There is a natural rule that unlike forces attract while like forces mutually repel (as described with electric charges) and so the North Pole of a compass needle should not point to the North Pole of the Earth.

    The confusion arises because the Earth's "North Pole" should properly be called the "North Geographic Pole" and the north pole of the compass needle is properly called the "North-seeking Pole";  the Earth's north magnetic pole is close to its South Geographic pole.

  ** To return to Fig. 3 that the compass needle moves when the switch is closed, shows that an electric current calls into being a magnetic force and that this dies away again when the current ceases to flow.  A magnetic force which is sustained by an electric current is called an electromagnetic force;   apart from its dependence on a current such a force is indistinguishable from the permanent magnetic force exerted by magnets such as the Earth or a compass needle. See also under 1.8: Fields of Force and under 1.9: Induction, Inductance and Inductors.

1.7  CAPACITANCE and CAPACITORS

1.7.1 Electron-flow in a Capacitor

  ** Capacitance is the ability to store electric charge;  sometimes it is a desirable property and sometimes it is a downright disadvantage. A component manufactured to provide a known or controllable amount of storage capacity is called a capacitor.

  ** A capacitor consists of two parallel plates made of conducting material which are separated by an insulating material referred to as the dielectric;   sometimes the dielectric is just air.  Such an arrangement is shown in Fig. 4 where it is connected to an electric cell.

  ** As chemical action removes electrons from the positive plate of the cell so electrons belonging to the attached capacitor plate are attracted and, for a short period, an electric current flows in the connecting wire. Equally, as the cell deposits electrons on its negative plate, so they are repelled and move to the other plate of the capacitor; for a similar short period a similar current flows in this connecting wire also.

Fig. 4 Charging a Capacitor

 ** The surplus of electrons accumulated on the negative plate of the capacitor cannot transfer to the positive plate and so return to the cell because they lack a conducting path and so +ve and -ve charges build-up on the capacitor plates.  When the p d. between the capacitor plates equals the p.d. between the cell terminals then electrons can no longer be forced either into or out-of the capacitor and so the action ceases.  This process is referred to as charging the capacitor.

  ** In the equilibrium condition, with the current in each connecting wire at zero, the capacitor is said to be fully-charged  but it is a purely relative term.  Should a larger-voltage battery be substituted for the single cell shown then the equilibrium would be destroyed and so the charging process must recommence.

      An instrument built to record the passage of an electric current (an amp meter) is known as an ammeter;  such an instrument set in either of the connecting wires would, during the charging process, record the passage of a direct current which started at some value and decayed to zero.

  ** The real usefulness of a capacitor is revealed however when the cell or battery is removed and an alternator substituted. While the capacitor does its best to acquire a charge it is continually frustrated because the alternator constantly reverses the polarity;  electrons are first pushed into the capacitor, then withdrawn only to be pushed back again and again with each successive cycle of the alternator.  The ammeter perceives only a stream of electrons which moves first one way and then the other and so it dutifully records the passage of an alternating current.  Note that electrons do not circulate as with a d.c.

  ** A capacitor therefore has a very-important property in that it blocks the passage of dc (after an initial charging current) but apparently permits free passage to ac.

    The oppositely-charged plates of a capacitor exhibit a strong mutual attraction as do the plates of a cell;  an instrument for measuring voltage can be made by allowing one plate to move against the return-action of a spring.

1.7.2 Dimensions of a Capacitor

      When the plates of a capacitor are allowed to approach each other the opposing charges which they carry tend more and more to cancel;  the resulting destruction of equilibrium means that the cell will re-commence charging the capacitor. The effect of reducing the gap between the plates of a capacitor therefore is that it can accept a greater charge;  i.e. reducing the gap increases the capacitance.

      More electrons can be stored on the surface of a large plate than is possible on a small plate and so capacitance (the ability to store charge) is dependent also on the surface area of the plates;  the greater that area the greater is the capacitance. The shape of the plates is not relevant.

      That capacitance depends only on the area of the plates and on their separation means that capacitance can exist between any surfaces which have sufficient conduction to allow electrons to spread over their total area.

  ** Capacitance is measured in units named after Faraday and called Farads;  however a capacitor that offers but a single Farad is about the size of a normal room and practical capacitors are measured in microfarads which are one-millionth of a Farad (mathematically 10^-6 Farads).  The practical range of capacitors goes down yet again by a million to what were once called micro-microfarads but which are now named picofarads (10^-12 Farads);   pF's are normally referred to as "puffs".

1.7.3 Stray Capacitance

  ** It was stated in the previous Section that capacitance exists between any conducting surfaces;  such all-pervading and unavoidable capacitances are referred to as stray capacitances or simply as strays. The alternating currents which these allow to leak in all directions are a curse.  Experience and careful design can keep their values to a few picofarads and, in general, they do not become a serious nuisance until the frequency range approaches a few MHz.

  ** There are four ways of dealing with stray capacitance once it has been reduced to a minimum:

(a)     make use of it by incorporating it into the design (see Distributed Amplifiers)

(b)     arrange the circuitry so that currents introduced through stray couplings cancel; see under 1.10: Transformers,

(c)     screening (shielding) techniques which are described in the next Section

(d)     deliberately introduce a similar current which cancels the flow through the stray coupling;   this technique is referred to as neutralising and is described in Part 4.2: Transmitters.

1.7.4 Static Electricity

      In Lesson-1 it was proposed that an electric current consisted of a drift of free electrons which are propelled away from a negatively-charged electrode (or terminal) toward one that is positively-charged .  In this Lesson it was proposed also that such a drift of electrons could flow "into" a capacitor ;   i.e. that free electrons would accumulate on one plate of the capacitor and that a dearth of electrons (a collection of "holes") would accumulate on the other plate.

      When the source of e.m.f. is disconnected from such a charged capacitor then clearly the electrons must tend to flow from the negatively-charged plate back to the positively-charged plate but this is not possible because they lack a conducting path along which to create that flow.

      The result of such actions is a capacitor which has:

(a) a surplus of electrons on one plate
(b) a dearth of electrons on the other plate
(c) an electric field in existence between the two plates (there is an electric force between the plates)
(d) a mechanic force which tends to pull the two plates together.

      Clearly the capacitor is involved with electricity but , even though the electrons are free from their parent atoms, they are not in the form defined in that they are not drifting.

      A stream of drifting electrons are mobile electrons or dynamic electrons   and the electricity which is ascribed to their action can be described as dynamic electricity.   Electrons which are sitting on one plate of a capacitor - electrons which are being stored in a capacitor - form a standing/stationary charge and are referred to as static electricity.

      Perhaps the most noticeable effect of static electricity is the mechanic force which it manifests. A useful party trick to amuse children while the jelly and cream is awaited is to rub a balloon on a woolen Jersey or some suit materials and then "stick" it to the ceiling.   Provided the room is dry then friction between the cloth and balloon will transfer electrons;  one of them acquires a positive charge and the other a negative charge. As the balloon approaches the (dry) ceiling so it induces an opposite charge on the surface and the mechanic force between the two charges is capable of supporting the weight of the balloon.

      Why the emphasis on the word "dry" ?   Water is not a very good conductor of electricity but it is sufficiently good to allow small static charges to leak away.  Hence, in a damp atmosphere, the balloon's surface is slightly moistened and any charge induced on the balloon may well return to the cloth as fast as it is formed.  Alternatively it may leak away via the performer's hand and finally it must leak away across the ceiling so that the balloon remains in place for a disappointingly short period.

    In a warm and dry room friction effects between clothing, furnishings, shoes, carpets and the air as we move about generate static charges of surprisingly large values;  they can be as high as 1,500 volts.  Such high voltages can be lethal - indeed under the right circumstances 50-volts can kill - but normally we are totally unaware of these static charges.  Most people however have experienced the crackle as they remove clothing or comb their hair which is generally described as "static"?

     The capacitor which stores these static charges is in fact our own bodies. Perhaps, to be more accurate, it is the capacitance between our bodies and surrounding objects. This stray capacitance has a very small value and so it requires but few electrons to charge it to high values. The transfer of such small amounts of charge represents but a very-small electric current and so we remain unaware of their presence - mostly !

      One area in which these small charge-transfers become important even to the point of disaster is when handling semiconductor devices of the type described in later Lessons as CMOS (complementary metal oxide semiconductor). These employ insulating layers of glass a few nanometres thick (virtually a few molecules in thickness) and this insulation is easily punctured by the static discharge. CMOS devices are destroyed simply by grasping them in the hand unless precautions are taken to discharge the static electricity.  Briefly this involves ensuring that the human operator, tools, bench surface, CMOS devices and the intended destination board are all at the same potential (most conveniently Earth potential).

      It is a dangerous practice however to connect yourself to Earth because an inadvertent contact with a point of high-potential could result in electrocution. So-called anti-static precautions involve connecting everything to earth via a high-resistance which, while allowing static charges to leak away,  limits accidental currents to safe values.

      Why do static charges produce visible flashes and audible crackles when clothing is removed or a comb is passed through the hair? The ability of a capacitor to store charge (its capacitance) is proportional to the area of its plates . As those plates are moved closer together, so the opposite charges tend to cancel and so the potential across that capacitor decreases. By the same argument, as the plates are pulled apart, so the cancellation effect is decreased and the potential across the capacitor increases.

      In removing an item of clothing the gap between cloth and body is increased and so the potential-difference between them is increased ;  the same effect occurs as a comb is withdrawn from the hair. The p.d. rises until the dielectric - the air - breaks down (it dissociates into positive arid negative ions) and the capacitor is discharged by a spark. The spark causes the flash;  its supersonic passage

through the air generates a shock-wave which is the crackle.   This is traditional thunder-and-lightning on a microscopic scale.

      These effects are most noticeable in homes which use central-heating which causes air to be re-circulated and so rendered dry. Where rooms are heated by open coal fires air is constantly vented through a chimney so that fresh air is drawn in from outside;  this raises the humidity and lessens both the static charges that accumulate and unpleasant bronchial effects.

      Static electricity has some intriguing habits but the subject is not examined for the R.A.E.  Here are three curios for the thoughtful:

1. Static charges always reside on the outer surface of a hollow body.  Hence it is possible to generate very large static voltages between two hollow metal spheres, with thunderous discharges occurring between them, while machinery and operators sit safely within.

2. An early capacitor, the Leyden Jar, consisted of two metal cups one inside the other separated by a glass cup (or jar). Once charged the Jar can be disassembled when tests will fail to reveal any sign of charge. Reassembled (with care) the jar is found to be still charged

3. The handrails/bars on public transport vehicles are bound with cord where passengers are likely to grip them. This is because friction between the moving vehicle and the air charges the bodywork and provides a nasty surprise for indignant would-be boarders. Similarly friction between clothing and leather/plastic upholstery can charge passengers and drivers of motor cars who receive similar unpleasant surprises when a foot touches the ground. To a large extent these effects are now nullified by including an amount of graphite in the tyre composition to provide a static leak.

      In connection with (1) above it should be noted that dangling leather straps and chains can do little to relieve the car-sickness of passengers which is often attributed to these static charges. Such charges - if present - will all be found on the outside surface of the car and so cannot affect the passengers.  A leak from car-body to ground will not discharge the electric field between passenger and seat;  the cure is to change the upholstery material

END OF LESSON 2

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QUESTIONS

1. Describe the basic difference in electron behaviour when
   (a) a direct current flows through a resistor

   (b) an alternating current flows through a capacitor.
   In what particular way is a capacitor useful in electronic circuits ?

2. What is the basic difference between conductors, insulators and semi-conductors?

3. Insulators are given to sudden failure under excessive voltage;  what is the reason for the destruction which usually follows ?

4. Why are Resistors manufactured in different sizes ?

5. What is an electromagnetic force 

6. Given two capacitors of different physical sizes and apparently of similar construction which would you expect to exhibit the larger capacitance ? Explain why.

7. What is the difference between a microfarad and a "puff". Why do we not encounter capacitors that offer 1 Farad ?

8. What do you understand by the term stray capacitances ?
    What is the principle that lies behind methods of eliminating their effects?

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LESSON  3

1.7.4  Time Constant

 **  To charge or to discharge a capacitor involves either depositing electrons or removing electrons from it and this takes time. The amount of time taken is determined by the current which flows because this is a measure of the rate at which the electrons move. In turn the magnitude of the current is determined by the resistance of the complete circuit.

**  Fig.4(A) shows a capacitor connected via a resistor to a source of emf. At the instant the emf is connected the capacitor is without charge and so its terminal voltage is zero;  this means that all the emf must appear across the resistor and so the current which flows is determined only by the values of the emf (E) and the resistance (R). The current I is given by E/R(Ohm's Law).        For the capacitor/resistor arrangement the time required at the initial rate is given by the product RC and so, as a practical rule-of-thumb, a full charge or discharge takes about 5RC seconds where R is in Ohms and C is in Farads. The quantity RC is known as the circuit Time Constant and it appears in any circuit where elapsed time is important.

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1.7.5.  Screening and Earthing

  ** Stray capacitive couplings (and the stray magnetic couplings described under 1.9) which occur between a signal circuit and any other piece of electrical equipment lead to the introduction of unwanted noise signals and also to the creation of unwanted feedback loops (see under 1.11: Amplification). Stray couplings are a fundamental part of Nature and cannot be avoided but the signal currents which they allow to pass can be diverted and rendered harmless.

  ** The great mass of the Earth can be regarded as an "electron sink"; to pass unwanted electrons to the Earth or to acquire needed electrons from the Earth does not change its potential in practical terms. Thus any conductor which is connected electrically to the Earth is held at a constant potential.

Fig. 5  Mechanism of Screening Against Capacitive Couplings

           Screening consists of placing an earthed piece of metal (or metal mesh) between the circuit to be protected and the source of interference.  As shown in Fig. 5 the stray coupling is divided into two parts: (a) stray capacitance between the circuit and the screen and (b) stray capacitance between the screen and the interfering source. Of itself the screen provides little or no protection because currents from the interfering source produce noise potentials on the screen and these potentials on the screen then act as a noise-source in their own right to drive noise currents through the circuit.

  **  Earthing the screen provides a path to earth for noise currents which flow toward the signal circuit and for signal currents which flow from it. Because the screen is now held at a constant potential it can no longer act as a source to relay unwanted signals either into or out-of the signal circuit.

  ** Where voltages in excess of 50 volts are present earthing is employed as a matter of safety;  equipment is enclosed either in insulated cases or in metal cases and these last must be efficiently earthed if they are to provide protection.  Earth connections must not be removed even when fault-finding or carrying-out maintenance routines unless the Operator thoroughly understands the procedure;  an unearthed metal case is coupled by stray capacitance to the equipment inside and is only too willing to use the unwary as a path to Earth.

1.7.6    Capacitive Reactance

  ** Although a capacitor does not provide a conducting path between its two connecting wires those two connections always carry identical currents. A capacitor thus provides a means of passing information in the form of an alternating current between two points which are not electrically connected.

  ** The magnitude (the size) of this current is a matter of the number of electrons which move into and out of the capacitor and is determined therefore by (a) the magnitude of the charging emf and (b) the capacitance (the storage capacity) of the capacitor. The capacitance-value thus controls the magnitude of an alternating current in the same manner as does a resistor.

  ** However there is a significant difference between the alternating current which flows through a capacitor and that which flows through a resistor. When a capacitor is fully charged (when the p.d. between its terminals equals the charging emf) then the current falls to zero; i. e. the current reaches a zero value when the p.d. reaches its maximum value.

