CHAPTER 8 RESISTANCE

8.1       CURRENT AND VOLTAGE

In the manual ‘Fundamentals of Electricity 1’ resistance in a circuit was likened to friction in a mechanical system.  It opposed any force attempting to cause motion, and it absorbed energy which showed in the form of heat.

Unlike inductance which caused a slow build-up of current, or unlike capacitance which caused a slow build-up of charge, resistance has an instantaneous effect on the current in a circuit. In a d.c. circuit Ohm’s Law states that:

 
                                                                                    

where V is the d.c. voltage applied and / is the current (in amperes) caused by that voltage. R is then defined as the ‘resistance’ of the circuit and is measured in the unit ‘ohm’.

FIGURE 8.1

RESISTIVE CURRENTS, D.C. AND A.C.

If Ohm’s Law is written in the form , the current (in amperes) in a d.c. circuit is equal to the voltage divided by the resistance (in ohms) and starts to flow virtually instantaneously the moment that the voltage is applied (see Figure 8.1(a)).  For any given circuit or sample the resistance is fixed (though it differs between samples), so that the current too is constant and proportional to the voltage.

The same argument applies to a.c. when it flows through a resistance.  Since at all times, and R is fixed, the current at any instant is directly proportional to the voltage at that instant.  As in an a.c. system the voltage is changing periodically, so also will the current change periodically and will bear a fixed ratio to the voltage at all times, as shown in Figure 8.1(b).

Because of this fixed ratio the current will reach its peaks at the same instants as the voltage, and it will also pass through zero at the same instants as the voltage.  This is shown clearly in Figure 8.1.  The current is then said to be ‘in phase’ with the voltage.

It follows that Ohm’s Law applies not only to d.c. but also to a.c. so long as the a.c. circuit consists only of resistance, and in the a.c. case the resulting current is in phase with the voltage.

The effect of inductance and capacitance on the current in an a.c. circuit, and the consequent modification of Ohm’s Law, is dealt with in Chapters 9 to 11.

8.2       HEATING

It was stated in the manual ‘Fundamentals of Electricity 1’ that, whenever a d.c. current is forced by pressure of voltage to flow through a conductor which has resistance (and all conductors do, even metals), heat is generated in that conductor.  The rate of heat generation is proportional to the resistance and to the square of the current (in amperes squared).  That is to say, the heat generated is I²R, and, since it represents a continuing loss of energy, it is expressed in the energy-rate unit ‘watts’ (W).

FIGURE 8.2

CURRENT HEATING EFFECT

Consider now the heating effect of an alternating current when flowing through a resistance R.  In Figure 8.2(a) is a pure sine-wave current trace with amplitude (or peak value) ‘A’. In Figure 8.2(b) is the corresponding ‘current squared’ wave, whose amplitude must be A².  Since the square of a quantity, whether positive or negative, is always positive, the current-

squared wave is wholly above the line.

The rate of heat generation depends on the resistance and the square of the current, so that the height of this curve at any instant indicates the heating rate I²R at that instant, and the area below the curve is the total heat generated over a given period.  The middle line (shown dotted) is then the average rate of heat generation.  It therefore represents the ‘mean square’ current, and its height is ½A².

It is shown in Chapter 5 that currents in a.c. systems are measured not by their amplitudes or peak values but by their ‘root mean square’ values, which were shown to be the square root of the mean square current, or ‘root mean square’ (or ‘rms’ current for short).  This has

the value .

So long as the current measured (I) js the rms current (which it normally is), then its square is the ‘mean square’ current which, when multiplied by R, determines the average heating rate.  Consequently with a.c. the average heating rate is given by the expression I²R (where I is the rms current), which is the same expression as used for d.c.