2012/9/17
Methods of Analysis
•Introduction
•Nodal Analysis
•Nodal Analysis with Voltage Sources
•Mesh Analysis
•Mesh Analysis with Current Sources
•Nodal Analysis vs. Mesh Analysis
•Applications
Introduction
•Nodal Analysis
–Based on KCL.
•Mesh Analysis
–Based on KVL.
•Linear algebra is applied to solve the resulting
simultaneous equations.
– Ax = B
or x = A-1B.
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Nodal Analysis
•Circuit variables = node voltages
–KVL is automatically satisfied.
•Steps to analyze an n-node network
–Select a reference node (as ground), assign voltages v1,
v2,
…,
vn-1 for the remaining n-1 nodes.
–Use Ohm’
sl
awto express currents of resistors.
–Apply KCL to each of the n-1 nodes.
–Solve the resulting equations.
Earth ground
Chassis ground
Case Study
* At node 1, applying KCL gives
I1 I 2 i1 i2
(1)
* At node 2, applying KCL gives
I 2 i2 i3
(2)
* Applying Ohm' s law gives
v 0
i1  1
or i1 G1v1
R1
v v
i2  1 2
R2
or i2 G2 
v1 v2 
v 0
i3  2
R3
or i3 G3v2
Assign vn
(1)  I1 I 2 G1v1 G2 
v1 v2  (3)
(2)  I 2 G2 
v1 v2 G3v2
G G2

 1
G2
(4)
G2 
v1  
I I 

1 2 



G2 G3 
v2   I 2 

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Nodal Analysis with Voltage Sources
•If a voltage source is connected
between a nonreference node and
the reference node (or ground).
–The node voltage is defined by the
voltage source.
–Number of variables is reduced.
–Simplified analysis.
Cont
’
d
•If a voltage source is connected between two
nonreference nodes.
–IS is difficult to define.
–I
t
’
sdi
f
f
i
c
ul
tt
os
ol
v
et
hepr
obl
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m byus
i
ngKCL.
I
IS
VS V
V VS


- I S 

I-V curve of a voltage source
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Cont
’
d
•Analysis Strategy
–The two nodes form a supernode (a closed boundary).
–Eq. 1: Apply KCL to the supernode.
–Eq. 2: Apply KVL to derive the relationship between the two
nodes.
Supernode
Case Study with Supernode
v1 10 V
(1)
Applying KCL to the supernode,
 i1 i4 i2 i3
v1 v2 v1 v3

2
4
v2 0 v3 0


(2)
8
6
Applying KVL to the supernode,

 v2 v3 5
(3)
3 variables solved by 3 equations.
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Example 1
Supernode
2 i1 i2 7
v 2 v1 2
Example 2
Supernode
Supernode
i 3 i1 i2
i1 i3 i4 i5
v1 v 2 20
v 3 v4 3v x
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What is a mesh?
•A mesh is a loop that does not contain any
other loop within it.
Mesh Analysis
•Circuit variables = mesh currents
–KCL must be satisfied. ( How ??? )
•Steps to analyze an n-mesh network
–Assign mesh currents i1, i2,
…,
in .
–Use Ohm’
sl
awto express voltages of resistors.
–Apply KVL to each of the n meshes.
–Solve the resulting equations.
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Mesh Analysis
•Applicable only for planar circuits.
•An example for nonplanar circuits is shown
below.
Case Study
For mesh 1, applying KVL gives
V1 R1i1 R3 
i1 i2 0

R1 R3 
i1 R3i2 V1
For mesh 2, applying KVL gives
R2i2 V2 R3 
i2 i1 0
R R3

 1
R3
R3 
i1  V1 

 


R2 R3 
i2 
V2 

 
 R3i1 
R2 R3 
i2 
V2
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Mesh Analysis with Current Sources
•If a current source exists only in one mesh.
–The mesh current is defined by the current source.
–Number of variables is reduced.
–Simplified analysis.
Cont
’
d
•If a current source exists between two meshes.
–VS is difficult to define.
–I
t
’
sdi
f
f
i
c
ul
tt
os
ol
vet
hepr
obl
e
m byus
i
ngKVLf
ore
a
c
h
mesh.
I
+
VS
_
IS
V
I I S

- VS 

I-V curve of a current source
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Cont
’
d
•Analysis Strategy
–A supermesh is resulted.
–Eq. 1: Apply KVL to the supermesh.
–Eq. 2: Apply KCL to derive the relationship between the
two mesh currents.
Supermesh
KCL : i2 i1 I S
Excluded
Example 1
i1 i2
i2 5 A
Applying KVL for mesh 1,
 10 4i 6
i1 i2 0
 i1 2 A
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Example 2
Supermesh
Applying KVL to the supermesh,
 20 6i1 10i2 4i2 0
Applying KCL to node 0,
 i2 i1 6
 6i1 14i2 20
 i1 3.2 A, i2 2.8 A
Example 3
Supermesh
=i2-5
=i2-3Io
• Applying KVL to the supermesh
• Applying KCL to node P
• Applying KCL to node Q
• Applying KVL to mesh 4
4 variables solved
by 4 equations
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How to choose?
•Nodal Analysis
–More parallel-connected elements, voltage
sources, or supernodes.
–Nnode < Nmesh
–If node voltages are required.
•Mesh Analysis
–More series-connected elements, current
sources, or supermeshes.
–Nmesh < Nnode
–If branch currents are required.
Applications: Transistors
•Bipolar Junction Transistors (BJTs)
•Field-Effect Transistors (FETs)
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Bipolar Junction Transistors (BJTs)
•Current-controlled devices
I E I B I C
(KCL)
VCE VEB VBC 0 (KVL)
VBE 0.7 V
I C I B
( ~ 100)
I C I E
(0 1)
I E 
1 I B


1 
DC Equivalent Model of BJT
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Example of Amplifier Circuit
I C I B
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