Methods of Analysis [相容模式] · 2012/9/17 2 Nodal Analysis •Circuit variables = node...
Transcript of Methods of Analysis [相容模式] · 2012/9/17 2 Nodal Analysis •Circuit variables = node...
2012/9/17
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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 resultingsimultaneous 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.–UseOhm’s lawto express currents of resistors.–Apply KCL to each of the n-1 nodes.–Solve the resulting equations.
Earth ground Chassis ground
Case Study
2
21
2
1
322
221
232122
2121121
2333
23
21222
212
1111
11
322
2121
(4))2(
(3))1(
or0
or
or0
giveslawsOhm'Applying*
(2)givesKCLapplying2,nodeAt*
(1)givesKCLapplying1,nodeAt*
III
vv
GGGGGG
vGvvGI
vvGvGII
vGiR
vi
vvGiR
vvi
vGiR
vi
iiI
iiII
Assign vn
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Nodal Analysis with Voltage Sources
•If a voltage source is connectedbetween a nonreference node andthe 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.–It’s difficult to solve the problem by using KCL.
V
I
VS
I-V curve of a voltage source
IS
- S
S
I
VV
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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
equations.3bysolvedvariables3
(3)5supernode,thetoKVLApplying
(2)6
08
042
supernode,thetoKCLApplying(1)V10
32
32
3121
3241
1
vv
vv
vvvv
iiii
v
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Example 1Supernode
212 vv72 21 ii
Example 2
Supernode Supernode
2021 vv xvvv 343
213 iii 5431 iiii
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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.–UseOhm’s lawto 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
223213
123222
123131
213111
0givesKVLapplying2,meshFor
0givesKVLapplying1,meshFor
ViRRiR
iiRViR
ViRiRR
iiRiRV
2
1
2
1
323
331
VV
ii
RRRRRR
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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.–It’s difficult to solve the problem by using KVL for each
mesh.
V
IIS
I-V curve of a current source
- S
S
V
II+VS_
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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.
ExcludedSIii 12:KCL
Supermesh
Example 1
A2
064101,meshforKVLApplying
A5
1
21
2
iiii
i
21 ii
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Example 2
201460410620
supermesh,thetoKVLApplying
21
221
iiiii
A8.2A,2.36
0,nodetoKCLApplying
21
12
iiii
Supermesh
Example 3
•Applying KVL to the supermesh•Applying KCL to node P•Applying KCL to node Q•Applying KVL to mesh 4
4 variables solvedby 4 equations
Supermesh
=i2-5
=i2-3Io
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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)
1
1)(01
100)~(
V0.7
(KVL)0
(KCL)
EC
BE
BC
BE
BCEBCE
CBE
IIII
II
V
VVV
III
•Current-controlled devices
DC Equivalent Model of BJT
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Example of Amplifier Circuit
BC II