116 Three-phase a.c. circuits
where VpH is the rms value of the phase voltage,/PH is the rms value of the phase
current, and cb is the angle between VpH and Ipu.
For the balanced star-connected system we have that VpH = VL/~/3 and that
Ipu = IL SO that P = 3(VL/X/3)IL COS cb = X/3 VEIL COS Oh.
For a balanced delta-connected system we have that Vpu = V L and that
IpH = IL/~/3 SO that P = 3 VL(IL/~/3) COS Cb = ~/3 VEIL COS 4~"
For any balanced three-phase system, therefore, whether it be star or delta
connected, the total power is given by
P = V'3 V~IL cos 05 watts (5.7)
In this equation, VL and IL are the rms values of the line voltage and current,
respectively, and cos 4~ is the power factor of the circuit.
From Equation (4.53), Chapter 4, the real power (P) is the apparent power
(S) multiplied by the power factor. It follows that S - P/cos 4~, so the apparent
power is then given by
S = ~/3 VEIL volt-amperes (5.8)
We also saw in Chapter 4 that the reactive power (Q) is the apparent power
(S) multiplied by the sine of the phase angle ~b so that
Q = ~3 VEIL sin 05 volt-amperes reactive (5.9)Example 5.4
A three-phase load takes a line current of 10 A at 0.8 power factor lagging from
a supply whose line voltage is 415 V. Determine (1) the power taken by the
load, (2) the weekly energy cost of operating the load for eight hours a day, six
days a week. The energy charge is 7.5 p per unit.Solution
1 From Equation (5.7) the power taken is given by P = ~/3VLIL cos ~b. In this
case VL = 415 V, IL = 10 A and cos ~b (the power factor) is 0.8. Therefore
P = "v/3 • 415 • 10 • 0.8 = 5750 W
2 The energy consumed by the load is measured in joules, which is watt-
seconds. This is a rather small unit, so that for most purposes supply
authorities use the kilowatt-hour (kW-hr) as the unit of energy. Energy
used per week is
P • (6 • 8) = 5750 • 48 = 276011 W-hr = 276.011 kW-hr
Cost of energy is 276.01.1 • 7.5 = s This is the cost of the electricity
consumed. In operating any piece of equipment there are also costs of
maintenance and depreciation.