ECE 1250 homework # P1
Read about complex numbers and phasors in your textbook (sections 2.26 & 2.27, starting on p.159).
1. For the complex numbers a) Does z 1 equal
4
.
and z 2 2
?
.
Determine the following b) Does c) Does z 1 z 2 z 1 z 2 equal equal z 1 z 2 z 1 z 2
?
2. a) Find the phasor for v( ) = )
?
Express in both forms, polar and rectangular.
b) The phasor representation of a current is I ( 5
.
)
.
µ
A Find the time-domain representation, i(t) .
f
3. Add or subtract the sinusoidal voltages using phasors. Draw a phasor diagram which shows all 3 phasors, and give your final answer in time domain form.
a) v 1( ) = b) v 1( ) =
.
.
ω
ω
.
.
t t
)
) v 2( ) v 2( )
=
=
.
.
ω
ω
.
.
t t
)
) you may add V
4
to the phasor diagram you've already drawn for part a).
Find
Find v 3( ) v 4( )
=
= t t
4. Express the impedance of a 5.2mH
inductor at 60 Hz in polar form.
5. a) A capacitor impedance has a magnitude of 240
Ω
at a frequency of 1.8kHz
. What is the value of capacitor?
b) What value inductor has the same impedance magnitude at the same frequency?
c) Find the reactance (magnitude of the impedance with + or - sign) of this capacitor and this inductor at 3.6kHz
?
d) What would be the total impedance of this inductance and this capacitance connected in series at 2.7kHz
?
6.
Find Z eq a)
in each case.
V( j
ω
)
.
j0 f
R
L b)
330
.
Ω
V( j
ω
)
.
j0
.
f
C
R 330
.
Ω c)
V( j
ω
)
.
j0 f
0.22
.
µ
F
R
L a
C d)
V( j
ω
)
.
j0 f
L
R
.
Ω
C 0.03
.
µ
F
7.
Find the current I(j
ω
) in each case above.
8.
a) Find Z . Hint: Find the total impedance (R+Z) first.
b) Which leads, current or voltage?
v s( ) =
.
ω .
t c) By how much?
I.E. what is the phase angle between the voltage and current?
ω
377
)
R 180
.
Ω i( ) =
.
ω .
t
Z = ?
ECE 1250 homework # P1 p.1
)

9. a) A resistor and a capacitor are connected in series to create an impedance of Z at a frequency f
.
Find R and C .
50
.
Ω .
e
.
= 50
Ω
/-66 o b) A resistor and a capacitor are connected in parallel to create an impedance of Z at a frequency f
.
Find R and C .
1
Hint: invert Z eq
, Instead of solving this: 50 /-66deg = solve this:
1 j
.
ω .
C
1
50
Z eq
.
Ω .
e
.
= 50
= 0.02/66deg =
Ω
1
R
/-66 o
R j
.
ω .
C
10. The phasor diagram at right shows the voltage and current in the circuit below
V s
Assume the load consists of a resistor in series with a reactive component and the frequency is 60 Hz .
V s ( 60
.
) a) What is the magnitude of the impedance?
b) What is the value of the resistor?
c) What is the reactive component (type and value)?
I
.
.
11. For the circuit shown, draw a phasor diagram showing V
S
, I , V
R
, and V
C
. Draw the voltages to scale so that you can show that they obey KVL.
V s f
.
.
Answers
1. a) 28.636
= 28.636
Yes c) 9.22
not = 10.875
No
.
2. a) 8.4 /-90 o = =
3. a) b) v 1( ) v 1( ) v 2( ) v 2( )
=
=
8.4j
ω .
t
ω .
t b) 1.432
= 1.432
Yes b) 13
)
.
.
)
µ
.
.
4.
1.96
Ω
/90 o
5. a) 0.368
.
µ
F b) c) 120
.
Ω
480
.
Ω d)
)
.
.
Ω
6. a) b)
7. a) b)
(
(
(
(
330
330
6.6
13.8
.
)
.
Ω
= 709.7
Ω
/62.29
o
.
)
.
Ω
= 489.6
Ω
/-47.63
o
.
)
.
mA = 14.1mA /-62.29
o
.
)
.
mA = 20.4mA /47.63
o c) c) d) 4.4mA /15.2
o
8. a)
9. a)
10. a)
259
20.34
.
Ω
19.5
.
Ω b) The current leads the voltage c) 20 o
1.16
.
µ
F b) 123
.
Ω
0.969
.
µ
F b) 11.2
.
Ω c) inductor
R
R
2
ω .
L
ω .
L
1
ω .
C
.
j
1
ω .
C
2
11.
ECE 1250 homework # P1 p.2
V s
C 12
.
µ
F
V C
R 40
.
Ω
I d) 1.82k
Ω
/-15.2
o angle: atan
ω .
L
R
1
ω .
C
V R