Engineering 43
MaxPower
SuperPosition
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-43: Engineering Circuit Analysis
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
OutLine: MaxPwr & SuperPose
 Work On WhtBd Student Suggest
HomeWork Problem
 Thevénin & Norton Review
• Example Problem (WhtBd)
 Maximum Power Transfer
Theorem Derivation
• MaxPwr Application Examples
 Thevénin & Norton Summary
Engineering-43: Engineering Circuit Analysis
2
0
d PL
d RL
PL 
0
PL 
V2
1   RL

VS 
R L R L   Rs  R L
d PL
d

d RL d RL
VS2
R s  R L 
2
2
 RL
V  VS 
 Rs  R L
V2
RL



2

VS2 R L
 
 R s  R L 2

 VS2 R L

  R  R 2
L
 s
2

VS2

  VS R L
 R  R 2   2 R  R 3
s
L
s
L

VS2 R L
Rs  RL 3
Rs  R L  2 R L
RL  Rs
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
OutLine: MaxPwr & SuperPose
 Linearity & Homogeneity
• Guess Solution, Work BackWards, Scale
Guess
• Comparative Case Study
 SuperPosition → Activate & DeActivate
• Example Problem (WhtBd)
Engineering-43: Engineering Circuit Analysis
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Thevénin’s Equivalence Theorem
LINEAR CIRCUIT
May contain
independent and
dependent sources
with their controlling
variables
PART A
RTH


vTH

i
a
vO
b
_

i
a
LINEAR CIRCUIT
vO
_
LINEAR CIRCUIT
May contain
independent and
dependent sources
with their controlling
variables
PART B
 vTH = Thévenin
Equivalent
VOLTAGE
Source
 RTH = Thévenin
Equivalent
SERIES
RESISTANCE
b
PART B
PART A
 Thevenin Equivalent Circuit for PART A
Engineering-43: Engineering Circuit Analysis
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Norton’s Equivalence Theorem
LINEAR CIRCUIT
May contain
independent and
dependent sources
with their controlling
variables
PART A

vO
_

iN
RN
i
i
 iN = Norton
a LINEAR CIRCUIT
Equivalent
May contain
independent and
CURRENT
dependent sources
Source
controlling
b with their
variables
 RN = Norton
PART B
Equivalent
PARALLEL
RESISTANCE
a
LINEAR CIRCUIT
vO
_
b
PART B
PART A
 Norton Equivalent Circuit for PART A
Engineering-43: Engineering Circuit Analysis
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Example: VOC, ISC, RTH = RN
 Use Thevénin and Norton for find the
OutPut Voltage in the Circuit Below
 Recall: VTH = VOC & IN = ISC
Engineering-43: Engineering Circuit Analysis
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
 VOC

A
B
I SC
A
Engineering-43: Engineering Circuit Analysis
7
B
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Engineering-43: Engineering Circuit Analysis
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Now Isc
Engineering-43: Engineering Circuit Analysis
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Engineering-43: Engineering Circuit Analysis
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Engineering-43: Engineering Circuit Analysis
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Engineering-43: Engineering Circuit Analysis
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Maximum Power Transfer
 Consider The Amp-Speaker Matching Issue
RTH
VTH
From PreAmp
(voltage )
Engineering-43: Engineering Circuit Analysis
13
+
-
To speakers
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Maximum Power Xfer Cont
 The Simplest Model for a
Speaker is to Consider it
as a RESISTOR only
 Since the “Load” Does the
“Work” We Would like to
Transfer the Maximum
Amount of Power from the
“Driving Ckt” to the Load
RTH
VTH
+
-
SPEAKER
MODEL
BASIC MODEL FOR THE
ANALYSIS OF POWER
TRANSFER
• Anything Less Results in
Lost Energy in the Driving
Ckt in the form of Heat
Engineering-43: Engineering Circuit Analysis
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Maximum Power Transfer
 Consider Thevenin Equivalent
Ckt with Load RL
 Find Load Pwr by V-Divider
2
L
V
RL
PL 
; VL 
VTH
RL
RTH  RL
RL
2
PL 
V
2 TH
RTH  RL 
 For every choice of RL we
have a different power.
• How to find the MAXIMUM
Power value?
Engineering-43: Engineering Circuit Analysis
15
R TH

