ELEC 202 Electric Circuit Analysis II -...
Transcript of ELEC 202 Electric Circuit Analysis II -...
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(a)
Complex Arithmetic and
Rectangular/Polar Forms
2
Real vs. Complex Numbers
1j
The product of a real number & the
operator j is an imaginary number. 2
3 , , , 5.17
j j j j
The sum of a real number & an imaginary
number is a complex number, z . 2 4 , 1j j
…where the real part is denoted Rea z
…and the imaginary part is denoted Imb z
Re 2 4 2, Im 2 4 4j j
rectangular
form
3
Addition & Subtraction
Graphical addition & subtraction
are performed like vector addition
(“tip-to-tail”).
3 1M j
2 2N j
Algebraic addition & subtraction
are performed piece-wise:
1 1
2 2
1 2 1 2
M a b j
N a b j
M N a a b b j
5 1M N j
4
Multiplication: Rectangular Form
5 3M j
2 4N j
Multiplication may be accomplished in rectangular form…
1 1 1z a b j
2 2 2z a b j
1 2 1 1 2 2
2
1 2 1 2 2 1 1 2
1 2 1 2 2 1 1 2
1 2 1 2 1 2 2 1
z z a b j a b j
a a a b j a b j b b j
a a a b a b j b b
a a b b a b a b j
2
5 3 2 4
10 20 6 12
22 14
M N j j
j j j
j
…but it is more easily accomplished in polar form.
5
Exponential Form
cos sin
cos sin
j
j
j
e j
z e z j z
z e a b j
cos
sin
a z
b z
assume |z| is
positive, real
2 22 2 2 2
2 22 2 2 2
2 2
cos sin
cos sin
a b z z
a b z z
a b z
sintan
cos
b
a
6
Rectangular Exponential Form
Re
Im
a z
b z
4 3M j
|z|
2 2
tanb
a
z a b
2 2
1
4 3 5
tan 3 4 37
z
375 jM e
|z| = magnitude of z
= phase/angle of z
7
Polar Form
jz a b j z e z
polar rectangular exponential
Polar form is a shorthand
for the exponential form.
374 3 5 5 37jM j e
|z| = magnitude of z
= phase/angle of z
8
Multiplication: Polar Form
3 4 5 53
3 33 45
2 2
M j
N j
Multiplication in polar form is carried out using exponentials…
1
2
1 1 1 1
2 2 2 2
j
j
z a b j z e
z a b j z e
1 2
1 2
1 2
1 2 1 2
1 2
1 2
j j
j j
j
z z z e z e
z z e
z z e
5 53 3 45
15 98
M N
1 1 1 2 2 1,z z z z 1 2 1 2 1 2z z z z
9
Division: Polar Form
1 2
1 2
2 2
1 1 1
2 2 2
j jj j
j j
z e z zee
z e z e z
1 1 1 2 2 1,z z z z
11 11 2
2 2 2
zz
z z
6 8 10 53
5 55 45
2 2
M j
N j
10 53 5 45
2 8
M N
10
Complex Conjugate
The complex conjugate of z is denoted z
then z a b j and if z a b j
3 1M j
3 1M j
The conjugate of z is the same
number, except that the
imaginary part is negated.
Graphically, the complex
conjugate of z is the
mirror image of z
across the Real axis.
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(b)
Complex Arithmetic Examples
and Matlab Scripts
12
Example: Complex Addition
9 2 Aj 3 1 Aj 2 6 Aj
>> i1 = 9 + 2*j;
>> i2 = -3 + j;
>> i3 = -2 + 6*j;
>> i = i1 + i2 + i3
i = 4.0000 + 9.0000i
13
9 2 Aj 3 1 Aj 2 6 Aj
4 9 Aj
>> i1 = 9 + 2*j;
>> i2 = -3 + j;
>> i3 = -2 + 6*j;
>> i = i1 + i2 + i3
i = 4.0000 + 9.0000i
Example: Complex Addition
14
Example: Complex Multiplication
Find v x i in rectangular form:
7 3 mV
5 4 mA
v j
i j
2
7 3 5 4
35 15 28 12
47 13 μW
v i j j
j j j
j
2 9 V , 3 5 Av j i j
2 9 3 5
6 27 10 45
51 17 W
v i j j
j j
j
>> V = 2 + 9*j;
>> I = -3 + 5*j;
>> p = V * I
p = -51.0000 -17.0000i
15
Find v x i in rectangular form:
7 3 mV
5 4 mA
v j
i j
2
7 3 5 4
35 15 28 12
47 13 μW
v i j j
j j j
j
2 9 V , 3 5 Av j i j
2 9 3 5
6 27 10 45
51 17 W
v i j j
j j
j
>> V = 2 + 9*j;
>> I = -3 + 5*j;
>> p = V * I
p = -51.0000 -17.0000i
Example: Complex Multiplication
16
Example: Complex Arithmetic
Determine the quantity va – vb
in polar form if vn = 0 .
va
vb
vn 110 0 V
110 120 V
110 240 V
vc
>> v_a = 110*exp(j*0);
>> v_b = 110*exp(j*-2*pi/3);
>> v = v_a - v_b;
>> abs(v)
ans = 190.5256
>> angle(v)*180/pi
ans = 30.0000
17
Determine the quantity va – vb
in polar form if vn = 0 .
