Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

Z
V
I
(
)
(
)
j
V
I
e
j
v
i
w
f f
=
=






−
m
m
and the phasor 
admittance function 
Y
I
V
(
)
)
j
I
V
j
i
v
w
f f
=
=






−
m
m
(
e
. Note that both impedance function and admittance 
function is represented as functions of j
w
. They are in fact complex functions of a real variable 
w
and 
not complex functions of an imaginary variable j
w
as indicated by the notation. The j in j
w
serves to 
remind us they are complex functions.
The magnitude of the complex 
Z
gives the ratio of amplitudes of voltage phasor and current phasor. 
The angle of 
Z
gives the angle by which the current lags the voltage phasor.
The real part of 
Z
 
is the resistance part of 
Z
and the imaginary part of 
Z
is defined as its reactance 
part. Similarly, the real part of 
Y
 
is the conductance part of 
Y
and the imaginary part of 
Y
is defined as 
its susceptance part. Thus 
Z
 
=
 
R 
+ 
jX and 
Y
=
G 
+ 
jB where X is the reactance and B is the susceptance. 
Reactance has ohms as its unit and susceptance has siemens as its unit.
We have already derived the relation between voltage and current phasors for R, L and C earlier. 
Thus we conclude the following with the help of Eqns. 7.4-1 to 7.4-3.
Z
(
j
w
) 
=
R
and 
Y
(
j
w
) 
=
1/
R
for a resistor of 
R
Ω
.
Z
(
j
w
) 
=
j
w
L
and 
Y
(
j
w
) 
=
1/
j
w
L
for an inductor of 
L
henries.
 
Z
(
j
w
) 
=
1/
j
w
C
and 
Y
(
j
w
) 
=
j
w
C
for a capacitor of 
C
farads.
The phasor equivalent circuit is formed by carrying out the following steps:
1. Convert all sinusoidal sources at a single frequency 
w
into their phasor representations and mark 
them near the source symbols. There is no change in the graphic symbols used. Cosine function 
is assumed in time-domain by default.
2. Replace all passive elements by their phasor impedance/admittance and linear dependent sources 
by their phasor relations. The graphic symbols used for all elements will be the same in the 
original circuit and in its phasor equivalent circuit.

---

Sinusoidal Steady-State Response from Phasor Equivalent Circuit 
7.17
The procedure is illustrated for the circuit in Fig. 7.5-2.
The first source function 200 sin 314t is expressed as 200 cos (314t 
-
90
°
). Then the first source 
voltage phasor is 200 e 
j9
0°
in exponential form, 200 
∠-
90
° 
in polar form and 0
-
j 200 in rectangular 
form.
The second source function 250 sin(314t
-
45
°
) is expressed as 250 cos(314t 
-
135
°
). Then the 
second source voltage phasor is 250 e 
-
j135
°
in exponential form, 200 
∠-
135
° 
in polar form and – 
176.77 
- 
j 176.77 in rectangular form.
The value of 
w
=
314 rad/s. Therefore the 4mH inductor will have an impedance of j1.256 
Ω
, the 
5mH inductor will have j1.57 
Ω
and the 10mH inductor will have j3.14
Ω
. The impedance of 100
m
F 
capacitor will be 1/j0.0314 
=
-
j 31.85 
Ω
.
We see that impedances of inductor and capacitor are purely reactive. The reactance of an inductor 
is a positive quantity, whereas the reactance of a capacitor is a negative quantity. Similarly, the 
susceptance of an inductor is a negative quantity, whereas the susceptance of a capacitor is a positive 
quantity.
The phasor equivalent circuit of the circuit in Fig. 7.5-2 is now completed as in Fig. 7.5-3.