  ** When a capacitor is in the middle of the operation of discharging and then recharging in the opposite direction then the current (the flow of electrons) is at its maximum value but, at that moment, the p.d. across the capacitor is zero as it reverses polarity; i.e. the current reaches a maximum value when the p.d. is zero.

  ** Where the ac applied to the capacitor is of sine-waveform (see Part 7:  Mathematics) the waveforms of voltage and current are always displaced by 90⁰ and are said to be in quadrature;  the effect of this is that the power-flow (the product of voltage and current) is zero.

  ** In fact a capacitor can pass fairly-large currents without getting warm while a resistor handling the same current could be destroyed by the heat generated.  For this reason ac which flows in a purely capacitive circuit is often referred to as wattless current (power-less current).  To distinguish this wattless current-controlling action of a capacitor from the power-dissipating action of a resistor it is given the name Reactance but it is still measured in ohms;  with certain provisos it obeys Ohm's Law as does resistance.

** Because a resistive current and a capacitive current have different phase relative to the emf which drives them it is not possible to add the two values directly and this is dealt with under 1.12: Resistance, Reactance and Impedance.

     Similar behaviour is exhibited by inductors (see under 1.9) but they produce a different quadrature relation between current and voltage; to distinguish between them the two reactances are described by the terms capacitive reactance and inductive reactance.

1.8   FIELDS OF FORCE

      In the domestic world a field is an area (2-dimensions) characterised by grass although sometimes it may be specified as (for example) "a field of corn" or "a cornfield".

      In Physics a field is a volume (3-dimensions namely length, breadth and height) throughout which a force acts;  it may be specified as an electric field (which surrounds an electric charge or fills the space between electric charges),  as a magnetic field (which fills the space between and around magnetic poles) or perhaps as a gravitation field (about which we know very little).

    Fields of force are represented graphically by drawing lines-of-force; these have the following purposes

(a) as with the contour lines which are drawn on maps (they join points of equal height) lines-of-force join points of equal field-strength.

(b)    at each point on a line-of-force the line indicates the direction in which the force acts;  for example, an electron placed in an electric field will be attracted to the positive pole and repelled by the negative pole and a line-of-force which passes through that electron shows the direction of the resultant force (the overall effect of these two forces when they act simultaneously).

(c)   many lines-of-force drawn close together are used to indicate a powerful field; this arises from their role as contour lines in that they indicate the gradient of the field  (the way in which the field-strength changes over a distance) although it is seldom that lines-of-force are allocated specific values.

  ** In an electric field the lines-of-force are defined as showing the path that would be traveled by an isolated positive charge;  it follows therefore that electric force is deemed to act from the positive pole to the negative pole.

  ** In a magnetic field the lines-of-force are defined as showing the path that would be traveled by an isolated North Pole;  in fact we cannot achieve an isolated magnetic Pole but the concept gives rise to the convention that magnetic force acts from the North Pole to the South Pole. (An isolated Pole can be approximated by using a very-long bar or rod magnet.)

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There is one very important difference between electric lines-of-force and magnetic lines-of-force; see Fig. 6.   Electric forces always form open lines which terminate at one end on a positive pole and at the other end on a negative pole;  magnetic lines always form closed loops which enter the magnetic source at the South Pole and continue through the source to exit at the North Pole.

Any attempt to close an electric line-of-force into a loop, or any attempt to open a magnetic loop, invariably decreases the magnitude of the force concerned.

   Lines-of-force often show graphically the behaviour of a system.  For example Fig.7 shows in diagram (a) the electric field between a negative pole and a positive pole while, in diagram (b), it shows the field between two opposing positive poles.  Diagram (a) is an ordered pleasing pattern which is believably trying to contract while diagram (b) is angular and the field patterns associated with each pole do appear to be pushing against each other.

END  OF  LESSON 3

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QUESTIONS

1.         Potential difference (p.d.) and electromotive force (emf) are both measured in volts. In what manner do they differ?

2.         What is the property known as Reactance?

            What do you understand by the term "wattless current ?

3.         Describe the principle used in screening electrical circuits against intruding electric fields.

4.         Why is it that resistive and reactive currents cannot be added directly using simple arithmetic ?

5.         What is an electric field ?

6.         What would you deduce from a diagram that showed many lines of electric force close together ?

7.         Why is it that the terminal voltage of a generator must fall before the generator is able to supply current ? 
            (Not needed for R .A. E.)

8.         What is the difference between a battery and a cell ?

9.         What is the effect of connecting a conducting body to Earth ?

** Paragraphs marked with a double asterisk contain material which is relevant to the RAE;  
 the remainder form a continuous explanation.

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LESSON  4

1.9   INDUCTION,  INDUCTANCE  AND  INDUCTORS

1.9.1  Induction

  ** Under 1.6: Electromagnetism with reference to Fig.3 it was explained that an electric current generates a magnetic force.  By exploring with a small compass needle it can be established that the force acts in a circular path that surrounds, and is concentric with, the conductor;  the magnetic field has the form of a cylinder pierced from end to end by the conductor.

  ** The direction of the circular lines-of-force is related to the direction of current-flow and this is the subject of a classical Law which states that, when an Observer looks along the conductor in the direction of classical current-flow (i.e. from positive to negative), then the magnetic force is always seen to act clockwise.

**  Fig.8 shows a development of Fig. 3 in which a second conductor has been introduced close to the first (or primary) conductor so that it lies within the magnetic field;  this secondary conductor  is connected to an ammeter.

** It is an experimental fact which we cannot explain that, at the moment the switch is closed, a current flows for a short period in the secondary conductor. The secondary conductor then remains quiescent until the switch is opened when again a current flows but this time in the opposite direction.

** That the secondary current-flow occurs whenever the primary current changes its value can be demonstrated by substituting an alternator for the cell;   the primary current now changes value continuously and a constantly-changing current of similar form is found to flow in the secondary conductor.

    This somewhat mysterious transfer of electrical power from the primary circuit to the secondary circuit is termed electro-magnetic induction; it is the basis of an important component called a transformer which is described in the next Section, but the mechanism of the power-transfer is not known.

      The current that is induced in the secondary circuit is exactly the same as any other current and so it too brings into being an attendant magnetic field;  like the primary field this magnetic field too surrounds both conductors and the two are said to be magnetically coupled.  A question arises as to whether the secondary field aids or opposes the primary magnetic field?

      To the ammeter the secondary conductor is a generator and so there must be a fall in terminal voltage (the p.d. across the ammeter) before current can flow and this fall must be dependent on the strength of the current-flow; see under 1.3.3.  Because the secondary conductor derives it power from the primary by means of the magnetic field such a fall in terminal voltage must be caused by a fall in the magnetic-field strength. It follows that the magnetic field generated by the secondary current must oppose the magnetic field generated by the primary current. Therefore, as drawn in Fig. 8 the secondary current must flow in a direction opposite to that of the primary current.

1.9.2  Self-Inductance

      The above argument can be pursued however without introducing the secondary conductor. When the primary current creates a magnetic field the primary conductor itself is encompassed by that field; the primary conductor therefore is in exactly the same situation as any secondary conductor. Hence, when the primary current varies, an opposing secondary current should appear in that primary conductor.

   ** Pursued to the limit this argument implies that any conductor will carry a somewhat smaller ac than would be expected from its performance with dc ?

     By definition a direct current does not change its direction of flow but it does often change its value especially when either building from zero at switch-on or collapsing to zero at switch-off.  If the above arguments are valid it is to be expected that, even with dc, these switch-on and switch-off periods are modified by the inter-action between a conductor and its surrounding primary magnetic field.

** In practice all these effects are observed;  in any given conductor, for a given emf, the value of alternating current is less than that of direct current.  Because of this reduced-current effect it takes an appreciable period of time for current to reach its final value at switch-on and to fall to zero after switch-off.

  ** This phenomenon is given the name Inductance.  It is not difficult to believe that inductance can be a nuisance but, as described in a later Lesson, it can also be of use and so components are manufactured to provide a specified amount of inductance. These are called Inductors and the amount of inductance provided is measured in units called Henrys (in honour of Henry); note the spelling.

  **The reduced value of alternating currents with respect to direct currents is interpreted as an increase in resistance and, quite reasonably, it is measured in ohms. However, as with capacitors, it is found that the alternating current does not vary exactly in time with the applied emf and so the apparent "extra resistance" of an inductor is referred to as its inductive reactance (see also under 1.7.5: Capacitive Reactance).

  ** Again, because of the quadrature relation between current and voltage, an inductor draws a wattless current and provides an efficient means of reducing a 

current-flow. In this application inductors are less efficient than capacitors but they are often used where it is required to preserve a dc-connection between terminals.

  ** The inductive reactance of a simple conductor is fairly small and is not noticed in practice until the frequency of the alternating current rises well above 1 MHz (1-million cycles per second). To achieve appreciable values of inductance it is necessary to use long lengths of conductor and, for convenience, such lengths are coiled. Fortunately the effect of coiling is to increase the inductance still further; as shown in Fig. 9, the magnetic field which surrounds each turn of the coil reinforces the magnetic fields of all the other turns and the effective increase in this field enhances the self-inductance.

  **Note that by winding half the turns in the opposite direction the fields can be made to cancel; if, by careful construction, the magnetic field is reduced to zero, then all that remains to control current-flow is the conductor's resistance. A resistor constructed in this manner is referred to as a non-inductive resistor. These are found for example in test meters (multi-meters) which are required to work equally with ac as with dc.

    Inductors are often referred to as "coils" (for obvious reasons).  Because they are used to oppose or "choke-off" a flow of alternating current, they are also called chokes.

  ** Note that the standard textbook approach to Inductors (and the Transformers described later) is that it is an emf which is induced into the secondary conductor and that this then drives an alternating current through any connected load.  Investigation into this theory shows that such an emf is proportional to the rate at which the primary current changes its value;  in particular that the more rapidly the primary current changes its value the greater is the value of the induced secondary emf.

  **Clearly if the number of current-reversals per second increases (the frequency rises) then the current must change at a greater rate;  thus the secondary emf increases with increasing frequency.   Therefore, as far as Inductors are concerned, (inductive-) reactance increases with increasing frequency; see under 1.12:   Resistance, Reactance, Impedance, Q-factor.

  ** Note that the physical property of Inductance, the degree to which the conductor reacts with its own surrounding magnetic field, does not change with frequency; this is a function of the number of turns, the shape and spacing of those turns and the magnetic properties of the core material (if any); see below.

  **The important characteristic of an inductor is its resistance to change of current-flow. Any attempt to increase the current calls into being a counter-current which mitigates that increase; any attempt to reduce the current calls into being a counter-current which mitigates that reduction. The inductor is the electrical equivalent of the mechanical flywheel in that it tries to maintain current-flow at a constant (the present) value.  It is in this role that it appears in the ripple filters which are described in Part 5 as a necessary part of Power Suppliers;  it plays a flywheel-like part also in the action of Transmission Lines which are described in Part 2 (under 2.3).

1.9.3 Magnetic Cores

  ** The magnetic field which surrounds an inductor, and hence the inductance, can be increased still further by introducing a magnetic material into the magnetic field (the path of the lines-of-force). A magnetic material is one which, under the influence of a magnetising force, will itself become magnetised and, of course, it adds its resulting magnetic field to the surrounding magnetic field. In turn this must increase the induced secondary current.

  So-called soft magnetic materials lose most of their induced magnetism when the magnetising force is removed;  hard magnetic materials retain most of the induced magnetism and so become permanent magnets. When soft magnetic materials are placed in the varying field of an alternating current their magnetic field-strength varies with the waveform of the alternating current and so they add an alternating effect which increases the inductance. This argument applies equally to a dc that is varying in value;  the inductor offers no extra "resistance" to the dc but the presence of a magnetic material greatly increases its "flywheel" opposition to the current variations.

  ** The usual material used to enhance inductance in this manner is iron which results in the name iron-cored chokes. There are other materials however some of which contain iron in finely-powdered form but others are specially prepared ceramics.

  ** Air-cored inductors are those without additional magnetic materials but others may be wound directly on to the magnetic core, on a bobbin which is then fitted on to a magnetic core or wound on a special former into which a specially-formed core can be screwed a variable inductor).

  ** Magnetic cores have a limitation however in that they magnetically saturate. Although the magnetic field which surrounds a current-path is always proportional to the current-flow, magnetic materials perform only up to a critical value after which their magnetism ceases to increase further. Saturation does not take place abruptly but, as the magnetic field fails to follow the variations in current, so the performance of any inductive device suffers.

      The chokes used in ripple filters (see Part 5: Power Suppliers) carry the direct current which is supplied to the associated equipment and this adds its quota of magnetising force;  as a result there is a risk that the iron core will saturate at the peaks of the ripple current. This is avoided by creating a small air-gap in the magnetic path through the core which increases the magnetising force required to cause saturation;  it also reduces the effective inductance but usually this is not a problem. Such components are referred to as gapped chokes.

  ** A more serious problem with iron-cored inductors arises from the fact that the core is made from conducting material which is magnetically coupled to the current-carrying coil; as a consequence the core acts like a secondary winding and eddy currents appear in it. These currents can have very large values because the core is both a secondary winding and its own short-circuiting connection. Eddy currents are destructive because they reduce the effective inductance and also dissipate signal energy in heating the core.

    The cure is to raise the resistance of the core material so reducing the value of the eddy currents. This is achieved by dividing the core into thin slices or laminations (see under 1.12.1: Ohm's-Law).    Laminations are assembled in various ways but the finished device needs to be clamped firmly to eliminate air-gaps and to prevent them vibrating in sympathy with the alternating magnetic field. Core material which uses finely-divided iron uses a non-conducting binder material to reduce eddy-current losses.

1.9.4  Skin Effect

  ** The magnetic field which surrounds the path of an electric current is strongest at its centre and diminishes with increasing distance from that centre. Because the generation of a secondary current depends on the strength of the magnetic field it follows that the strength of the (opposing) secondary current must be strongest at the centre of the field;  i.e. it will be strongest at the centre of the conductor.

  ** From this it follows that alternating-current does not flow evenly throughout the cross-section of a conductor;  it is weakest in the centre and has its greatest value in the surface layer of the conductor.

  ** This tendency for ac to flow only through the surface layers of a conductor is termed skin effect;  it does not occur at all with dc but, as frequency rises, it causes the current to crowd more and more into an ever thinner layer at the surface.  Thus, quite apart from inductive effects, it is found that the resistance of a conductor increases as frequency rises;  in practice this means that a conductor becomes increasingly lossy.

      Skin effect becomes appreciable above 20 kHz (i.e. just above the audio range). Up to about 1 MHz its effects are minimised by using Litz  wire which consists of a number of fine insulated wires interwoven so that small individual currents are forced to flow in all parts of the conductor's cross-section.   At higher frequencies the technique is rendered useless by stray capacitance between the fibres which, as it were, allows current to flow radially and so negates the effect of the Litz wire.

      For high-frequency working thick conductors are used so as to obtain the maximum possible surface area and often they are silver-plated because silver does not so readily oxidise in air. Unplated copper conductors become coated with a high-resistance layer of oxide into which skin effect crowds the current.

1.9.5   Core Material for HF Inductors

        Increasing losses limit the use of soft-iron cores to the audio band.  From around 20 kHz to a few MHz dust-iron cores are used;  as implied by the name these consist of finely-divided iron particles held in an insulating binder and molded to required shapes (this is an extreme form of lamination). "Ferrite", an iron-carbon alloy, is used in a similar manner.  Such moldings may take the form of pot-cores which, when assembled, completely enclose the component thus acting as both magnetic core and magnetic screen (see below).

      Ferrite is also molded into small rings or annuli (singular annulus) which can be slipped over a conductor. They are made in various grades;  some are intended to increase the inductance but are useful also in making small current-transformers

see next Section) while others, by increasing the losses, are intended to dissipate the power of unwanted (spurious) signals.

      At very-high frequencies, and also for high-power working, core material has to be abandoned.   Coils are wound either on insulating formers or are made self-supporting by using thick wire;  they are usually referred to as air-cored coils.

1.9.6 Magnetic Screening

      Under 1.7.4 it was shown that screening against electric fields required the interposition of a metallic sheet or screen between the circuit to be protected and any source of interference and also that the screen must be earthed.

  ** Magnetic screening also requires the interposition of a conducting sheet but, although it does not need to be earthed, it must be made from a material with good conductivity, it must completely surround the circuit to be protected and it must be continuous (i.e. either seamless or with all seams soldered). The purpose is to ensure that eddy-currents can flow freely in the screen and produce magnetic fields which cancel any ambient fields. As with electric screening this works in both directions;  magnetic fields set up by currents in the protected circuit are cancelled outside the screen and, simultaneously, intruding fields from outside the screen are cancelled within.

  ** It is usual however to earth a screening can because it then acts as both a magnetic and an electric screen. The canceling action of the eddy-currents' magnetic fields reduces the inductance of an enclosed circuit and so circuit adjustments must always be made with the screen in place.

      A mesh screen provides very good electric screening but it allows magnetic fields to penetrate. Such a Faraday Screen enables the effects of stray capacitance to be removed while still permitting magnetic coupling between two circuits.  Such a device is to be found in the output stage of some transmitters where a balanced tank-circuit coil is coupled to an unbalanced output circuit; (see next Section on Transformers and in Part 4.2: Transmitters).

1.9.7 Variable Inductors

      The inductance of air-cored coils is adjusted by adding or removing turns but, for fine trimming, the coils are either stretched or compressed;  this is because the inductance of a coil depends not only on the number and size of the turns but also on the "form-factor" which is the ratio between the length and the diameter of the coil.

      Iron-cored inductors are not so easy to adjust but, where the core is fitted with a gap, the inductance can be adjusted laboriously by fitting shims of differing thickness in the gap and then re-clamping.  Sometimes a sliding "slug" of magnetic material is fitted over a too-big gap where it forms an adjustable magnetic short-circuit that partially removes the effect of the gap.

      Small-signal high-frequency inductors are wound on special formers with hollow centres within which a screw-thread is formed. Ferrite or dust-iron cores are molded in the form of a threaded rod so that, as the core is screwed into the coil-former, so the inductance of the coil increases. (It must be remembered that, as the core passes the centre of the coil, so the inductance falls again; this peak is often mistaken for a tuning point.)

      By substituting a silver-plated copper core the inductance of a coil can be reduced as the core is screwed-in;   this technique is another example of eddy-currents being used to cancel a magnetic field.

END OF LESSON 4

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QUESTIONS

1. What is the property known as reactance ?  
    What do you understand by the term "wattless current" ?

2. How does electromagnetism differ from other forms of magnetism?
    A conductor is connected to a source of electricity and is in close proximity to a second conductor. When would you expect to draw current from the second conductor ?

3. What is "skin effect" ?

4. Describe the principle used to protect electrical circuits from the effects of intruding magnetic fields.
   In what ways does this resemble and differ from the technique used to protect against intruding electric fields ?

5. Why are coils often wound on magnetic cores ?   When would you use a copper core?  How does such a core work ?

6. How does (a) inductance and (b) inductive reactance change with frequency?

Paragraphs marked with a double asterisk contain material which is relevant to the RAE; 
the remainder form a continuous explanation.

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LESSON  5

1.10 THE TRANSFORMER

1.10.1 Turns Ratio

      In the previous Section it was shown that, when a primary current varies, secondary conductors to which it is magnetically-coupled carry alternating currents whose waveforms copy the variations. Each of these secondary currents flows in a direction which causes their associated magnetic fields to oppose the changes in the primary field.

   ** On the assumption that each secondary conductor is coupled to the primary conductor with equal efficiency they should all carry identical currents and so it is possible to connect all secondary conductors in series. Fig. 10 shows this arrangement in the diagrammatic form which is normally used although here the diagram is being taken more literally than is normal.

     Each primary and secondary "conductor" referred to above is shown as a single turn of wire wound around a common iron core which is indicated by the solid line drawn between the primary and secondary coils. The purpose of the core is to concentrate the magnetic field to ensure that all fields pass through all turns of wire a condition which is termed maximum coupling between the conductors.

      The designations primary and secondary are ours and are introduced to identify the different coils;  as far as the transformer is concerned they are all just lengths of wire wound around the core and so it is to be expected that,  irrespective of which turn actually carries the primary current,  ALL TURNS carry equal secondary currents.

  **   If the turns are equal in every way and they carry equal currents then it follows that equal p.d.s must be developed across them.  Thus, if the primary consists of a single turn and a secondary consists say of three turns, it is to be expected that the p.d. developed across the secondary is three-times that which appears across the primary?

         **  There is a general rule therefore that the ratio

secondary p.d./primary p.d.

               must equal the ratio

no. of turns on secondary winding/no. of turns on primary winding.  

This is always referred to as the turns-ratio of a transformer and it is usual to quote it as Output/Input as given above;   sometimes, either for clarity or convenience,  transformers are referred to as step-up or step-down transformers which is intended to show the manner in which the voltage changes as the signal progresses from primary to secondary.

      The above is a very simplified approach to the transformer and full treatments in every degree of detail are given in many textbooks.  The discussion here is sufficient for the purpose of the R.A.E.

      As shown already the magnetic fields of secondary currents tend to cancel the magnetic field of the primary current.   In a transformer where the secondary winding has more turns than the primary winding there is an obvious difficulty in that the sum of the secondary fields is likely to either obliterate the primary field (transformer action would cease) or even reverse it.  In that a transformer works it has to be concluded that, as the turns ratio is increased (and the voltage ratio also is increased),  so the current ratio between secondary and primary must decrease.

  **  In Lesson-1 under 1.3.2: Work and Power it is stated that the power dissipated in an electrical circuit is obtained by multiplying the voltage by the current.  A transformer is an inactive device and does not have any means of generating additional power;  it does not get particularly hot and so it does not dissipate any appreciable power.  If the power-output is to be equal to the power-input then the product of volts and amps in the primary circuit must equal the product of volts and amps in the secondary circuit.

  ** A most important aspect of transformer behaviour therefore is that, as it steps the circuit voltage either up or down, so it must step the circuit current either down or up in an exactly inverse ratio. This is written in mathematical terms as follows:

      Let the turns-ratio be n ;  let the primary voltage be Vp ;  let the secondary voltage be Vs ; let the primary current be Ip ; let the secondary current be Is

THEN      n = Vs/Vp  = Ip/Is

This matter is referred to again when dealing with Impedance Matching

1.10.2   Balanced Circuits

 **  Although alternating or varying signals can pass between the windings of a transformer the lack of a conducting path between those windings means that dc signals are blocked.  As a consequence it is possible for the different windings of a transformer to be at different potentials.  Such a potential difference is not confined to direct potentials and it may be either a varying direct voltage or an unwanted alternating signal.

      Unwanted or noise signals appear in circuits through stray capacitive or magnetic couplings (see under 1.7.3, 1.7.4, 1.9.6) and it is usual to minimise these couplings by earthing a circuit at some convenient point.

 **  The transformer offers a useful technique in the form of the balanced circuit.   Fig.11 shows the two conductors of an electric circuit (perhaps a telephone cable) connected between two transformers;   the input to the first transformer and the output from the second each have one leg earthed and so, in those parts of the circuit, the signal information is carried only on the one (non-earthed) wire.   Such circuits are said to be unbalanced.

Fig. 11  Use of Screened Centre-tapped Transformers to Create a Balanced Circuit

      The remaining two transformer windings are each centre-tapped and those centre connections are taken to earth. Signal currents in this part of the circuit are carried simultaneously on both wires but, at any instant in time, the two currents flow in opposite directions. Such an arrangement is sometimes said to be operating in push-pull;   one leg is "pushing" the signal while the other is "pulling".  Each wire carries the same value of current and the same voltage but, with respect to earth, the voltages are of opposite polarity and the circuit is described as balanced.

  **  There are many reasons for requiring a balanced circuit but, in terms of noise suppression, it results in stray signals being injected equally into both legs of the circuit. The two noise signals travel along the balanced cable in the same direction and are sometimes described as being in push-push.  At the delivery end of the balanced circuit the wanted signal (in push-pull) is transferred from the primary to the secondary winding but the noise signals (in push-push) flow in opposite directions through the halves of the primary winding and so cancel.

      In telephone cables such transformers are known as repeating-coils;   they are carefully balanced and their purpose is to prevent cross-coupling between the many pairs of wires in the telephone cable.

 **   Transformers of various kinds which are used to couple balanced circuits to unbalanced circuits are known as bal to unbal transformers and have acquired the

composite name balun.   They incorporate a Faraday Screen (see under 1.9.6) between the two windings to eliminate stray capacitive couplings that otherwise would destroy their balance;  this must be earthed.

1.10.3   High-frequency Transformers

 **  In the above discussion transformer windings were said to be wound on an iron core to ensure efficient coupling between them.  However, to magnetise an iron core and to constantly reverse that magnetisation, consumes power and the consequent hysteresis losses increase as the frequency increases. Coupled with eddy-current losses this restricts the usefulness of iron-cored transformers roughly to the audio band; i.e. 50 Hz to 20 kHz.

  ** The comments on iron-cored Inductors under 1.9.3 apply equally to iron-cored transformers.   For high-frequency working it is usual for one or both transformer windings to be resonated as described later under Resonance and Bandwidth.   As with variable inductors the inductance of each winding may be adjusted by use of screw-in ferrite or copper cores.

 **  For very-high frequencies and for high-power working air-cored self-supporting windings are used which can be fine-tuned by either stretching or squeezing the coils;  the distance between windings may also be adjusted to control the degree of coupling between primary and secondary; see also under Transmitters.

 **  Note that the coils are self-supporting because they are made from large-diameter wire to reduce skin effect (see under 1.9.4.).

1.10.4 The Autotransformer       The discussion of Inductance under 1.9.1 was extended to Self-inductance under 1.9.2 by eliminating the secondary winding.  The same argument can be applied to transformer action in that a single winding can serve both as primary and secondary.   **  Fig. 12  shows at  (a) a step-down transformer in which one leg of  each  winding  has  been   " commoned".   In diagram (b) the secondary winding has been omitted and the output is taken from a tapping point at y on the primary.  This arrangement is known as an autotransformer;   diagram (C) shows a balanced form which is often met in transmitters.

      Because the primary turns carry a current which is common to them all it can be deduced that the applied emf is divided equally across them. Thus the ratio output-voltage/input-voltage is given by:

number of turns yz / number of turns xz.

  **  The autotransformer is suitable only for small step-up or small step-down ratios but it does provide a saving in the amount of copper required and in the copper-losses in comparison with an equivalent double-wound transformer.    It is necessary however to be aware of its shortcomings especially when used with mains power supplies:

1.    There is a direct connection between input and output circuits.

2.    Should a break occur in the common part of the winding then the input emf is applied directly to the output (another reason for limiting the step-down ratio).

3.    Part of the winding must be wound with wire that is thick enough to carry the larger current while the whole device must be insulated to withstand the larger voltage.

4. A very dangerous situation arises in mains applications if the output is inadvertently taken from across xy .

  ** The Autotransformer is used a great deal in aerial-matching circuits and in the final (P.A.) stages of transmitters.   It is also the basis of the "Variac" transformer which supplies a variable-voltage supply from the mains by making the tapping-point into a sliding contact.  When using a Variac always check its coil for continuity,  make sure that the output connections are taken from the correct points and do not exceed the recommended current loadings.

1.10.5   The Toroidal Transformer

      This consists of a small-diameter helix of wire which is wrapped (like a collar) around a conductor;  the conductor forms the primary winding of a transformer and the helical coil forms the secondary.  The magnetic field of the primary follows the same path around the conductor as does the helix and so they are linked.

      The device forms a useful way of measuring ac without interrupting the primary circuit and is the basis of the many instruments often described as claup meters which are simply clamped around any conductor in which the current-flow is to be measured.   In these meters the coil is wound in standard fashion on a two-part soft-iron core which forms the jaws of the clamp.

      The toroidal transformer, often wound on a toroidal (ring) core,  is also the basis of the so-called bridge arrangements used in standing-wave-ratio meters which are described in Part 7 of this Course.

      A version of the toroidal transformer is sometimes created in which both primary and secondary windings consist of a single conductor;  such a current transformer is achieved by laying the two wires side by side and threading a suitable ferrite bead over them.  It can be a useful means of checking alternating current-flow where space is more important than absolute accuracy.

Paragraphs marked with a double asterisk contain material which is relevant to the RAE; the remainder form a continuous explanation.

1.11   AMPLIFICATION

 **  Amplification is the technical term for deliberately increasing the strength (the amplitude) of a signal.   The reverse operation, that of starting with a too large signal and cutting it down to a practical value, is called attenuation.

      There are two ways of increasing size:

(i) a snowball set rolling downhill gathers snow and steadily increases its bulk

(ii) a marine architect may first build a scale model of his ship and then, with the problems under control, set about building the full-scale version;  to do this he takes a supply of new materials and ends up with both the model and the finished ship.

In engineering the process of amplification uses the second method.

      The first steam engines were built in Cornwall to raise flood water from the mines;  these engines did not operate continuously but required constant driving.  Basically steam was raised at high pressure in a boiler and was then allowed to enter a cylinder in which it pushed a piston to and fro;  ultimately the piston movement, with the power of the steam boiler behind it, was used to raise water to the surface.

      Steam was guided through the engine by a man who used the relatively-puny power of his hands to operate steam-control valves. Thus the result of simple low-power hand movements was that tons of water were raised through several hundred feet.  Such power-controlling work systems are known as servo systems.   It could be said that the power of the steam boiler through the agency of the steam engine amplified the power of the man;  in addition the man provided a controlling intelligence that, in turn, was guided by signals received from the engine through his eyes and ears. Modern engines move their own valves but the principle remains.

 **  An electronic amplifier performs the same function.  It requires an auxiliary source of power from which it fashions a new higher-power output signal;  the form of the output signal is determined by copying the form of a low-power input signal (the control signal).   The device on which the input signal operates to control the power-flow is called an active device because it actively does something;   in the steam-engine system described above the active device was the steam valve; in the overall water-raising system the active-device was the steam engine.   In electronic amplifiers there are a variety of active devices but the most notable are the thermionic valve and the transistor which are described in detail in the Lesson on Electronic Amplifiers .

 **  All amplifying systems, from the engine through horse-and-cart to the domestic Hi-Fi system,  incorporate the basic requirements:

1. A source of power sufficient to meet the required demand

2. An active device to control the flow of power from the auxiliary source

3. A control signal - an input signal -which decides the form of the power-flow by controlling the action of the active device

4. A source of information which originates the control signal (a generator) ..

  **  Fig.13 shows the basic layout of an amplifier and Fig. 14 shows a very important variation.  The variation was mooted above when it was stated that modern engines move their own valves.  Some of the work output is tapped-off and fed back to the control-signal to provide a secondary control of the power-flow.  Such feedback control takes many forms and often goes unrecognised.

      For example:  at the end of the piston's stroke the steam must be shut-off and an exhaust valve opened to allow the spent steam to escape as the piston returns for the next power stroke.  A lever system which links the valves and the piston can do this job and so dispense with the bored engineer and his twiddling hands;  properly engineered against mechanical failure this feedback lever-system will never get bored and so will never make mistakes.

      Yet again, without the engineer to watch and control it, the engine might overspeed and so destroy itself.   As  shown in in  Fig. 14 a sensor can be fitted which measures the speed of the engine (against a reference which is provided for the purpose) and generates an error signal;    this low-power signal is fed back to control the steam input to the engine in such a manner that it corrects any tendency to depart from the required speed.  This "correct" speed is set by the reference signal which now becomes the speed control or throttle.

 **  The kind of feedback system described above, which operates so as to maintain control and order,  is referred to as negative feedback because the error-signal is used to cancel or to negate any errors.   Negative-feedback (n.f.b.) systems at their most simple are exemplified by the mechanical flywheel which

(a)  is difficult to start spinning
(b) once spinning is difficult to stop
(c) has a preferred spinning rate namely the one it already possesses.

    In short, flywheels like to preserve the status quo - they dislike change and oppose acceleration.  Indeed  n.f.b. systems not only behave like flywheels but, in electromechanical systems such as video-recorders, they are actually used as flywheels because they can be switched-off when their conservative nature becomes inconvenient. They are also more compact and less lethal than mechanical types.

 **  In the example given above it was the engine speed that was measured and the result was used to control the engine speed.  This circular action gives rise to the technical term "feedback loop".  Note that, whatever is sampled in the measuring process, it is the sampled parameter that is controlled.

  **   In electronic equipment the output may take the form of either a voltage waveform (with negligible ability to supply current) or a current waveform (with negligible ability to develop voltage);   if both voltage and current waveforms are required then it is necessary to use a power-amplifier.  If a feedback sample is derived from the voltage waveform then the voltage waveform is maintained accurately at the expense if need be of the current waveform.  If a feedback sample is derived from the current waveform then the current waveform is maintained accurately at the expense if need be of the voltage waveform.

  **  To talk of negative-feedback implies that there is a positive-feedback (p.f.b.) system and it is so;  where the feedback loop is connected so that the error-signal helps an error instead of negating it then chaos results.  Hard perhaps to believe but there are conditions under which p.f.b. can be helpful;  for example it is a most effective way of speeding-up some operations but the Designer must be careful to keep matters under control. Positive feedback is the essence of systems which continuously oscillate (such as a clock pendulum or an electronic oscillator );   control is maintained by a system that restricts operation to a narrow band of frequencies (see Lesson on Electronic Oscillators).  In particular most p.f.b. systems are controlled by a n.f.b. loop.

END  OF  LESSON   5

* * * * * * * * * * * *

QUESTIONS

1. What is meant by the "Turns Ratio" of a transformer ?  How does this ratio relate: to

(a) the input and output voltages
(b) the primary and secondary currents ?

2. What is the thinking behind a balanced circuit?

3. What is a Balun?

4.  In what manner do Audio Transformers and HF Transformers differ?

5. Why would you not use an Autotransformer in the power supply circuits of a piece of equipment?

6. An amplifier increases the level of a signal and a transformer can be used to step-up the voltage of a signal. Why is the transformer never referred to as an amplifier?

7. In feedback systems what is it that is fed-back? What is the purpose of such a technique?

8. Can you state briefly the different effects of positive and negative feedback?

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LESSON 6

1.12   Resistance, Reactance, Impedance, Q-factor

1.12.1 Resistance and Ohm's Law

    In the previous Sections it was shown that conductors have a natural resistance to the flow of electric current and that, when a current varies, the natural resistance apparently increases because of inductive reactance.

   Although all conductors must exhibit inductive effects there are so-called non-inductive resistors.  One popular method of constructing these is to take a length of wire which has the required resistance,  fold it at the centre and then wind the doubled wire on to a former taking care that the twinned conductors are kept in close proximity.  The idea is that current flows in opposite directions along the two parts of the wire and so their magnetic fields cancel.  Apart from their awkward bulk such components can be very effective and are often found for example in multimeters (test-meters) which function over a range of frequencies.

This Section is concerned with true (non-inductive)resistance and so it is applicable only to circuits which handle dc, very-low-frequency ac or to non-inductive components.

   Consider several conductors which are alike in every detail;  they are made from the same material, they have identical widths (W), identical thickness (T) and identical length (L).  When such identical conductors are all connected across the same source of emf it is to be expected that they will all draw the same current I amps.

   Fig.14(a) shows one of these conductors connected across a source of emf;  the current I sets-up a p.d. across the Resistor and that p.d. must be equal to the value of the emf (else the current increases or decreases until equilibrium is attained).

 Fig.14(b & c) shows two graphs;  the upper one shows the variation of the applied emf with time and the lower shows the variation of  the p.d.   At the moment the switch is closed the applied emf leaps instantaneously from zero to the value of the cell's emf but the p.d. builds slowly to this final value.  This is because the current is prevented from changing instantaneously by the delaying effects of both inductance and stray capacitance

      In a similar manner, when the switch is opened, the applied emf falls instantaneously to zero but the current - and therefore the p.d. across the resistor - falls slowly. There is a philosophical problem here in that, after the switch has been opened, the current has nowhere to flow and so it should also fall to zero instantaneously.  Consider that, for the current to fall at all, a mechanism is required to discharge both the stray capacitance and the magnetic field.  This is usually explained in terms of the spark that jumps the switch contacts as an inductive circuit is broken.

**  Fig.15(a.)shows two of the identical conductors connected across the cell in a manner that requires the current to flow first through one and then through the other.  The conductors are said to be connected in series or to be series-connected.   Because the one current flows through the identical resistors it must develop the same p.d. across each and so equilibrium is established when each p.d. is equal to half the applied emf.  It follows that the current which flows through two identical resistors in series must be half that which flows through one of them when connected to the same emf (Fig.14).       The exercise is repeated in diagram (b) with three series-connected resistors to show that the current has to fall to a value I/3 so as to produce a p.d. across each resistor of emf/3.  From this it is possible to deduce a general rule that, when n conductors are connected in series, the current falls to I/n so that the individual p.d.'s fall to emf/n.   ** This simple relationship is the subject of Ohm's Law which states that in association with any given conductor,  the ratio of voltage-to-current is always constant;  that constant is known as the resistance of that conductor. The Law is expressed in symbols as: E/I = R where E is the voltage in volts developed across the conductor,  I is the current in amps which flows through the conductor and R is the resistance of the conductor which is expressed in ohms.  **  Given any two of these quantities the third can be calculated by manipulating the expression (see Lesson on Algebra). E = I.R     or     I = E/R Again it is best to remember only one of these relations and to obtain the others by cross-multiplication as required.

Fig.16 shows two identical conductors connected to a source so that a current flows through each of them independently;  connected in this way they are said to be connected in parallel or to be parallel-connected   Once again the p.d.'s developed across each conductor are identical (equal to the source emf) but the source is called upon to supply twice as much current. When three conductors are connected in parallel the current in each stabilises at a value I, the p.d. across each conductor stabilises at the value of the emf and the total load current drawn from the source becomes 3.I.

   **  Because the current demanded increases with the number of parallel-connected conductors it would appear to the source that the load resistance presented by n conductors each of resistance R looks like R/n.

  ** In practice however it is fairly unusual to be connecting identical resistors in parallel or in series. With series connection the resulting total resistance is straightforward. Although they are of different values the resistors carry the same current and so they develop different values of p.d.; these individual p.d.^,s sum to the value of the emf.  Draw a diagram similar to Fig.15 and you will see that the total effective resistance Reff is given by   R1  +  R2  +  R3  +  . . . .

  ** Different resistors connected in parallel produce a more complicated calculation because it is now the currents which sum instead of the p.d.'s and current (I) is on the bottom line of the equation E/I = R.   For the RAE you are not expected to derive these formulae only to be able to quote them and use them.  For resistors in parallel the total effective resistance is given by

1/Reff   =   1/R1     +    1/R2     1/R3     +  . . . . .

For two resistors in parallel this is usually remembered as product-over-sum

Reff    =   R1.R2 
               R₁+ R₂

  **  There is sometimes a requirement to find the equivalent resistance of a network which combines parallel-connected resistors and series-connected resistors. The rule is:

(i)   reduce all parallel-connected groups to single equivalent resistors
(ii)  collect all series combinations to single equivalent resistors
(iii) repeat steps (i) & (ii) until only one equivalent resistor remains.

      Under 1.3.2 it was shown that the power dissipated in an electrical circuit is obtained by multiplying together the values of electrical pressure (volts) and electrical flow-rate (amps);   for a purely-resistive (i.e. non-reactive) circuit this is expressed in symbols as

P =  E x I

where P is the power in watts,  E is the voltage applied across the component (or the p.d. developed across that component) and I in amps is the current which flows through the component.

  **  However the two parameters E and I form part of the expression for Ohm's Law and so it is possible to express power in terms of the circuit resistance;   in the above expression for power the product I.R can be substituted for E or the ratio E/R can be substituted for I which yields either

P =   I.R x I   =    I² .R    or    P  =  E x E/R  =  E²/R

 (These two variations should be remembered.)

1.12.2   Inductive Reactance

      Under 1.9 it was stated that, when the current which flows in an electric circuit varies in value, then there is interaction between the electrical path and the surrounding magnetic field.  This interaction causes the current to fall to a value below that predicted by Ohm's Law. This result is usually interpreted as an increase in the apparent resistance of the conducting path but matters are not at all so delightfully simple.

      A basic experiment in the early study of electricity and magnetism shows that current can be induced in a coil of wire by moving a permanent magnet in the vicinity of the coil. It is easily established that, although the strength of the magnet is important, the major factor which controls the rate of current-flow is the speed of the magnet. relative to the coil.  The greater the rate of relative-movement (the greater the rate at which the magnetic field changes) so the greater is the induced electric current.

  ** In a circuit which carries ac the rate at which the associated magnetic field varies is controlled by the frequency of the ac;  it follows therefore that, as frequency is increased, so also does the induced (the opposing) current. Thus the apparent increase in resistance to an alternating current becomes more pronounced as frequency rises.

  ** Note that the "proper" resistance of the conductor remains unchanged and this is usually referred to as the dc resistance.    It is only the apparent increase in resistance - the inductive reactance - which is proportional to frequency;   this reactance must be added to the dc-resistance to obtain a  prediction  of  the  overall  behaviour  of  the  electric  circuit   ( see  below under  1.12.4: Impedance)

  ** The inductive reactance of a conductor (or of an Inductor) is given the symbol XL and is expressed in ohms. Where the inductance has a value L (Henrys) and the frequency is f (Hertz or cycles-per-second) the inductive reactance is given by

XL    =   2חfL

      The constant 2ח arises because frequency is measured as the number of revolutions per second of a vector and each complete revolution is expressed as 2.ח radians (see under 7.2: Trigonometry).   The combination 2חf recurs frequently in ac theory and it is convenient to replace it with the Greek symbol omega (lower case)  ω .   Thus the above expression is more usually written as

XL  =  ωL

(ω is known as the angular velocity in that it defines the number of radians traced out every second -  it defines the rate of change).

      The above expression for inductive reactance is shown in Fig.17 as a plot of reactance against frequency (see under 7.3:  Graphs).  Because all terms are in linear measure (i.e. the expression does not include squared or higher-order terms) the graph is a straight line although we still refer to it as a curve.  We say that reactance,  inductance and frequency are linearly related.   The full line shows the variation of  XL with frequency; note that at zero frequency (i.e. dc) the reactance is zero. The broken line shows the practical result when dc-resistance is added.

  **  The difference between dc-resistance and inductive reactance lies in the relative timing of current and voltage waveforms.. In a truly resistive circuit the current. is always related to the voltage by Ohm's Law.  If the current and voltage values are sampled at any instant of time the ratio always yields the same figure.  In a reactive circuit, either inductive or capacitive, this is not so.

      The flywheel-like action of inductance (see under 1.9.2) causes a circuit to resist changes in current flow and this means that a period of time is required for current either to increase or to decrease in value.  In an ac circuit the applied emf constantly cycles and the resulting current, although it varies in value with the same waveform,  is delayed with respect to the voltage waveform.

      In Lesson-2 of Trigonometry (under Mathematics) a sinewave is derived as a result of rotating a vector;  in this waveform the amplitude is proportional to the sine of the angular displacement.  Such a sinewave is shown in Fig.18 where, in diagram (a), it represents the alternating emf applied to an inductor.

      Diagram (b) shows the waveform of the current which flows through the Inductor;   it is shown as delayed by a ¼-cycle or 90⁰.   In fact this is the delay (or the relative phase-angle) that would be caused by a theoretically-perfect Inductor which would have zero dc-resistance.   In a practical Inductor the presence of resistance reduces the relative phase-angle;   as the resistance is increased so the phase-difference moves toward zero as the component changes from an effective inductor to an effective resistor.

      Note that, because inductive reactance is proportional to frequency so the relative phase-angle moves towards 90⁰ as frequency rises. However the general result of increasing the frequency is that an Inductor becomes less efficient (or more "lossy") and this reduces the phase angle. (It is shown below that power cannot be dissipated in pure reactances and so losses must be represented as the presence of resistance.)

      In a pure inductor the 90⁰ relative-phase between current and voltage is often described by saying that the two parameters are in time quadrature;   the expression simply means that they are displaced in time by 90^0.  . (The electric and magnetic components of the field radiated from an aerial are mutually at right-angles and are said to be in space-quadrature;   see under 2.4: Aerials.)    It is usual to drop the adjective and simply refer to them as being "in quadrature".

      Note that current maxima occur where the voltage is zero and is undergoing its maximum rate-of-change;   current minima occur where the voltage is at maximum value with zero rate-of-change.

  **  Under 1.3.2 it was shown that the power dissipated in a circuit is obtained by multiplying together the values of voltage and current.   When using alternating current this can be done only on a point-by-point basis;   for each value of angle (along the horizontal axis) the instantaneous values of current and voltage are selected and multiplied with the result shown in Fig 18(c).    (See also in Mathematics under RMS Va1ues).   When two positive values or two negative values are multiplied the result has a positive sign but, when either the voltage or the current (but not both) are negative, the product has a negative sign.

  **  Fig.18 (c) appears to indicate that power is first drawn from the source for a ¼-cycle and then returned to the source during the following ¼-cycle.   However this is a mathematical result and requires interpretation;  clearly the average power during each whole cycle is zero.   For this reason a purely reactive component - one in which the voltage and current are in quadrature - is said to draw a wattless current.

  **  All practical components possess dc-resistance and this causes the voltage and current waveforms to depart from exact quadrature;  current which flows through the dc-resistance causes the dissipation of power.  This problem is dealt with in calculation by treating each practical  component as two separate "pure" components namely a perfect inductor in series with a perfect resistor;   the voltage waveform developed across the pure resistor is in phase with the current and so power is dissipated;  the voltage waveform developed across the pure Inductor is in quadrature and the theoretical component does not dissipate power.

  **  Clearly the two voltages cannot be added directly although they must sum to the value of the applied emf;   this matter is dealt with under 1.12.4 Impedance.

END OF LESSON 6

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QUESTIONS

1. What is a non-inductive resistor ?

2. , Three resistors each of value 300 ohms are connected (a) in series and (b) in parallel;  what is their effective value In each arrangement ?

3. Two of the resistors are connected in parallel and the third in series with the combination. What now is their effective resistance?

4. What is the voltage developed across the 50-ohm feedpoi.nt of an aerial which is accepting 50 watts ?

5. What is the difference between a purely-resistive circuit and a practical circuit (one that contains reactance) when they carry ac ?

6. Given that you have a normal multimeter (measures amps, volts or ohms either ac or dc) what can you measure about an inductor ?

7. What effects would you expect to observe when an inductor is connected to a dc source ?

8. An emf is applied to a resistor;   if the emf is doubled in value what do you expect to happen to the current flowing through the resistor ?

Which Law of Physics enables you to solve this question simply ?

 9. CATCH QUESTION If the emf applied to resistor is doubled in value what do you expect to happen to the power that is dissipated by that resistor ?

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LESSON  7

1.12.3 Capacitive Reactance

  **   A capacitor of course does not pass a direct current and its ability to pass alternating current is more a matter of effect than reality. However, as with an inductor, the current and voltage waveforms do not vary in step;   in a loss-free capacitor they are in quadrature. There is a significant difference however in that capacitive current is advanced with respect to the voltage as distinct from inductive current which lags behind the voltage.

      It is reasonable to ask how the current is able to lead on the voltage;  how could the current know what the voltage is going to do in the immediate future ?  The answer is that this relationship is possible only with recurrent waveforms;  information which controls current flow comes from the preceding cycles.  In effect the current is not so much a quarter-cycle in advance as three-quarter cycle lagging behind and this can be deduced as follows:

** Alternating current flows in the connections to a capacitor and alternating p.d.s build-up between the capacitor plates. The current in the connecting leads falls to zero as the capacitor p.d. reaches its peak value; current in the connecting leads reaches its maximum value as the capacitor p.d. falls to zero (as the p.d. reverses polarity) and the capacitor begins to re-charge with the opposite polarity.

  **  As with inductors this quadrature relation between current and voltage means that the term capacitive-reactance must be substituted for resistance.

  **  Alternating current which flows in the connections to a capacitor represents the charge (the number of electrons) which is being moved in and out of the capacitor ;  thus the larger the capacitance the greater the amount of charge to be moved and so the greater must be the alternating current. Such an increase of current (as a result of an increase of capacitance) is interpreted through measurement as a decrease of reactance.

      As the frequency of an ac is increased so the capacitor is allowed less and less time in which to charge and discharge and so the alternating p.d. which develops across it is reduced - given that the value of the ac remains constant.  According to Ohm's Law the p.d. is the result of multiplying current and resistance or, in this instance, current and reactance. If the value of the ac remains constant as the p.d. decreases then the fall of p.d. with increasing frequency has to be interpreted as another fall in reactance.

 **  Thus capacitive reactance (Xc) falls when either the capacitance or the frequency is increased and this is expressed in mathematical symbols as

Xc  =  1/(2πfC)

where the capacitive reactance (Xc) is in ohms,  the frequency is expressed in circular measure as 2πf radians per second and the capacitance (C) is in Farads    Again it is usual to insert the Greek symbol w for the angular frequency and the above formula becomes

Xc   =   1/ωC

      The expression for Xc. is plotted in Fig.19.  Although all the terms are linear the result is not a straight line because of the inverse relationship. As frequency falls toward zero (dc) so the capacitive reactance increases toward an infinitely-large value (an open-circuit) ;   at the other end of the curve, where frequency is rising toward an infinitely-large value, so the value of the reactance is falling toward zero (a short-circuit).

      Note that at each end of the curve there is a gradual approach to the relevant axis although, in theory, contact never occurs. We say that the curve is asymptotic* to (in this instance) both axes. In practice the actual point of contact is of no importance because, as the interval becomes smaller, so the curve gets lost in the vagaries of measurement - we say that it falls below the noise level

      As with inductors the quadrature relation between current and voltage causes capacitors to draw a wattless current. Capacitors too have dc-resistance, both in their connecting leads and in the plates themselves, and there are other losses associated with the dielectric ;  all these combine to reduce the relative phase-angle below 90⁰ degrees.  In terms of Fig.19 this causes the right-hand end of the curve to be asymptotic to a line slightly above zero ohms and this line represents the effective dc-resistance (the losses) of the capacitor.

  ** Because the current that flows through a capacitor leads the applied voltage by approximately 90⁰ while that which flows through an inductor lags the applied voltage by approximately 90  then, in any circuit which contains both components, the two currents must be nearly 180-degrees out of phase; in other words they must tend to cancel.  The opposing nature of capacitive and inductive reactances
____________________________________________________________________
  *  Pronounced  a - simp - totic

are shown mathematically by placing a minus(negative) sign before the symbols or figures for capacitive reactance.  Hence the expression given above for capacitive reactance should be written as

Xc  =  - 1/wC

      The negative sign ensures that, when an expression is written to describe the overall behaviour of an ac circuit, the inductive and capacitive reactances cancel each other automatically;   for example, a circuit which  contains  resistance,  inductive-reactance  and  capacitive-reactance  is  described  as  R + wL - 1/wC

  **  Although resistance and both forms of reactance inevitably occur in the same circuit their overall effect cannot be expressed simply by adding together their individual values in ohms. For the series resistive circuit considered under 1.12.1 (Lesson-6)  the overall resistance was found by adding the individual p.d.'s and dividing by the common current ;  for the parallel circuit the overall resistance was found by adding together the various currents and then dividing the total into the common p.d. For an ac circuit neither voltages nor currents can be added directly unless they vary in sympathy;  i.e. either the circuit is free of all reactance or the inductive and capacitive reactances exactly cancel (see later under 1.13: Resonance)

      The solution to this problem is geometric using vector addition.    A vector quantity is simply one which indicates direction as well as size (magnitude).  For example it is correct to say that a car travels at a speed of 30 m.p.h. but, to be credited with a velocity, it must be given a direction such as 30 m.p.h. due east. Speed is a scaler quantity - it indicates only scale or size. Velocity is a vector quantity.  In this example the motion of the car could be depicted graphically by an arrow (called a vector) of length 3-inches pointing to the right where the scale chosen for the diagram is 1-inch represents 10 m.p.h.

  ** The overall effective "a-resistance" of a circuit which contains both dc-resistance and either or both forms of reactance is given the name impedance.

      The effective resistance or reactance which is present in any circuit is measured by the amount of current which it permits to flow for a given applied emf.   Fig.20 (a) shows three vectors:

(i) an arrow AS whose length represents the dc-resistance (magnitude of the direct current) which, by choice, is drawn horizontally

(ii) a second arrow BC whose length represents the inductive reactance (magnitude of the alternating current);  current which flows through this reactance leads by 90⁰ and so the vector is drawn at 90⁰  to the resistance vector but rotated anticlockwise because this is the standard agreed in mathematics for positive-rotation (inductive-reactance is given a +ve sign).

(iii) a third arrow AC which shows the overall effect of the first two and which is therefore a vector that indicates both the magnitude of the resulting impedance (Z) and the phase-angle between the applied voltage and the resulting alternating current.

      It is perhaps easier to justify this vector-addition if the scenario is changed;  consider AB as the velocity of an aeroplane flying due east while BC represents the motion of the air (the wind) which is blowing toward the north. The aeroplane may travel from A to C either by:

(a) traveling along AB in still air for a given time and then remaining stationary with respect to the air while the wind blows for the same period of time

(b) traveling directly from A to C by flying vector AB at the same time as the wind blows vector BC;  the arrow AC represents the track over the ground as the aeroplane progresses simultaneously along both AB and BC.  Thus AC is a vector whose length represents the true speed over the ground and whose direction shows the true direction of travel.

      This technique can be used to add resistance to reactance, reactance to reactance or to find the resultant waveform when two out-of-phase waveforms are added;  it can be extended to sum any number of such quantities. Fig.20(b) shows the vector diagram which is used to add resistance to capacitive-reactance and diagram (c) shows the construction to add resistance to a combination of both kinds of reactance.  Note that  the reactances and the resistances (all "like" quantities) are summed first before vector addition of unlike quantities is attempted.

      The above method using vector diagrams is a valid way of adding vector quantities and, where it is required to add several such quantities, it may well prove to be the best method.  It can also be the best method when the quantities involved are not in exact quadrature;  see under Mathematics: Vector Diagrams. For calculations involved in electrical work, where practical problems are usually

concerned with essentially pure components, all that is required is the solution to a right-angled triangle.  As shown in Mathematics under Lesson-1 of Trigonometry this can be achieved by use either of Pythagoras Theorem or by using Trigonometry Tables.

  ** Pythagoras Theorem states that, in any right-angle triangle, the square on the hypotenuse (the longest side) equals the sum of the squares on the other two sides;  with reference to Fig.20 this relation is expressed as

AC^{2   =}  AB² + BC²

which translates as

Z²  =  R² + X²

Thus the square of the Impedance can be calculated by adding together the squares of resistance and reactance. Of the three quantities resistance, reactance and impedance,  if any two are known then the third can be calculated.

  **  A vector diagram also shows the relative phase-angle between quadrature components.  As shown in Fig.21(a) a perfect inductor connected in series with a perfect resistor (which represents a practical inductor) has a common-current I, a voltage across the inductor VL, a voltage across the resistor VR and a voltage across the whole component VZ ;   this last must equal the applied emf.

      In diagram (b) the current-vector I is shown as a reference for direction;  the voltage VR is drawn in phase with I and the voltage VL is shown leading the common-current by 90⁰.   Because the current is common the lengths of the two voltage vectors represents also the magnitudes of the resistance and reactance;   thus their vector sum VZ represents both the applied emf and the magnitude of the impedance.  The angle φ (Greek letter phi) shows the angle by which the emf leads the current which flows through the practical inductor (if preferred, it shows the angle by which the current lags behind the applied voltage.

  **From Fig.21 it can be deduced that the relative-phase angle φ is given by 

cos φ  =  VR/Vz  =  R/Z

tan φ  = VL/VR = XL/R

(see Lesson on Trigonometry)

  **  The quantity cosφ  is known as the Power Factor of that particular component.   Remember that the perfect inductor draws a wattless current ;  any power that the practical inductor dissipates must be consumed in the perfect resistor.  The Power Factor expresses the ratio between apparent power (the applied voltage  x  the current ....known as volt-amps) and the true power (p.d.-across-R  x  current).  To obtain the actual power dissipated in a reactive circuit therefore it is necessary to multiply the apparent power in volt-amps by the Power Factor ;   i.e.

Power (Watts)  =  V x I x cosφ

** The quantity tan φ is the ratio between perfect reactance and perfect resistance and so expresses the "goodness" of the component or the degree by which it departs from being a perfect inductor. The ratio is known as the Q-factor ;  for example a Q of 1 indicates equality between the reactive and resistive components while a Q of 1,000 shows that the inductor is close to being a perfect component.

END OF LESSON 7

*  *  *  *  *  *  *  *  *  *  *  *

QUESTIONS

1. What is the basic difference between the ac which flows through an inductor and the ac which flows through a capacitor with respect to the applied emf?

2. What other difference is there between these two currents in terms of frequency ?

3. In what way does resistance modify inductive and capacitive currents ?

4. Why is capacitive reactance written mathematically with a negative sign ?      What is impedance ?
    A circuit contains 3-ohms resistance and 4-ohms inductive-reactance;   what is its impedance ?

6. What is the significance of the Power Factor?

** Paragraphs marked with a double asterisk contain material which is relevant to the RAE ; the remainder form a continuous explanation.

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LESSON  8

1.12.5    Notation   (not essential for RAE)

      It is perhaps easiest to write " R  +  X  =  Z "  and to remember (hopefully) that the addition must be made vectorially as discussed in Lesson 7 ; i.e. first square R and X, then add the results and then find the square-root to obtain  Z.

      This is not too difficult when looking at the symbols R and X but, once numbers are inserted in place of those symbols, then the likelihood of error becomes very great.  As a reminder it is possible (and indeed sometimes done) to add some kind of symbol to indicate that the quantities are vectors:  for example

R +  X =  Z

      Such a notation however is clumsy and does not readily lend itself to algebraic manipulations (where the requirement is to solve a problem in general terms rather than to obtain an arithmetical answer). The mathematical tool used to handle these expressions is the j-notation which is described in detail in the Mathematics Lesson.

      At its most simple a letter j is placed before each item which is reactive thus for example

Z   =   52   +  j67   - j15   +   12   +   ,j6

indicates that a series circuit consists of a 52-ohm resistance followed by a 67-ohm inductive-reactance followed by a 15-ohm capacitive-reactance followed by a 12-ohm resistance followed by a 6-ohm inductive-reactance.

      Had the frequency been given these figures could be interpreted in terms of inductors and capacitors but, without such information, all that can be derived from the expression is the overall impedance (at whatever the frequency may be).  The first step in this process is to collect together all the similar terms;  to make life even easier we put all the reactive terms together inside a set of brackets and use a single j in front of the brackets

Z   =   52   +   12   +   j(67 - 15 + 6)

=   64 + j58

From this can be calculated:

(a) the impedance Z  =  √(64² + 58²)

(b) the phase-angle  φ  between current and voltage  =  tan-¹ (58/64)

(The notation tan^-1 is mathematicians' shorthand which simply means   " ... the angle whose tangent is ... ")

      A full treatment is given in Mathematics but here it is noted only that j is more than just an indicator of a quadrature relation;   it can be used very effectively in manipulating equations and is known as an Operator.  A quantity

expressed as R  +   JX is called a complex quantity.   These are sometimes expressed in what are known as polar co-ordinates which consist of a figure that gives the magnitude (Z) followed by an angle that represents the relative phase angle;   for example the above expression 64 + j58  becomes

√ (64² + 58²),  tan^-1 (58/64)

      The above discussion deals with a set of components connected in series.  When components are connected in parallel the formulae derived from Ohm's Law deal in inverse quantities (see under 1.12.1 in Lesson-6).  For ease in handling these formulae a set of names has been given to the inverse of resistance and reactances namely

    1/resistance     (1 /R)    is called     conductance (G)
    1/reactance     (1/X)     is called     susceptance (B)
    1/impedance    (1/Z)     is called     admittance (Y)

Thus in place of  l/Z = 1/R + 1/jX  we can write Y =  G + JB.

      The benefits of this alternative notation may not be immediately evident until it is realised that Instruments for measuring resistance/reactance can be built to give their results in either the normal or inverted form.   An Impedance Bridge gives its results in terms of R and jX (a series-connected circuit) but an Admittance Bridge gives results in terms of G and jB (a parallel-connected circuit). Where measurements are made with an Impedance Bridge the circuit is adjusted by adding extra components in series (often difficult);  where measurements are made with an Admittance Bridge the circuit is adjusted by adding extra components in parallel.

1.12.6 Impedance Matching

  **  Consider two circuits both of which carry a signal at 10 watts. Circuit A draws a current of 1-amp and so has 10-volts developed across it (power = volts x amps) while circuit B draws a current of 0.1 amps (100 mA) and so has 100-volts developed across it.  The application of Ohm's Law shows that circuit A has an impedance of 10 ohms but that circuit B has an impedance of 1,000 ohms.

      A problem arises when it is required to transfer the 10-watt signal from circuit-A into circuit-B.  Because circuit-A can supply only 10 volts from which circuit-B will accept only a maximum of 10/1,000 A (10 mA) it follows that the maximum signal power that can be transferred from A into B is 100 mW or 0.1 W .    Additionally circuit-A is left with a surplus of  990 mA equivalent to 9.9 W which it has to dump somewhere.

  **  This inability to transfer power between two circuits because they operate at different impedances is known as an impedance mismatch ;   to transfer the maximum possible power between two circuits it is essential that their impedances match. Such a transfer can be achieved between two un-matched circuits however by use of a suitable coupling arrangement and the process is referred to as  Impedance Matching ;   basically the trick is to find a device whose input impedance matches the impedance of the source-circuit but whose output impedance matches that of the receiving-circuit (the load).

  **  One type of matching device, the Transformer, has been discussed already in Lesson-5.   Note that the transformer cannot amplify signals because it lacks any form of auxiliary power. Thus, when a transformer steps-up the circuit voltage,  it must at the same time step-down the circuit current ;  an increase (or fall) in the circuit voltage accompanied by a decrease (or rise) in the circuit current corresponds to a change of impedance as the signal is transferred across the transformer from primary winding to secondary winding.

  **  In the example above a matching transformer required to transform the 10-volts of circuit-A to the 100-volts of circuit-B would require a turns-ratio  (n) of 1:10.   However the transformer is matching between a 10-ohm circuit and a 1,000-ohm circuit;   thus a turns-ratio of 10 produces an impedance-ratio of 100 and, in general terms, we can write that 

impedance-ratio  =   n²

  **  Matching can be achieved also by using networks of inductors and capacitors which are referred to generally as matching filters, matching sections or simply as networks.    It can be achieved also by using networks of resistors although these produce a reduction in the signal level, known as attenuation, that may sometimes be a disadvantage.   These resistive networks are variously called attenuators or pads and they may also be introduced into a system to provide only attenuation or to isolate circuits that otherwise would mutually interfere.  There are occasions when all three functions are indistinguishable.

      Detailed knowledge of filters and attenuators is not required for the RAE but their use to cure or to avoid interference is dealt with under 6.4: Measurements and EMC. They are also introduced under Transmitters.

      In general terms (i.e. not usually used in Amateur Radio) impedance matching can be achieved also by using a feedback-amplifier whose input and output impedances can be tailored to the required values;   these circuits are described in later Sections of this Course.

      There is a  Maximum Power-transfer Theorem  which states that the maximum possible power is transferred between circuits when their impedances are matched.  As described under 1.15: Internal Impedance this means that a maximum 50% of the signal power can be transferred; the remainder is dissipated in the source circuit.

1.12.7   Component Q-factor

  **  Although a "pure" inductor as defined draws a wattless current a practical inductor is wound with wire that has dc-resistance;   together with eddy-currents and stray couplings these cause losses.

      Given the experimental technique it is possible to measure each of these power-losses but, in general, such information is of little use.   Interest centres on the total power consumed by the inductor and so all losses are treated as though caused by dissipation as heat in a single theoretical resistor.

      One way of determining this total loss is to measure the relative phase- angle between the current and voltage waveforms.  A more accurate method is to use an ac instrument called a Bridge which, as already discussed, will show a reactance value in series with a resistance value (an Impedance Bridge) or as a reactance value in parallel with a resistance value (an Admittance Bridge). For a high-quality inductor the values of reactance will not differ appreciably between the two methods of measurement but that is not so for types with appreciable losses.

  **  An efficient (low-loss) inductor is indicated by either a low-value series resistor or by a high-value parallel resistor but, with either method of measurement, the ratio between resistance and reactance is the same.  This ratio is in effect a quality rating and is known as the Q-value.  As discussed above the ratio also describes the relative phase between voltage and current waveforms.

  **  So far the Q-factor has been defined in terms of an inductor but there is no reason why the idea should not be extended to a capacitor. Although a capacitor does not provide a dc path between its plates it does permit alternating currents to flow in its connections and also to flow  across the plate surfaces;   where current flows there is a heating effect which represents loss of signal power.  Additionally the dielectric dissipates a small amount of power. All these losses are usually amalgamated into a single (theoretical) resistor which again can be calculated as either a series resistor or a parallel resistor. Once again an efficient component is characterised by a high Q-value.

Q-factors are involved also in resonant circuits ; see next Section.

1.13  RESONANCE

1.13.1 Introduction

  **  Resonance occurs in electric circuits which contain both inductive and capacitive reactances ;   inevitably resistance must be present also but, although it modifies the behaviour of the circuit, it does not itself play a significant part in the basic phenomenon of resonance.

      Thus the charged capacitor is now free to discharge through the Inductor;   in doing so, it causes a current to flow in the reverse direction and so sets-up a new flywheel action to maintain that reversed flow.

      Once again, with the reverse current established, the inductor remains in control until all the circuit energy is again stored in the capacitor but this time with the polarity reversed.

      This cycle must repeat and would do so endlessly but for the practical reality of circuit resistance (and other small losses) which slowly convert the stored energy into heat which, in the form of infra-red radiation, is then lost from the circuit.

  **  This constant interchange of energy between two different storage media is the mechanism of resonance;  for electrical resonance it is the interchange between:

(a) static or stationary charge held in the electric field between the plates of a capacitor and
(b) dynamic or moving charge (an electric current) where the energy is held in the accompanying magnetic field.

      The rate at which this interchange takes place is controlled by the amount of storage available and clearly must be a function of both the capacitive storage and the inductive storage.  The larger the amount of storage so the longer is the period required to discharge one of them in the process of charging the other;  in other words, if either the capacitance or the inductance is increased,  the circuit must operate at a slower speed and there are fewer cycles of operation per second.

  **  For any combination of inductance and capacitance there is a preferred (or natural) rate of oscillation at which the interchanging action takes place and this, reckoned as the number of complete cycles per second, is known as the natural frequency or as the resonance frequency of the circuit.

      (It is common to see this frequency referred to as the "resonant"  frequency which implies that it is the frequency which is resonating.  The correct term is derived from the full expression " the frequency of resonance ". )

  **  Resonance frequencies like all frequencies are expressed in Hertz (Hz) where 1-Hz is defined as 1 cycle per second.   In an electrical circuit  the frequency at which a combination of inductance and capacitance resonates is given by the expression

f  =  1/(2π.√LC)

where f is in Hz (cycles per second), L is in Henrys and C is in Farads.

 NOTE   This formula is required for the R.A.E.

  **  The resonance frequency is found to be that frequency at which the inductive reactance is numerically equal to the capacitive reactance ;  if XL is equal to -XC  then the overall reactance in a resonant circuit is zero and so, at resonance, the current that circulates between the two components rises to a maximum value which is controlled only by the circuit resistance RL + RC .

  **  Thus the resonance frequency is that frequency at which it is most easy to cause current to circulate between the inductor and capacitor.

      It is worth remembering that resonance is not purely an electrical phenomenon but occurs in almost every aspect of everyday life.  The well-known leg is a pendulum hanging from the pelvic girdle and it is moved with minimum effort by swinging it at its natural frequency (see next Lesson).  Hence, in developing the technique of locomotion,  life forms varied their speed by changing the length of each stride rather than trying to change the striking frequency of the leg(s).

      When walking,  each leg remains in contact with the ground for the majority of the time ;  for two-legged locomotion, such as is used by humans, one leg supports the body while the other moves through to catch the body as it falls off the supporting leg.  In the trot only one leg remains in contact with the ground in a series of hops from leg to leg. In the run both legs lose contact with the ground over most of the running period as the runner leaps from leg to leg to gain the maximum possible length of stride.

      Quite simple tests will convince you that changes in the striking rate of the legs are brought about by bending the knees so shortening leg-length.

END OF LESSON 8

*  *  *  *  *  *  *  *  *  *  *  *

QUESTIONS

1. What is the purpose of using the letter j in calculations ?
    What is the approximate impedance of a circuit comprised of 7 + j7 ?

2. Why do we want to match one circuit to another ?
    Given one circuit of 50 ohms and another of 200 ohms what must be the turns-ratio of a transformer that will produce a satisfactory match ?

3. What is meant by the term Q-factor as applied to an inductor ?

4. Describe the basic mechanism of Resonance

5. What is the natural resonance frequency of a circuit which contains on inductance of 1 mH and a capacitance of 0.001 microfarad (i.e. 1/1000 Henrys and 1/1,000,000 000 Farad)

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LESSON  9

1.13.1 Introduction (continued)     

Examination of the mechanism responsible for resonance in a pendulum shows that its behaviour is very similar to that of the resonant electric circuit. Energy-transfer in the two systems is compared in Fig.23.

d the current-flow has created a magnetic field ;  see diagram (c).   In diagram (f) the pendulum is shown to have swung beyond the starting position and to have climbed to a second position of potential energy-storage.  By virtue of its mass and velocity it has, like the child's swing, been reluctant to stop at the bottom of its fall and has imitated the flywheel-like action of the inductor. This reluctance to stop is referred to as momentum.

      The similarity between the electrical circuit and the mechanical pendulum is very real and it is common practice to solve mechanical design problems by drawing an equivalent electrical circuit in which inductors, capacitors and resistors represent mechanical parameters such as mass, compliance and loss.  The problems involved in the design say of a loudspeaker can thus be solved using standard ac theory.

** That the resonance frequency is the frequency at which things happen most easily can be deduced by inspecting the impedance expressions derived in the previous Section.   It was said that inductive reactance is written as ωL and capacitive reactance as -1/ωC ; in any circuit which involves both kinds of reactance there results the combination ωL - 1/ωC (or XL - Xc).       Figs. 17 & 19 show that XL  increases with frequency while XC decreases;  there has to be a point where the two curves intersect thus showing that,  at that (resonance) frequency, the reactances have the same value and so cancel. At this frequency therefore the value of the current is determined only by the circuit resistance.       Fig. 22 showed a practical inductor and a practical capacitor connected together. The two resistors RL and Rc represent the losses of the two components which consist mainly of so-called ohmic losses that arise from the passage of current through their dc- resistance.  In practice the losses associated with capacitors are low relative to those of inductors (it is said that the capacitor-losses are swamped by the inductor- losses) and it is legitimate either to ignore Rc or to combine it with RL.   This leads to the more practical representation of Fig. 24(a).

** Impedance is measured with an instrument called a Bridge which, being without intelligence, has to be instructed as to whether the load presented to it is a series combination of components or a parallel combination.    Fig.24(b) shows the arrangement of diagram (a) broken open so that measurements can be made as a series circuit ;  note particularly that a common-current flows through all three components and also through the Bridge.

      There is a difficulty here;  in diagram (b) the Bridge is clearly presented with a series combination of L, C and R but in diagram (c) it is presented with a mixture of series and parallel components.  If told that arrangement (c) was a parallel circuit the Bridge would interpret its measurement results in the form shown in diagram (d).

      The fact that both diagram (b) and diagram (d) can represent the same combination of components is a bit startling but it is a most useful result. To convert from one to the other form is a straightforward mathematical process known as a series-parallel transformation but it is not required for the R.A.E.   A point to remember however is that such transformations are valid only for the particular frequency for which they were calculated.

Both Rs in the series arrangement and Rp in the parallel arrangement represent the circuit losses which means that, whatever their calculated or measured values, they must dissipate equal powers.  They differ in that Rs carries the common-current - the circulating current - that circulates between inductor and capacitor while Rp has its own private current which is driven by an emf that is common to itself, the inductor and the capacitor.

** The circulating current is controlled by the small-value Rs and so a large current can be expected.  This large circulating current must set-up large p.d.'s across the reactors but, because the reactances cancel (i.e. the voltage waveforms cancel), it is only the applied emf that appears across the series resistor.

**  When the circuit is measured as a parallel combination the very large p.d. across the L and C becomes apparent but the overall current (the make-up current) is small because cancellation now occurs between the currents which flow through the inductor and the capacitor.

** Thus a series measurement shows the losses as a large resistive current with a small voltage which requires a low-value resistor. A parallel measurement shows the losses as a small resistive current with a large voltage which requires a high-value resistor.

When measurements are made as shown in diagram (b) an LCR circuit is referred to as a series-resonant circuit;  when measurements are made as shown in diagram (d) an LCR circuit is referred to as a parallel-resonant circuit.  The two "arrangements" are essentially the same circuit but in their applications they are very different.

1.13.2   The Series-resonant Circuit

      When an LC resonant circuit is used in the series mode the alternating circulating current is determined according to Ohm's Law by the applied emf and the overall impedance at that particular frequency.   In the general form this is given by

I  =  E/Z  =  E/(R + XL - XC)

      At the resonance frequency, where the inductive reactance equals the capacitive reactance,  this reduces to E/R.  When the frequency is reduced below the resonance frequency the capacitive reactance Xc  is increased but the inductive reactance XL is reduced and so the two reactances no longer cancel.  Similarly if the frequency is increased above the resonance frequency the capacitive reactance decreases but the inductive reactance increases;  it is only at the frequency of resonance that complete cancellation is possible.

  **  Fig.25 is a graph on which is re-plotted the inductive and capacitive reactance curves of Figs. 17 & 19  together with a curve that represents the difference (XL -  Xc:).  This shows that the impedance is reduced to Rs  alone at resonance with the curve rising toward the capacitive-reactance curve at lower frequencies and toward the inductive-reactance curve at higher frequencies.

** Because, at resonance, the series impedance falls to its lowest value then the circulating current must reach its highest value;  as already discussed the p.d. developed across each reactor must also reach its maximum value. Note that the impedance curve is not symmetrical because the two reactance curves have different forms.       Although at resonance the series circuit appears to be purely resistive and to present its lowest impedance it is important to remember that nevertheless it does contain reactors and that there is a relatively large voltage across each of them.   If the given inductor and capacitor are removed from the circuit and replaced by components of the same value but of higher quality (i.e. with lower losses) then the value of the equivalent resistor Rs. is reduced and so the circulating current is increased ;   in turn this leads to an even higher p.d. developing across the reactors.   With components of even reasonable quality these voltages can be 10-times the value of the applied emf and are often much greater.

WARNING    When dealing with transmitters these voltages can be lethal.

  **  This voltage-multiplication action is one of the most useful properties of resonant circuits.   If an applied Source delivers a mixture of signals at different frequencies (as does a receiving aerial) then that signal whose frequency is closest to the resonance frequency develops a much larger voltage across both the inductor and the capacitor than do the other signals. This provides a means of selecting a wanted signal from amongst many. The action of adjusting a circuit so that its resonance frequency equals the frequency of a wanted signal is known as tuning the circuit to that signal and so these resonant circuits have acquired the name tuned circuits.

  ** A practical arrangement which enables a wanted signal to be selectively enhanced before being passed to a receiver is shown in Fig.26(a) ;  the arrow drawn through the capacitor symbol indicates that the component can be adjusted mechanically to provide a means of tuning the circuit. Similar tuning arrangements can be provided by varying the inductor but, as a general rule, it is less easy and more expensive to achieve than the variable capacitor.

Fig. 26   Use of Resonant Circuits In Aerial Arrangements

(a) series-tuned Aerial Transformer
  (b) parallel-tuned Aerial Transformer
                 (c) Rejector and Acceptor Traps or Notch Filters

Note that, although the wanted signal is magnified relative to unwanted signals, this is not an example of amplification;  that term implies an increase in signal power and requires the use of an auxiliary power source.

      The degree of magnification obtained is determined by the current-flow at resonance compared with the current-flow at off-resonance frequencies. The current-flow at resonance is controlled by the circuit resistance,  the current-flow below the resonance frequency is controlled by the capacitive reactance and the current-flow above the resonance frequency is controlled by the inductive reactance.

  **  Signal magnification therefore depends on the ratio of reactance to resistance and this was the criterion used to define the quality (the Q-value) of a reactor. Thus the performance of tuned circuits is also given in terms of Q.

      In general the losses connected with capacitors are negligibly small and so the magnification of a resonant circuit is usually taken as the Q-value of the inductor.   For example:  a resonant circuit which contains an inductor with a Q of 10 would be expected to provide a signal magnification of 10 times at the resonance frequency.  In most practical circuits a Q of 10 is about the lowest likely to be acceptable although, in the final stages of a transmitter, the output circuit may have a Q as low as 2 because it is required to be "lossy" - it is required to lose its power to the aerial

      A very important point which has implications in receiver design and also in oscillator circuits arises from the tuning arrangements outlined above. Suppose for example that it is required to raise the resonance frequency of a tuned circuit;  the value of the capacitor is decreased so that its reactance at the new frequency will equal the higher reactance of the inductor at that new frequency and this adjustment will not appreciably alter the value of the circuit losses Rs.   However, because the reactances have increased while the value of Rs is unchanged, the circuit Q-factor has been increased.

  **  This means that the magnification of a variable tuned-circuit varies as its resonance-frequency is adjusted.

** An effect of much greater importance however is shown in Fig.27  where the magnification of resonant circuits is shown as it varies with changes of frequency. At low values of Q E the magnification is almost independent of frequency and we say that the selectivity of the circuit is low.   At high Q-values the magnification peaks significantly at the resonance  frequency and such circuits are said to offer high selectivity.    **  If the  Q-value of adjustable circuits varies with frequency then clearly the magnification also varies with frequency and this is a serious handicap for equipment which is required to operate over a band of frequencies.  For this reason the simple arrangement of Fig.26(a) is seldom found by itself ;  see under Bandwidth in the next Section and also in the Lessons dealing with Receivers.

  ** Series-tuned circuits are characterised by a large flow of current at the resonance frequency and they are regarded as being willing to accept signals at that frequency while rejecting all others.  From this they have acquired the name Acceptor Circuits .  Their stock-in-trade is a reduction of impedance at resonance.

END OF LESSON 9

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QUESTIONS

1. What particular characteristic of a reactive circuit indicates that it is operating at its resonance frequency ?

2. Explain in simple terms the difference between a so-called series-resonant circuit (or Acceptor circuit) and a so-called parallel-resonant circuit (or Rejector circuit).

3. At what frequency will a 2 μH choke resonate with stray capacitances measured as 10 pF ?   See Lesson-8.  (This question tests your ability to work with powers of 10 and is the type that can earn easy marks in the exam.   If you have problems then please ask for help.)

4. Why are very large voltages present in tuned circuits which are handling power (such as the final stages of a transmitter) ?

5. What is the basic purpose of an aerial transformer as used in a receiver?

6. Why is a simple tuned-circuit arrangement not very satisfactory in equipment which is required to operate over a fairly large band of
frequencies ?

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LESSON  10

1.13.3 The Parallel-resonant Circuit

     An alternative way of measuring a resonant circuit is simply to connect a Bridge across the combination of components as shown in Fig.24 (c). From the measurement point of view the Inductor, the Capacitor and the (theoretical) Resistor are now seen as being connected in parallel as shown in Fig.22(d).

  **  When a bridge is connected in this manner it is no longer possible to determine the common circulating current and only the common voltage across the components can be measured.  To the Bridge the capacitive and inductive currents now flow in parallel and, as already discussed, they cancel (exactly at the resonance frequency).   Thus the bridge measures a relatively small current which is known as the make-up current ; this represents the losses in the circuit.

  **  In the series circuit discussed in the previous Section there is a common current and the two reactive voltages differ;  in this parallel circuit there is a common voltage and it is the two reactive currents that differ.  As the frequency of resonance is approached so the current-canceling action becomes total and the make-up current consists of that which flows through the resistor alone.  Hence, in this circuit, resonance is indicated by a minimum value of current which is interpreted as a maximum value of impedance.

  **  Where the series circuit accepts signals at or close to the resonance frequency, this parallel circuit rejects such signals and is accordingly known as a Rejector Circuit.

  ** Remember however that, although the bridge cannot measure the current which circulates between the Inductor and Capacitor,  that current nevertheless still flows and it still reaches it maximum value (controlled by the effective series resistance) at the resonance frequency. Thus the parallel circuit rejects current at resonance but develops a magnified voltage at that frequency. The use of such a circuit in selecting a wanted signal from an aerial is shown in Fig.26 (b).

  **  The impedance presented by an aerial may be relatively-low or extremely reactive so that it either reduces the Q of the tuned-circuit or  de-tunes it. This effect is off-set by using the tuned inductor as an autotransformer (see Lesson-5) and so the two arrangements of Fig.26 (a & b)  are perhaps less different than it may appear.

 ** Fig.26 (C) shows another use for both acceptor and rejector circuits ;  here they are used as traps to reduce the passage of powerful unwanted signals. A rejector connected in series with the aerial feed reduces the level of signals at or close to the frequency to which it is tuned ; an acceptor connected across the terminals which feed the receiver tends to short-circuit signals at or close to the frequency to which it is tuned.

      Because these circuit arrangements result in a "hole" in the range of receivable frequencies they are also called notch filters.

  ** The two different methods of measuring resonant circuits in fact result in slightly different resonance frequencies and reference books will be found to give different formulae.  In practice however when dealing with circuits of

reasonable Q-value the difference is negligible and for all practical purposes the standard formula, as already given, is

f   =  1/(2π.√(LC)

where f is in Hertz, C is in Farads, L is in Henrys.

1.13.4   Dynamic Impedance

      The impedance thus presented by a rejector circuit at resonance is known as its dynamic impedance;  at resonance the impedance should be exactly resistive and so is sometimes referred to as the dynamic resistance.

      With the application of Ohm's Law the dynamic impedance is obtained by dividing the applied voltage by the make-up current which, in turn, depends on the Q-factor.  A low Q implies that the components are lossy and that perfect cancellation between the reactive currents is not possible;   in turn this means that the make-up current is not small and so the dynamic impedance is low.

      With similar reasoning a high-Q circuit presents a large dynamic impedance.

      As already discussed the Q of a resonant circuit varies with

(a)   frequency - because the ratio of resistance to reactance changes

(b)   the value chosen for the inductor/capacitor (L/C) ratio combination because this too changes the ratio of resistance to reactance ;  the resonance frequency is determined by the product L x C.

Thus the value of the dynamic resistance can be controlled by the Designer because he is free to choose the combination of  L and C to suit the value of resistance that is imposed by the circuit losses.  He is also faced with a compromise decision because the ratio X/R determines the circuit Q which controls the band of frequencies over which the tuned circuit operates.

      This matter is usually dealt with in a somewhat rule-of-thumb manner by considering the ratio of inductance to capacitance (i.e. L/C) which is called the L-C Ratio ;   a high L/C ratio results in a high dynamic resistance.

See also the later Sections on Bandwidth, Bandpass Circuits, Amplifiers, Oscillators, Receivers and Transmitters.

1.14   BANDWIDTH

1.14.1  Definitions

  ** The term bandwidth refers to the band of frequencies over which a particular piece of equipment will operate satisfactorily.  Bandwidth is limited by a number of factors:

** The coupling capacitor in Fig. 28(a) is inserted to separate an ac signal from dc with which it is mixed but at low frequencies the rising reactance of the capacitor reduces the wanted signal also.  **  In Fig.28(b) a transformer is used to separate the wanted signal from a dc.  However the transformer windings appear as Inductances connected across the signal path and, as the frequency decreases, so the falling reactance diverts signal current away from the intended destination.   ** (ii)  Another cause of loss at low frequencies is the increase of feedback through the failure of by-pass capacitors (see Part 2 Electronic Amplifiers. (iii)  At the high end of the frequency range stray capacitances, shown in Fig. 28(a) in broken line, provide increasingly-effective alternative paths for signal currents which therefore fail to reach their intended destination as the frequency increases.   **  (iv) In designing transformers it is very difficult to ensure that all the magnetic field from one winding interacts with the other winding and this "lost" magnetic coupling is referred to as leakage inductance.   As shown in the equivalent circuit (C) leakage inductance appears as an inductive reactance in series with the signal path and this limits the high-frequency response.   ** (v)  In tuned (radio-frequency) amplifiers the bandwidth is determined solely by the characteristics of their resonant circuits ;  see under  2.6  and  under 1. 13 in the previous Lesson.

Bandwidth however, expressed as a lower-to-upper frequency range, can be a deceptive quantity.  For example:
an audio amplifier ranges from                         50 to 15,000 Hz
a video (television) amplifier ranges from           0 to 5MHz
a radio-frequency amplifier might range from 100 to 120 MHz

In the order shown the bandwidths may appear to increase from 15 kHz  to 20 MHz yet the difficult tasks are to extend the bandwidths of the audio amplifier and to restrict the bandwidth of the rf amplifier.

** Another way of expressing bandwidth is to give the upper and lower limits as variations on the centre frequency.  Thus the rf amplifier, which has a centre frequency of 110-MHz and extends both upward and downward by 10 MHz,  can be described as having a bandwidth of 110 ± 10 MHz.  Alternatively this amplifier is said to have a relative bandwidth of (approximately) 9% which shows the frequency range as a fraction of the centre (or average) frequency.

** Again there is a difficulty in that at 1 MHz a relative bandwidth of 10% is only 100 kHz which is at variance with the same relative bandwidth at 100 MHz.  Clearly, as the centre frequency is increased, so it becomes ever more difficult to restrict the relative bandwidth.

      As a description of performance in audio and video amplifiers the term relative bandwidth becomes something of a joke. For example, the video amplifier would be said to cover a range 2.5 ± 2.5 MHz ;  expressed as relative bandwidth this becomes +100% - ??% .

      Where, as in video amplifiers, the bandwidth approaches zero frequency (or dc) then bandwidth is described in terms of the number of octaves covered.    Mathematically an octave describes a frequency interval in which the upper limit is twice the lower frequency-limit;  to the Human ear an octave is not exactly 2:1.  For the above audio amplifier the bandwidth is approximately 8 octaves;  for the video amplifier the practical bandwidth is about 17 octaves;  for the rf amplifier the bandwidth is about 0.2 octaves.  This method of expressing bandwidth makes clear the difficulty of  extending the range of audio and video equipment and of compressing the range of rf amplifiers.

      Apart from the limitations described above all three methods are valid and the choice is governed by the particular requirements of the moment.

      It must always be remembered however that a given bandwidth is purely a matter of definitions because electronic equipment does not suddenly cut-off at the edges of the band.   For amplifiers it is usual to define the band as the range between those frequencies where the gain has dropped to 0.7 of the maximum gain (known as the 3-dB points).  For aerials the bandwidth is usually defined in terms of the standing-wave ratio (see under 2. 3: Transmission Lines) which is a measure of the accuracy to which an aerial system is tuned and matched.

1.14.2   The Significance of Bandwidth

  ** The sinewave (see Trigonometry in Part 7 : Mathematics) is the simplest, indeed the purist, of repetitive waveforms.  All waveforms which repeat can be broken down into a series of sinewaves which are harmonically-related to the repetition frequency (or frequencies).  The repetition frequency is called the first harmonic or the Fundamental,  the frequency which is twice that of the Fundamental is called the second harmonic, the frequency which is three times that of the Fundamental is called the third harmonic and so on.   For any one specific repetitive waveform each harmonic has a specific amplitude and a specific phase relative to the Fundamental.

      The process of deriving these component sinewaves from a given repeating waveform is known as Fourier Analysis after the French mathematician who first derived it but knowledge of this is not required for the R.A.E.

  **  It is the presence of these extra frequencies that makes it necessary to consider bandwidth in the performance of any equipment.  In general the ear is not fussy about phase relationships and so, in audio-amplifier design,  phase-distortion (disturbance of phase relations) is not directly of concern;  the important thing is to preserve the relative amplitudes if the original sound is to be reproduced faithfully.  However, when feedback is  used in an amplifier, phase-distortion must be controlled else negative-feedback turns into positive-feedback with unfortunate results. (Note that the amplifier must be well designed before feedback can be applied.

      The eye is not so accommodating and, in video amplifiers, it is essential to preserve both the amplitude and phase relations.

  **  In general rapid phase shifts are associated with rapid changes of gain as frequency is varied and a design which eliminates too-rapid changes of gain at the edges of the pass-band should not suffer problems from phase-distortion,

  ** In communications engineering, and especially applicable to Radio Amateur working, bandwidth is an important consideration in limiting interference which a transmitter may cause to other installations. A signal which varies its amplitude slowly and only by small amounts generates few harmonics and they have relatively small amplitudes;  a signal which varies rapidly and by an appreciable amount generates a large number of large-amplitude harmonics.

  **  In particular those signals which change amplitude abruptly contain high-order harmonics which can have very large amplitudes and intrude into a large slice of the frequency spectrum.  Thus to keep interference to a minimum - to restrict the frequency-band over which the signal spreads - the basic requirement is to ensure that signals do riot change too rapidly.

  ** Of course, when transmitting in Morse code, it is not possible to either close or open a key slowly;  when speaking into a microphone it is not possible to avoid those sharp explosive sounds (expletives) which cause transients.   These matters are dealt with electronically by "restricting the bandwidth";   when high-order harmonics are removed then sharp-edged waveforms become softened.

  ** For W/T working it is mandatory to fit so-called key-click filters which prevent a too-rapid change between transmitter-on and transmitter- off.   For R/T working the audio amplifier must be suitably restricted in its bandwidth to ensure that it neither passes nor produces those high-order harmonics. See also under 4.2.5 Low-pass Filters.

1.14.3   Bandpass Arrangements

      In radio-frequency amplifiers tuned circuits are used as impedances and these have a bandwidth which is determined by the circuit Q-factor ;  see under 1.13, 4.2 and 4.6 .   High-Q circuits are relatively loss-free and produce high dynamic impedances but they restrict the bandwidth.  Low-Q 

circuits are lossy and produce low dynamic impedances but they offer wide bandwidths.

      Apart from the obvious compromise required there is a further complication mentioned under 1. 13. 2 that the circuit Q - and therefore the bandwidth - varies as tuning adjustments are made.  One expensive cure for this problem is to arrange that the tuning control varies simultaneously both the capacitor and the inductor.  If the trick is made to work then the LC product is varied to change the frequency while the L/C ratio is varied to correct the bandwidth and/or the dynamic impedance.

      A cheaper method is to use so-called bandpass circuits;  see under 2.7.3 .  Basically there are two types;

      In the first of these the signal is passed through a succession of resonant circuits whose resonance frequencies are staggered across the required band. The alignment of such circuits is not easy but they can offer a substantially constant response and they are relatively independent of Q-values.   Because most of the tuned circuits are not operating at their resonance frequencies, and therefore not providing their maximum magnification, the signal suffers attenuation as it passes through.  These are known as stagger-tuned circuits.

      The second type is the bandpass-coupled circuit in which two or more resonant circuits, each tuned to the centre frequency of the required band,  are coupled together.  The coupling may be magnetic (transformer) or via a common impedance which can be either inductive, capacitive or a combination of both.   A common-resistor coupling is possible but this lowers the Q.

In these circuits not only is energy circulated between the capacitor and inductor of each resonant circuit but it is also interchanged between the coupled circuits. The effect is to produce a constant response over the required bandwidth but, with each circuit providing maximum magnification,  the signal suffers less attenuation.  The Q-values of the resonant circuits must be well matched.

      Circuit Q-values are dependent on circuit losses and these are not easily predicted;  it is usual to design the resonant circuits with too-high a Q and to add damping resistors as necessary.

      Although bandpass circuits produce a substantially-constant response over a band of frequencies the price exacted is that the response falls rapidly at the edges of the band. The resultant phase-distortion could cause problems but these steep skirts in the response curve is a desirable feature in receivers where they effectively remove unwanted signals.

  **  The nature of bandpass circuits means that they are only really practical for fixed-frequency working;  to make them tunable over any useful range requires constant control of Q, dynamic impedance, bandwidth and coupling factors. The problem is overcome in radio receivers by using the superheterodyne principle;  in such receivers the majority of the circuit operates at a fixed frequency (known as the Intermediate Frequency or IF)  and any wanted incoming signal is converted to that frequency in a frequency-changer stage. See also under 2. 7 and  4.6.

END OF LESSON  10

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QUESTIONS

1. An Inductor and a Capacitor are connected together. Say briefly why this arrangement is often considered as two different circuits ?/p>

2. Why and when would you beware of dangerously high voltages in such an arrangement?

3. There is a requirement for a trap to attenuate a signal at a frequency of 1-MHz in a circuit which operates at an impedance of 1,000 ohms.  Two combinations of L and C are given below both of which resonate at (approx) 1 MHz ;  state which you would choose to form a high-Q trap.

(a) 0.003 μF and 8 μH each offer 50 ohms at 1-MHz 
(b) 5 pF and 5 mH each offer 30,000 ohms at 1-MHz.
 Give the Q-factor you would expect to achieve.

(Think carefully about this one - you are unlikely to meet this in the R.A.E.)

4. What is the purpose of a notch filter ? 5. What problems arise from attempts to make a resonant circuit adjustable to a range of resonance frequencies ?

8. Constant performance over a given bandwidth is achieved by use of bandpass circuits. What are the major advantages and disadvantages of using either stagger-tuning or coupled circuits ? (Not for R. A. E.)

9. Variable-tuning circuits are a particular disadvantage in receiver design. How is this problem overcome?

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LESSON  11

1.15     INTERNAL  RESISTANCE

1.15.1 Introduction

      Under 1.3.3. in Lesson-1 it was shown that a generator cannot deliver current until its terminal voltage has fallen because current cannot flow between points of equal potential;  the terminal voltage falls according to the current-demand. This phenomenon introduces a problem because the current that will flow through a given load depends on the value of the terminal voltage while that voltage itself depends on the current-flow.

  **  The solution to this problem is elegant and yet absurd. A circuit is invented which behaves in exactly the same manner as the practical  generator but which uses only a theoretically-perfect generator.  Such a circuit (which cannot possibly exist in practice) is known as an equivalent  circuit.  The generator used in this equivalent circuit is theoretically untroubled by problems of internal equilibrium and so its terminal voltage does not fall when current is demanded.  To form the equivalent circuit the theoretical generator is degraded (i.e. made normal) by adding theoretical losses until the performance of the combination matches that of the practical generator.

      The idea of the perfect generator causes the problem to split into two separate solutions which, in fact, are but two ways of looking at the same phenomenon.  Either the perfect generator can be made to supply any required current without its terminal voltage changing or it can be made to supply any required voltage without the current changing.  It cannot be asked to keep both parameters constant because that would severely limit the usefulness of the generator.

      Perfect generators of the first type are called constant-voltage generators (or cv-generators) and the second type are called constant-current  generators (or cc-generators)

1.15.2  The Constant-voltage Generator

Fig.29  shows a cv-generator (note the symbol) connected through a resistor labeled rint to the output terminals of a practical generator;  a load resistor RL is shown connected externally. The resistor rint which is shown "inside" the generator is always known as the internal resistance of the practical generator.       Without the connected load the supplied current I must be zero and so the p.d. developed across rint must be zero also.  Under no-load conditions therefore the voltage at the terminals of the practical generator is that of the constant-voltage generator. This voltage is referred to as the emf (electromotive force) of the practical generator and is given the symbol E.  The symbol V is reserved for the terminal voltage of a generator when current is flowing and also for the p.d. developed across a load (caused by the transfer of electrons).

      The purpose of the resistor r_int in the equivalent circuit is to ensure that, as current is demanded, so the terminal voltage (of the equivalent circuit) falls in imitation of the terminal voltage of the practical generator which it represents.

  **  The only way to assign a value to r_int  is to measure it experimentally;  the values of V are noted for different values of current-flow when the value of  r_int can be found using Ohm's Law as

r_int   =   change of V
            change of  I

 **  The value of E can be measured at the terminals with zero load current thus, once the value of r_int has been measured, it is possible to calculate the current for any given value of load resistor (RL) from the equation

E  =   I x (RL  +  r_int)

(the current multiplied by the total resistance around the circuit).

  ** Once the value of I has been determined it is possible to calculate the p.d. across the internal-resistance;  the terminal voltage is then found by subtracting that p.d. from the emf  E.   Alternatively the value of V can be found directly by multiplying  RL x I .

  **  Yet again the two resistors RL and  r_int  form a potential divider across the emf;  part of the total voltage appears across r_int  and the  remainder appears across RL.   From this it is an exercise in fractions to deduce the voltage across the load

V  =  E  x  RL/(RL  +  r_int)

  **  For generators which produce ac the internal resistance becomes the internal impedance which has the complex form (r_int  +  jX_int); see  Lesson-8.    It is important to remember that, although the effect of internal resistance is real enough, its existence is pure fiction - a mathematical convenience.  For active systems such as amplifiers r_int may be referred to as the output impedance or as the source impedance of the system.

      (The same idea in reverse leads to the concept of input impedance which describes the current which flows into a device as a result of applying a given input voltage.)

      In separating the cv-generator from the failings of the real generator it has been stipulated that the cv-generator shall have an internal-resistance of zero; this is the reason its terminal voltage does not fall under load.   It is this aspect of the equivalent circuit which is absurd.  The Maximum Power-transfer Theorem states that a generator delivers maximum power into a load which exactly matches its internal impedance; this  cv-generator therefore must 

deliver its maximum power into a short-circuit (zero ohms) and zero output power does not make it an ideal power source.  Nevertheless the trick provides a legitimate way to calculate the performance of circuits and that is a prerequisite for design work.

      Consider again the equality above in which the e.m.f. is equated with the product I  x  (RL + R_int) ;  in mathematicians' language we say that, if the value of r_int  is small compared with the value of the load, then r_int can be neglected .  In that event the terminal voltage does not fall when current flows.  We say that r_int is swamped by the load;   it has negligible effect on the current.

      In practice therefore, as long as a load is considerably greater than the internal resistance of the generator, then the generator tends to behave as though it were a perfect constant-voltage generator (zero internal impedance).   This is an important consideration in circuits where it is required to maintain an undistorted voltage-waveform ;  before pursuing the matter further however it is best to consider the alternative equivalent circuit which arises when the internal-resistance is the greater and the load becomes negligible.

1.15.3 The Constant-current Generator

      Fig. 30 shows a cc-generator (again note the symbol) which is connected directly to the output terminals of the practical generator which it represents.   Be reminded that this theoretical generator produces an unvarying current without the intervention of an emf. 

    For the internal resistance to be much larger than any load it must have an infinitely-large value. But an infinitely-large value means that r_int is not there and the generator would be required to produce an output when it is not connected to the load.

**  The solution (in this theoretical circuit) is to place the internal resistor across the generator ;  its function now is to absorb, or to divert, any of the constant-current that the load cannot accept. This perhaps odd arrangement is perfectly legitimate because we are trying to invent an equivalent circuit that will exactly imitate the behaviour of the real circuit.       In this equivalent circuit the distinction between E and V is meaningless because this theoretical generator produces a current (as distinct from producing an emf) . The current from the generator (I) divides according to the relative impedances of the two available paths.       Using the parallel version of Ohm's Law the load current IL is this total current I multiplied by  rint/ RL +  rint) . If RL is negligible then all the current is flowing through the load. The voltage that is developed across the Load is the product  I  x  RL

      In practice as long as the internal resistance is much larger than the load then the generator tends to behave as though it were a perfect constant-current device with infinite internal impedance.  This is an important consideration in circuits where it is required to maintain an undistorted current waveform.

The concept of a cc-generator is in fact as absurd as the cv-generator as can be seen by again referring to the Maximum Power-transfer Theorem. This generator must deliver its maximum power into a load that is not there but again it provides a means of performing circuit calculations.

1.15.4   Voltage and Current Sources

      Any practical generator behaves either as a cv device or as a cc device determined by the circuit into which it is driving.   A signal source is referred to as a voltage source if its operation tends to that of a cv- generator because its output impedance is low in comparison with the load when variations in the load impedance do not affect the voltage waveform.

      The same signal source would be referred to as a current source if its operation tended to that of a cc-generator because its output impedance was high in comparison with the load ;  variations in the load impedance would not affect the current waveform.

      A practical application of this is shown in Fig. 31 where a variable resistor is connected as a pot'meter to vary the drive to an amplifier stage.

      Behaviour of a valve is determined by the voltage-waveform which is impressed between its grid and cathode but the similar behaviour of a (bipolar) transistor is determined by the current which is injected into its base-emitter path.

Diagram (a) shows that the input to a valve has the nature of a capacitor and the reactance of this reduces as the frequency is increased. To avoid distorting the voltage waveform at the grid of the valve it is imperative that it be driven from a source whose internal impedance is low relative to the overall impedance between grid and cathode. The gain control must therefore be a potential divider which taps-off a portion of the applied voltage.

      Diagram (b) shows that the input to a transistor offers a dc path and so the device can be expected to draw current from the source. What the diagram fails to show is that the resistance of the base-emitter path depends on the magnitude of the applied signal-current and so the input resistance varies in sympathy with the signal waveform. To avoid distorting the current waveform between base and emitter it is imperative that the transistor is driven from a source whose internal impedance is relatively high. The gain control for this simple amplifier therefore must be a current divider.

      As shown the signal is split between two alternative paths and the pot'meter is connected as a differential control which shunts unwanted signal away from the transistor.  The series resistor R_int is inserted as necessary to prevent the pot'meter short-circuiting the source at minimum-gain setting.

      The arrangement of (a) is sometimes seen controlling a transistor stage and the error has two consequences:

(a) operation of the control to increase the gain causes the gain first to rise and then to fall

(b) the transistor is driven from a source-impedance represented by the lower half of the pot'meter ;  consequently at low-gain settings the drive is converted to cv operation and distortion ensues.

      Note however that the application of feedback to the transistor stage may convert it effectively to voltage-operation and so use of arrangement (a) would be correct.

1.15.5   Practical Significance of Internal Impedance

      The concept of internal impedance was introduced to make it possible to perform circuit calculations.  To demonstrate its impact in the practical world consider a unit intended for test purposes which is known as a Signal Generator (or S.G.) . This produces a signal at a frequency which can be adjusted and also at a level which can be adjusted accurately over a range of values.

      From what has been said above it is clear that the actual signal which the test equipment injects into a load must depend on both the input-impedance of that load and the output-impedance of the S.G. and this could make nonsense of any attempt to calibrate the output level of the generator.  In practical work all generators are considered as voltage generators unless otherwise specified and so manufacturers state S.G. outputs in terms of an emf and internal resistance.

      For example it might be given as an emf of 1 volt at 50 ohms.   Because an emf' is stated we know that, connected to a 50-ohm resistor, this generator will provide an output of 0.5 volts when all the output attenuators are set for zero attenuation (maximum output).   It is a bit inconvenient to work relative to 0.5 volts and so most S.G.'s are built with an emf of 2 volts.

      Sometimes a generator may be quoted as "delivering 1-volt across 50 ohms";    this is a statement of terminal voltage when the Instrument is correctly matched and so we can deduce that it affords an emf of 2-volts with an Internal resistance of 50 ohms.

      However not all equipment offers an input impedance of 50 ohms and it is an expensive hobby to keep a flock of S.G.'s to provide a range of output impedances.  It is perfectly legitimate to connect a 50-ohm generator to say the 75-ohm input of a receiver -  the Designer will have ensured that this does not affect the operation of the device - but the Operator must re-calculate the voltages that will appear across the receiver  input-terminals. This is a simple matter ;   go back to the value of the emf, note the output impedance (r_{int )} and call up the equation given above wherein

V  =  emf   x   RL/(RL + r_int )

Once this value has been established for any convenient setting of the output-attenuator(s) all other outputs are changed in the same ratio.

      An alternative is to connect the S.G. to the receiver via a matching pad which is a network of resistors whose input resistance matches  r_int and whose output resistance matches the input resistance of the receiver.  Of course allowance must be made for the attenuation caused by the pad which is most easily done by directly measuring the voltage across 50-ohms without the pad and the voltage developed across (say) 75-ohms with the pad.

1.15.6   The Constant-resistance Network

      In the above discussion the internal impedance of the S.G. has been assumed to be a pure resistance but the argument is not in any way affected if that internal impedance is complex.  Most definitely however that is not true for a practical situation ;   if the internal impedance is not resistive it is imperative to know its exact value for all possible output frequencies so that the true terminal voltage can be calculated.   To prepare a full calibration is not practicable.

      Fortunately it is possible to make the output impedance substantially resistive for all frequencies by adding corrective components which turn the output circuit into a constant-resistance network.   The underlying mathematical derivation is not required here but the circuit arrangement is shown in Fig. 32.   For example, if the output is measured as an inductor in series with a resistor,  then that output circuit is shunted with a capacitor in series with a resistor of the same value such that

L/C   =  R²

where L is in Henrys, C is in Farads and R is in Ohms.  When this is done correctly then the output impedance is resistive at all frequencies.

      Such a correcting network is built into the instrument which is why its r_int can be quoted as (say) 50-ohms. (Note that the equation does not include a frequency term;  i.e. neither w nor 2πf  appear

      This same circuit trick is called upon to obviate the effects of reactances where they become a nuisance.  For example electrical equipment causes radio interference because sparks are produced when an inductive circuit is interrupted.  When the device is incorporated into a suitable constant-resistance network the circuit inductance is neutralised and interruptions of the current no longer cause sparking and consequent radiation.

1.15.7   The Maximum Power-transfer Theorem

      This theorem states that a source delivers the maximum possible power to a load which has exactly the same value as the source's internal impedance;   i.e. to a properly-matched load.

      With reference to the cv equivalent circuit of Fig. 29 the current which flows through a voltage generator and its load is given by E/ (RL + r_int ) which is mathematical talk for emf divided by the total circuit resistance. The power dissipated in the equivalent circuit comes in two packages:

(a) power in the load  =  I^2.RL

(b) power in the internal resistance = I².r_int  

When the load has a very low value the voltage developed across it becomes very small and the power it dissipates becomes negligible. When the load has a very high value the current which flows through it becomes small and so once again the power dissipated by the load becomes negligible.

      Somewhere between these two extremes the power dissipation rises to a maximum value. For those who enjoy algebra it is possible to show that this maximum occurs when RL  = r_int  .     If load and internal-resistance have the same value (they are matched) then they dissipate equal powers and so it follows that a generator can deliver only half the. theoretical generated-power and only then when the output is properly matched.

END OF LESSON  11

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QUESTIONS

3. Constant-voltage and constant-current generators are both theoretically perfect generators. In practice the concept gives rise to both voltage-sources and current-sources.   In what way - if any - do these two sources differ. (Not a question for R.A.E.)

4. A battery is to be used in the design of a circuit but its internal resistance is not known. On test it yields the following figures:

5. The internal impedance of a generator is given as 2 ohms and its open- circuit terminal voltage is 10 volts.
When a lamp is connected to the generator the terminal voltage falls to 8 volts. What is the power dissipated by the lamp?

6. The input impedance of an amplifier is 30 ohms.  The amplifier is driven from a signal generator with a source impedance quoted by the manufacturer as 75 ohms.   Given that the S.G. produces 1-volt across 75-ohms what is the voltage it will produce at the input to the amplifier ?

END  OF  FUNDAMENTALS  PART  1