va
vb
vn 110 0 V
110 120 V
110 240 V
190.5 30 V
22 1
110 0 110 120
110cos0 110sin 0
110cos 120 110sin 120
110 0 110 1 2 110 3 2
165 55 3
165 55 3 tan 55 3 165
a bv v
j
j
j j
j
vc
>> v_a = 110*exp(j*0);
>> v_b = 110*exp(j*-2*pi/3);
>> v = v_a - v_b;
>> abs(v)
ans = 190.5256
>> angle(v)*180/pi
ans = 30.0000
Example: Complex Arithmetic
18
Example: Complex Division
Determine the ratio of VL to IL:
LLVL IL
L L
VV
I I
VL
+
–
IL
3 3 3 mVj 1 3 mAj
>> V_L = -3*sqrt(3)+3*j;
>> I_L = 1+j*sqrt(3);
>> Z = V_L / I_L
Z = -0.0000 + 3.0000i
>> abs(Z)
ans = 3.0000
>> angle(Z)*180/pi
ans = 90.0000
19
2 2 1
22 1
3 3 3 mV
1 3 mA
3 3 3 tan 1 3 6 150
2 601 3 tan 3
L
L
j
j
V
I
3 90
Example: Complex Division
Determine the ratio of VL to IL:
VL
+
–
IL
3 3 3 mVj 1 3 mAj
LLVL IL
L L
VV
I I
>> V_L = -3*sqrt(3)+3*j;
>> I_L = 1+j*sqrt(3);
>> Z = V_L / I_L
Z = -0.0000 + 3.0000i
>> abs(Z)
ans = 3.0000
>> angle(Z)*180/pi
ans = 90.0000
20
Example: Complex Conjugate
Write the quantity V x I* in polar form, given 3 5 V
6 7 mA
j
j
V
I
>> V = 3 - 5*j;
>> I = 6 + 7*j;
>> S = V * conj(I)
S = -17.0000 -51.0000i
>> abs(S)
ans = 53.7587
>> angle(S)*180/pi
ans = -108.4349
21
Write the quantity V x I* in polar form, given
3 5 6 7
18 30 21 35
17 51 mW
j j
j j
j
V I
>> V = 3 - 5*j;
>> I = 6 + 7*j;
>> S = V * conj(I)
S = -17.0000 -51.0000i 2 2 117 51 tan 51 17
53.8 108 mW
V I
>> abs(S)
ans = 53.7587
>> angle(S)*180/pi
ans = -108.4349 34 59 85 49
34 85 59 49
53.8 108 mW
V I
3 5 V
6 7 mA
j
j
V
I
Example: Complex Conjugate
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(c)
Sinusoids and
Sinusoidal Steady-State
23
Alternating Current (Sinusoidal)
0 0sin , 0mv t V t
Vm = amplitude (in Volts), 0 = phase (in radians)
= frequency (in radians/second)
T = period (in seconds)
f = frequency (in cycles/second) = 1/T = / 2
0 0sin cos 2m mV t V t
+ – v t
A
B
A
B
24
Review of Sinusoids
v2 “leads” v1 by
v1 “lags” v2 by
1 2sin , sinm mv t V t v t V t
25
Sinusoids & Exponential Form
1 2sin , sinm mv t V t v t V t
cos sinj t
m m mV e V t j V t
cos sinj
e j
Im sinj t
m mV e V t
Re cos
j t
m mV e V t
26
RL Circuit with a Sinusoidal Source
i
0 cosVd R
i t i t tdt L L
0 cosV t
-- oscillates forever
-- never settles to a DC value (e.g. zero)
0 cosi t I t It’s possible that the solution is of the form
Substituting i(t) into the differential equation…
Solving for I0 and substituting back into i(t) yields
00 0sin cos cos
VRI t I t t
L L
10
2 2 2cos tan
V Li t t
RR L
amplitude scaling,
phase shift
27
RL Circuit with a Sinusoidal Source
vs vs
iL
iL
The RL circuit’s transient response
is negligible after 5t .
The remaining response is sinusoidal.
400 μH
5 5 5 1μs2 kΩ
L
Rt
tran
sien
t
28
RC Circuit with a Sinusoidal Source
vs
vs vC
+
–
tran
sien
t
vC
The RC circuit’s transient response
is negligible after 5t .
The remaining response is sinusoidal.
5 5
5 2 kΩ 100 pF 1μs
RCt
29
RLC Circuit with a Sinusoidal Source
is
vC
vC
+
–
tran
sien
t 5 1.6 msst
1 2 3cos sint
c d dv t e V t V t V
1 1 rad3125
2 2 10 16 μF sRC
The RLC circuit’s transient response
is negligible after ts .
The remaining response is sinusoidal.
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(d)
Phasors
31
2cosmI t
1cosmV t
Phasor Notation
2mI
1mV
1 1 1
1 1cos Re Rej t j jj t
m m m m mV t V e V e e V e V
2 2 2
2 2cos Re Rej t j jj t
m m m m mI t I e I e e I e I
Assume all voltages & currents
oscillate with frequency = 2 f …
Pick off the amplitude & phase for each v/i ;
write each in polar form.
phasor notation
(longer derivation provided in
textbook section 10.3)
32
Let all phasors be referenced to a cosine (zero phase).
Convert the following time-domain functions to the phasor domain :
(a) 40cos(t + 30°) mV, = 100 rad/s
(b) 25cos(t – 75°) A, = 400 rad/s
(c) 70cos(2 f + 45°) V, f = 20 MHz
(d) 36sin(2 f + 110°) mA, f = 8 MHz
sin cos 2t t
(a)
(b)
(c)
(d)
Example: Phasor vs. Time Domain
33
Let all phasors be referenced to a cosine (zero phase).
