Homework #7 Solution

Summer 2009 1

1. Plot each of the following complex numbers as vector in the complex plane:

a. (3-3j) b. 4j

eπ

c.

1 1

2 4

j j+ −−

d. 3

1

j

e.

1

1

j

j

−

− +

2. Give numerical answers to each of the following questions:

a. Rationalize 80 60

28 96

j

j

− −

−. Express your answer in rectangular form.

2.5

5

30

210

60

240

90

270

120

300

150

330

180 0

Real

Ima

gin

ary

X: 3

Y: -3

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

Real

Ima

gin

ary

X: 0.7071

Y: 0.7071

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

Real

Ima

gin

ary

X: 0.25

Y: 0.75

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

X: 6.123e-017

Y: 1

Real

Ima

gin

ary

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

Real

Ima

gin

ary

X: -1

Y: 0

Homework #7 Solution

Summer 2009 2

b. Find the polar form of ( )3 3

1 1 12 2

j j

∗

∗ + + − −

(Note: The asterisk means

conjugate.)

c. Find the following phasor: 3sin(25 120 )okt Ρ − .

d. Find the magnitude of ( ) 10

90

1 7 2

1

o

o

j

j

j e

e

−−

−.

Homework #7 Solution

Summer 2009 3

e. Find the imaginary part of 30

1 3o

j

j

e−

+.

3. a. Write phasors (as both j

Aeφ

and A φ∠ ) for each of the following signals:

i. v(t)=9cos(2kt+30o)V => V=9e

j30

ii. i(t)=2sin(ωt+10 o

)mA=2cos(ωt-90 o

+10 o

)mA => I=2e-j80

iii. v(t)=cos(5t+30 o

)V+5sin(5t-30 o

)V=

V=ej30

+5ej-120

=cos(30)+jsin(30)+5cos(-120)+jsin(-120)= -1.6340 - 3.8301j

=> V=4.2ej247

b. Given w=3krad/sec, write inverse phasors for each of the following signals:

i. I=2034

oj

e A => i(t)=34cos(3kt+20o)A

ii. V=3j− V => v(t)=cos(3kt+90

o)V

iii. I=203

oj

eπ+ −

A => i(t)= ( )3 cos 3 20oe ktπ+ − A

Homework #7 Solution

Summer 2009 4

4. Given 1kω = rad/sec, find abZ .

1Ω

500 Fµ

b

a

500 Fµ

500 Fµ 6 mH 4 mH

2 mH

4Ω

2Ω

abZ

Homework #7 Solution

Summer 2009 5

5.

Given 447ω = rad/sec, find abZ .

ab1 1

Z 1|| j2.24 || j2.241 1 1 j2.24 1 1

1 j2.24 j2.24 j2.24 j2.24 j2.24

11

j2.24

j2.24

= − = =− −

+ + + +− − − −

= =−

−

6 and 7.

5 mH 1 mF 1Ω abZ

a

b

50 Fµ

40 mH

20Ω

+

120sin(377t 60 )V+ o

V(t) =

i(t)

5 mH

j(447)(5m)=j2.24

1 mF=-j(1/447*1m)=-j2.24 1Ω abZ

a

b

a. Find the phasor value for V(t).

b. Draw the frequency-domain circuit

diagram, including the phasor

value for V(t) and the impedance

values for components.

Homework #7 Solution

Summer 2009 6

~ + 2 Fµ

15Ω

4mF

i(t)

V(t) 120cos(31t)=

8 and 9.

a. Find the phasor value for V(t).

b. Draw the frequency-domain circuit diagram,

including the phasor value for V(t) and the

impedance values for components.

9. Find the phasor value for i(t) for the circuit in

Problem 8.

Homework #7 Solution

Summer 2009 7

10. Find i1(t).

+ _

+

1i

1(30 )iΩ

1 Fµ

20Ω

10Ω

3mH 130 cos(10kt)

Homework #7 Solution

Summer 2009 8