Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

Z
1
is seen to consume 300 W of active power and 
the load 
Z
2
is found to deliver 300 VAr of reactive power. (i) Find the currents delivered to the 
impedances as functions of time. (ii) the impedance values.
Z
1
Z
2
110
V
 
rms
f
= 60 Hz
+
–
45º
Fig. 7.12-13 
45. A motor connected to a 50 Hz, 230 V rms supply line draws 9 A rms and consumes 1.6 kW. (i) 
Find the apparent power, reactive power and power factor. (ii) Find the capacitance of a parallel 
capacitor that will make the power factor at the source unity. (iii) Find the source current with this 
capacitor installed.
46. Find the complex power delivered by the current source in the circuit in Fig. 7.12-14.
5 
Ω
5 
Ω
j
5 
Ω
2 A rms
1.5
i
x
i
x
Fig. 7.12-14 
47. Find the value of angular frequency such that v
o
(t) 
=
v
S
(t) in the circuit in in Fig. 7.12-15 if v
S
(t) 
=
V
m
cos
w
 t V. Assume ideal opamp model for the Opamp..
10 k
Ω
10 k
Ω
10 k
Ω
20 k
Ω
1 
µ
F
1 
µ
F
+
–
+
+
–
–
v
o 
(
t
)
v
S 
(
t
)
Fig. 7.12-15 

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7.62
The Sinusoidal Steady-State Response
48. Using the approximate power flow equations for a synchronous link, find out the complex power 
delivered by the two sources, currents delivered by the sources and power factor at which they are 
operating in the circuit shown in Fig. 7.12-16. Also find the voltage phasor at node C. (Hint – Let 
the voltage at C be V
∠
d
. Then AC and BC are two synchronous links.) 
1 
Ω
j
0.1 
Ω
j
1 
Ω
j
0.1 
Ω
A
B
C
+
–
+
–
1
4º
1.05
5º
Fig. 7.12-16 
49. The current source at C in Fig. 7.12-17 can deliver only Q. Q delivered by it is adjusted such 
that the voltage at C is 1.0 V rms. (i) Find the Q that it has to deliver. (ii) Find the 
S
delivered by 
the two voltage sources. (iii) Find the angle of voltage phasor at node C. Use synchronous link 
equations.
1 
Ω
j
0.1 
Ω
j
0.1 
Ω
A
B
C
+
–
+
–
0.95
1.05
5º
4º
Fig. 7.12-17 

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8.1
S i n u s o i d a l  S t e a d y -
S t a t e  i n  T h r e e - P h a s e
C i r c u i t s
CHAPTEROBJECTIVES
• To define and explain balanced and unbalanced three-phase systems and to extend the 
concepts of active power, reactive power and complex power to such systems.
• To develop and employ relations between line and phase quantities in Y-connected and 
D
-connected three-phase circuits.
• To emphasize constraints on line quantities in three-phase three-wire and four-wire systems.
• To develop a systematic procedure to analyse sinusoidal steady-state in three-phase balanced 
circuits using Y-equivalent and single-phase equivalent circuits.
• To show how to use phasors, phasor equivalent circuits and phasor diagrams for solving 
three-phase circuits under sinusoidal steady-state condition.
• To introduce and illustrate the problem of neutral-shift voltage in four-wire systems.
• To illustrate circuit analysis procedures for unbalanced three-phase circuit analysis through 
solved examples.
• To introduce and explain symmetrical components in detail and use it in analysis of 
unbalanced circuits.
• To develop power relations using sequence components.