3.1 PURELY RESISTIVE OR REACTIVE CIRCUITS
In Chapter 1 it was shown that in a purely resistive
circuit, where the load current is completely in phase with the applied
voltage, the power output takes the form of a double-frequency wave which is
wholly on the positive side of the zero axis.
The wave has an average or net value equal to half its peak-to-peak
height; this average value represents the net power transmitted and is equal to
V x I, where V is the rms
value of the applied voltage and I
the rms current in amperes.
In Chapter 2 it was shown that in a purely reactive
circuit, whether inductive or capacitive and where the current lags or leads 90° on the applied voltage, the power output takes the form of a
double-frequency wave which is wholly symmetrical about the zero axis and
therefore has a mean or net value of zero.
That is to say, in purely reactive circuits no net power is transmitted;
the power going in during one half-cycle is returned during the next.
It was further shown that, although no active
power (watts) was transmitted, the product of rms volts and rms amperes is
still a perfectly good figure which represents the magnetic energy stored, but
not consumed, in the system. This product
of volts and amperes does not represent true power (watts) but is given the
name ‘vars’. It is termed ‘reactive
power’ as distinct from the true or active power measured in watts. Active and reactive power can be separately
indicated on switchboard wattmeter and varmeter instruments.
3.2 GENERAL (INDUCTIVE) CASE
We have considered until now only power in purely
resistive and purely reactive (inductive and capacitive) circuits. The general case occurs when a circuit is
partly resistive and partly reactive, which is much more common.
Figure 3.1 shows the general case of a
resistive/inductive circuit. The resistive part of the load draws in-phase
current, and the reactive part a current lagging 90°. Between them they draw a
single current somewhere between in-phase (0° lag) and 90° lag, as shown on the second curve of
the figure. The actual phase angle
between current and voltage is usually written ‘j’ (Greek ‘phi’ for ‘phase’); it is considered positive when current
is lagging and negative when leading.
If the same process is used, as before, of multiplying
the voltage by the current at each instant of time, the power wave so produced
(bottom of the figure) will again be double-frequency but will now be neither
wholly symmetrical about the zero axis nor wholly asymmetrical above it. It will be partly asymmetrical, and its average value (half-way between its
upper and lower peaks) will be positive and will lie somewhere between zero and
the half-way value shown in Figure 1.1.
This means that, in the general case, the average active power (watts)
will always be less than the maximum value which occurs in the purely resistive
case, where the net power was shown to be VI
watts (V and I being rms values).
Because in the d.c. days power was always the simple
product of V and I, with the advent of a.c. people continued with this outlook and
preferred still to think of power as the product of V and I (rms values) but
to insert a ‘correcting factor’ to make it apply to a.c. This correcting factor was given the name
‘power factor’ (‘pf’).
For the general case therefore:
P = V
x I x (power factor) (watts) ….(i)
the power factor being in general
less than 1. In the special case where the circuit is resistive only, the power
factor equals 1, and where the circuit is reactive only it equals zero.
FIGURE
3.1
A.C. POWER - GENERAL CASE
Reverting to the ‘impedance triangle’ shown in Figure
3.2 the angle between the impedance vector Z and the current
vector I is ‘j’. Now Z is the overall impedance across which the voltage is applied, so
the voltage vector V lies along Z,
just as the current vector I lies
along R, and j is then also the angle between voltage
and current - that is, the ‘phase angle’.
By Ohm’s Law for a.c:
and the uncorrected power (volts x amperes) is V x I;
or, substituting for V, it is IZ x I
or I2Z
FIGURE 3.2
IMPEDANCE AND POWER TRIANGLE
In the impedance triangle of Figure 3.2 multiply all
three sides by the same quantity I2 (which will not alter its
shape). The hypotenuse is now I2Z
and the horizontal side I2R.
I2Z has just been shown to be the uncorrected power (V
x I); I2R is the
active power absorbed by the resistance; and I2X is the
reactive power in the inductance. The
impedance triangle has now become a power triangle.
But by ordinary trigonometry 
Therefore
Therefore
active
power (watts) = uncorrected power 
This shows that 
is in fact the ‘power factor’ of equation (i).
is in fact the ‘power factor’ of equation (i).
Therefore in an a.c. system, where
the phase angle between applied voltage and load current is j, the active power is obtained by the formula:
where V and I are rms values, and 
is the power factor.
is the power factor.
Switchboard instruments are provided
which show the power factor direct. They
used to be marked ‘POWER FACTOR’ or simply ‘PF’, but modern instruments are now
generally marked ‘’.
The direct uncorrected product ‘VI’,
referred to above as ‘uncorrected power’, is more properly called ‘apparent
power’. It is given the symbol ‘S’ and is measured in volt-amperes (VA).
’.
The direct uncorrected product ‘VI’,
referred to above as ‘uncorrected power’, is more properly called ‘apparent
power’. It is given the symbol ‘S’ and is measured in volt-amperes (VA).
3.3 GENERAL (CAPACITIVE) CASE
It has already been shown that the phase angle between load current and voltage is considered to be positive when
the current is lagging, and negative when it is leading.
3.4 POWER-FACTOR METERS
Power-factor meters, which are basically only j-indicators calibrated to read cos j, always show the power factor as a positive number whether the
current is lagging or leading, but they are arranged to indicate lagging and
leading power factors in opposite directions (j positive and negative respectively). This can be seen on both examples of Figure
3.3, where lagging pf’s are to the left and leading to the right.
(a) ALL-ROUND SCALE (b) SHORT SCALE
FIGURE 3.3
POWER-FACTOR METERS
The older type of power-factor meter, shown in Figure
3.3(a), has an all-round (360°) scale. The upper two quadrants are the ones normally
used, but if power can flow in either direction (for example in a ring main or
interconnector) the upper two quadrants are used for the forward direction and
the lower two for the reverse direction of flow. These directions are sometimes marked
‘Export’ and ‘Import’.
Where only one direction of power flow is involved, a
‘short-scale’ instrument is nowadays more generally used, as shown in Figure
3.3(b). Lagging pf is to the left, as
before, and leading to the right, but the scales are limited from 1 down to
about 0.5 in either direction. Sometimes
the ‘1’ point, instead of being in the centre, is biased one way to give a
longer lagging scale and a shorter leading scale. This type of power-factor meter, which is
transducer-operated (see Chapter 9), is almost universally used on platform
switchboards.
3.5 REACTIVE POWER FACTOR
It was shown above that in the impedance triangle of
Figure 3.2, if all three sides are multiplied by I2, the hypotenuse OP represents the uncorrected power (V
x I) and that ON is the active power (I2R) in watts,
3.6 SUMMARY
To sum up: in a circuit with applied rms voltage V and rms load current I, whose resistance is R, reactance X and impedance Z
ohms, the expression:
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