Convert the following time-domain functions to the phasor domain :
(a) 40cos(t + 30°) mV, = 100 rad/s
(b) 25cos(t – 75°) A, = 400 rad/s
(c) 70cos(2 f + 45°) V, f = 20 MHz
(d) 36sin(2 f + 110°) mA, f = 8 MHz
(a) 40 30 mV
(b) 25 75 A
(c) 70 45 V
(d) 36 20 mA sin cos 2t t
Example: Phasor vs. Time Domain
34
Example: Phasors and Voltage/Current
Let = 2000 rad/s with phasors be referenced to a cosine (zero phase).
Determine the instantaneous value, at t = 1 ms ,
of the current corresponding to this phasor: 20 + 10j A .
omega = 2000;
I = 20 + 10*j;
T = 2*pi/omega;
delta_t = T/1000;
t = 0:delta_t:2*T;
i_t = abs(I)*cos(omega*t + angle(I));
plot(t,i_t)
grid
axis([0 2*T -Inf Inf])
ylabel('Current (A)')
xlabel('Time (s)')
– 17.5 A
35
2 2 120 10 20 10 tan 10 20 22.4 26.6 Aj
22.4 cos 2000 0.464 A
1 ms 22.4cos 2.464
i t t
i
omega = 2000;
I = 20 + 10*j;
T = 2*pi/omega;
delta_t = T/1000;
t = 0:delta_t:2*T;
i_t = abs(I)*cos(omega*t + angle(I));
plot(t,i_t)
grid
axis([0 2*T -Inf Inf])
ylabel('Current (A)')
xlabel('Time (s)')
Example: Phasors and Voltage/Current
Let = 2000 rad/s with phasors be referenced to a cosine (zero phase).
Determine the instantaneous value, at t = 1 ms ,
of the current corresponding to this phasor: 20 + 10j A .
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(e)
Phasors & Ohm’s Law, KVL/KCL
37
Phasor Voltage vs. Current: R, L, C
v t R i t
For this equation to be true,
1 2andm mV I R
L L
dv t L i t
dt
1cosmv t V t 2cosmi t I t
1 2 90 , m
m
VL
I 2 1 90 , m
m
IC
V
C C
di t C v t
dt
For this equation to be true, For this equation to be true,
V = R I V = jL I
v leads i
by 90°
v and i are
in phase I = jC V i leads v
by 90°
38
Example: KCL, Phasor Domain
+
–
Determine v(t).
v t
2 nF 70
10
mH
68cos 2 10 30 mAsi t t
39
V
+
–
Determine v(t).
v t
• Convert to phasor form… +
–
IR IL IC
8 30 mAs I
2 nF 70
2 nF
10
mH 70
• Employ the appropriate Kirchhoff Law(s)…
0s j CR j L
V V
I V
6 9
6 68 30 2 10 2 10 0
70 2 10 10 10j
j
V VV
10
mH
68cos 2 10 30 mAsi t t
Example: KCL, Phasor Domain
40
+
–
Determine v(t).
v t
2 nF
10
mH 70
• Convert between rectangular & polar forms as necessary…
6 9
6 68 30 2 10 2 10 0
70 2 10 10 10j
j
V VV
1 10.0126 8 30
70 62.8j
j
V
0.0143 0.0159 0.0126 8 30j j V
0.0143 0.0033 8 30j V
0.0147 13 8 30 V
68cos 2 10 30 mAsi t t
Example: KCL, Phasor Domain
41
+
–
68cos 2 10 30 mAsi t t
Determine v(t).
v t
2 nF
10
mH 70
omega = 2*pi*10^6;
I = 8*exp(j*30*pi/180);
R = 70;
L = 10e-6;
C = 2e-9;
Y = (1/R + 1/(j*omega*L) + j*omega*C);
V = I / Y;
abs(V)
ans = 545.2174
angle(V)*180/pi
ans = 43.1941
Example: KCL, Phasor Domain
42
+
–
68cos 2 10 30 mAsi t t
Determine v(t).
v t
2 nF
10
mH 70
• Convert between rectangular & polar forms as necessary…
0.0147 13 8 30 V
8 30 mA544 43 mV
0.0147 13
V
omega = 2*pi*10^6;
I = 8*exp(j*30*pi/180);
R = 70;
L = 10e-6;
C = 2e-9;
Y = (1/R + 1/(j*omega*L) + j*omega*C);
V = I / Y;
abs(V)
ans = 545.2174
angle(V)*180/pi
ans = 43.1941
• Convert from phasors to time domain…
6544cos 2 10 43 mVv t t
Example: KCL, Phasor Domain
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(f)
Impedance
44
Impedance
Impedance, Z is the ratio of phasor voltage to phasor current
for an electrical element or network. like resistance, but it is complex
-- It is a measure of an element/network’s ability to impede current flow.
V
ZI
For a resistor, RR R V I Z
For an inductor, Lj L j L V I Z
For a capacitor, 1Cj C j C
j C
I V Z
-- current and voltage are always in-phase
-- there is no frequency dependence
-- voltage always leads current by 90°
-- at higher frequencies, less current is passed (for constant V )
-- current always leads voltage by 90°
-- at higher frequencies, more current is passed (for constant V )
45
Impedance vs. Frequency
0
R
R
R
R
R
Z
Z
Z
R = 100
90
L
L
L
j L
L
Z
Z
Z
2 f
1
1
90
C
C
C
j C
C
Z
Z
Z
L = 3 mH
C = 300 pF
46
Impedance vs. Frequency
100 , RR R Z
300 pF, 1CC j C Z
freq = 2e6:1e3:10e6;
omega = 2*pi*freq;
R = 100;
Z_R = R*ones(1,length(omega));
L = 3e-6;
Z_L = j*omega*L;
C = 300e-12;
Z_C = 1./(j*omega*C);
2 MHz 10 MHz, 2f f
3μH, LL j L Z
figure(1)
subplot(2,1,1)
plot(freq/10^6,abs(Z_R))
axis([-Inf Inf 0 300])
set(gca,'Xtick',[2:1:10])
ylabel('| Z_R | (ohms)')
title('{\bf Resistor Impedance ...
vs. Frequency}')
subplot(2,1,2)
plot(freq/10^6,phase(Z_R)*180/pi)
axis([-Inf Inf -180 180])
set(gca,'Xtick',[2:1:10])
ylabel('phase[ Z_R ] (degrees)')
xlabel('Frequency (MHz)')
figure(2)
subplot(2,1,1)
plot(freq/10^6,abs(Z_L))
axis([-Inf Inf 0 300])
set(gca,'Xtick',[2:1:10])
ylabel('| Z_L | (ohms)')
title('{\bf Inductor Impedance ...
vs. Frequency}')
subplot(2,1,2)
plot(freq/10^6,phase(Z_L)*180/pi)
axis([-Inf Inf -180 180])
set(gca,'Xtick',[2:1:10])
ylabel('phase[ Z_L ] (degrees)')
xlabel('Frequency (MHz)')
figure(3)
subplot(2,1,1)
plot(freq/10^6,abs(Z_C))
axis([-Inf Inf 0 300])
set(gca,'Xtick',[2:1:10])
ylabel('| Z_C | (ohms)')
title('{\bf Capacitor Impedance ...
vs. Frequency}')
subplot(2,1,2)
plot(freq/10^6,phase(Z_C)*180/pi)
axis([-Inf Inf -180 180])
set(gca,'Xtick',[2:1:10])
ylabel('phase[ Z_C ] (degrees)')
xlabel('Frequency (MHz)')
47
Impedances in Series & Parallel
1
N
s n
n
R R
1
N
s n
n
Z Z
Impedances in series
are combined
like resistors in series.
Z1
Z2
Z3
ZN
V1
V2
V3
VN
Vs
I
1
s
N
n
n
VI
Z
Impedances in parallel
are combined
like resistors in parallel. 1
1
s
N
n
n
IV
Z
1
1 1N
np nR R
1
1 1N
np n
Z Z
48
Example: Equivalent Impedance
Determine the equivalent
impedance of the network at
terminals A–B if = 5 rad/s. 2
100
mF
1.6 H 33.3
mF
4
A
B
49
Determine the equivalent
impedance of the network at
terminals A–B if = 5 rad/s. 2
100
mF
2 –2j
1.6 H
8j –6j
33.3
mF
4
4
• Convert all resistances,
inductances, capacitances
into impedances…
A
B
A
B
R
L
C
R
j L
j C
Z
Z
Z
Example: Equivalent Impedance
50
• Combine impedances in series &
parallel, starting away from A–B
and working towards A–B…
1
2 2 4 902 45 1
2 2 8 45
jj
j
Z
2 1 8 6
1 8 6 1
j j
j j j j
Z ZZ1
Example: Equivalent Impedance
51
• Combine impedances in series &
parallel, starting away from A–B
and working towards A–B…
Z2
2
2
4 1 4
4 1 4
4 4 32 45.0
5 26 11.4
A B
j
j
j
j
ZZ
Z
1.1 33.6
If a test source were applied at terminals A–B : test 1 0 A I
then the voltage across A–B would be test 1 0 1.1 33.6 1.1 33.6 V V
Example: Equivalent Impedance
52
Determine the equivalent
impedance of the network at
terminals A–B if = 5 rad/s. 2
100
mF
1.6 H 33.3
mF
4
A
B
omega = 5;
C1 = 100e-3;
C2 = 33.3e-3;
R1 = 2;
R2 = 4;
L1 = 1.6;
Z_C1 = 1/(j*omega*C1);
Z_C2 = 1/(j*omega*C2);
Z_R1 = R1;
Z_R2 = R2;
Z_L1 = j*omega*L1;
Z_eq1 = Z_C1*Z_R1/(Z_C1 + Z_R1);
Z_eq2 = Z_eq1 + Z_L1 + Z_C2;
Z_ab = Z_R2*Z_eq2 / (Z_R2 + Z_eq2)
Z_ab =
0.9217 + 0.6120i
abs(Z_ab)
ans = 1.1063
phase(Z_ab)*180/pi
ans = 33.5837
Example: Equivalent Impedance
53
Example: Phasor Analysis
(a) Determine vC(t).
(b) Plot vC(t) and the source
versus time for 2 cycles. 3 cos(2×106t + 20°) V
1 k
1 nF
+
vC
54
Example: Phasor Analysis
(a) Determine vC(t).
(b) Plot vC(t) and the source
versus time for 2 cycles.
vS
vC
3 cos(2×106t + 20°) V
1 k
1 nF
+
vC
+
55
• Convert the circuit from the time domain
to the phasor domain.
VC 6 92 10 10 160 ΩC j C j j Z–160j
1 k 3 20
V
• Use KVL/KCL to solve for V/I
in the phasor domain.
3 20 1000 160 0
160C
j
j
I I
V I
I
(a) Determine vC(t).
(b) Plot vC(t) and the source
versus time for 2 cycles.
Example: Phasor Analysis
3 cos(2×106t + 20°) V
1 k
1 nF
+
vC
56
• Perform complex algebra to find V/I…
• Convert back to the time domain…
6472cos 2 10 60.9 mVCv t t
2 2 1
160 160 903 20 3 20
1000 160 1000 160 tan 160 1000
160 903 20 0.472 60.9
1013 9.1
C
j
j
V
(a) Determine vC(t).
(b) Plot vC(t) and the source
versus time for 2 cycles.
Example: Phasor Analysis
+
VC –160j
1 k 3 20
V
I
57
Example: Plotting Sinusoids, Matlab
(a) Determine vC(t).
(b) Plot vC(t) and the source
versus time for 2 cycles.
omega = 2*pi*10^6;
T = 2*pi/omega;
delta_t = T/100;
t = 0 : delta_t : 2*T - delta_t ;
v_S = 3.000 * cos( omega*t + 20.0*pi/180 );
v_C = 0.472 * cos( omega*t - 60.9*pi/180 );
figure(1)
plot(t/10^-6, v_S, 'r--', ...
t/10^-6, v_C, 'b-', 'LineWidth', 2)
ylabel('Voltage (V)')
xlabel('Time (\mus)')
legend('v_S','v_C')
grid
6472cos 2 10 60.9 mVCv t t
3 cos(2×106t + 20°) V
1 k
1 nF
+
vC
58
Example: Transient AC Circuit, PSpice
6472cos 2 10 60.9 mVCv t t
Amplitude is 472 mV
as determined by
written analysis.
59
Example: Frequency Response, Matlab
9
9 1010 ΩC
jj C j
Z
9
9
103 20 3 20
1000 1000 10
CC
C
j
j
ZV
Z
Plot the amplitude of vC(t) versus f0
for 500 kHz < f0 < 2 MHz.
V_s = 3 * exp( j*20*pi/180 );
R = 1000;
C = 1e-9;
f = 500e3 : 1e3 : 2e6 ;
omega = 2 * pi * f;
Z_C = -j ./ (omega * C);
V_C = V_s * Z_C ./ (R + Z_C);
plot(f/10^3, abs(V_C), 'LineWidth', 2)
ylabel('|V_C| (volts)')
xlabel('Frequency (kHz)')
grid
6 6
6
for 10 Hz ( 2 10 rad s) ...
472cos 2 10 60.9 mVC
f
v t t
Frequency (kHz)
3 cos(2×106t + 20°) V
1 k
1 nF
+
vC
60
Example: Frequency Response, PSpice
Plot the amplitude of vC(t) versus f0
for 500 kHz < f0 < 2 MHz.
“VAC” part,
“SOURCE”
library
61
Example: Frequency Response, PSpice
Plot the amplitude of vC(t) versus f0
for 500 kHz < f0 < 2 MHz.
“VAC” part,
“SOURCE”
library
6 6
6
for 10 Hz ( 2 10 rad s) ...
472cos 2 10 60.9 mVC
f
v t t
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(g)
Nodal & Mesh Analysis
in the Phasor Domain
63
Nodal Analysis with Phasors
Analysis Steps
(1) Choose a reference node (usually ground or the bottom node) to have a voltage of zero.
(2) Assign a unique voltage variable to each node that is not the reference (v1, v2, v3, … vN–1).
(3) For independent & dependent voltage sources, identify a supernode
and write the voltage across the supernode in terms of node voltages.
Write a KCL equation at all N – 1 nodes including the supernode
(and not the reference, or a supernode which includes the reference).
(4) Solve the N – 1 node equations + source equations simultaneously.
0
v1 v2
0
V1 V2 V3 v3
SAME as with DC circuits. Now use complex arithmetic.
64
Example: Nodal Analysis, Phasors
Write a valid matrix equation whose
solution includes the phasor form of v(t).
+
–
v(t)
0.5iA
iA
65
Write a valid matrix equation whose
solution includes the phasor form of v(t).
+
–
V
0
V1
V2
V3 V4
320 70 10 1.4LZ j L j j
320 250 10 0.2CZ j C j j
1.4 j
0.2 j 0.2 j
9 0 V 9 90 V
• Convert to phasor form…
• Identify supernode(s)… 1 2 0.5 AV V I
• Write KCL equations…
• Identify v sources next to the reference…
3 1 3 2 3 4 3 00.2 1.4 0.2 3
V V V V V V V
j j j
1 49, 9V V j
• Write equations governing
dep. src. ctrl…
IA
0.5IA
0
v1
v2
v3 v4
0.5iA
iA
3
3A
VI
+
–
v(t)
Example: Nodal Analysis, Phasors
66
Write a valid matrix equation whose
solution includes the phasor form of v(t).
+
–
V
0
V1
V2
V3 V4
0.5IA
1.4 j
0.2 j 0.2 j
9 0 V 9 90 V IA
• Rearrange into matrix form…
3 1 3 2 3 4 3 00.2 1.4 0.2 3
V V V V V V V
j j j
1 49, 9V V j 1 2 0.5 AV V I
1 2 3 4
15 5 5 5 0
1.4 1.4 3
j jj V V j j V j V
1
4
9
9
V
V j
1 2 0.5 0AV V I
3
3A
VI
2V V
3 3 0AV I
1
3
4
105 9.3 5 0
1.4 39
1 0 0 0 09
0 0 0 1 00
1 1 0 0 0.50
0 0 1 3 0 1 A
jVj j j
V
V j
V
I
8.3 5.5 VV
8.3cos 20 5.5 Vv t t
Example: Nodal Analysis, Phasors
67
Write a valid matrix equation whose
solution includes the phasor form of v(t).
