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5. Handy Circuit Analysis Techniques
The nodal and mesh analysis require a complete set of equations to describe a
particular circuit, even if only one current, voltage, or power quantity is of interest
In this chapter, we investigate several different techniques for isolating specific
parts of a circuit in order to simplify the analysis
This chapter covers the following sections:
1. Linearity and Superposition
2. Source Transformations
3. Thévenin and Norton Equivalent Circuits
4. Maximum Power Transfer
5. Delta-Wye Conversion
6. A Summary of Various Techniques
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5.1 Linearity and Superposition
Linear Elements and Linear Circuits
A linear circuit is a circuit composed of independent sources, linear
dependent sources, and linear elements (resistors)
The sources are often called forcing functions, and the nodal voltages are
termed response functions
The response of linear circuits can be given as a weighted linear combination
of the forcing functions
Example
Using node voltage analysis, we have
The response, v1, can be given as
Linear circuit
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The Superposition Principle
The superposition method uses the linearity property of the linear circuit to
simplify the analysis
We remove all sources (independent only) expect one, and analyze the
circuit
We repeat the procedure for another source, and so on
Finally, the net result is found by summing all the single-source responses
Example (cont.)
1- We reduce a current source to zero (open
circuit), we have
2- We reduce a voltage source to zero (short
circuit), we get
3- The solution is
K. A. Saaifan, Jacobs University, Bremen
Use superposition to compute the current ix
1- We reduce a current source to zero (open circuit), we
have
2- We reduce a voltage source to zero (short circuit), we
get
3- The solution is
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K. A. Saaifan, Jacobs University, Bremen
Use superposition to compute the current ix
1- We reduce a current source to zero (open circuit), we
have
2- We reduce a voltage source to zero (short circuit), we
get (
)
3- The solution is
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5.2 Source Transformation
A practical voltage source is defined as an ideal voltage source connected in
series with an internal resistance Rs
A practical current source is defined as an ideal current source in parallel with an
internal resistance Rp
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Equivalent Practical Sources
With transformation, we can simplify a complex circuit so that in the
transformed circuit, the devices are all connected in series or in parallel
Changing the practical voltage source to an equivalent current source (or vice
versa) requires the following conditions:
The internal resistors must be equal in both sources Rs=Rp
The source transformation must be constrained by vs=Rsis=Rpis
The conversion of a voltage source into an equivalent current source can be
given as follows:
Electrical
Circuit
Electrical
Circuit
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Equivalent Practical Sources
The conversion of a current source into an equivalent voltage source can be
given as follows:
Electrical
Circuit
Electrical
Circuit
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Compute the current IX through the 47 k resistor after
performing a source transformation on the voltage source
1- We replace the voltage source with an equivalent current source
2- The current sources are divided as
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Compute the current the labeled current
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K. A. Saaifan, Jacobs University, Bremen
Compute the current the labeled current
The current I can now be found using KVL:
where
. Thus,
68
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5.3 Thévenin and Norton Equivalent Circuits
Efficient methods to simplify a complex circuit containing a load resistor into a
very simple equivalent
circuit
A Thévenin equivalent
circuit
A Norton equivalent
circuit
Thévenin’s theorem replaces the complex part of the circuit, except the load
resistor, with an independent voltage source in series with a resistor
The voltage and current seen by the load resistor will be unchanged
Norton’s equivalent circuit consists of an independent current source in parallel
with a resistor
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Using repeated source transformations, determine Thévenin and Norton
Equivalent Circuits
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The circuit is much simpler, and we can now easily compute the equivalent
circuits
A Thévenin equivalent
circuit
A Norton equivalent
circuit
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1. Thévenin’s equivalent circuit (independent sources)
1. Given any linear circuit, rearrange it in the form of two networks, A and B,
connected by two wires
2. Disconnect network B and determine a voltage Voc, which appears across the
terminals of network A
3. If the network A contains only independent sources
1. Replace the voltage source with open circuit and a current source with short
circuit
2. Compute the equivalent resistance Roc seen by the terminals of network A
4. Finally, Vth=Voc and Rth=Roc
+
Network A
Voc Network B
-
Network A
vc⇾oc
cs⇾ sc
Roc Network B
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2. Thévenin’s equivalent circuit (dependent/independent sources)
1. Given any linear circuit, rearrange it in the form of two networks, A and B,
connected by two wires
2. Disconnect network B and determine a voltage Voc, which appears across the
terminals of network A
3. If the network A contains dependent and independent sources
1. Leave the dependent and independent sources unchanged
2. Short circuit the terminals of network A and determine Isc
4. Finally, Vth=Voc and Rth=Voc/Isc
+
Network A
Voc Network B
-
Network A
Isc
Network B
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Determine the Thévenin equivalent of the shown circuit
+
Voc
-
1- Since the circuit is linear, we use a superposition
method to compute Voc
+
Voc
-
+
Voc
-
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2- We kill the independent sources to determine the
equivalent resistor
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Determine the Thévenin equivalent of the shown circuit
1- We note that vx=Voc and the current of dependent
passes only through the 2 k resistor, since no current can
flow through the 3 k resistor. Using KVL, we have
Voc
Then,
2- To determine Rth, we short-circuit the output
terminals. Since vx=0, then Isc is given by
3- Finally,
Isc