Measurements on dc circuits are usually carried out using either digital or moving-coil meters. With these it is a simple matter to measure the constant-value of the voltage and also the constant-value of the current; by multiplying these two figures the constant-value for the dissipated power can be ascertained. Equally, by dividing the current into the voltage the constant-value of the circuit resistance is obtained.
Measurements on ac circuits however are not so simple because, even though the circuit resistance remains constant, should the circuit contain reactances then everything else changes with frequency. Furthermore the current variations do not keep in step with the voltage variations (there is a phase discrepancy) and even that varies with frequency.
In terms of meters there is an immediate difficulty. Digital meters are of little use to measure varying parameters because of their plodding action. Analogue meters suffer from mechanical inertia which prevents them from following a waveform which varies at a rate much above 1-Hz.
To be of any use at all a m-c meter must have a centre-zero so that it can show both positive and negative values. As frequency is increased the meter does not have sufficient time to reach the peak value of the waveform before the value begins to decrease once more; it follows that with increasing frequency the deflection of a m-c meter is reduced until, above about 10-Hz, it simply remains stationary at the average value. The one thing which is obvious about the symmetrical sinewave is that its average value is zero.
(Note that, in a Class-A amplifier, a meter always records the value of the dc in an anode or collector circuit irrespective of the (ac) signal because the average value of the varying direct-current is that of the dc on which the ac is superimposed. Indeed this fact can be useful in that a kicking m-c meter indicates the presence of distortion due to overloading an amplifier stage.)
Given that the circuit is in a "steady-state" there is one parameter which remains constant and that is the power which is being dissipated. This provides a basis for measuring voltage and current in an ac circuit. For example, if a lamp is set up in front of a photometer it is possible to measure the brightness of the lamp ; in turn this is a measure of the temperature at which the filament is running and this, of course, is a function of the power which it is dissipating.
First feed the lamp with the ac-source which it is desired to measure and then measure the brightness of the lamp. Now remove the ac-source and substitute a variable dc-source. Increase power until the photometer indicates the same brightness and then note the values of direct-current and direct-voltage. The product of these two gives the value of the ac-power.
Not only is the above experiment a legitimate way of measuring an ac but it. is a method that is actually used where appropriate and which is the basis of many measuring techniques. Instead of using a photometer and lamp it is possible to measure filament temperature by noting the expansion of the filament wire this is the basis of the Hot-wire Meter which is described in Part-6. The Thermocouple Meter measures temperature by the p.d. which is developed across a non-uniform junction.
However, to measure power by such means is not very convenient and it is desirable to have an ac-voltmeter and an ac-ammeter which will give indications such that the direct multiplication of their readings yields power directly. Such meters give a value for an ac-waveform which is equivalent, in its heating. effect, to a dc-waveform.
This value is derived theoretically by a mathematical process which results in the name The Root-mean-square value which is always shortened to the RMS Value. However, it is important to remember that RMS values apply only to pure sine-waveforms from which the definition has been developed; to measure any other waveform it is necessary to use a power-averaging meter such as the hot-wire or thermocouple meters mentioned above.
As already described the average value for a sinewave is zero ; what we are seeking here is the average heating effect as the waveform rises from zero-to a peak value and then falls again to zero. It is only necessary to consider one half-cycle because, although the voltage and current reverse, the heating effect is the same in the following half-cycle. (Do not confuse this with the reactive circuit in which the current and voltage waveforms are displaced in time and the average power is zero.)
The mathematical study of quantities that vary continuously is known as The Calculus and its use gives a very quick and simple result for this problem.. However it is possible to derive the RMS value for a sinewave without resorting to Calculus.
Fig.1 shows the plot of a sinewave and it could equally represent either the voltage or the current waveform. The power in the circuit is proportional to the square of either parameter and so, at any particular moment, power is proportional to the square of the value indicated by the sinewave.
In Fig.1 the plot has been divided into vertical strips which divide the sinewave into small sections. The argument is that, if these strips are very narrow, then each segment of the sinewave remains at a substantially constant value and the use of Calculus is not necessary.
The idea is to find the power dissipated during the time of each strip, add all the powers together to obtain the total amount of power dissipated over the half-sinewave and then divide by the total time so as to get the average power throughout the half-sinewave. The power dissipated in each strip (each time-segment) is proportional to the square of the current (voltage) value for that strip. Once the average effective power is known then, by taking the square-root, a value is found for a current (voltage) that would dissipate the same overall amount of power.
The method therefore is to:
(i) square the instantaneous values to get a figure proportional to the instantaneous power in each strip
(ii) add all the powers and divide by the number of strips to get an average value for each strip
(iii) take the square root of this average-power figure to obtain an effective value for the current (or for the voltage).
In plain English we square it, find the average value (the mean value) of the squares and then take the square-root which leads to the name "root of the mean of the squares" - the Root-mean-square or RMS value.
To derive a method of defining the RMS value does not of itself solve the problem of measuring that value. A hot-wire or a thermocouple meter can be calibrated in terms of RMS voltage or RMS current if the impedance of the heating element is known. The meter is used to measure a known direct-current power and then the value of the current (or voltage) is marked on the scale — such a calibration is valid for a hot-wire meter but, for a thermocouple meter, it is only valid for that particular combination of thermocouple and moving-coil meter used during the calibration process. For either meter the calibration holds only at low frequencies because, at high frequencies, stray capacitances tend to by-pass the heating element and so the meter reading progressively falls below the correct value.
Most m-c meters calibrated in RMS values are in fact dc meters fitted with a rectifier. The time-constant of the smoothing circuit is adjusted to make the meter register the peak value of the waveform — this is the only part of the waveform which can be identified easily. The scale however is marked with a set of figures that represent that peak-value divided by root-2 (1.414) because it can be shown that the ratio between peak and RMS values is √2:1.
Rectifier circuits also suffer from high-frequency limitations and so, although such meters provide a performance better than that of the thermocouple meter (and their response is quicker), they still have a limited range. The range can be extended to high frequencies by employing a more complex form of rectifier circuit which is fed by either an audio-amplifier (up to 20-kHz) or by a radio receiver; the response of these devices can be tailored to correct any frequency-related errors. However they all employ the technique of peak-waveform detection and (fiddled) calibration ; some may be calibrated both in terms of peak and RMS values.
An advantage of this arrangement is that the scale can be extended as desired by inserting a calibrated attenuator at the input. Yet again negative-feedback can be used to raise the input impedance and also to make the meter response logarithmic so that the scale can be calibrated in linear steps marked in decibels (see Part 7).
Such meter systems in the audio band are known as Amplifier-detectors (shortened to "Amp-Det"). In old books or Articles you may find them referred to as VVM's (valve-voltmeters). They are the basis of the "electronic multimeter". When the amplifier takes the form of a tunable radio receiver and, together with an aerial and feeder, it is calibrated overall in a standard radiated field, the result is a "field-strength meter".
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