Maximum Power Transfer and Thevenin-Norton Equivalents, Slides of Electrical Circuit Analysis

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Engineering 43

MaxPower

SuperPosition

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

      ^        L s

s L L

s S L sS LL

LL L sS LL s SL sS LL

L L L S s L L sS LL

LL L L S s LL

R R

R R R

RVR RV RR

RP R RVRR RVR RVRR

P VR R V RRR RVRR

RP P VR V V RRR



 

  

       

       

     

2

2

(^0) dd dd 2

1

(^0) dd

(^2223)

(^222223)

(^22) 2 2

2

Thevénin’s Equivalence Theorem

LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A

LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART B

a

_ b

v O



i

LINEAR CIRCUIT PART B

a

_ b

v O



i





R TH

v TH

PART A

 Thevenin Equivalent Circuit for PART A

 vTH = Thévenin Equivalent VOLTAGE Source  RTH = Thévenin Equivalent SERIES RESISTANCE

Norton’s Equivalence Theorem

 Norton Equivalent Circuit for PART A

 iN = Norton Equivalent CURRENT Source  RN = Norton Equivalent PARALLEL RESISTANCE

LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A

LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART B

a

_ b

v O



i

LINEAR CIRCUIT PART B

a

_ b

v O



i

iN R N

PART A

Maximum Power Transfer

• Consider The Amp-Speaker Matching Issue

From PreAmp (voltage ) (^) To speakers

+-

RTH

VTH

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

• Anything Less Results in Lost Energy in the Driving Ckt in the form of Heat

+-

RTH

VTH SPEAKER MODEL BASIC MODEL FOR THE ANALYSIS OF POWER TRANSFER

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

    

 

 

 

 

   (^2 )

2

TH L

TH L L TH L

L R R

R R R V dR

dP

 Solving for “Best” (Pmax) Load RL  RTH

 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

   

  (^20)

0

2 3

2

   

  

 

 

 

  

TH L L

TH L

TH L L TH L R

L

R R R

R R

R R R V dR

dP

L

Max Power Quantified

• By Calculus we Know

RL for PL,max

RL  RTH

 Recall the Power

Transfer Eqn

 

2 2 TH TH L

L L V R R

R P 



 Sub RTH for RL

 

2 , max 2 TH TH TH

TH L V R R

R P 



 

2 2

2 ,max (^24) 2

TH TH

TH TH TH

TH L V R

R V R

R P  

 So Finally

TH

TH L R

V P

2

, max 4

1 

Max Pwr Xfer Example cont

• To Find VTH Use Meshes
• The Eqns for Loops 1 & 2

 Solving for I 2

8 [ ] 2 [ ] 10 [ ]

3

1 4 * 2 6 *

4 * 1 6 * 2

V V V

k mA k mA

VOC k I k I

  

   

 

I 1  2 mA

3 k ^ I 2  I 1 ^  6 kI 2  3 V  0

[ ]

9 [ ]

3 [ ]

2 k I 1 mA

V

I  

 Now Apply KVL for VOC

 Recall

2 2 TH TH L

L L (^) R R V

R

P

 At Max: PL = PMX, RL = RTH

TH

MX TH R

P V 4

2 

[ ] 6

25 4 * 6

100 [ 2 ] mW k

P V MX   

Max Pwr Xfer

• Determine RL and Max Power Transferred
• Find Thevenin Equiv. At This Terminal-Set

 Recall for Max Pwr Xfer

a

b

RL  RTH TH

MX TH R

V P 4

2 

 This is a MIXED Source Circuit

• Analysis Proceeds More Quickly if We start at c-d and Adjust for the 4kΩ at the end

c

d  Use Loop Analysis

 Eqns for Loops 1 & 2 I 1  4 mA 4 k (I 2  I 1 ) 2 kI (^) X'   2 kI 2  0

6 I 2  2 I (^) X'   4 I 1  16 mA

I 1 I 2

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. mA, then Calculate the other, say VP
• VTH = IN = 0
• RTH = RN = VP/ IP

WhiteBoard Work

 Let’s Work this nice

Max Power Problem

 Find Pmax for Load RL

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

T 1 u 1  2 u 2   1 Tu 1    2 Tu 2 

• For All Possible
  • Input Pairs: u 1 & u 2
  • Scalars α 1 & α 2  AN Alternative, And Equivalent, Linearity & Superposition Definition
• The Model T(u) is Linear IF AND ONLY IF It Exhibits
• ADDITIVITY
• HOMOGENEITY

Linearity cont.

• Linearity

Characteristics

• 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

T u 1 u 2  Tu 1  Tu 2 

T(u )  T u

• Homogeneity
  • a.k.a. Scaling