0195136047.pdf

3.1 SINUSOIDAL STEADY-STATE PHASOR ANALYSIS 113

The instantaneous powerp(t) supplied to the network by the source is

p(t)=v(t)·i(t)=

√

2 Vrmscos(ωt+φ)·

√

2 Irmscosωt (3.1.35)

which can be rearranged as follows by using trigonometric relations:

p(t)=VrmsIrmscosφ( 1 +cos 2ωt)−VrmsIrmssinφsin 2ωt

=VrmsIrmscosφ+VrmsIrmscos( 2 ωt+φ) (3.1.36)

A typical plot ofp(t) is shown in Figure 3.1.2, revealing that it is the sum of an average component,
VrmsIrmscosφ, which is a constant that is time independent, and a sinusoidal component,VrmsIrms
cos(2ωt+φ), which oscillates at a frequency double that of the original source frequency and has
zero average value. The average component represents the electric power delivered to the circuit,
whereas the sinusoidal component reveals that the energy is stored over one part of the period
and released over another, thereby denoting no net delivery of electric energy. It can be seen that
the powerp(t) is pulsating in time and itstime-averagevaluePis given by

Pav=VrmsIrmscosφ (3.1.37)

since the time-average values of the terms cos 2ωtand sin 2ωtare zero. Note thatφis the angle
associated with the impedance, and is also the phase angle between the voltage and the current.
The term cosφis called thepower factor. An inductive circuit, in which the current lags the
voltage, is said to have alagging power factor, whereas a capacitive circuit, in which the current
leads the voltage, is said to have aleading power factor. Notice that the power factor associated
with a purely resistive load is unity, whereas that of a purely inductive load is zero (lagging) and
that of a purely capacitive load is zero (leading).
Equation (3.1.37), representing the average power absorbed by the entire circuit, known as
thereal poweror active power, may be rewritten as

P=VrmsIrmscosφ=Irms^2 R (3.1.38)

whereφis the phase angle between voltage and current. Equation (3.1.38) can be identified as
the average power taken by the resistance alone, since

pR(t)=vR(t)i(t)=i^2 R=

(√

2 Irmscosωt

) 2

R=Irms^2 R( 1 +cos 2ωt) (3.1.39)

One should recognize that the other two circuit elements, pure inductance and pure capacitance,
do not contribute to the average power, but affect the instantaneous power.
For a pure inductorL,

pL(t)=vL(t)i(t)=iL

di

dt

p(t) = VRMSIRMScos φ + VRMSIRMScos(2ωt + φ)

Pav

p(t)

t, s

Instantaneous power

0

average component sinusoidal component

p, W Figure 3.1.2Typical plot of in-

stantaneous powerp(t) and average

powerPav.