4.2 BALANCED THREE-PHASE LOADS 207the same as multiplying the average power in any one phase by 3, since the average power is the
same for all phases. Thus one has
P= 3 VphIphcosφ (4.2.12)whereVphandIphare the magnitudes of any phase voltage and phase current, cosφis the load
power factor, andφis the power factor angle between the phase voltageV ̄phand the phase current
I ̄phcorresponding to any phase. In view of the relationships between the line and phase quantities
for balanced wye- or delta-connected loads, Equation (4.2.12) can be rewritten in terms of the
line-to-line voltage and the line current for either wye- or delta-connected balanced loading as
follows:
P=√
3 VLILcosφ (4.2.13)whereVLandILare the magnitudes of the line-to-line voltage and the line current.φis still the
load power factor angle as in Equation (4.2.12), namely, the angle between the phase voltage and
the corresponding phase current.
In a balanced three-phase system, the sum of the three individually pulsating phase powers
adds up to a constant, nonpulsating total power of magnitude three times the average real
power in each phase. That is, in spite of the sinusoidal nature of the voltages and currents,
the total instantaneous power delivered into the three-phase load is a constant, equal to the
total average power. The real powerPis expressed in watts when voltage and current are
expressed in volts and amperes, respectively. You may recall that the instantaneous power in
single-phase ac circuits absorbed by a pure inductor or capacitor is a double-frequency sinusoid
with zero average value. The instantaneous power absorbed by a pure resistor has a nonzero
average value plus a double-frequency term with zero average value. The instantaneous reactive
power is alternately positive and negative, indicating the reversible flow of energy to and
from the reactive component of the load. Its amplitude or maximum value is known as the
reactive power.
The total reactive powerQ(expressed as reactive volt-amperes, or VARs) and the volt-
amperes for either wye- or delta-connected balanced loadings are given by
Q= 3 VphIphsinφ (4.2.14)or
Q=√
3 VLILsinφ (4.2.15)and
S=∣
∣S ̄
∣
∣=√
P^2 +Q^2 = 3 VphIph=√
3 VLIL (4.2.16)where the complex powerS ̄is given by
S ̄=P+jQ (4.2.17)In speaking of a three-phase system, unless otherwise specified, balanced conditions are
assumed. The terms voltage, current, and power, unless otherwise identified, are conventionally
understood to imply the line-to-line voltage (rms value), the line current (rms value), and the total
power of all three phases. In general, the ratio of the real or average powerPto the apparent
power or the magnitude of the complex powerSis the power factor, which happens to be cosφ
in the sinusoidal case.