+
-
VTH
VL

RL
(LOAD)
SOURCE
 Consider PL as a
FUNCTION of RL and find
the maximum of such a
function  have at left!
• i.e., Take 1st Derivative
and Set to Zero
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Power Xfer cont
 Find Max Power Condition Using Differential Calculus
 RTH  RL   2 RL 
dPL
2

 VTH 
3

dRL


R

R
TH
L


 Solving for “Best” (Pmax)
Load
*
 Set The Derivative
To Zero To Find
MAX or MIN Points
RL  RTH
• For this Case Set To Zero
The NUMERATOR
 This is The Maximum
Power Transfer Theorem
 RTH  RL   2 RL 
0
 V 
3



R

R
*
TH
L


RL
 RTH  RL   2 RL  0
dPL
dRL
2
TH
Engineering-43: Engineering Circuit Analysis
16
• The load that maximizes the
power transfer for a circuit is
equal to the Thevenin
equivalent resistance
of the circuit
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Power Quantified
 By Calculus we
Know RL for PL,max
R  RTH
*
L
 Recall the Power
Transfer Eqn
RL
2
PL 
V
2 TH
RTH  RL 
Engineering-43: Engineering Circuit Analysis
17
 Sub RTH for RL
RTH
2
PL,max 
V
2 TH
RTH  RTH 
PL,max
RTH
RTH 2
2

V 
VTH
2 TH
2
4 RTH
2RTH 
 So Finally
PL ,max
2
TH
1V

4 RTH
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Pwr Xfer Example
 Determine RL for
Maximum Power
Transfer
a
b
 Need to Find RTH
• Notice This Ckt
Contains Only
INDEPENDENT
Sources
 Thus RTH By
Source Deactivation
RTH  4k  3k 6k  6k
 This is Then the RL For
Max Power Transfer
Engineering-43: Engineering Circuit Analysis
18
 To Find the AMOUNT of
Power Transferred Need
the Thevenin Voltage
 Then use RTH = 6kΩ
along with VTH
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Pwr Xfer Example cont
 To Find VTH Use Meshes
 The Eqns for Loops 1 & 2
I1  2mA
3k * I 2  I1   6k * I 2  3V  0
 Solving for I2
3[V ] 1
1
I2  
 I1  [mA]
9[k] 3
3
RL
2
 Recall PL 
V
RTH  RL 2 TH
 Now Apply KVL for VOC
VOC  4k * I1  6k * I 2
1
 4k * 2mA  6k * mA
3
 8[V ]  2[V ]  10[V ]
 At Max: PL = PMX, RL = RTH
Engineering-43: Engineering Circuit Analysis
19
PMX
VTH2

4 RTH
PMX
100[V 2 ] 25

 [mW ]
4 * 6k 6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Pwr Xfer
 Determine RL and Max
Power Transferred
 Find Thevenin Equiv.
At This Terminal-Set
d
 Recall for Max Pwr Xfer
VTH2
RL  RTH PMX 
4 RTH
 This is a MIXED Source
Circuit
• Analysis Proceeds More
Quickly if We start at c-d
and Adjust for the 4kΩ at
the end
Engineering-43: Engineering Circuit Analysis
20
a
c
b
 Use Loop Analysis
I1
I2
 Eqns for Loops 1 & 2
I1  4mA
 
4k ( I 2  I1 )  2k I X'  2kI2  0
 
6I 2  2 I X'  4I1  16mA
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Max Pwr Xfer cont
 The Controlling Variable
I X'  I 2
so
6 I 2  2I 2   4 I1 so
I 2  I1  4mA and
VOC  2kI2  8V
 Remember now the
partition points
c
 Now Short Ckt Current
• The Added Wire Shorts
the 2k Resistor
I X"  0  2000 I X"  0
 I SC  4mA
 Then RTH
RTH 
VOC
8V

 2k
I SC 4mA
Engineering-43: Engineering Circuit Analysis
21
8V
d
a
b
 The RTH for ckt at a-b =
2kΩ+4kΩ; So RL*  6k
Pmax
82
8

[mW ]  [mW ]
4*6
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Thevenin & Norton Summary
 Independent
Sources Only
 Mixed INdep
and Dep Srcs
• RTH = RN by
Source
Deactivation
• VTH
– = VOC or
– = RN·ISC
• IN
– = ISC or
– = VOC/RTH
Engineering-43: Engineering Circuit Analysis
22
• Must Keep
Indep &
dep Srcs
Together in
Driving Ckt
• VTH = VOC
• IN = ISC
• RTH = RN
= VOC/ ISC
 DEPENDENT
Sources Only
• Must Apply V
or I PROBE
– Pick One,
say IP = 1.00
mA, then
Calculate the
other, say VP
• VTH = IN = 0
• RTH = RN
= VP/ IP
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
WhiteBoard Work
 Let’s Work this nice
Max Power Problem
 Find Pmax for Load RL
Engineering-43: Engineering Circuit Analysis
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Previous Equivalent Circuits
 Series & Parallel
Resistors
 [Independent Srcs]
• Vsrc’s in Series
• Isrc’s in Parallel

Engineering-43: Engineering Circuit Analysis
24
The Complementary Configs
are Inconsistent with Source
Definitions
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Linearity
 Models Used So Far
Are All LINEAR
• For All Possible
– Input Pairs: u1 & u2
– Scalars α1 & α2
• Mathematically This
 AN Alternative, And
Implies That They
satisfy the principle
Equivalent, Linearity
of SUPERPOSITION
& Superposition
 The Model T(u) is
Definition
Linear IF AND
• The Model T(u) is
Linear IF AND ONLY
ONLY IF
IF It Exhibits
T 1u1   2u2   1T u1    2T u2 
– ADDITIVITY
– HOMOGENEITY
Engineering-43: Engineering Circuit Analysis
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Linearity cont.
 Linearity
Characteristics
• Additivity
T u1  u2   T u1   T u2 
• Homogeneity
– a.k.a. Scaling
T (u)  T u 
Engineering-43: Engineering Circuit Analysis
26
 NOTE
• Technically, Linearity
Can Never Be Verified
Empirically on a
System
• But It Could Be
Disproved by a
SINGLE Counter
Example.
• It Can Be Verified
Mathematically For
The Models Used
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Linearity cont.
 Using Node
Analysis For
Resistive Circuits
Yields Models Of
The Form
 The Model Can Be
Made More Detailed
Av  Bs
• Where
– A and B are Matrices
– s Is A Vector Of All
Independent Sources
Av  f
• Where
– v Is A Vector
Containing All The
Node Voltages
– f Is a Vector
Containing Only
independent Sources
Engineering-43: Engineering Circuit Analysis
27
 For Ckt Analysis Use
The Linearity
Assumption To
Develop Special
Analysis Methods
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Past Techniques  Case Study
 Find Vo


 Redraw the Ckt to
Reveal Special Cases
• After Untangling



 Solution Techniques
Available?
Engineering-43: Engineering Circuit Analysis
28

Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Case Study cont.
 Loop Analysis for Vo
 Node Analysis
•
-
Out → Positive
-
-
3
Engineering-43: Engineering Circuit Analysis
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Case Study cont.
 Series-Parallel
ResistorCombinations
 In other Words
REQ  R3 || R1  R2 
 By VOLTAGE Divider
Engineering-43: Engineering Circuit Analysis
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Use Homogeneity Analysis
 Find Vo by Scaling

REQ
V1

 Assume That The
Answer Is KNOWN
 If Vo is Given Then V1
Can Be Found By The
Inverse Voltage Divider
R2
R1  R2
VO 
V1  V1 
VO
R1  R2
R2
 Now Use VS As a 2nd
Inverse Divider
VS 
R4  REQ
REQ
R4  REQ R1  R2
V1 
VO
REQ
R2
• How to Find The Input
In A Very Easy Way ?  Then Solve
Engineering-43: Engineering Circuit Analysis
31
for Vo
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Homogeneity Analysis cont

The Procedure Can Be Made
Entirely Algorithmic
1. Give to Vo Any Arbitrary Value (e.g., V’o = 1V )
2. Compute The Resulting Source Value
and Call It V’s
'
'
'
'
3. Use linearity V0  VS  kVS  VS  kV0  V0
4. The given value of the
source (Vs) corresponds to
5. Then The Desired Output
Engineering-43: Engineering Circuit Analysis
32
VS
k '
VS
VS '
V0  kV  ' V0
VS
'
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Homogeneity Comment
 This is a Nice Tool For Special Problems
 Normally Useful When
• There Is Only One Source
• Best Judgment Indicates That Solving The
Problem BACKWARDS Is Actually Easier
Than the Forward Solution-Path
Engineering-43: Engineering Circuit Analysis
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration  Homogeneity
 Solve Using Homogeneity (Scaling)
 Assume
 Then By Ohm’s Law
• V’out = V2 = 1volt
Engineering-43: Engineering Circuit Analysis
34
I1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration  Homogeneity cont
 Solve Using Homogeneity
 Scaling Factor
VS 12V
k ' 
2
VS
6V
'
Vout  kVout
 2 1V
 Again by Ohm’s Law
VO
 Then
 Using Homogeneity
• Scale from Initial
Assumption:
VO  6[V ]  Vout  1[V ] so :
VO  12[V ]  Vout  2[V ]
Engineering-43: Engineering Circuit Analysis
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Source Superposition
 This Technique Is A
Direct Application
Of Linearity
 Normally Useful When
The Circuit Has Only
A Few Sources
Engineering-43: Engineering Circuit Analysis
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration  Src Superposition
VS
 Consider a Circuit With
Two Independent
Sources: VS, IS
+ -
IL
 Now by Linearity
circuit
VL  a1VS  a2 I S
IS
VL2
 Calculated By Setting The CURRENT Source To
ZERO (OPEN ckt) And Solving The Circuit
 Calculated By Setting The VOLTAGE Source To
ZERO (SHORT ckt) And Solving The Circuit
Engineering-43: Engineering Circuit Analysis
37
VL
_
CONTRIBUTION BY VS
CONTRIBUTION BY I S
1
VL
V L2
VL1
+
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration cont
I
I L2
1
L
=
V
1
L
 Circuit With Current
Source Set To Zero
• OPEN Ckt
IL  I  I
2
L
Engineering-43: Engineering Circuit Analysis
38
VL2
 Circuit with Voltage
Source set to Zero
• SHORT Ckt
 By Linearity
1
L
+
VL  V  V
1
L
2
L
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration cont.
I
=
I L  I L1  I L2
Engineering-43: Engineering Circuit Analysis
39
VL1
VL  VL1  VL2
 The Above Eqns
Illustrate
SUPERPOSITION
I L2
1
L
+
VL2

This approach will be useful if
solving the two, 1-Src circuits is
simpler, or more convenient, than
solving a circuit with two sources

We can have any combination of
sources. And we can partition
any way we find convenient
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Example  Solve for i1
=
+
Req  3  3 || 6 [k ]
 v2
 v2
i 

Req
15k 2
Loop equations
"
2
 Alternative for i1(t)
By SuperPosition:
i1 t   i1' t   i1'' t 
– Find i1’’ by I-Divider
Contribution of v1
 Once we know the “partial circuits” we need
to be able to solve them in an efficient manner
Engineering-43: Engineering Circuit Analysis
40
Req  6  (3 || 3) [k ]
Contribution of v2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Numerical Example
 Find Vo By SuperPosition
 Set to Zero The V-Src
• i.e., SHORT it
Current division
3 6
2
I 0  2 mA   mA  mA
9 9
3
Contribution by Isrc →
Engineering-43: Engineering Circuit Analysis
41
V  I 0  6k  4[V ]
'
0
Ohm’s law
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Numerical Example cont.
 Find Vo By SuperPosition
 Set to Zero The I-Src
• i.e., OPEN it
 By V-Divider
 2[V ]
 Yields
Voltage
Divider
(UNtangle)
Contribution by Vsrc
6k
3V
+
-
3k
Engineering-43: Engineering Circuit Analysis
42
V0"
 Finally,
Add by SuperPosition
V0  V  V
'
0
"
0
V0  4  2  6[V ]
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
WhiteBoard Work
 Let’s Work this Nice
SuperPosition Problem
 Find IO
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Example  SuperPosition
 Find Vo Using Source
SuperPosition
 Set to Zero The I-Src
• i.e., OPEN it
 Set to Zero The V-Src
• i.e., SHORT it
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Example cont

 Define V1 on the V-Src ckt
V1

 If V1 is known then V’o is
obtained using the 6&2
Voltage-Divider
• V1 can be obtained by series
parallel reduction and divider
6k
+
+
-
V1
_
2k
V1
_
4k||8k
V'0
_
2k
8/3
V1 
(6)
2  8/ 3
Engineering-43: Engineering Circuit Analysis
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+
+
VO' 
6k
18
V1  [V ]
6k  2k
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Numerical Example cont.2
 Determine
I2
• Current I2 By Current Divider
• V”o Using Ohm’s Law
 When in Doubt REDRAW
2k||4k
2mA
+
I2
6k
 Finally The SuperPosition
Addition
V"0
2k
_
 The Current Division
I2 
2k  (2k || 4k )
5
 2mA  mA
6k  2k  (2k || 4k )
7
Engineering-43: Engineering Circuit Analysis
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30
V  6kI2  V
7
18
30
'
"
VO  VO  VO  V  V
7
7
48
VO  V  6.857V
7
"
O
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Sample Problem
 Determine Io by Source
SuperPosition
 First Consider Only the
Voltage Source
• Yields
 Second Consider Only
the 3 mA I-Source
• Yields Current Divider
 Then I   9V  1.5mA
01
6k
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Sample Prob cont
 Determine Io by Source
SuperPosition
 By IO2 Current Divider
3k
I 02  3mA
 1.5mA
6k
 Third Consider 4mA Src
 The Current will Return on
the Path of LEAST
Resistance; Thus
I 03  0
 So by Source Superposition
I 0  I 01  I 02  I 03  3mA
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration
 Use Source Superposition
to Determine Io
 Open the Current Source
I1
 Next Short the V-Source
 By Equivalent Resistance
I1 
VS
V
I
I O1   1   S
6 R  3R || 3R 
2
15 R
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Illustration cont
 Looks Odd & Confusing → REDRAW
2
2
2
1
2
2
3
1
2
 Now Use I-Divider
3
 Finally By Linearity
 2R
I 
IS
2 R  R  3R || 6 R 
2
2
IO   I S
5
2
O
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
All Done for Today
SuperPosition
Of
Plane-Polarized
Light
Cyan = Red + Green
 Red & Green Light-Waves are Polarized
in Perpendicular Planes
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
Engineering 43
Appendix:
Wheatsone
Bridge P2.103
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
The WheatStone Bridge
 When the Wheatstone Bridge is Balanced:
ig  0 and
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vab  0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx