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:
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