CHAPTER 11 IMPEDANCE; OHM’S LAW FOR A.C.

11.1     PURE RESISTANCE, INDUCTANCE OR CAPACITANCE

It is shown in Chapters 8, 9 and 10 that opposition to the flow of current when an alternating voltage is applied to a circuit can be caused by any one or more of the following:

·                     resistance (R)

·                     inductive reactance (X_L)

·                     capacitive reactance (X_C)

All are measured in ohms, and, provided that only one of the three is present at any one time - that is, that the resistance or reactance is ‘pure’ - the current which flows is governed by Ohm’s Law, namely:

where V and I are rms values.

In this chapter will be considered how the current is affected if more than one of these factors are present.

11.2     RESISTANCE PLUS INDUCTIVE REACTANCE

Suppose that there is a series circuit containing a resistance R and an inductance L.  If the frequency f is known, the inductive reactance X_L = 2pfL, so X_L also is known.

FIGURE 11.1

RESISTIVE/INDUCTIVE CIRCUIT

If the pair of elements R and X_L are fed in series from an a.c. supply voltage V, they will have a common current I.  This is shown as the red vector I in Figure 11 1.

Since current I passes through the pure resistance R, it will be in phase with the voltage IR developed across it, the magnitude of this voltage being determined by Ohm’s Law V = IR.  This is shown as voltage vector IR (full line) in Figure 11.1, where it is in phase with the current I.

The same current I also passes through the pure inductive reactance X_L.  The current I through X_L will lag 90° on the voltage IX_L across it, the magnitude of this voltage being determined by Ohm’s Law V = IX_L for a pure inductance.  As the current I lags 90° on voltage IX_L, voltage IX_L leads on the current I and is shown as voltage vector IX_L (full line) in Figure 11.1.

Thus the total voltage across R and X_L is the vector sum of IR and IX_L, which has been shown in Chapter 7 as being the diagonal of the rectangle formed by IR and IX_L - that is, the line OP.  It can be written:

 
                                                                        

This combined voltage is of course the same as the applied voltage V, so that:

 

                                                                    V  = I (R + X_L)

This shows that, for a combined resistive/inductive circuit, Ohm’s Law applies if the vector sum of R and X_L is substituted for R.  This vector sum is called the ‘impedance’ of the circuit; it has the symbol ‘Z’ and is measured in ohms.

Numerically) from the right-angled triangle formed by the diagonal OP and the vectors IR and IX_L,

FIGURE 11.2

IMPEDANCE TRIANGLE

This triangle OPN is called the ‘impedance triangle’ (Figure 11.2) of the circuit and enables the impedance to be calculated, or measured directly, if R and X_L are both known.

11.3     RESISTANCE PLUS CAPACITIVE REACTANCE

Suppose that there is a series circuit containing a resistance R and a capacitance C.  If the frequency f is known, the capacitive reactance , so X_C also is known.

FIGURE 11.3

RESISTIVE/CAPACITIVE CIRCUIT

If the pair of elements R and X_C are fed in series from an a.c. supply voltage V, they will have a common current I.  This is shown as the red vector I in Figure 11.3.

The argument from here on is exactly as for the inductive case above, except that the capacitive reactance X_C, being regarded as negative, is drawn downwards.  This has no effect on the magnitudes of the various quantities, and the circuit’s impedance Z is still the vector sum of R and X_C and has the magnitude given by .  The impedance triangle OPN is similar but inverted.

Ohm’s Law for a resistive/capacitive circuit is, as for the resistive/inductive case, ohms, where .

11.4     GENERAL CASE - OHM’S LAW FOR A.C.

For a circuit containing resistance and reactance, whether inductive or capacitive reactance, or both, Ohm’s Law applies in the form:

 

The reactance X is, in the general case, the sum of all reactances in the circuit, whether inductive or capacitive, remembering that capacitive reactances are regarded as negative.

In the special case where the inductive and capacitive reactances are numerically equal (but opposite), X_L = -X_C, or X_L + X_C  = 0.  In that case the expression (ii) for Z reduces to:

Z has then its minimum value and behaves as a simple resistance.  With Z at a minimum, equation (i) shows that I is at its maximum - we have a condition known as ‘resonance’.

11.5     PHASE ANGLE

In Figure 11.2 the angle between OP and ON (the Z and R vectors) is called the ‘phase angle’ and is given the Greek symbol ‘j’ (phi, for ‘phase’).  From the trigonometry of the impedance triangle:

so that, if any two of R, X and Z are known, the phase angle j can be determined.  Also, if X is capacitive (= X_C), it is by convention negative, so that j too is negative and below the line.  If j above the line represents a lagging phase angle, below the line it represents a leading phase angle.

FIGURE 11.4

IMPEDANCE - GENERAL CASE

In the special cases (a) where there is only resistance and no reactance, X = 0 and therefore = 0; (b) where there is only reactance and no resistance, R = 0 and therefore

= 90°.  That is to say, in these two special cases of pure resistance and pure reactance the phase angles are 0° and 90° (lagging or leading) respectively, which are precisely the situations shown in Figures 8.1, 9.2 and 10.2 respectively, where the current is in phase or 90° lagging, or 90° leading, on the voltage.

In the general case where there is both resistance and reactance, the phase angle lies somewhere between 0° and +90° for inductive circuits, and between 0° and -90° for capacitive circuits.

Figure 11.4 shows the two general cases of a partly inductive and a partly capacitive circuit, where it will be seen that the current lags, or leads, on the voltage by an angle which is less than 90° and which is, in fact, the phase angle .