+
–
V
0
V1
V2
V3 V4
0.5IA
1.4 j
0.2 j 0.2 j
9 0 V 9 90 V IA
3 1 3 2 3 4 3 00.2 1.4 0.2 3
V V V V V V V
j j j
1 49, 9V V j
3
3A
VI
>> A = [ -5*j j/1.4 9.29*j+1/3 -5*j 0 ;
1 0 0 0 0 ;
0 0 0 1 0 ;
1 -1 0 0 -0.5 ;
0 0 -1/3 0 1 ];
>> B = [0 ; 9 ; -9*j ; 0 ; 0 ];
>> x = A^-1 * B
>> V = x(2);
>> abs(V)
>> angle(V) * 180/pi
x =
9.0000 - 0.0000i
8.2703 + 0.7913i
4.3784 - 4.7477i
0.0000 - 9.0000i
1.4595 - 1.5826i
ans =
8.3080
ans =
5.4653
1 2 0.5 AV V I
Example: Nodal Analysis, Phasors
8.3 5.5 VV
68
Mesh Analysis with Phasors
Analysis Steps
(1) Draw a mesh current for each mesh.
(2) Identify supermeshes.
(3) Write KVL around each supermesh,
then KVL for each mesh that is
not part of a supermesh.
(4) Express additional unknowns
(e.g. dependent-source V/I)
in terms of mesh currents.
(5) Solve the simultaneous equations.
SAME as with DC circuits.
Now use complex arithmetic.
i1 i2
I1 I2
69
Example: Mesh Analysis, Phasors
Write a valid matrix equation whose
solution includes the phasor form of v(t).
v(t)
+
–
0.5iA
iA
70
320 70 10 1.4LZ j j
320 250 10 0.2CZ j j
• Convert to phasor form…
• Identify supermesh(es)…
• Write KVL equations…
1 2 1 4
2 3 2 1
3 2 3 3 4
4 1 4 3
9 0.2 3 0
0.5 1.4 0.2 0
1.4 5 9 0.2 0
3 0.2 9 0
A
j I I I I
I j I I j I I
j I I I j j I I
I I j I I j
• Write equations governing
dep. src. ctrl… 1 4AI I I
v(t)
+
–
i2 i3
i1 i4
0.5iA
iA
I2 I3
I1 I4
+
–
V
9 0 V
9 Vj
0.5IA
IA
Write a valid matrix equation whose
solution includes the phasor form of v(t).
0.2 j 0.2 j
1.4 j
Example: Mesh Analysis, Phasors
71
Write a valid matrix equation whose
solution includes the phasor form of v(t).
• Rearrange into matrix form…
1
2
3
4
0.2 3 0.2 0 3 0 0 9
0.2 1.2 1.4 0 0.5 0 0
0 1.4 1.2 5 0.2 0 0 9
3 0 0.2 3 0.2 0 0 9
1 0 0 1 1 0 0
0 0 5 0 0 1 0
A
j j I
j j j I
j j j I j
j j I j
I
V
8.3 5.5 VV 8.3cos 20 5.5 Vv t t
I2 I3
I1 I4
+
–
V
9 0 V
9 Vj
0.5IA
IA
0.2 j 0.2 j
1.4 j
1 2 1 4
2 3 2 1
3 2 3 3 4
4 1 4 3
9 0.2 3 0
0.5 1.4 0.2 0
1.4 5 9 0.2 0
3 0.2 9 0
A
j I I I I
I j I I j I I
j I I I j j I I
I I j I I j
1 4AI I I 35V I
1 2 4
1 2 3
2 3 4
1 3 4
1 4
3
0.2 3 0.2 3 9
0.2 1.2 1.4 0.5 0
1.4 1.2 5 0.2 9
3 0.2 3 0.2 9
0
5 0
A
A
j I j I I
j I j I j I I
j I j I j I j
I j I j I j
I I I
I V
Example: Mesh Analysis, Phasors
72
Write a valid matrix equation whose
solution includes the phasor form of v(t).
I2 I3
I1 I4
+
–
V
9 0 V
9 Vj
0.5IA
IA
0.2 j 0.2 j
1.4 j
1 2 1 4
2 3 2 1
3 2 3 3 4
4 1 4 3
9 0.2 3 0
0.5 1.4 0.2 0
1.4 5 9 0.2 0
3 0.2 9 0
A
j I I I I
I j I I j I I
j I I I j j I I
I I j I I j
1 4AI I I 35V I
A = [ -0.2*j+3 0.2*j 0 -3 0 0 ;
0.2*j 1.2*j -1.4*j 0 0.5 0 ;
0 -1.4*j 1.2*j+5 0.2*j 0 0 ;
-3 0 0.2*j 3-0.2*j 0 0 ;
1 0 0 -1 -1 0 ;
0 0 -5 0 0 1 ];
B = [ 9 ; 0 ; -9*j ; 9*j ; 0 ; 0 ];
x = A^-1 * B
V = x(6);
abs(V)
angle(V) * 180/pi
x =
-18.1368 +20.4776i
5.6121 - 2.6198i
1.6540 + 0.1583i
-19.5970 +22.0609i
1.4602 - 1.5833i
8.2699 + 0.7916i
ans =
8.3077
ans =
5.4679
Example: Mesh Analysis, Phasors
8.3 5.5 VV
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(h)
Thevenin Equivalence
in AC Circuits
74
Review of Thevenin Equivalents
• allows us to replace a large, complicated circuit with a much
simpler 2-element series/parallel circuit
• the simpler circuit allows for rapid calculations of V, I, P that the
original circuit can deliver to a load
• helps us to choose the best value of load resistance to maximize
the power delivered (e.g. from an amplifier, to a speaker)
75
Review of Thevenin Equivalents
only ind.
src.?
To determine VTH ...
To determine ZTH ...
• source transformation
• find VTH = VOC
• find VTH = VOC
YES NO
only ind.
src.?
• deactivate src., find ZTH = Zeq
• source transformation
• find ISC, use ZTH = VOC/ISC
• insert test source, find ZTH = Vtest/Itest
YES NO
VTH = 0 ?
• find ISC, use ZTH = VOC/ISC
• insert test source, find ZTH = Vtest/Itest
NO
• insert test source,
find ZTH = Vtest/Itest
YES
76
Determine the Thevenin
equivalent of Network A using
open-circuit voltage and
short-circuit current.
OC TH
6 612 12 8 V =
6 3 9V
V V
+
–
VOC
SC
1 7 84 A
1 2 1 7 9
I
OCTH
SC
89
8 9
VZ
I
ISC
Review of Thevenin Equivalents
77
Determine the Thevenin
equivalent of Network A
by using a test source.
TH8 V V
+
–
VOC
Itest
Vtest
Vtest
Itest
2
testtest
2 7
VI
testTH
test
9 V
ZI
Review of Thevenin Equivalents
78
Example: Thevenin & Sinusoids
Determine the phasor voltage
difference V1 – V2 (with –j10 ) .
Use the Thevenin equivalent
at V1 , V2 (without –j10 ) .
79
TH OC A B 1 0 4 2 0.5 90 2 4
4 2 0.5 2 4 4 2 2 6 3 V
j j
j j j j j j
V V V V
VA VB
TH 4 2 2 4 6 2j j j Z
1 2
106 3
6 2 10
3 6 6.7 63.4 V
jj
j j
j
V V
Example: Thevenin & Sinusoids
Determine the phasor voltage
difference V1 – V2 (with –j10 ) .
Use the Thevenin equivalent
at V1 , V2 (without –j10 ) .
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(i)
Phasors &
Superposition
81
Linearity
linear
network
N
1x t 1y t 2x t 2y t
1 1x t t 1 1y t t
1 1 1 2 2 2A x t t A x t t
linear
network
N
N
1 1 1A x t t 1 1 1A y t tN
1 1 1 2 2 2A y t t A y t t
x = input or
source or
stimulus
y = output or
response
N
N
82
Superposition: Voltage Sources
2 k
2 k
4 V
8 V
16 V
32 V
Determine the current i using superposition.
16 4 4 mAi 32 4 8 mAi
8 4 2 mAi 4 4 1mAi
deactivate all but 1 solve sum
60 V
60 4 15 mAi
4 8 2 1
15 mA
i i i i i
83
Determine the voltage v
using superposition.
deactivate all but 1
solve sum
3 6 A 6 12 A 24 A
6 0 12 V3 6
v vv
12 0 24 V3 6
v vv
24 0 48 V3 6
v vv
12 24 48
60 Vv
Superposition: Current Sources
84
Example: Superposition & Sinusoids
Determine the phasor
voltage difference V1 – V2 .
Use superposition.
85
Example: Superposition & Sinusoids
1V 2
V 2V
1V
1 2 1 2 2
1V V V V V V
1 2
4 21 10 2 4 V
4 2 2 6
jj j
j j
V V
2
2 40.5 10 1 2 V
2 4 4 12
jj j j
j j
1
V V
1
1 2 3 6 9 36 tan 2
6.7 63.4 V
j
V V
Determine the phasor
voltage difference V1 – V2 .
Use superposition.
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(x,1)
Thevenin Equivalent
Example
87
Determine the Thevenin equivalent
circuit with respect to terminals a–b.
Example: Thevenin & Sinusoids, #2
88
Determine the Thevenin equivalent
circuit with respect to terminals a–b.
TH eq
5 10 55 || 10 5
5 10 5
25 502.5 5
10
j jj j
j j
jj
Z Z
TH OC
1 10 53 30 10 5
1 10 5 1 5
13.0 10.6 V
RLs RL
RL C
jj
j j
j
YV V I Z
Y Y
Example: Thevenin & Sinusoids, #2
89
Determine the Thevenin equivalent
circuit with respect to terminals a–b.
TH OC SC
13.0 10.6 3 30
2.5 5 j
Z V I
2.5 5 j
13.0 10.6 Vj
Example: Thevenin & Sinusoids, #2
90
Determine the Thevenin equivalent
impedance with respect to terminals a–b.
TH 2.5 5 j Z
Example: Thevenin & Sinusoids, #2
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(j)
Phasors &
Source Transformation
92
Review of Source Transformation
If these two circuits provide the same v/i characteristics at their
outputs (vL, iL), the two circuits are equivalent.
sV
max s sI V R
max sV V
max sI I
max s pV I R
ss p
s
VR R
I
condition
for equivalence
sI
93
Source Transformation & Phasors
If these two circuits provide the same V/I characteristics at their
outputs (VL, IL), the two circuits are equivalent.
sV
max s sI V Z
max sV V
max sI I
max s p V I Z
ss p
s
V
Z ZI
condition
for equivalence
sZ LI
LV
LI
LVpZsILZ LZ
94
Example: Src. Transform. & Sinusoids
Determine the phasor
voltage difference V1 – V2 .
Use source transformation(s).
95
Example: Src. Transform. & Sinusoids
VS1 VS2
Z2 Z1
1 2 S1 S2
1 2
10
10
j
j
V V V V
Z Z
1
2
4 2
2 4
j
j
Z
Z
S1 1 0 4 2
4 2 V
j
j
V
S1 0.5 90 2 4
0.5 2 4 2 V
j
j j j
V
1 2
104 2 2
10 4 2 2 4
106 3 3 6
8 6
6.7 63.4 V
jj j
j j j
jj j
j
V V
Determine the phasor
voltage difference V1 – V2 .
Use source transformation(s).
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(k,1)
PSpice for
AC Circuits
97
• From nodal analysis…
3 1 3 3 4 3 00.2 1.4 0.2 3
V V V V V V V
j j j
1 49, 9V V j 1 2 0.5 AV V I
3
3A
VI
1
3
4
105 9.3 5 0
1.4 39
1 0 0 0 09
0 0 0 1 00
1 1 0 0 0.50
0 0 1 3 0 1 A
jVj j j
V
V j
V
I
8.3 5.5 VV
8.3cos 20 5.5 Vv t t
Example #1: Written Analysis
+
–
V
0
V1
V2
V3 V4
1.4 j
0.2 j 0.2 j
9 0 V 9 90 V IA
0.5IA
0.5iA
iA
+
–
v(t)
Solve for v(t) .
98
+
–
V V1
V2
V3 V4
0.5IA
1.4 j
0.2 j 0.2 j
9 0 V 9 90 V IA
• Rearrange into matrix form…
3 1 3 2 3 4 3 00.2 1.4 0.2 3
V V V V V V V
j j j
1 49, 9V V j
3
3A
VI
1 2 0.5 AV V I
8.3 5.5 V V
8.3cos 20 5.5 Vv t t
Example #1: Matlab
Solve for v(t) .
>> A = [ -5*j j/1.4 9.29*j+1/3 -5*j 0 ;
1 0 0 0 0 ;
0 0 0 1 0 ;
1 -1 0 0 -0.5 ;
0 0 -1/3 0 1 ];
>> B = [0 ; 9 ; -9*j ; 0 ; 0 ];
>> x = A^-1 * B
>> V = x(2);
>> abs(V)
>> angle(V) * 180/pi
x =
9.0000 - 0.0000i
8.2703 + 0.7913i
4.3784 - 4.7477i
0.0000 - 9.0000i
1.4595 - 1.5826i
ans =
8.3080
ans =
5.4653
99
Example #1: PSpice
Amplitude is 8.4 V
as determined by
written analysis.
Plot v(t) using PSpice.
8.3 5.5 V V
8.3cos 20 5.5 Vv t t
100
Determine the phasor
voltage difference V1 – V2 .
Confirm this answer
using PSpice.
VS1 VS2
Z2 Z1
1 2 S1 S2
1 2
10
10
j
j
V V V V
Z Z
1
2
4 2
2 4
j
j
Z
Z
S1 1 0 4 2
4 2 V
j
j
V
S1 0.5 90 2 4
0.5 2 4 2 V
j
j j j
V
1 2
104 2 2
10 4 2 2 4
106 3 3 6
8 6
6.7 63.4 V
jj j
j j j
jj j
j
V V
Example #2: Written Analysis
+
–
+
–
101
1 2 6.7 63.4 V V V
Example #2: PSpice
“IAC” part,
“SOURCE”
library
Determine the phasor
voltage difference V1 – V2 .
Confirm this answer
using PSpice.
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(k,2)
Op Amps
in AC Circuits
103
Example #3: Phasors & Op Amps
200
2 k
125 nFSketch |Vout/Vin| vs. for 1 rad/s < < 100 krad/s .
Determine vout(t) for vin(t) = cos(4x104t) V .
104
200
out
in
||1 1
1
1 400010
1 4000
f f f f f
i i i f f
f f f
i f f
R j C R j C
R R R j C
R R C
R j R C j
ZV
V Z
2 k
125 nF
out
2 2in
400010
4000
V
V
4000
Sketch |Vout/Vin| vs. for 1 rad/s < < 100 krad/s .
Determine vout(t) for vin(t) = cos(4x104t) V .
4
4
out in3
4 10 104 10 7.1 135 V
4 10 4000 1j j
V V
4
out 7.1cos 4 10 135 Vv t t
Example #3: Phasors & Op Amps
THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA
171 Moultrie Street, Charleston, SC 29409
Dr. Gregory J. Mazzaro
Spring 2016
ELEC 202 – Electric Circuit Analysis II
Lecture 10(k,3)
AC Thevenin Equivalent
w/ a Dependent Source
106
Example #4: Thevenin, Dependent Src
A
B
Determine the Thevenin equivalent
of this circuit at terminals A–B .
107
Example #4: Thevenin, Dependent Src
A
B
Determine the Thevenin equivalent
of this circuit at terminals A–B .
OC
10 30 30 5 25 0
25
a a a
a
j
j
I I I
I V
OC
10 300.28 75
25 25
25 0.28 75 7.07 15 V
aj
j
I
V
SC
SC
10 30 30 5 25 0
25 50 0
a a a
a
j
j
I I I I
I I
SC
25 25 30 10 30
25 50 0
aj
j
I
I SC 106 2 mA I7.07 15 V
TH
7.07 15
0.106 2
66.7 13
Z
A
B