İzmir University of Economics
CE 205 Fundamentals of Electrical Circuits Lab
LO-8
Analyze R-L-C circuits using phasors
Preliminary
1. Prove that
e-jθ = cos θ + j sin θ
using the following Taylor series expansion of ex, cos x, and sin x,
ex = 1+ x +
!
+
+ + + + ……
!
!
!
!
!
cos x = 1 -
x2
2!
+
x4
4!
-
x6
+
6!
……
sin x = x -
x3
3!
+
x5
5!
-
x7
+
7!
……
and
2. The complex number a + j b may be
expressed in polar form as:
Im
a + j b = r e-jθ = r ∠θ
a+jb
jb
r
a. Express a and b in terms of r and θ.
b. Express r and θ in terms of a and b.
θ
a
Re
3. Determine the polar form of the following complex numbers.
a. 1 + j √3
b. 1 + j
f. -2
g. 4
c. √3 + j
h. 1 - j √3
d. √3 + j √3
e. j 3
i.
j.
- 1 - j √3
-1 +j
4. Do the following multiplication of complex numbers and understand the multiplication on the
complex plane.
i. Locate the numbers on the complex plane.
ii. Obtain the result using in cartesian form.
iii. Find also the result using the polar form.
iv. Locate the result on the complex plane and interpret the multiplication operation.
a. (1 + j) ( 1 - j)
b. (√3 + j ) (1 + j √3)
c. (√2 + j √2) (1 - j √3)
d. (- 1 - j √3) (- 1 + j)
2-1
5. Do the following division of complex numbers and understand the division operation on the
complex plane.
i. Locate the numbers on the complex plane.
ii. Obtain the result using in cartesian form.
iii. Find also the result using the polar form.
iv. Locate the result on the complex plane and interpret the multiplication operation.
√3- j
1 -j
a.
c.
1 - j√3
1+j
b.
d.
1-j
1+j
1+j
1 + j√3
6. Do the following arithmetic operations of complex numbers.
-√3. j/-0 + j√3/
a.
c.
2 + j2
1
-√3. j/-0 + j√3/
b.
d.
(0 . j2-0 + j√3/
(01 j2 -10 + j√3/
-√3. j/-10 + j√3/
-√31 j/-101 j√3/
Basic
1. Determine the phasors corresponding to the following sinusiodal signals.
(a) 5 cos ω0 t
π
(b) 3 cos 6ω0 t + 8
(c) - 2 cos ω0 t
(d2 10 cos -ω0 t - 600 /
(e2 sin ω0 t
π
(f2 4 sin 6ω0 t - 6 8
4
2. Consider the circuit given on the right.
(a) Determine the impedance of the
capacitor (ZC) at the frequencies given
(b) Determine the total impedance
ZT = R + ZC.
(c) Determine the output voltage phasor VC
at these frequencies.
ω0 (rad/sec)
1
10
100
1,000
10,000
1Ω
+
cos ω0t
volt
ZC
ZT
+
vS (t2 = 8 cos 2t volts
Determine the phasor VS (or @@@A
VS ) of vs.
Draw the phasor equivalent circuit.
@@@AL ).
Determine the phasors I and VL (or IA and V
Determine the voltage vL(t2 at steady state.
VC
2Ω
3. Consider the circuit given. The voltage source
waveform is a cosine waveform given as
(a)
(b)
(c)
(d)
VC
10 mF
vS
i
1H
vL
-
2-2
4. Consider the circuit given. The current source
waveform is a cosine waveform given as
iS (t2 = 10 cos 10 t Ampere
(a)
(b)
(c)
(d)
Determine the phasor IS of iS .
Draw the phasor equivalent circuit.
Determine the phasors IC and VC.
Determine the voltage vC(t2 at steady state.
10 Ω
iS(t)
(a)
(b)
(c)
(d)
+
Draw the phasor equivalent circuit.
Determine the phasors VR, VC, and VL.
Determine also the voltage phasor VS.
Determine the voltages vR(t2, vC(t2,
vL(t2, ans vS(t2 at steady state.
iS
+
0.01 F
vC
-
5. Consider the circuit given. The current
source is given as
iS (t2 = 10 cos 5 t volt
iC
+ vR −
+ vC −
2Ω
0.2 F
vS
+
1H
-
vL
-
Intermediate
15 Ω
6. Consider the circuit given. The voltage
sources are given as
30 Ω
VO
vS1 (t2 = 10 cos 2 t volt
vS2 (t2 = 2 cos -2 t + 450 / volt
vS1
vS2
0.05 F
(a) Draw the phasor equivalent circuit.
(b) Determine the phasor VO of vO , using
any approach (superposition, mesh
currents or node voltages) you
prefer.
(c) Determine the voltage vO(t2 at steady
state.
7. Consider the circuit given. The current
source waveform is a cosine waveform
given as
iS (t2 = 10 cos 5 t Ampere
(a) Draw the phasor equivalent circuit.
(b) Determine the phasor VO of vO .
(c) Determine the voltage vO(t2 at
steady state.
+
iS
2Ω
0.1 F
0.4 H
vO
-
2-3
15 Ω
8. Consider the circuit given. The sources
are given as
VO
30 Ω
vS (t2 = 10 cos 2 t Volt
iS (t2 = 2 cos 2 t Ampere
iS
vS
0.05 F
(a) Draw the phasor equivalent circuit.
(b) Determine the phasor VO of vO , using
any approach (superposition, mesh
currents or node voltages) you
prefer.
(c) Determine the voltage vO(t2 at steady
state.
Advanced
9. Consider the circuit given. Assume the
input sources is given as
R1
R2
+
vS (t2 = A cos ωt volt
(a) Draw the phasor equivalent circuit.
(b) Determine the transfer function
H(jω) as
V
H(jω2 = Vo
vi
C
L
vO
-
Vi
(c) Determine the magnitude and phase
of H(jω).
15 Ω
10. Consider the circuit given. The voltage
sources are given as
VO
30 Ω
vS1 (t2 = 5 cos 2 t volt
vS2 (t2 = 4 cos 4 t volt
vS1
0.05 F
vS2
(d) Draw the phasor equivalent circuit.
(e) Determine the phasor VO of vO , using
any approach (superposition, mesh
currents or node voltages) you
prefer.
(f) Determine the voltage vO(t2 at steady
state.
2-4