1. MaxPower SuperPosition
Engineering 43
MaxPower SuperPosition
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. OutLine: MaxPwr & SuperPose
Work On WhtBd Complex Thevénin Problem → not enough time 09Feb16
Thevénin & Norton Review
Example Problem (WhtBd)
Maximum Power Transfer Theorem Derivation
MaxPwr Application Examples
Thevénin & Norton Summary

3. OutLine: MaxPwr & SuperPose
Linearity & Homogeneity
Guess Solution, Work BackWards, Scale Guess
Comparative Case Study
SuperPosition → Activate & DeActivate
Example Problem (WhtBd)
09Feb16 → SKIP Today

4. Thevénin’s Equivalence Theorem
vTH = Thévenin Equivalent VOLTAGE Source
RTH = Thévenin Equivalent SERIES RESISTANCE
Load
Driving Circuit
Thevenin Equivalent Circuit for PART A

5. Norton’s Equivalence Theorem
iN = Norton Equivalent CURRENT Source
RN = Norton Equivalent PARALLEL RESISTANCE
Driving Circuit
Load
Norton Equivalent Circuit for PART A

6. Find 𝑽 𝑻𝒉 , 𝑰 𝑵, , 𝑹 𝑻𝒉 = 𝑹 𝑵 Then One circuit problem
1. Determine the
Thevenin equivalent
source
Remove part B and
compute the OPEN
CIRCUIT voltage
Second circuit problem
2. Determine the
SHORT CIRCUIT
current
Remove part B and
compute the SHORT
CIRCUIT current
Then

7. Example: VOC, ISC, RTh = RN
Use Thevénin and Norton for find the OutPut Voltage in the Circuit Below
Recall: VTh = VOC & IN = ISC
09Feb16 → SKIP This One

10. Now Isc

14. Maximum Power Transfer
Consider The Amp-Speaker Matching Issue
From PreAmp
(voltage )
To speakers

15. Maximum Power Xfer Cont
The Simplest Model for a Speaker is to Consider it as a RESISTOR only
BASIC MODEL FOR THE
ANALYSIS OF POWER
TRANSFER
Since the “Load” Does the “Work” We Would like to Transfer the Maximum Amount of Power from the “Driving Ckt” to the Load
Anything Less Results in Lost Energy in the Driving Ckt in the form of Heat

16. Maximum Power Transfer
Consider Thevenin Equivalent Ckt with Load RL
Find Load Pwr by V-Divider
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
For every choice of RL we have a different power.
How to find the MAXIMUM Power value?

17. Max Power Xfer cont
Find Max Power Condition Using Differential Calculus
Set The Derivative To Zero To Find MAX or MIN Points
For this Case Set To Zero The NUMERATOR
Solving for “Best” (Pmax) Load
This is The Maximum Power Transfer Theorem
The load that maximizes the power transfer for a circuit is equal to the Thevenin equivalent resistance of the circuit

18. Max Power Quantified By Calculus we Know RL for PL,max Sub RTH for RL
Recall the Power Transfer Eqn
So Finally

19. Max Pwr Xfer Example Determine RL for Maximum Power Transferb
Determine RL for Maximum Power Transfer
Need to Find RTH
Notice This Ckt Contains Only INDEPENDENT Sources
Thus RTH By Source Deactivation
To Find the AMOUNT of Power Transferred Need the Thevenin Voltage
Then use RTH = 6kΩ along with VTH
This is Then the RL For Max Power Transfer

20. Max Pwr Xfer Example cont
To Find VTH Use Meshes
The Eqns for Loops 1 & 2
Solving for I2
Recall
Now Apply KVL for VOC
At Max: PL = PMX, RL = RTH

21. Max Pwr Xfer Determine RL and Max Power Transferredb c d
Determine RL and Max Power Transferred
Find Thevenin Equiv. At This Terminal-Set
Use Loop Analysis
Recall for Max Pwr Xfer
This is a MIXED Source Circuit
Analysis Proceeds More Quickly if We start at c-d and Adjust for the 4kΩ at the end
Eqns for Loops 1 & 2

22. Max Pwr Xfer cont The Controlling Variable
Remember now the partition points
Now Short Ckt Current
The Added Wire Shorts the 2k Resistor c d a b
The RTH for ckt at a-b = 2kΩ+4kΩ; So
Then RTH

23. Thevenin w/ Dependent Srcs cont
VTH = 0 is a BIG Simplification
But Need A Special Approach To Find the Thevenin Equivalent Resistance
Since The Circuit CanNOT Self Start, PROBE It With An EXTERNAL Source
The PROBE Can Be Either A VOLTAGE Source Or A CURRENT Source Whose Value Can Be Chosen ARBITRARILY
Which One To Choose Is Often Determined By The Simplicity Of The Resulting Circuit

24. Voltage Probe
If a VOLTAGE Probe is Chosen, Then Must Find the CURRENT Supplied by The Probe V-source
Since VP is Arbitrary, Usually Set it to

25. Current Probe
If a CURRENT Probe is Chosen, Then Must Find the VOLTAGE Generated by The Probe I-source
The Value for IP is Arbitrary, Usually Set it to

26. Numerical Example Find the Thevenin Equivalent Circuit at A-B
Use a CURRENT or VOLTAGE Probe?
Using a Voltage Probe Results In Only One Node Not Connected to GND Through a Source
Apply the V-Probe, and Analyze by KCL at V1

27. Numerical Example cont.
The Controlling Variable
Solving the Eqns
Calc RTH using VP & IP
To Determine RTH need to Calc Probe Current
933Ω

28. Dependent Source Example
Find the Thevenin Equivalent circuit at A-B
Only Dependent Sources, Thus VTH = 0
Apply a CURRENT Probe to Determine The equivalent resistance
Have a“Conventional” circuit with dependent sources
use node analysis
Let IP = 1 mA = 1x10-3
By KCL at V1 and V2

29. Dependent Source Example cont
Now the Controlling Variable
Sub for Ix in V1 KCL
Multiply LCD Against Both Resulting KCL Eqns
Then
And From the Ckt Observe
VP = V2
With IP = 1 mA
Eliminate V1
1.43 kΩ

30. Thevenin & Norton Summary
Independent Sources Only
RTH = RN by Source Deactivation
VTH
= VOC or
= RN·ISC IN
= ISC or
= VOC/RTH
Mixed INdep and Dep Srcs
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
Dependent Source Prob in Hambley = 2.60 & 2.61

31. WhiteBoard Work Let’s Work this nice Max Power Problem
Find Pmax for Load RL

32. Previous Equivalent Circuits
Series & Parallel Resistors
[Independent Srcs]
Vsrc’s in Series
Isrc’s in Parallel
Can’t have
V sources in Parallel
I sources in series
The Complementary Configs are Inconsistent with Source Definitions

33. Linearity Models Used So Far Are All LINEAR
Mathematically This Implies That They satisfy the principle of SUPERPOSITION
The Model T(u) is Linear IF AND ONLY IF
For All Possible
Input Pairs: u1 & u2
Scalars α1 & α2
AN Alternative, And Equivalent, Linearity & Superposition Definition
The Model T(u) is Linear IF AND ONLY IF It Exhibits
ADDITIVITY
HOMOGENEITY

34. Linearity cont. Linearity Characteristics NOTE
Additivity
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
Homogeneity
a.k.a. Scaling

35. Linearity cont.
Using Node Analysis For Resistive Circuits Yields Models Of The Form
The Model Can Be Made More Detailed
Where
A and B are Matrices
s Is a Vector Of All Independent Sources
For Ckt Analysis Use The Linearity Assumption To Develop Special Analysis Methods
Where
v is a (Soln) Vector Containing All The Node Voltages
f is a Vector Containing Only independent Sources

36. Past Techniques  Case Study
Find Vo
Redraw the Ckt to Reveal Special Cases
After Untangling     
Be FAITHFUL to the nodes in untangling
Solution Techniques Available? 

