Circuits Lecture 8: Thevenin and Norton Theorem (1) 李宏毅 Hung-yi Lee.
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Transcript of Circuits Lecture 8: Thevenin and Norton Theorem (1) 李宏毅 Hung-yi Lee.
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
v
i
Computing i-v characteristics
i
v
Add a voltage source v
Find i• v is a unknown variable
• Represented by v
Obtain the relation of i and v
Add a current source i
Find v• i is a unknown variable
• Represented by i
Obtain the relation of i and v
vi
Computing i-v characteristics
i
vVoltage source v
Independent Sources x1, x2, x3 ……
i is the weighted sum of the voltage (or current) of the sources.
n
nnxaavi
The relation of i and v is linear!
bavi
constant
Computing i-v characteristics• The v-i characteristics is linear!
v
iCase 1: current source Case 2: voltage source Case 3: resistor
i=mv
R
vi
Resistorwith resistance 1/m Ω
m>0, normal resistorm<0, ?
Negative resistor
Computing i-v characteristics
ocv
tR
iRvv toc
t
oc
t R
v
R
vi
v
i
ocv
t
oc
R
v
Case 4: resistor +voltage source
Computing i-v characteristics• The v-i characteristics is linear!
v
iCase 1: current source Case 2: voltage source Case 3: resistorCase 4: resistor +voltage source
Thevenin Theorem
TwoTerminalNetwork
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 - Rt
(As we have done in Superposition)
• Textbook P72 - 73
Voltage SourceSuppress the independent sources:
Current SourceShortOpen
v
i
i
vt R
Why?????
v
i
Thevenin Theorem - Rt
n
nnxaavi
Superposition:
v
Independent Sources in network: x1, x2, x3 ……
i
Voltage source v
v
i
toc iRvv
t
oc
t R
vv
Ri
1
0
0
i
vt R
Norton Theorem - isc
TwoTerminalNetwork
tRsci
Let two terminals short
scisci
If we already know voc, Rt=voc/isc
After we find isc,
If we already know Rt, voc=iscRt
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
• Short Circuit
xv5vx 0vx 0ix
mA3
Then “open circuit” or “source set to zero”?
sci tRmA3
Example 2.14
• Source set to zero
Find equivalent resistance Rt
t
tt i
vR
txx vvv 5xt vv 4
i
iii xt k
v
k
v
k
v
k
v
k
vi xxxxxt 10
4
10240
4
2
k
kvv
i
vR
x
x
t
tt 10
1044
sci tRmA3
k10
Example 2.14
• Check: Open circuit
k10
Vvoc 30
vvv xx 5
xvv 4
k
v
k
v
k
v
k
v
k
vmA xxxxx
10
4
10240
4
23
4
30xv
304
3044
xvv
Example
• Find the current and voltage on RL
Three-terminal networks?
Thevenin Theorem
Circuit Analysis
Obliterate RL
Example
• Find the current and voltage on RL
Open circuit Short Circuit Set to ZeroFind voc Find isc Find Rt
Example
• Find the current and voltage on RL
Open circuit
Find voc
𝑣𝑜𝑐=𝑣𝑎−𝑣𝑏
¿𝑉 𝑠
𝑅3
𝑅1+𝑅3
−𝑉 𝑠
𝑅4
𝑅2+𝑅4
So easy!
Example
• Find the current and voltage on RL
𝑅𝑒𝑞=𝑅1𝑅2
𝑅1+R2+𝑅3𝑅4
𝑅3+R4
𝑖𝑉 𝑠=𝑉 𝑠
𝑅𝑒𝑞
Short CircuitFind isc
R1, R2, R3, R4:
𝑖𝑅1=𝑉 𝑠
𝑅𝑒𝑞
𝑅2
𝑅1+𝑅2
𝑖𝑅3=𝑉 𝑠
𝑅𝑒𝑞
𝑅4
𝑅3+𝑅4
𝑖𝑠𝑐=𝑖𝑅1− 𝑖𝑅3
=𝑉 𝑠
𝑅𝑒𝑞( 𝑅2
𝑅1+𝑅2
−𝑅4
𝑅3+𝑅4)
𝑣𝑜𝑐=𝑉 𝑠
𝑅3
𝑅1+𝑅3
−𝑉 𝑠
𝑅4
𝑅2+𝑅4𝑅𝑡=
𝑣𝑜𝑐
𝑖𝑠𝑐=…
(last page)
Example
• Find the current and voltage on RL
Set to ZeroFind Rt
Simple (but hard to figure out)
𝑅𝑡=𝑅1𝑅3
𝑅1+R3+𝑅2𝑅4
𝑅2+R4
𝑅𝑡=𝑣𝑜𝑐
𝑖𝑠𝑐? Check by yourself
Example
• Find the current and voltage on RL
𝑅𝐿
+ -i
𝑣𝑜𝑐=𝑉 𝑠
𝑅3
𝑅1+𝑅3
−𝑉 𝑠
𝑅4
𝑅2+𝑅4
𝑅𝑡=𝑅1𝑅3
𝑅1+R3+𝑅2𝑅4
𝑅2+R4