CHAPTER 12 IMPEDANCES IN SERIES AND PARALLEL

In the manual ‘Fundamentals of Electricity 1’, Chapter 5, the behaviour series and in parallel was considered when they carried direct current.

FIGURE 12.1

SERIES AND PARALLEL RESISTANCES

The conclusion was reached that, when a number of resistances R₁, R₂, R₃ ……etc were placed in series, they behaved as a single equivalent resistance R, where:

                                    R = R₁ + R₂ + R₃ + …. etc.                                                           …. (i)

When however a number of resistances R₁, R₂, R₃ …. etc. were placed in parallel they behaved as a single equivalent resistance R, where R is given by:

                                    …. etc.                                                                            …. (ii)

That is to say, each resistance element must first be inverted, the inverses added together, and the sum so obtained inverted again to obtain the equivalent resistance R.

Exactly the same rules apply when the resistances carry alternating current.  When a current flows in a pure resistance it was shown in Chapter 8 that it is in phase with the applied voltage.  In a network containing nothing but resistances, all currents are in phase with all voltages and therefore with each other.  It was shown in Chapter 7, Figure 7.3, that quantities which are in phase simply add or subtract numerically and so behave just as in the d.c. case.  Therefore Figure 12.1 and equations (i) and (ii) apply equally to the a.c. case.

If the network consisted only of pure reactances, then all currents would lag (or lead) 90° on the applied voltages and so would be in phase with each other.  They would therefore add and subtract numerically just as if they were resistances, and the series and parallel counterparts of equations (i) and (ii) for the equivalent reactance in an all-reactance network are:

                 Series:          X = X₁ + X₂ + X₃ + .… etc.                                                         …. (iii)

                 Parallel:        .… etc.                                                      …. (iv)

(Note that, when a reactance is capacitive, it is regarded as negative.  Provided that care is taken with the sign of X, the above expressions apply equally to inductive or capacitive reactances, or to a mixture of both.)

FIGURE 12.2

PARALLEL REACTANCES

When both resistance and reactance are present together, the problem is less simple.  It was shown in Chapter 7, Figure 7.4, that quantities which are not in phase can only be added or subtracted vectorially.  When two parallel limbs of a circuit are pure resistance and pure reactance (Figure 12.2(a)), the current in the resistive limb is in phase with the common applied voltage, whereas that in the reactive limb lags 90° on that voltage; the currents are therefore 90° out of phase with each other.

Similarly, when a resistance is in series with a reactance (Figure 12.2(b), the common current is in phase with the voltage across the resistive element but lags 90° on the voltage across the reactive one.  The former therefore has a voltage in phase with the common current, and the latter has a voltage which leads 90° on that current.  The two voltages, which are thus 90° out of phase with each other, together add up to the total applied voltage V.  They can therefore only be added vectorially.

Figure 12.3(a) (bottom) is the ‘impedance triangle’ for the series case; I is the common current, R is in phase with it and X is at 90°.  Z is the total impedance given by the vector addition of R and X - that is, Z = R + X, or numerically:

                                  

FIGURE 12.3

SERIES AND PARALLEL IMPEDANCES

If all three sides of the triangle are multiplied by I, it becomes a voltage triangle, IR being the voltage developed across R by the common current (in phase with I), and IX being the voltage developed across X by that current and leading 90° on I, and IZ being the applied voltage across the whole impedance.

If there are several resistances and several reactances, R and X in expression (v) are respectively the sums of all the resistances and all the reactances as given by the series expressions (i) and (iii).

The phase angle so of the equivalent impedance Z is given by:

and the power factor .

Figure 12.3(b) (bottom) is the ‘impedance triangle’ for the parallel case.  We have already seen from expressions (ii) and (iv) that with parallel circuits it is the inverses of resistances or reactances which must be added together.  So also when they are mixed, except that the addition must be vectorial.  (Note that in this case the impedance triangle is drawn downwards, since the current in the reactance limb must lag 90° on the common voltage.)

The impedance triangle is therefore drawn not for the actual resistance and reactance but for their inverses 1/R and 1/X.  They can then be added vectorially to give the inverse of the equivalent impedance Z.  Thus: 

 

As before, if there are several resistance and several reactance limbs, in expression (vi) are the sums of the inverses of all the resistances and all the reactances as given by the parallel expressions (ii) and (iv).

The phase angle  of the equivalent impedance Z is given by: