CHAPTER 3 POWER FACTOR

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

active power (watts) = uncorrected power

This shows that 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.

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).

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: