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Circuits
Lecture 8: Thevenin and
Norton Theorem (1)
李宏毅 Hung-yi Lee
Textbook
• Chapter 2.5
Network
• A two-terminal network is a function represented
by i-v characteristics
• Given a network, computing its i-v characteristics,
and then label it with simpler equivalent network
i
v
Computing i-v characteristics
i
i
v
v
 Add a voltage source v
• v is a unknown variable
 Find i
• Represented by v
Obtain the relation of
i and v
 Add a current source i
• i is a unknown variable
 Find v
• Represented by i
Obtain the relation of
i and v
Computing i-v characteristics
i
Voltage
source v
v
Independent
Sources
x1, x2, x3 ……
i is the weighted sum of the voltage (or current) of the
sources.
i  av   an xn
n
constant
i  av  b
The relation of i and v
is linear!
Computing i-v characteristics
• The v-i characteristics is linear!
i
i=mv
Case 1: current source
Case 2: voltage source
Case 3: resistor
v
i
v
R
Resistor
with resistance 1/m Ω
m>0, normal resistor
m<0, ?
Negative resistor
Computing i-v characteristics
Case 4: resistor
+voltage source
Rt
i
voc
voc
Rt
voc
v
v  voc  Rt i
v voc
i 
Rt Rt
Computing i-v characteristics
• The v-i characteristics is linear!
Case 1: current source
Case 2: voltage source
Case 3: resistor
Case 4: resistor +voltage source
i
v
Thevenin Theorem
Two
Terminal
Network
 Two terminal network consists entirely of
independent source, resistor and controlled
sources.
 If controlled sources are present, then the
control variables is within the same network.
Find voc and Rt directly without drawing i-v characteristics?
Thevenin Theorem - voc
Two
Terminal
Network


voc
voc


Keep two terminals open
Thevenin Theorem - Rt
• Textbook P72 - 73
i
i
v
Suppress the independent sources:
Voltage Source
Short
Open
Current Source
(As we have done in Superposition)
v
v
Rt 
i
Why?????
Thevenin Theorem - Rt (Example)
Suppress the
independent sources
• Refer to lecture 7
3
3
2
4V
2
Thevenin Theorem - Rt
i
i
v
v
v  voc  iRt
voc
1
i  v
Rt
Rt
Superposition:
Voltage source v
Independent Sources in
network: x1, x2, x3 ……
i  av   an xn
n
0
v
Rt 
i
0
Norton Theorem
Thevenin
Theorem
Two
Terminal
Network
voc
i sc 
Rt
Rt
Norton
Theorem
Norton Theorem - isc
Two
Terminal
Network
i sc
i sc
Rt
Let two terminals short
After we find isc,
If we already know voc, Rt=voc/isc
If we already know Rt, voc=iscRt
i sc
Thevenin Parameters
• Voc, Rt and isc are Thevenin parameters
• voc: keep the two terminals open
• Rt: suppress the independent sources
• isc: let the two terminals short
Know any of two can find the last one
Example 2.14
i sc
Rt
Example 2.14
Open circuit
Find voc
Short Circuit
Find isc
Set to Zero
Find Rt
Example 2.14
i sc
3mA
Rt
• Short Circuit
3mA
v x  5v x
vx  0
ix  0
Then “open circuit” or “source set to zero”?
i sc
3mA
Example 2.14
• Source set to zero
i
v x  5v x  vt
Rt
 10k
vt
Rt 
it
Find
equivalent
resistance Rt
vt  4v x
v x  4v x v x
vx
4v x
it  ix  i
it 




2k 40k
2k 10k 10k
vt  4v x
Rt  
 10k
4v x
it
10k
Example 2.14
 10k
i sc
3mA
Rt
 10k
voc  isc Rt
 3mA 10k   30V
Example 2.14
• Check: Open circuit
 10k
voc  30V
v x  5v x  v
v  4v x
v x  4v x v x
vx
4v x
3mA 




2k 40k
2k 10k 10k
 30 
v  4v x  4   30
 4
30
vx 
4
Example
• Find the current and voltage on RL
Circuit Analysis
Three-terminal networks?
Obliterate RL
Thevenin Theorem
Example
• Find the current and voltage on RL
𝑅𝐿
i
+
-
Find Thevenin Parameters
Example
• Find the current and voltage on RL
Open circuit
Short Circuit
Set to Zero
Find voc
Find isc
Find Rt
Example
• Find the current and voltage on RL
𝑣𝑜𝑐 = 𝑣𝑎 − 𝑣𝑏
𝑅3
𝑅4
= 𝑉𝑠
− 𝑉𝑠
𝑅1 + 𝑅3
𝑅2 + 𝑅4
So easy!
Open circuit
Find voc
Example
• Find the current and voltage on RL
R1, R2, R3, R4:
𝑖𝑉𝑠
𝑉𝑠
=
𝑅𝑒𝑞
𝑖𝑅1
𝑉𝑠
𝑅2
=
𝑅𝑒𝑞 𝑅1 + 𝑅2
𝑖𝑠𝑐 = 𝑖𝑅1 − 𝑖𝑅3 =
Short Circuit
Find isc
𝑅𝑒𝑞 =
𝑣𝑜𝑐 = 𝑉𝑠
𝑅1 𝑅2
𝑅3 𝑅4
+
𝑅1 + R 2 𝑅3 + R 4
𝑖𝑅3
𝑉𝑠
𝑅4
=
𝑅𝑒𝑞 𝑅3 + 𝑅4
𝑉𝑠
𝑅2
𝑅4
−
𝑅𝑒𝑞 𝑅1 + 𝑅2 𝑅3 + 𝑅4
𝑅3
𝑅4
− 𝑉𝑠
𝑅1 + 𝑅3
𝑅2 + 𝑅4
(last page)
𝑅𝑡 =
𝑣𝑜𝑐
=⋯
𝑖𝑠𝑐
Example
• Find the current and voltage on RL
Simple (but hard to figure out)
𝑅1 𝑅3
𝑅2 𝑅4
𝑅𝑡 =
+
𝑅1 + R 3 𝑅2 + R 4
Set to Zero
Find Rt
𝑅𝑡 =
𝑣𝑜𝑐
?
𝑖𝑠𝑐
Check by yourself
Example
• Find the current and voltage on RL
𝑅𝐿
i
+
𝑅3
𝑅4
− 𝑉𝑠
𝑅1 + 𝑅3
𝑅2 + 𝑅4
𝑅1 𝑅3
𝑅2 𝑅4
𝑅𝑡 =
+
𝑅1 + R 3 𝑅2 + R 4
𝑣𝑜𝑐 = 𝑉𝑠
Homework
• 2.62, 2.64
Homework
Homework
Thank you!
Answer
• 2.62
• Rt=5k, isc=vs/8k, voc=5vs/8
• 2.64
• Rt=-60, isc=is, voc=-60is
Homework
Rt=100k,isc=-20m
Homework
Rt=-1k/60, voc=2
Acknowledgement
• 感謝 徐瑞陽(b02)
• 糾正錯誤的作業答案
• 感謝 林楷恩(b02)
• 糾正錯誤的作業答案