Power flow and measurement

Single-phase

Suppose we have a single-phase load as in Figure 2.7 supplied with a sinusoidal voltage whose instantaneous value is . The RMS value is and the phasor value is V. If the load is linear (i.e. its impedance is constant and does not depend on the current or voltage), the current will be sinusoidal too. It leads or lags the voltage by a phase angle , depending on whether the load is capacitive or inductive. With a lagging (inductive) load, ; see Figure 2.29.

The instantaneous power is given by , so

(2.30)

This expression has a constant term and a second term that oscillates at double frequency. The constant term represents the average power : we can write this as

  (2.31)

is equal to the product of the rms voltage , the RMS current , and the power factor . The amplitude of the oscillatory term is fixed: i.e. it does not depend on the power factor. It shows that the instantaneous power varies from to and back to twice every cycle. Since the average power is VmIm/2, this represents a peak-peak fluctuation 200% of the mean power, at double frequency. The oscillation of power in single-phase circuits con- tributes to lamp flicker and causes vibration in motors and transformers, producing undesirable acoustic noise.

 

Fig. 2.29  Instantaneous current, voltage and power in a single-phase AC circuit.

Two-phase

Suppose we have a two-phase load with phases a and b, with υ_a = V_m cos ωt, i_a = I_m cos (ωt – ϕ) and υ_b = V_m sin ωt, i_b = I_m sin (ωt - ϕ). This system is said to be balanced, because the voltages and currents have the same RMS (and peak) values in both phases, and their phase angles are orthogonal. The total instantaneous power is now given by

The oscillatory term has vanished altogether, which means that the power flow is constant, with no fluctuation, and the average power P is therefore equal to the instantaneous power p. Note that if the phases become unbalanced, an oscillatory term reappears.

Three-phase
Suppose we have a three-phase load as in Figures 2.20 and 2.22, with phases a, b and c, with

This system is said to be balanced, because the voltages and currents have the same RMS (and peak) values in all three phases, and their phase angles are equi-spaced (i.e. with a 120^o symmetrical phase displacement). The total instantaneous power is now given by

As in the two-phase system, the oscillatory term has vanished. The power flow is constant, with no fluctuation, and the average power P is equal to the instantaneous power p. If the phases become unbalanced, an oscillatory term reappears.

The voltages and currents in equation (2.34) are phase quantities. In terms of line quantities, for a wye connection we have   and I_L = I_ph, whereas for a delta connection we have and V_LL = V_ph. In both cases, therefore,

where ϕ is the angle between the phasors V_ph  and I_ph.
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