§ 2-1 Nodal analysis
We must apply KCL to nodes 1 and 2
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22120
32070
21
21
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We select one node as a reference node,and then
define a voltage between each remaining node and the
reference node.
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31)(2)(3:2 23212 ????? ?????n o d e
25)(4)(25:3 13233 ????? ?????n o d e
11437 321 ???? ???or
3263 321 ???? ???or
251124 321 ???? ???or
By Cramer's rule and determinants,we have
1
191
191
1124
263
437
11225
263
4311
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32 32 ?? ?? a n d
Now let us increase the number of nodes by one.
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2? 3?
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A8?
A3?
S1
S5
S4
S3 S2
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I1
- 8 A
R4
0,2 5
R2
1
3,0 0 0 V
2,0 0 0 V
I2
- 3 A R5
0,5
R1
0,3 3 3 3
1,0 0 0 V
R3
0,2
0V
0
I3
- 2 5 A
)( 52223 ?? ??
)()()( 42894732 321 ?????? ???
031)(3:2 212 ????? sin o d e ???
025)(45:3 133 ????? sin o d e ???
)1(11437 321 ???? ???or
)2(343 21 ???? sior ??
)3(2594 31 ???? sior ??
Equations (1),(4),(5) may be solved
54
42
189
110
947
437
1122
9428
4311
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1,Make a neat,simple,circuit diagram,Indicate all
element and source values,Each source should have its
reference symbol.
2,Assuming that the circuit has N nodes,choose one
of these nodes as a reference node,Then write node
voltages v1,v2,… vN-1 at their respective nodes,
remembering that each node voltage is under-stood to
be measured with respect to the chosen reference.
Let us summarize the method by which we
obtain a set of nodal equations for any resistive
circuit:
3,If the circuit contains only current sources,apply
Kirchhoff 's current law at each non-reference node,To
obtain the conductance matrix if a circuit has only
independent current sources,equate the total current
leaving each node through all conductance to the total
source current entering that node,and order the terms
from v1 to vN-1,For each dependent current source
present,relate the source current and the controlling
quantity to the variables v1,v2,…,v N -1,if they are not
already in that form.
4.If the circuit contains voltage sources,temporarily
modify the given circuit by replacing each such source
by a short circuit,thus reducing the number of nodes
by one for each voltage source that is present,The
assigned node voltages should not be changed,
Using these assigned node-to-reference voltage,apply
Kirchhoff 's current law at each of the nodes or super-
nodes in this modified circuit,Relate each source
voltage to the variables v1,v2,…,v N-1,if it is not already
in that form.
Example 2,
(A) If X=2S.Find the power supplied by the 10A source;
(B) If X=10V,Find the power supplied by the 10A source;
(C) If X=v1/3,Find v1 ;
(D)If X= 4v1/3,Find v1.
Solution:
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1v
3/4 1v
DP,use nodal analysis to find if elementX?
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X)B(
AX)A(
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321,,::2 vvvF i n dDP ?
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2,3,1 321 ???? ???
(与电流源串联的电阻不介入节点方程 )
Find:P2V,
Find,P2A.
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2111 13212 ????? ?????
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A2
1
1
1
1
1
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