Download PWA_Mod08_Prob04_v05

Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Problems With Assistance
Module 8 – Problem 4
Filename: PWA_Mod08_Prob04.ppt
This problem is adapted from Quiz #6 from the fall of 1998 in ECE 2300 Circuit
Analysis, in the Department of Electrical and Computer Engineering at the University
of Houston
Go
straight to
the First
Step
Go
straight to
the
Problem
Statement
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Overview of this Problem
In this problem, we will use the following
concepts:
• Phasor Analysis
• Thevenin’s Theorem
• Equivalent Circuits in the Phasor Domain
Go
straight to
the First
Step
Go
straight to
the
Problem
Statement
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Textbook Coverage
The material for this problem is covered in your textbook in
the following chapters:
• Circuits by Carlson: Chapter 6
• Electric Circuits 6th Ed. by Nilsson and Riedel: Chapter 9
• Basic Engineering Circuit Analysis 6th Ed. by Irwin and
Wu: Chapter 8
• Fundamentals of Electric Circuits by Alexander and
Sadiku: Chapter 9
• Introduction to Electric Circuits 2nd Ed. by Dorf: Chapter
11
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Coverage in this Module
The material for this problem is covered in
this module in the following presentations:
• DPKC_Mod08_Part01,
DPKC_Mod08_Part02, and
DPKC_Mod08_Part03.
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Problem Statement
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
L1=
10[mH]
b
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Solution – First Step – Where to Start?
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
How should we start this
problem? What is the first step?
L1=
10[mH]
b
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Problem Solution – First Step
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
How should we start this problem?
What is the first step?
R1=
1[kW]
R2=
1[kW]
a)
Apply superposition
+
b)
Convert the circuit to the
phasor domain
vC
c)
Find the open-circuit voltage
d)
Find the short-circuit current
vS=
5vC
+
C1=
5[mF]
-
a
L1=
10[mH]
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Your Choice for First Step –
Apply superposition
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
L1=
10[mH]
-
This is not a good choice for the first step.
Superposition will not help with this problem. Superposition can be applied when
we have more than one independent source. We do not even have a single
independent source. Go back and try again.
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Your Choice for First Step –
Find the open-circuit voltage
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
L1=
10[mH]
-
This is not a good choice.
If we were to find the open-circuit voltage in this circuit, we would get zero. There are no
independent sources. This would not be helpful, even if we were going to solve in the time
domain, which we are not. Go back and try again.
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Your Choice for First Step –
Find the short-circuit current
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
L1=
10[mH]
-
This is not a good choice.
If we were to find the short-circuit current in this circuit, we would get zero. There are no
independent sources. This would not be helpful, even if we were going to solve in the time
domain, which we are not. Go back and try again.
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Your Choice for First Step –
Convert the circuit to the phasor domain
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as
seen at terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors,
inductors and capacitors.
R1=
1[kW]
R2=
1[kW]
a
+
vC
vS=
5vC
+
C1=
5[mF]
-
L1=
10[mH]
-
This is a good choice for the first step.
Indeed, this should be an easy choice, since we are told to do this explicitly in the problem.
Even if we had not been told this, we should know it, since we are asked for an
equivalent impedance, which means the phasor domain. Let’s convert.
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Converting to the Phasor Domain
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
ZR1=
1[kW]
ZR2=
1[kW]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
b
We have converted the circuit to the phasor domain, assuming that the angular frequency is
2,000[rad/s]. Notice that the variable for the dependent source, VC, has also been
converted to a phasor. Next, we need to find the Thevenin impedance at these two
terminals, a and b.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Finding the Thevenin Impedance
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
b
We need to find the Thevenin impedance at the two terminals, a and b. How should we
find this value?
a) Find the open-circuit voltage and short-circuit current, and then take the ratio.
b) Apply a test source, and find the ratio of the voltage to current.
c) Replace all the components with their equivalent impedances, and combine using series
and parallel impedance rules.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Finding the Thevenin Impedance – Voc and Isc
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
b
You suggested that to find the Thevenin impedance at the two terminals, a and b, we
should find the open-circuit voltage and short-circuit current, and then take the ratio.
Unfortunately, this will not work for this circuit. There are no independent sources, and
therefore both of these values will be zero, and the ratio will be undefined. This method
will not work for this circuit. Go back and try again.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Finding the Thevenin Impedance -- Equivalent Z’s
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
b
You suggested that to find the Thevenin impedance at the two terminals, a and b, we
should replace all the components with their equivalent impedances, and combine using
series and parallel impedance rules.
Unfortunately, this is not a straightforward process for this circuit. There is a dependent
source present in the circuit, and it is fairly complicated to come up with an accurate
equivalent impedance for this source. Generally, this is not a recommended approach.
There is an easier, and more straightforward method. Go back and try again.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Finding the Thevenin Impedance – Apply a Test Source
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
b
You suggested that to find the Thevenin impedance at the two terminals, a and b, we
should apply a test source, and find the ratio of the voltage to current.
This is the approach that we recommend for this kind of circuit. While more
complicated than the other methods of finding a Thevenin impedance, this approach will
always work. Don’t forget that the first step in this approach is to set all independent
sources equal to zero. There are no independent sources in this circuit, so we can apply the
test source.
Dave Shattuck
University of Houston
Applying a Test Source – 1
© Brooks/Cole Publishing Co.
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
ZR1=
1[kW]
ZR2=
1[kW]
It
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
We can apply the test source. Here, we have chosen to apply a 1[V] voltage source.
Any source will do; this is simply a convenient choice for this problem. Our goal is to find
the current It, and then take the ratio,
ZThev 
Vt
It
.
Next slide
Dave Shattuck
University of Houston
Applying a Test Source – 2
© Brooks/Cole Publishing Co.
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
ZR1=
1[kW]
ZR2=
1[kW]
It
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
To solve, we first find Vc. We can write one equation for Vc using KCL, to write
Vc  5Vc Vc  1[V]
Vc


 0.
1[kW]
1[kW]
100 j[W]
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Applying a Test Source – 3
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
It
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
Solve for Vc.
Vc  5Vc Vc  1[V]
Vc


 0. We can collect terms,
1[kW]
1[kW]
100 j[W]
 4
 1[V]
1
1
Vc 


, or

 1[kW] 1[kW] 100 j[W]  1[kW]
Vc  95.8  107[mV].
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Applying a Test Source – 4
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
It
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
With this solution for Vc, we can find It.
1[V]  Vc
1[V]

 I t , which comes from KCL at a. Solving,
1[kW]
20 j[W]
1[V]  95.8  107[mV]
1[V]

 I t , or
1[kW]
20 j[W]
I t  49.9  88.8[mA].
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Applying a Test Source – 5
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
It
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
Finally, we plug in this value for It to get ZThev, which is
ZThev
Vt
1[V]
 
 2088.8[W]  0.412  20 j[W].
I t 49.9  88.8[mA]
Next slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Finding a Circuit Model – 1
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
Z =
Z =
R1
R2
1[kW]
1[kW]
It
a
+
ZThev  0.412  20 j[W].
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
Vt=
1[V]
-
b
Now, to solve part b, we examine the nature of the equivalent circuit. We found ZThev,
and it turned out that the imaginary part was positive. Thus, we can model this impedance
with a resistor and an inductor in series. The resistor, Rm, value will be simply the real part
of the impedance, and thus Rm = 0.412[W]. This is true because the impedance of a resistor
is just equal to the resistance. Now, we need to find the inductance of the inductor.
Rm  0.412[W].
Next slide
Dave Shattuck
University of Houston
Finding a Circuit Model – 2
© Brooks/Cole Publishing Co.
The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s].
a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at
terminals a and b.
b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and
capacitors.
ZR1=
1[kW]
ZR2=
1[kW]
It
Rm=
0.412[W]
a
+
Vc
Vs=
5Vc
+
ZC1=
-100j[W]
-
ZL1=
20j[W]
+
a
Vt=
1[V]
-
-
Lm=
10[mH]
b
ZThev  0.412  20 j[W].
Rm  0.412[W].
To find the inductance of the inductor, we recognize
that the impedance of the inductor is ZLm = jwLm =
20j[W]. Since we know that w = 2,000[rad/s], we can
solve this for Lm and we get
Circuit Model
b
20 j
Lm 
 0.01[H].
2000 j
Go to Comments Slide
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Does it matter about the polarity of the
test source?
• Yes, just as it mattered how we applied the
test source in the time domain, it matters
how we apply it in the phasor domain.
• The voltage and current must be assigned in
the passive convention for the circuit,
because that is how Ohm’s Law is
defined. This means that they
must be in the active convention
Go back to
Overview
for the source you attach.
slide.