MaxPower SuperPosition Engineering 43 MaxPower SuperPosition Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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

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

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

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

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

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

Now Isc

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

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

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?

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

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

Max Pwr Xfer Example Determine RL for Maximum Power Transfer b 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

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

Max Pwr Xfer Determine RL and Max Power Transferred b 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

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

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

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 1.00000

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 1.00000

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

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

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

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

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

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

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

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

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

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

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? 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

SuperPosition Of Plane-Polarized Light All Done for Today SuperPosition Of Plane-Polarized Light http://www.enzim.hu/~szia/cddemo/edemo4.htm Cyan = Red + Green Red & Green Light-Waves are Polarized in Perpendicular Planes

Find Pmax for RL

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

When the Wheatstone Bridge is Balanced: