We normally count in what is known as the Decimal System or the system of tens. Most probably this system was selected simply because, with five digits on each hand, we have a total of ten counting aids.
The decimal system is written using a total of ten symbols namely zero (or nought) and the numerals 1 to 9. When the count is greater than 9 we simply start all over again from 0 but place a 1 to the left to indicate that we have been round the circle once already.
Thus the hieroglyphic one-eight (18) indicates that the total is once-round plus eight;
i.e. (10 + 8).
Three-seven (37) indicates that we have been round the circle 3 times plus 7;
i.e. 10 + 10 + 10 + 7.
When the number to the left reaches 9 and still the count continues past nine-nine (99) then we return both figures to zero and place a 1 to the left again. It is not surprising that a short-speak was invented that makes three-six-four (364) read as three-hundred-and-sixty-four.
Each figure, starting from the right, has been given a name thus ; units, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, tens-of- millions, etc. Note that in Britain we call one-million-million a billion while, in America, this name is given to a thousand-million.
Mathematically we write these in terms of powers of ten. For example:
ten lots of ten we write as one-hundred or 10 x 10 or 102
ten lots of one-hundred is one-thousand or 10 x 10 x 10 or 103
ten lots of one-thousand is ten-thousand or 10 x 10 x 10 x 10 or 104
The first of these in the right-hand column we read as ten-squared and the second is ten-cubed.
From then on we speak of ten-to-the-power-of-four
and ten-to-the-power-of-five etc.
The basis of this system is the number 10 and so we say that we are working in Base-1O.
Numbering however is not limited to the Base 10 ; as already mentioned base-10 probably became popular because we have 10 fingers. In fact any number can be used as a base and perhaps the best known is Base-2 which gives rise to the Binary Arithmetic used almost exclusively in computers.
In Base-2 only two symbols are required namely zero and one. The first figure is still units, the second now becomes two's, the next fours (2 x 2), the next eights (2 x 2 x 2), 16's, 32's, etc. In place of "ten to the power of" we now have "two to the power of".
Binary arithmetic may look a lot more simple than decimal arithmetic (and in many ways it is) but the price lies in the size of binary numbers. The following Table compares numbers in Base-10, Base-2 and Base-3
| Number | Decimal | Binary | Base-3 |
| zero | 0 | 0 | 0 |
| one | 1 | 1 | 1 |
| two | 2 | 10 | 2 |
| three | 3 | 11 | 10 |
| four | 4 | 100 | 11 |
| five | 5 | 101 | 12 |
| six | 6 | 110 | 20 |
| seven | 7 | 111 | 21 |
| eight | 8 | 1000 | 22 |
| nine | 9 | 1001 | 100 |
| ten | 10 | 1010 | 101 |
A quick look at your cheque book will indicate the lengths to which binary numbers can grow. The Table shows that, as the base-number increases, so the required number of digits decreases.
WARNING Don't fall into the habit of thinking in decimal-number names when dealing with other bases - it can get you horribly confused. For example the binary-equivalent of seven is one-one-one (NOT one-hundred-and-eleven) ; the tertiary equivalent of six is two-zero (NOT twenty).
It can be shown mathematically that the optimum base in which to carry out arithmetical work lies somewhere between base-2 and base-3 and this argues that computers could use either . The choice of a binary system however is a matter of practical reality. A transistor can be made either to conduct or not to conduct (to be cut-off) and so it forms an ideal 2-state device which can represent 0's and 1's. Normally cut-off would represent a zero-condition but, within any particular system, you are of course at liberty to define the states as you wish.
The same transistor could be made into a 3-state device by having a third condition in which the transistor is neither cut-off nor conducting as heavily as possible - a "half-way" state. The problem is that the half-way state cannot be maintained exactly ; it must be defined within limits (it requires a tolerance) and the system becomes unreliable as the operating conditions drift toward those limits.
Manufacturers of electronic calculators and computers are pleased to give this name to the mathematical practice of expressing large numbers in powers of 10. They have a problem in that, given (say) an 8-digit display on their product, they cannot show any number greater than 99 999 999 or any number which is smaller than .000 000 1 (at least one digit is required to show the decimal point).
Even with a pen and paper it is hazardous to perform arithmetic with such large numbers because it is only too easy to make a mistake and very difficult to sort it out afterward.
The technique of course is varied to suit the problem of the moment but the basic idea is to reduce all numbers to a single digit before the decimal point; i.e. all numbers are expressed in terms of units only plus a multiplying factor which indicates by how much the figure must be either increased or decreased to obtain the correct notation.
For example:
10 becomes 1 x 10
156 becomes 1.56 x 102
12 423.78 becomes 1.242378 x 104
As a simple rule-of-thumb the small raised number which indicates the power of 10 shows how many places the decimal point has been shifted. As already discussed the "10" is known as the Base; the raised figure is known as the Index (Plural indices).
For quantities less than 1 (fractions of 1) the same rules apply. The decimal is moved until there is a single unit; however the decimal point is now being moved in the opposite direction and so we indicate this by placing a minus-sign in front of the index:
0.1 becomes 1 x 10-1
0.054 becomes 5.4 x 10-2
0. 006010 becomes 6.01 x 10-3
There is one loose end to be cleared up before we can use this notation to do simplified arithmetic. Between positive and negative induces there must be an index equal to zero but what does it represent ? Compare the two Tables above and you will see that a number is missing between (1 x 10+1) and (1 x 10-1) ; i.e. between 10 and 1/10. That number is 1 and so the expression 100 must indicate the value 1 ? In fact the argument holds in any Base - where x is any number we can write x0 = 1
Remember the rule-of-thumb that the index shows how many places
the decimal point has been moved?
It follows that 100 indicates that the point has NOT been moved.
Powers of 10 are easy to handle as can be seen by writing the problem out in full: for example:
100 x 10 000 = 10x10 x lO x lO x lO x lO = 1-million (102 x 104 = lO6)
10 000 / 100 = 10 x 10 x 10 x 10 / 10 x 10 = 100 ( 104 / 102 = 102 )
The rule is that, for multiplication
add the indices.
for division
subtract the indices
This rule of thumb gives us a valuable idea in using indices. The operation "divide-by-one-hundred" ( /102 or subtract the index) is the same as the operation "multiply-by-one-hundredth" ( x10-2 or add the negative index). In other words we can transfer a power-of-ten on the bottom line of any fraction to the top line of that fraction if we change the sign of the index. (The reverse operation is valid too.)
In this way any arithmetical problem is reduced to the multiplication or division of two numbers which are expressed in units and parts of 1. For example:
172.834 x 19.726 = 1. 72834 x 102 x 1.9726 x 10
= 1.72834 x 1. 9726 x 103To help us on our way this is roughly 1¾ x 2 x 103 (approximately 3,500)
OR
85.012 x 460. 373 / 19.47 = 8.5012 x 101 x 4. 60373 x 102 x 10-1 /1. 947
which is roughly 8.5 x 4.5 x 102/2
= 20 x 100 (approximately)
So far we have considered only indices which are whole numbers; in Base-10 this limits us to the real numbers in the sequence 1/100, 1/10, 1/1, 10, 100, and so on.
The number 6 lies between 1 and 10 while the number 60 lies between 10 and 100. It should be possible therefore to fill-in the gaps between 100, 101 and 102 with a range of indices which represent the numbers from 1 to 10 and from 10 to 100. Exactly how mathematicians work-out these numbers is not of importance here; all we need to know is that it can be done, it has been done and that the result can be very useful. When dealing with the full range of indices, which represent a full range of real numbers, we change the name from indices to Logarithms.
Remember the rule :
multiply two numbers
add the indices
divide two numbers
subtract the indices
The logarithm of every possible number is presented in tabular form in what are known as Log Tables. To multiply or to divide two (or more) difficult numbers all that is required is to extract from the log-tables the appropriate logarithms and then add or subtract them as required. This is a very straightforward computation and should not lead to error no matter how formidable the original numbers may have been.
Addition (subtraction) of the logarithms (in common usage logs) results not in the final answer but in the log of that answer. It is now necessary to go back into the Tables and find the original number of which this answer is the logarithm. In fact that is not a difficult task but, to speed the process, mathematicians have prepared the Tables "inside-out" as it were in a form known as Anti-log Tables; in these you enter with the log and find the number (the anti-log).
Instructions on the use of Log Tables are given in detail in an Appendix at the end of this Lesson (to come).
In practical work, when in the middle of a long calculation, it can be tiresome and distracting to be obliged to write "find the log of 100 and multiply it by 6 " ; equally we may well wish to write "find the number whose log is 2 and multiply that number by 6". It matters very much that there should not be room for confusion because the answer to the first proposition is 12 while the answer to the second proposition is 600.
A shorthand notation has been generally accepted to avoid such ambiguities. The first of the propositions would be written as 6.log10100; this is read as six-log-to-the-Base-10-one-hundred or more simply as six-log-Base-10-one-hundred. In practice however most work is carried out in logs to base-10 and it is customary to omit the Base and to assume that the Base is 10 unless otherwise specified. Thus the first proposition would be written as 6.log 10.
The second proposition "find the number whose log is 2 and multiply that number by 6" is written as 6.antilog2 (again assuming Base-10). Even more conveniently it can be written as 6.log-12. Remember that this is no more than an agreed convenient shorthand; that index (-1) is entirely without mathematical significance or complication.
(This style of notation is found again in the Lesson on Trigonometry.)
For normal calculations accuracy is not required beyond three decimal places and so standard log-tables are calculated to four decimal places. Navigation, for example, requires much greater accuracy and it is necessary to employ 7-figure tables. The extremely useful and fast slide rule is simply an adding-stick whose scales are calculated in logarithmic intervals; this machine has been discarded in favour of the electronic calculator but, in truth, for many purposes in practical engineering it is superior to the calculator.
Logarithms based on 10 are known as Natural Logarithms. Other Bases are used but their study would not serve any useful purpose here.
Survival, whether as hunter or hunted, depends on the sensitivity of our senses. An eye that functions by starlight, or an ear that detects an earthworm below the surface, both have serious disadvantages during a thunderstorm. For this reason all the senses have inhibitors which are no more than feedback systems that reduce sensitivity according to the degree of stimulation.
As a result the response of each sense versus the magnitude of stimulation is not a linear relationship but is close to being logarithmic. For example, under laboratory conditions, if you present the ear alternately with two sounds which differ only in sound-level it will not register any difference in loudness unless one sound is (at least) twice the power of the other.
For scientists and engineers, who attempt to rationalise their work through measurement and tabulation, this dependence of relative loudness on absolute loudness creates problems. For any just-discernible change of loudness the power level must increase by a factor of two. For example if the first sound results from dissipating 10 mW then the second must be at least 20 mW. BUT. if the first sound results from dissipating 1-watt, then the second must be at least 2 watts. At the lower of these two levels the apparent increase is caused by an extra 10-mW while, at the higher level, exactly the same result is obtained with an increase of 1-watt.
The discussion above on indices and logarithms provides a solution to this problem of measuring power levels as they sound to the human ear. When numbering in Base-10, each multiplication by 10 increases the index by one. Using the index only - such a power ratio (10/1) is referred to as 1 Bel; we say that the second sound is louder (or quieter) than the first by 1-Bel.In practice however the Bel is found to be too large a unit and so it is divided into ten intervals called decibels. Hence, by definition, a power ratio is expressed in bels as
log P1/P2
(where P1 and P2 are the two power-levels to be compared)
or, in practical units as
10. log P1/P2 decibels. (dB)
Although it sounds complicated the decibel is truthfully a very simple soul. Any changes of power-level are first reduced to a ratio (by dividing the smaller into the larger) and then, via the simple formula above, to a number of decibels. From then onward overall changes in power levels can be obtained very simply by adding (and subtracting) decibels. At the end of the exercise, if an answer is required in absolute terms, then the final overall figure expressed in decibels is converted to a power-ratio by using the formula in reverse.
It does not matter what the absolute power level may be because the dB is concerned only with power ratios.
Suppose for example that your transmitter is developing 50 watts but the operator at the other end of the circuit is complaining that he can barely read you. He has no way of measuring your output power but his receiver is fitted with an S-meter. Because you are a considerate person you increase your power to 100 watts which is a two-fold increase. Log 2 is approximately 0.3 and so you have increased the power by 10 x 0. 3 or by 3-dB. Although your contact is not aware of your radiated power his S-meter will register an increase of 3-dB in signal (about one-half an S-point) and (if he is lucky) his ear will just about register an increase in loudness.
It could be that the 3-dB increase is just enough to make the a.g.c. circuit of his receiver reduce the gain and so reduce the background noise in which case he will report an improvement in readability. It is more likely that he will continue to ask you to raise the power. (He might be aware of the increase and so make a greater effort to read you and thus report a much better signal - we are all human).
Now raise your power by a further 50-watts to a total of 150 watts (the legal limit). The increase, from 100 to 150 watts, is a ratio of 1.5. In decibels the extra increase is given by
l0 x log 1.5 = l0 x 0.176 = 1.8dB
which will not make any noticeable difference at the receiving end although, once again, it might reduce the background noise through the a,g.c. action.
Had the power been raised from 50-watts directly to 150 watts (i.e. by a factor of 3) the increase expressed in dB's would have been
l0 x log3 = l0 x 0.4771 = 4.8dB
an answer we could have obtained more easily from the previous calculations by adding 3-dB to l.8-dB.
Note that QRP (or low-power) operators who insist on keeping their power-levels below 5 watts are not by any means cranks. The purchase of Linears and the constant escalation of output power is a true instance of diminishing returns. The only purpose served by raising power levels is to overcome background noise; when several people raise their radiated-power levels they are doing a very good job of raising the general background noise.
Unfortunately the availability of high power produces a tendency to lazy engineering and most of that extra power is wasted.
As explained above the decibel scale relates strictly to power ratios. To actually measure power it is necessary to place a voltmeter across the circuit and an ammeter in series with the circuit and then to multiply the two readings. Although perfectly valid this technique is cumbersome.
Power measurements are usually required in arrangements where the value of the load is known and this permits a simplified technique using Ohm's Law. As discussed in Lesson-6 of Fundamentals-1 the power dissipated in a resistor of value R is given by
Power = E2/R
and so most power-meters utilise a dummy load of known value across which is connected a voltmeter whose scale is calibrated not in Volts but directly in Watts. The squared term means that the scale cannot be linear but is cramped toward the lower end.
Where signal levels are measured in this manner it is necessary to modify the expression for decibels; a ratio of power-levels must be replaced by a ratio of voltages-squared (or currents-squared) and, to square a quantity in logarithmic form, the logarithm is multiplied by 2. Hence, when measuring in terms of voltage (or of current) the ratio of two signal-levels E1 and E2 becomes
20. log. E1 /E2 decibels
The difference in these two expressions is important. To check the output power of a transmitter a dummy-load is connected. If this is just a dummy load then it is necessary to connect a rf-voltmeter in parallel with that load or an rf-ammeter in series with the load. Any measured power differences are expressed in dB's using the second expression (x20). If the dummy-load is a commercially-produced instrument then it may have a voltmeter built-in which is calibrated in terms of watts; any measured power differences are expressed in decibels using the first expression (xl0).
To measure the strength of an incoming signal the power delivered from the aerial is in the range from microwatts to milliwatts and the option of a true power reading is not available. Here a receiver is used as a voltmeter and signal differences are turned into dB's using the second expression.
Note that decibels always refer to the ratio between two power levels but a signal level or a noise level is often seen expressed in terms of dB's. This is possible because the level is being compared with a standard level BUT it is essential to state that reference level. Statements such as "Your signal strength is 24 dB ..." or "The noise-level from your car exhaust is 60-dB..." are meaningless. A value of 24-dB reference 1-V can be turned into a signal strength in volts; 24-dB reference 1mW can be turned into a signal strength in milliwatts. Those two measurements cannot be related however unless the impedance of the circuit is given; the noise-level from your exhaust cannot be calculated unless a standard reference-level is known.
Because of mankind's love of jargon a convention has arisen to avoid having to state the reference level. In this a letter is added after "dB" to indicate the nature of the reference level - and it works fine as long as everybody knows, understands and respects the symbols.
dBV indicates that the reference is 1-volt
dBm indicates that the reference is 1 milliwatt
dBW indicates that the reference is 1-watt
dBV/m indicates that the reference is 1-volt-per-metre
In the Amateur Licence the power levels permitted on each Amateur Band are stated in dBW. It would of course be much less open to error if the levels were simply given in watts. Clearly it is essential to know the difference in meaning attributed to upper and lower case letters,
You will frequently see noise levels (aircraft, vehicle, fan, etc) given as so many decibels. You can be certain that the writer is not sure of what he is writing ; in expressing noise levels an extra suffix is added to indicate the type of frequency-weighting (frequency-correction) under which the measurements are made. It is interesting to know that a document issued by the Department of the Environment as guidance for Town Planners claims that 0-dB represents zero noise-level and that the minimum change of level which is detectable by the human ear is 10-dB . If nothing else it makes life much easier for Local Councils but never buy a house close to an airport or a motorway.
QUESTIONS
1. Write down the (decimal) numbers 1 to 20 as they would appear in Base-5
2. A pocket calculator shows an answer as 2. 78E4; what do you think this means?
3. Using the power-of-ten notation work out the answers to the following
(a) 2.5 x 61.29 x 319.4 x 0.006
(b) 17.43 x 1.116 x 42.38 ÷ 71
(c) 17.43 x 1.116 X 42.38 ÷ 0.71
(d) 1/2 x 3/8 x 3/4 ÷ 7/8
4. Obtain an approximate answer to 1,792.856 x 31.45 ÷ 29.017
5. If logA = .386 and logB = .614 what is the value of A x B?
6. A communications receiver has had its S-meter calibrated and found to
be
very close to 6-dB per division from S3 to S7. An incoming signal
is
causing an indication which, for most of the time, is S6 but which fades
down to S4.
When the receiver is connected to a signal generator it is found that an input
of 1.5 mV gives an indication of S7.
What are the maximum and minimum levels between which the signal is fading?
7. The input voltage to an audio amplifier is 10 mV. The voltage which it is developing across an 8-ohm resistive dummy load is 1-volt. What is the gain of the amplifier expressed in dB ?
What extra information would you require to calculate the input power? Beware - I'm trying to catch you.
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