# Constancy of Power Flow in a Three-Phase System

For both the Y and А – connected systems it will be shown that the total of the instantaneous power p(t) for all three phases of a balanced three-phase system is constant and does not vary with time, as the voltages and currents do.

For a balanced three-phase system one can write for the phase voltages 

and the phase currents leading the phase voltages by an angle of f

The total instantaneous power of all three phases is

p(t) = pa(t)+pb(t)+pc(t);

with cosacosp = (1/2){cos(a-p)+cos(a+p)} one obtains

pa(t) = eaN(t)ia(t) = EphIph{cos(2rnt + f)+ cosf}, (8.9a)

Pc(t) = ecN(t)ic(t) = EphIph{cos(2ffit + f – 480°) + cosf}, (8.9c)

or

p(t) = 3EphIphcos f + EphIph{cos(2ot + ф) + cos(2ot + f — 240°)

. …. (8.10a)

+ cos(2ot + ф — 480 )}.

It will be shown below that the second term of (8.10a) adds up to zero!

Proof:

{cos(2ot + f)+ cos(2ot + ф — 240°) + cos(2ot + ф — 480°)} = 0, (8.10b)

with cos(a + p) = cosacosp — sinasinp and

cos(a + p + g) = cosacospcosg – sinasinpcosg – sinacospsing – cosasinpsing

Equation 8.10b becomes in an expanded manner:

cos2otcosf — sin2rntsinf + cos2ot cosfcos120° — sin2rnt sinf cos120°— sin2ot cosf sin120°-cos2ot sinf sin120° + cos2ot cosf cos120° — sin2ot sinfcos120° + sin2ot cosf sin120° + cos2ot sinf sin120° = 0,

or with cos120° = —(1/2):

cos2ot cosf — sin2ot sinf — (1/2)cos2ot cosf + (1/2)sin2ot sinf — (1/2) cos2otcosf + (1/2) sin2rntsinf = 0.

This shows that the time-dependent powers of a balanced three-phase system add up to zero and the power flow within such a system is

P = p(t) = 3EphIphcos f = independent of time! (8.11)

The total instantaneous power for a balanced three-phase system is constant and is equal to 3 times the average power per phase. This is of particular advantage in the operation of three-phase (or poly-phase) motors, for example, because it means that the shaft-power output is constant and that no torque pulsations arise due to the time-dependent sinusoidal variations of AC currents and voltages.

In the following  the time-dependent power relations of single, two, and three-phase systems will be visualized. Figure 8.7a, b illustrate the voltage, current and power at unity-power factor, and lagging power factor (consumer notation), e. g., cosF = 0.707 corresponding to Ф = —45°, of a single-phase sys­tem, respectively. Figure 8.8 shows the voltage, current and power at unity-power factor of a two-phase system. Figure 8.9a, b depict the voltage, current and power at unity-power factor, and lagging power factor (consumer notation), e. g., cosF = 0.80 corresponding to Ф = —36.87°, of a three-phase system, respectively. The influence of a 5% imbalance is studied in Fig. 8.10 and that of loss of phase c in Fig. 8.11.