37. Case Study cont.
Loop Analysis for Vo
Node Analysis
Out → Positive - -

38. Case Study cont. Series-Parallel Resistor-Combinations In other Words
By VOLTAGE Divider

39. Use Homogeneity Analysis
Find Vo by Scaling
If Vo is Given Then V1 Can Be Found By The Inverse Voltage Divider
Now Use VS As a 2nd Inverse Divider
Assume That The Answer Is KNOWN
How to Find The Input In A Very Easy Way ?
Then Solve for Vo

40. Homogeneity Analysis cont
The Procedure Can Be Made Entirely Algorithmic
Give to Vo Any Arbitrary Value (e.g., V’o = 1V )
Compute The Resulting Source Value and Call It V’s
Use linearity
The given value of the source (Vs) corresponds to
Then The Desired Output

41. 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

42. Illustration  Homogeneity
Solve Using Homogeneity (Scaling)
Assume
V’out = V2 = 1volt
Then By Ohm’s Law

43. Illustration  Homogeneity cont
Solve Using Homogeneity
Scaling Factor
Again by Ohm’s Law
Using Homogeneity
Scale from Initial Assumption:
Then

44. Source Superposition
This Technique Is A Direct Application Of Linearity
Normally Useful When The Circuit Has Only A Few Sources

45. Illustration  Src Superposition
Consider a Circuit With Two Independent Sources: VS, IS
Now by Linearity
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

46. Illustration cont + By Linearity =
Circuit With Current Source Set To Zero
OPEN Ckt
Circuit with Voltage Source set to Zero
SHORT Ckt
By Linearity

47. Illustration cont. + The Above Eqns Illustrate SUPERPOSITION =
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
The Above Eqns Illustrate SUPERPOSITION

48. Example  Solve for i1 = Alternative for i1(t) By SuperPosition: +
Loop equations
Contribution of v1
Alternative for i1(t) By SuperPosition:
Find i1’’ by I-Divider
Contribution of v2
Once we know the “partial circuits” we need to be able to solve them in an efficient manner

49. Numerical Example Find Vo By SuperPosition Set to Zero The V-Src
i.e., SHORT it
Current division
Contribution by Isrc →
Ohm’s law

50. Numerical Example cont.
Find Vo By SuperPosition
Set to Zero The I-Src
i.e., OPEN it
By V-Divider
Contribution by Vsrc
Yields Voltage Divider (UN- tangle)
Finally, Add by SuperPosition

51. WhiteBoard Work Find IO Let’s Work this Nice SuperPosition Problem

52. 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

53. Example cont Define V1 on the V-Src ckt
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
A good ckt to UnTangle

54. Numerical Example cont.2
Determine
Current I2 By Current Divider
V”o Using Ohm’s Law
When in Doubt REDRAW
Finally The SuperPosition Addition
The Current Division

55. 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

56. Sample Prob cont Determine Io by Source SuperPosition
By IO2 Current Divider
The Current will Return on the Path of LEAST Resistance; Thus
Third Consider 4mA Src
So by Source Superposition

57. Illustration Use Source Superposition to Determine Io
Open the Current Source
Next Short the V-Source
By Equivalent Resistance

58. Illustration cont Looks Odd & Confusing → REDRAW Now Use I-Divider2 2 2 2 2 1 3 1 3 2
Now Use I-Divider
Finally By Linearity

59. SuperPosition Of Plane-Polarized Light
All Done for Today
SuperPosition Of Plane-Polarized Light
Cyan = Red + Green
Red & Green Light-Waves are Polarized in Perpendicular Planes

60. Find Pmax for RL

61. P5-109

62. 5-109

76. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engineering 43
Appendix:
Wheatsone Bridge P2.103
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

77. When the Wheatstone Bridge is Balanced: