Electric Circuits (Fall 2015)
Pingqiang Zhou
Discussion 3
10/21/2015
Discussion 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Outline
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Linearity property
Superposition
Source transformation
Thevenin and Norton’s theorems
Examples of applications
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Linear Circuit
Linearity: homogeneity and additivity
 Linear circuit: consists of only linear elements and linear
sources.
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 Linear resistors, capacitors, inductors
 Linear sources: DC or AC?
Question: The current through a branch in a linear network is 2 A when the input source
voltage is 10V. If the voltage is reduced to 1V and the polarity is revised, the current
through the branch is?
Answer: -0.2 A
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Superposition
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What is superposition?
When to use superposition?
How to use superposition?
Problem 1 Use superposition to find i.
Answer: 1.875 A
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Applying Superposition
Problem 2 Use superposition to find 𝑖0 and 𝑣0 .
Do we turn off the dependent source?
No
Can we turn off the voltage source by open circuit?
No
Answer: 17.99 V, 1.799A
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Source Transformation
equivalent circuit: has identical terminal I-V characteristics
 If the sources are turned off, resistance at the terminals
are both R
 If the terminals are short circuited, the currents need to
be the same.
 What principle can we get?
vs
vs  is R or is 
R
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Source Transformation
Problem 3 Use source transformation to find 𝑖0 .
Answer: 1/9 A
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Dependent Sources
Source transformation also applies to dependent sources
 The same relationship between the voltage and current
holds here:

vs
vs  is R or is 
R
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Dependent Sources
Problem 4 Use source transformation to find 𝑖𝑥 .
Answer: 12/47.1 = 254.8 mA
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Electric Circuits (Fall 2015)
Pingqiang Zhou
New problem
Problem 5 Find the voltage drop between 𝑅𝐿 when
𝑅𝐿 =10 Ω, 20Ω, 30 Ω, and 40 Ω.
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Reanalyze the circuit for each change in the load?
Discussion 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Thevenin’s Theorem and Norton’s Theorem

A linear two terminal circuit may be replaced with an
equivalent circuits:
 a voltage source
in series with a resistor
(Thevenin’s theorem)
 a current source
in parallel with a resistor
(Norton’s theorem)
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Electric Circuits (Fall 2015)
Pingqiang Zhou
How to find the Thevenin/Norton equivalents?
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Find the equivalent source 𝑽𝑻𝒉 / 𝑰𝑵
 Thevenin equivalent: 𝑽𝑻𝒉 = terminal open circuit voltage
 Norton equivalent: 𝑰𝑵 = terminal short circuit current
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Electric Circuits (Fall 2015)
Pingqiang Zhou
How to find the Thevenin/Norton equivalents?

Find Thevenin/Norton equivalent resistance 𝑹𝑻𝒉 /𝑹𝑵
 In fact, 𝑅𝑇ℎ = 𝑅𝑁 (source transformation)
 Step 1: turn off all the independent sources;
 Step 2: apply a voltage source 𝑉𝑜 to the terminals and
determine the current 𝐼𝑜 , then 𝑅𝑇ℎ = 𝑅𝑁 = 𝑉𝑜 /𝐼𝑜 .
 Can we apply a current source 𝐼𝑜 and determine 𝑉𝑜 ?
 If there are no dependent source, can step 2 be simplified?
Discussion 3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Practice
Problem 6 Obtain the Thevenin equivalent circuit at
terminals a-b of the circuit.
Answer: 20 Ω, -49.2 V
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Practice
Problem 7 Find the Thevenin equivalent as seen by R.
Answer:
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Practice
Problem 8 Use Norton’s theorem to find 𝑉𝑜 .
Answer: 𝑰𝑵 = −𝟎. 𝟑 𝒎𝑨, 𝑽𝒐 = −𝟐𝟖𝟓. 𝟕𝒎𝑽
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Practice
Problem 9 Find the Thevenin and Norton equivalent circuit
across terminal a and b.
Answer:
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Maximum Power Transfer
Thé venin equivalent circuit
RTh
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–
+
VTh
+
iL
vL
RL
–
VTH and RTH are fixed. By varying the load resistance RL, the
power delivered to the load varies.
Maximum power transferred is when RL=RTH.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Practice
Problem 10 Determine the maximum power delivered to
the variable resistor R in the circuit.
Answer: 21.48 mW (𝑹𝑻𝒉 = 𝟏𝟎𝟒. 𝟕𝟓 𝛀, 𝑽𝑻𝒉 = 𝟑𝑽)
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Application of Equivalent Theorem
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Resistance Measurement
𝑅𝑥 to be measured.
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Source Modeling
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Summary
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Superposition
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Leave one independent source on at a time
Sum over all responses
Voltage off  SC
Current off  OC
Source Transforms:
 Voltage sources in series with a resistance can be converted to current
source in parallel with a resistance.
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Thevenin and Norton Equivalent Circuits
 Solve for OC voltage
 Solve for SC current
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Electric Circuits (Fall 2015)
Pingqiang Zhou
End
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