ECE 2006
Fall 2011
Homework #08 Solution
Problems in the textbook:
9.2, 9.6(a, b), 9.11, 9.14(a,b), 9.18, 9.31, 9.60, and solution to an additional problem
Chapter 9, Problem 2.
A current source in a linear circuit has
is = 8 cos(500πt - 25°) A
(a) What is the amplitude of the current?
(b) What is the angular frequency?
(c) Find the frequency of the current.
(d) Calculate is at t = 2 ms.
Chapter 9, Solution 2.
(a)
amplitude = 8 A
(b)
ω = 500π = 1570.8 rad/s
(c)
f =
ω
= 250 Hz
2π
(d)
Is = 8∠-25° A
Is(2 ms) = 8 cos((500π)(2 × 10 -3 ) − 25°)
= 8 cos(π − 25°) = 8 cos(155°)
= -7.25 A
Chapter 9, Problem 6 (a, b).
For the following pairs of sinusoids, determine which one leads
and by how much.
(a) v(t) = 10 cos(4t - 60°) and i(t) = 4 sin(4t + 50°)
(b) v1(t) = 4 cos(377t + 10°) and v2(t) = -20 cos377t
Chapter 9, Solution 6 (a, b).
(a)
v(t) = 10 cos(4t – 60°)
i(t) = 4 sin(4t + 50°) = 4 cos(4t + 50° – 90°) = 4 cos(4t – 40°)
Thus, i(t) leads v(t) by 20°.
(b)
v1(t) = 4 cos(377t + 10°)
v2(t) = -20 cos(377t) = 20 cos(377t + 180°)
Thus, v2(t) leads v1(t) by 170°.
Chapter 9, Problem 11.
Find the phasors corresponding to the following signals.
(a) v(t) = 21 cos(4t-15o) V
(b) i(t) =-8 sin(10t+70o) mA
(c) v(t) = 120 sin (10t –50o ) V
(d) i(t) = -60cos(30t +10o) mA
Chapter 9, Solution 11.
(a)
V = 21 < −15o V
(b) i (t ) = 8sin(10t + 70o + 180o ) = 8cos(10t + 70o + 180o − 90o ) = 8cos(10t + 160o )
I = 8 < 160o mA
(c ) v(t ) = 120sin(103 t − 50o ) = 120 cos(103 t − 50o − 90o )
V = 120 < −140o V
(d)
i (t ) = −60 cos(30t + 10o ) = 60 cos(30t + 10o + 180o )
I = 60 < 190o mA
Chapter 9, Problem 14.
Evaluate the following complex numbers:
(a)
(b)
( 5 − j 6 ) − ( 2 + j8 )
( −3 + j 4 )( 5 − j ) + ( 4 − j 6 )
( 240∠ 75
+ 160∠ − 30
) ( 60 − j80 )
( 67 + j84 )( 20∠ 32 )
Chapter 9, Solution 14.
(a)
3 − j14
14.318∠ − 77.91°
=
= 0.7788∠169.71° = − 0.7663 + j0.13912
− 7 + j17 18.385∠112.38°
(62.116 + j 231.82 + 138.56 − j80)(60 − j80)
24186 − 6944.9
=
= − 1.922 − j11.55
(67 + j84)(16.96 + j10.5983)
246.06 + j 2134.7
Chapter 9, Solution 18.
(b)
(a)
v1 ( t ) = 60 cos(t + 15°)
(b)
V2 = 6 + j8 = 10∠53.13°
v 2 ( t ) = 10 cos(40t + 53.13°)
(c)
i1 ( t ) = 2.8 cos(377t – π/3)
(d)
I 2 = -0.5 – j1.2 = 1.3∠247.4°
i 2 ( t ) = 1.3 cos(103t + 247.4°)
Chapter 9, Problem 31.
A series RLC circuit has R=80 Ω, L=240 mH, and C = 5mF. If the input voltage is v(t)=
10cos2t, find the current flowing through the circuit.
Chapter 9, Solution 31.
jωL = j 2 x 240 x10−3 = j 0.48
1
1
→
=
= − j100
C = 5mF
jωC j 2 x5 x10−3
Z = 80 + j 0.48 − j100 = 80 − j 99.52
L = 240mH
→
V
10 < 00
=
= 0.0783 < 51.206o
Z 80 − j 99.52
i (t ) = 78.3cos(2t + 51.206o ) mA = 78.3cos(2t+51.26˚) mA
I=
Chapter 9, Problem 60.
Obtain Zin for the circuit shown in Fig. 9.67.
50 Ω
j30 Ω
–j100 Ω
40 Ω
60 Ω
j20 Ω
Figure 9.67
Chapter 9, Solution 60.
Z in = (50 + j30) + (40 − j100) (60 + j20) = 50 + j30 + 52.19 − j10.244 = 102.19 + j19.776Ω
Zin = (102.19 + j19.756) Ω
Additional Problem
Assuming that vs = 16 cos(2t - 40°) V in the circuit, find the average power
delivered to each of the passive elements.
Solution to additional problem
Converting the circuit into the frequency domain, we get:
+
−
16∠ − 40°
= 3.36∠ − 25.38°
j 6( 2 − j 2)
1+
j6 + 2 − j 2
1
P1Ω = (3.36) 2 1 = 5.64 W
2
I 1Ω =
P3H = P0.25F = 0
I 2Ω =
P2 Ω =
j6
3.36∠ − 25.38° = 4.516
j6 + 2 − j 2
1
(4.516) 2 2 = 20.39 W
2