CHAPTER 5 PEAK, RMS AND MEAN VALUES

Figure 5.1(a) shows, at the top, a current/time curve for a pure alternating quantity.  Figure 5.1(b) is the corresponding (current)² /time curve for the same quantity, and, shown dotted through it, is the average value of the (current)² - that is, where areas above and below the dotted line are equal.  Projected back onto the current/time curve of Figure 5.1(a), and shown chain-dotted, is the square root of this average (current)².  Since the height of the peaks of the curve of Figure 5.1(b) is A², its mean value must be ½A². 

  The square root of this average quantity is (= 0.707A).  This then is the ‘root mean squares (‘rms’) of the quantity whose peak value is A.

FIGURE 5.1

PEAK, RMS AND MEAN QUANTITIES

Since the heating of a conductor due to its resistance is at any instant I²R, it is proportional to the square of the current, and the total heat developed during a cycle is proportional to the average of the square of the current over that cycle - that is, to the ‘mean square’.  So, as an a.c. ammeter reads the root mean square current, that reading, squared and multiplied by the resistance R, gives the net heating rate in an a.c. circuit.  So the rms current in a circuit causes the same heating rate as would a d.c. current of the same value.  In other words, when speaking of ‘I²R’ losses in a circuit, it applies equally to d.c. as to a.c. so long as we are speaking (as is usually the case) of rms currents.

All a.c. ammeters and voltmeters are calibrated to indicate rms values, and current- and voltage-operated relays are set to operate at certain predetermined rms values.

For certain applications, notably in rectifier work (see manual ‘Fundamentals of Electricity 3’) mean (or average) values of the rectified a.c. over the period are required rather than rms values.  Figure 5.1(c) shows the average value of the ‘rectified’ current curve of Figure 5.1(a), where areas above and below the average line are equal.  For a pure sine-wave its value is 0.637 of the amplitude (i.e. less than the rms).

Care must be taken not to confuse peak, rms and mean values of an alternating quantity.  Since almost all instruments indicate, and relays operate on, rms quantities, these are the ones that are generally used and are intended to be understood when voltages and currents are discussed.  However, it must be remembered that in the mathematical expression for an alternating quantity, A sin 2pft, ‘A’ is here the amplitude or peak value.

If it is necessary to distinguish between peak, rms and mean values of an alternating quantity ‘A’, they can be written respectively Â, A and A_m.

Whereas rms current values are important when considering the thermal current-carrying capacity of cables and items of plant, the peak values of current become important when considering the mechanical strength of such items under conditions of short-circuit when currents may be very high.  The peak values of voltage are specially significant when considering questions of insulation.

Examples

If a voltmeter reads 33 000V, this is an rms value, and the peak value is Ö2 times this, namely about 47 000V.  This is the voltage against which the system must be insulated, since it occurs twice every cycle.

If an ammeter reads 2 000A, this is an rms value, and the peak current is Ö2 times this, namely 2 800A.  The peak value, which occurs twice every cycle, does not affect the heating, which is based on the rms value, but it does affect electromagnetic forces in the neighbourhood of the conductor, and therefore the mechanical bracing of the current-carrying conductors.  This is described further in the manual ‘Electrical Protection’.

If an ammeter reads 40A (rms) in a circuit of resistance 5 ohms, then the heating rate, or losses, would be 40² x 5 watts, or 8 000W, or 8kW.  This would be equally true if it were a d.c. circuit and the d.c. ammeter read 40A.