Circuit Theorems Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung Universit y Hsinchu, Taiwan, R.O.C.
Jan 18, 2016
Circuit Theorems
Instructor: Chia-Ming TsaiElectronics Engineering
National Chiao Tung UniversityHsinchu, Taiwan, R.O.C.
Contents• Introduction
• Linearity Property
• Superposition
• Source Transformation
• Thevenin’s Theorem
• Norton’s Theorem
• Maximum Power Transfer
• Summary
Introduction
• To develop analysis techniques applicable to linear circuits
• To simplify circuit analysis and help handling the complexity
Linearity Property
• Homogeneity (Scaling) + Additivity
• Homogeneity property:
• Additivity property:
• A linear circuit follows the relationship below
kiRkviRv
212121
2211 and
vvRiRiRiiv
RivRiv
InputConstantOutput
Superposition• Based on linearity property
• Statement– The voltage across (current through) an element
is the algebraic sum of the voltage across (current through) that element due to each independent source acting alone.
),...,0;0,...,0(...)0,...,;0,...,0(
)0,...,0;,...,0(...)0,...,0;0,...,(
),...,;,...,(
1
1
11
M
N
MN
IvIv
VvVv
IIVVv
Example 1
+
=
V 1082 11
vvv
Example 2
+=
"0
'00 iii
Example 3
321 iiii
Source Transformation
i1 i2
R
Vii
R
Vvi
abS
abS
2
1
RivR
vi
ii
SSS
S
or
Let 21
Applicable to Dependent Sources
RivR
vi SS
SS or
i1 i2
Example 1
Example 2
V 5.7
A 5.4
013
01853
x
x
x
v
i
vi
vi
Thevenin’s TheoremA variableelement
Thevenin equivalent circuit
Continued
ThLTh
LLLL
LTh
ThL
VRR
RIRV
RR
VI
A simple voltage divider
=
Proof of Thevenin’s Theorem
+
=
=
All indep.sources on
All indep.sources off
1v+_
+_
2v
2111 )0,...,0;0,...,0;(),...,;,...,;0( vvivIIVVvv MN
Finding VTh and RTh
• If no dependent sources included
Continued• If dependent sources inclu
ded, two methods can be applied to determine RTh
– External voltage source method
– External current source method
Example 1
=RTh
VTh
Example 2
=RTh
voc
1
0iRTh
Norton’s Theorem
Norton equivalent circuit inTh RRRN
Proof of Norton’s Theorem
+
=
=
All indep.sources on
All indep.sources off
1211 ),...,;,...,;0()0,...,0;0,...,0;( iiIIVViVii MN
Example
= RNIN
Maximum Power Transfer
Th
Th
ThL
LTh
LThTh
LTh
LThLLThTh
L
LLTh
ThL
R
Vp
RR
RR
RRV
RR
RRRRRV
dR
dp
RRR
VRip
4
0
2
sfer,power tran maximum achieve To
2
max
3
2
4
22
2
2
A linear circuit
Applications: Source Modeling
sLs
LL v
RR
Rv
s
Ls
LL v
RR
Rv
sLs
LL v
RR
Rv
Voltage source
Current source
sLs
LL v
RR
Rv
sLp
pL i
RR
Ri
0sR
pR