0195136047.pdf

3.1 SINUSOIDAL STEADY-STATE PHASOR ANALYSIS 109

I ̄C=jωCV ̄C=jBCV ̄C (3.1.26)

whereXL=ωLis theinductive reactance,XC=− 1 /ωCis thecapacitive reactance,BL=
− 1 /ωLis theinductive susceptance, andBC=ωCis thecapacitive susceptance. (Notice that
the fact 1/j=−jhas been used.)
The general impedance and admittance functions withs=jωfor sinusoidal excitation are
given by

Z ̄(jω)=R+jX=^1

Y ̄(jω)

(3.1.27)

Y ̄(jω)=G+jB=^1

Z ̄(jω) (3.1.28)

where the real part is either the resistanceRor the conductanceG, and the imaginary part is either
thereactance Xor thesusceptance B.
A positive value ofXor a negative value ofBindicates inductive behavior, whereas capacitive
behavior is indicated by a negative value ofXor a positive value ofB. Further, the following
KVL and KCL equations hold:

V ̄=Z ̄I ̄=(R+jX)I ̄ (3.1.29)

I ̄=Y ̄V ̄=(G+jB)V ̄ (3.1.30)

Phasor Method

For sinusoidal excitations of thesame frequency, the forced or steady-state responses are better
found by the technique known as thephasor method. Time functions are transformed to the phasor
representations of the sinusoids. For example, current and voltage in the time domain are given
by the forms

i=

√

2 Irmscos(ωt+α)=Re

[√

2 Irmsejαejωt

]

(3.1.31)

v=

√

2 Vrmscos(ωt+β)=Re

[√

2 Vrmsejβejωt

]

(3.1.32)

and where Re stands for the “real part of”; their corresponding phasors in the frequency domain
are defined by

I ̄=Irmsejα=Irms α (3.1.33)

V ̄=Vrmsejβ=Vrms β (3.1.34)

Notice that the magnitudes of the phasors are chosen for convenience to be the rms values of the
original functions (rather than the peak amplitudes), and angles are given by the argument of the
cosine function att=0. The student should observe that phasors are referenced here tocosine
functions. Therefore, the conversion of sine functions into equivalent cosine functions makes it
more convenient for expressing phasor representations of sine functions. The phasor volt-ampere
equations forR, L, andCare given by Equations (3.1.21) through (3.1.26), whereas thephasor
operatorsZ ̄andY ̄are given by Equations (3.1.27) and (3.1.28). The KVL and KCL equations
(3.1.29) and (3.1.30) hold in phasor form.