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Circuits Lecture 2: Node
Analysis李宏毅 Hung-yi Lee
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DC Circuit - Chapter 1 to 4
KCL, KVL, Element Characteristics
Node Analysis
Mesh Analysis
Controlled Sources
Equivalent
Thevenin Theorem
Norton Theorem
Lecture 1
Lecture 2 Lecture 3 Lecture 4
Lecture 5&6Lecture 7
Lecture 8
Lecture 9
Superposition
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Review – Lecture 1
A B+ -
i
v
If v<0, then actually ……
If i<0, then actually ……A B
A B+-Resistor with resistance R:
R
vi
reference current should flow from “+” to “-”
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Review – Lecture 1
• Voltage defined for two points• Potential defined for one point• Voltage between the point and the reference
A B+ -v
v A B+ -
v
v-
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Review – Lecture 1
• KCL:
• KVL
321 iii
svvv 21Loop 1:
Loop 1
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Review – Lecture 1
Find the current and voltage of all elements.Systematic Solution:
Step 1. List all unknown variables and reference directionsStep 2. Use (a) Element Characteristics, (b) KCL and (c) KVL to list equations for unknown variables
How to reduce the number of unknown variables?
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Textbook
• Chapter 4.1
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Terminology
• Node: any connection point of two or more circuit elements (Textbook, P23)• Essential node: more than two elements• Non-essential node: two elements• Use “node” to represent “Essential node”
• Branch:• Circuit between nodes
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Node Analysis
Current + Voltage Voltage
Only consider the voltage as unknown variables• Reduce the number of unknown variables
Usually it is easy to find current if the voltages are known
A Bv+ -
R
vi
Resistor with resistance R
How about ……v
i??????
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Node Analysis
Current + Voltage Voltage
Voltages are not independent• If we know the voltage of some elements, we can
know the rest easily (KVL)• Maybe we only have to consider some of the
voltages as unknown variables • How to determine the voltage taken as unknown
variables?
+ -
-
+
+ -
+
-
v1
v2
v3
v4 = v1 + v2 – v3
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Node Analysis
Current + Voltage Voltage Node Potential
(Node Voltage)
The potentials are independent 10V15V
Target: node potential
A B+ -
Av Bv
BA vvv • Can know voltage immediately
Any potential value can satisfy KVL
+ -
-
+
+ -
+
-
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Node Analysis
• Find node potentials• 3 unknown variables
KVL:
+
-
+ -
+
-
1vvb 21 vvvc 2vvd
dcb vvv
Represent vb, vc and vd by node potentials
2211 vvvv KVL is automatically fulfilled!
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Node Analysis
• Find node potentials• 3 unknown variables
KCL:
cc R
vvi 12
b
b R
vi 1
a
s
a
aa R
vv
R
vi 1
0 scba iiiiRepresent ia, ib and ic by node potentials
Can we always represent current by node potentials (discuss later)?
Node v1:
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Node Analysis
• Find node potentials• Need 3 equations
KCL:
01211
scba
s iR
vv
R
v
R
vvNode v1:
Node v2:
Node v3:
023221
edc R
vv
R
v
R
vv
sfe
iR
v
R
vv
332
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Node Analysis
• Target: Find node potentials• Steps• 1. Set a node as reference point• 2. Find nodes with unknown node potentials• 3. KCL for these nodes• Input currents = output currents• Represent unknown current by node potentials • Always possible?
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8 Kinds of Branches
• There are only 8 kind of branches • 1. None• 2. Resistor• 3. Current • 4. Current + Resistor• 5. Voltage• 6. Voltage + Resistor• 7. Voltage + Current • 8. Current + Resistor + Voltage
Represent i by node potentials
i
branch
xvyv
R
R
vvi xy
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xvyv
sy vv
s
xsy
R
vvvi
Branch: Voltage + Resistor
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Branch: Voltage + Resistor - Example
1v
Find vo301 v V20
0
4
0
5
20
2
300 111
k
v
k
v
k
v
Vv 201 Vvo 20
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Branch: Voltage
sv xvyv
i
sxy vvv
Method 1: Beside node potential, consider i also as unknown variable as well
1i
2i
3i
4i5i
6i
0: 321 iiiivxiiiivy 654:
Represent i1 to i6 by node potential
One more unknown variable i, need one more equation
sxy vvv
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Branch: Voltage
sv xvyv
i
Method 2: Consider vx and xy as supernode
1i
2i
3i
4i5i
6i
0: 654321 iiiiiiSupernode
Represent i1 to i6 by node potential
Bypass i
sxy vvv
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Branch: None
xvyv
i
yx vv
Supernode
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Example 4.5
• Use node analysis to analyze the following circuit
v50
1v
2v301 v
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Example 4.5
• Use node analysis to analyze the following circuit
v50
1v
2v301 v
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Example 4.5
01
2
30
105
50 12121
vvvvv
KCL for Supernode:
KCL for v2:
07
1
0
2
30
1022121
vvvvv
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Node Analysis – Connected Voltage Sources
1v 101 v 101 v
0
5
10
5
10
10
100
10
1004 111111
k
vv
k
vv
k
v
k
vm
201 v
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Node Analysis – Connected Voltage Sources
1v 101 v 101 v
0
5
10
5
10
10
100
10
1004 111111
k
vv
k
vv
k
v
k
vm
If a branch starts and ends at the same super node
Put it into the supernode
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Node Analysis – Reference Points
2v
10 20
mk
v
k
v4
10
20
10
0 22
We don’t have to draw supernode.
Select the reference point carefully
302 v
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Homework
• 4.18• 4.22
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Thank you!
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Answer
• 4.18• V1=-6, v2=12, i1=2, i2=3, i3=2.4
• 4.22• V1=-16.5, v2=30, i1=2, i2=0.5
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Branch: Voltage – Special Case!
sv xvyv
i
If vy is selected as reference
1i
2i
3i
4i5i
6i
vx is equal to vs The node potential is known
Eliminate one unknown variables
Which node should be selected as reference point?
Ans: The node connected with voltage source