INC 112 Basic Circuit Analysis
Week 6
Other Techniques
Special Techniques
• Superposition Theorem
• Thevenin’s Theorem
• Norton’s Theorem
• Source Transformation
Source Transformation
RLRNINRL
RTH
VTH
THNTH
NTH
RIV
RR
When we do not need to know the current and the voltage inside the box, we can interchange the circuit inside without affecting other parts in the circuit.
Example
Use source transformation to find Ix
1Ω 2Ω
3Ω1A1V Ix
1Ω 2Ω
3Ω1A1V Ix
1Ω
2Ω
3Ω1A Ix1A
1Ω
2Ω
3Ω1A Ix1A
1Ω
2Ω
3Ω Ix2A
1Ω 2Ω
3Ω Ix2V
AI X 3
1
6
2
321
2
I
VVOC
ISC
Maximum Power Theorem
High RL: high voltage, low currentLow RL: high current, low voltage
Maximum Power Theorem
From Thevenin or Norton equivalent circuit, the RL that
gives the maximum power is RL that equals Rth.
RL that gives the maximum power is RL = Rth
Proof
RL
RTH
VTH
2
22
2
)( LTH
LTHL
LTH
TH
L
RR
RVR
RR
VP
RIP
LTH
TH
RR
VI
where
Substitutein theequation
0)(
)(2)(4
222
LTH
LTHLTHTHLTH
L RR
RRRVVRR
dR
dP
LTH
LLTH
LTHLTHTHLTH
LTH
LTHLTHTHLTH
L
RR
RRR
RRRVVRR
RR
RRRVVRR
dR
dP
2)(
)(2)(
0)(
)(2)(
222
4
222
Therefore, RL = Rth is the point that yields maximum power
Find the first derivative. Let it equal zero to find critical points.
Example
3Ω
2Ω
10V
10Ω
RL
2Ω
Find RL that gives the maximum power and find that power.
13.2
6V RL
Thevenin’s equivalent circuit
RL that gives the maximum power is 13.2Ω
Maximum Power is WR
V68.0
2.13
)2/6( 22
Voltage Sources in Real World
3V 3V
0.01Ω
I
V
I
V
Ideal Voltage Source Real Voltage Source
Current Sources in Real World
I
V
I
V
Ideal Current Source Real Current Source
1A 1A 10M
Delta-Wye Conversion (∆-Y)
Ra
Rb
RcR1 R2
R3
1
133221
3
133221
2
133221
R
RRRRRRRc
R
RRRRRRRb
R
RRRRRRRa
RcRbRa
RcRaR
RcRbRa
RbRcR
RcRbRa
RaRbR
3
2
1
Example
4Ω
2Ω
1Ω
5Ω
3Ω
3/2Ω
2Ω
3/8Ω
5Ω
1/2Ω
13/2Ω19/8Ω
1/2Ω
159/71Ω