Bruce Mayer, PE Regsitered Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 43 Chp 5.4Maximum PowerTransfer Bruce Mayer, PE Regsitered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. ReCall Thévenin Equivalent • vTH = Thevenin Equivalent VOLTAGE Source • RTH = Thevenin Equivalent SERIES RESISTANCE • Thevenin Equivalent Circuit for PART A

3. Recall Norton Equivalent • iN = Norton Equivalent CURRENT Source • RN = Norton Equivalent PARALLEL RESISTANCE • Norton Equivalent Circuit for PART A

4. Recognize Mixed sources Must Compute Open Circuit Voltage, VOC, and Short Circuit Current, ISC The Open Ckt Voltage Example • Solve for VTH • Use V-Divider to Find VX • The Short Ckt Current • Note that Shorting a-to-b Results in a Single Large Node • For Vb Use KVL • Now VTH = Vx - Vb

5. Need to Find Vx Single node Example cont • KCL at Single Node • Then RTH • Solving For Vx • The Equivalent Circuit • KCL at Node-b for ISC

6. Using Excel Spreadsheet Numerical Analysis • Short INDEPENDENT Sources to Find RTH • And VOC by 12V Source and V-Divider for V across RX

7. Numerical Analysis - Plot

8. Numerical Analysis - Limits

9. ENGR43_Chp5_Rth_Voc_Analysis_MATLAB_0602.m Plot using MATLAB Script File % ENGR43_Chp5_Rth_Voc_Analysis_MATLAB_0602.m % Bruce Mayer, PE % ENGR43 * 27Feb06 % Rx = [0:0.1:20]'; %define the range of resistors to use Voc = 12-6*Rx./(Rx+4); %the formula for Voc. Notice "./" Rth = 4*Rx./(4+Rx); %formula for Thevenin resistance. plot(Rx,Voc,'bx', Rx,Rth,'mv') title('USING MATLAB'), grid, xlabel('Rx (kOhm)'), ylabel('Voc (V), Rth (kOhm)') legend('Voc [V]','Rth [kOhm]')

11. Typical Interpretation Looks Like Series Resistance Thevenin Theorem – General View • The General View

12. Thevenin General - Comments • VTH Becomes the Sole Equivalent Power Source/Sink for the “Part-A” (a.k.a. Driving) Circuit • It’s Value is Set to Maintain The Open Ckt Voltage at vo • This Interpretation Applies Even When The Passive Elements Include INDUCTORS and CAPACITORS

13. Amplifier Driving Speaker • Consider an Amplifier Circuit connected to a Speaker Speakera.k.a. the“LOAD” DrivingCircuita.k.a. the“SOURCE”

14. Thévenin’s Equivalent Circuit Theorem Allows Tremendous Simplification of the Amp Ckt Circuit Simplification R S Thevenin + V S 

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

16. The Simplest Model for a Speaker is to Consider it as a RESISTOR only BASIC MODEL FOR THE ANALYSIS OF POWER TRANSFER Maximum Power Xfer Cont • 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

17. Consider Thevenin Equivalent Ckt with Load RL Find Load Pwr by V-Divider Maximum Power Transfer • 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?

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

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

20. Determine RL for Maximum Power Transfer Need to Find RTH Notice This Ckt Contains Only INDEPENDENT Sources a b Max Pwr Xfer Example • Thus RTH BySource Deactivation • To Find the AMOUNT of Power Transferred Need the Thevenin Voltage • Then use RTH = 6kΩalong with VTH • This is Then the RL For Max Power Transfer

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

22. Determine RL and Max Power Transferred Find Thevenin Equiv.At This Terminal-Set a c b d Max Pwr Xfer • 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

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

24. 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 Thevenin & Norton Summary • 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

25. WhiteBoard Work • Let’s Work Problem 5.109 • Find Pmax for Load RL

26. What’s an “Algorithm” • A postage stamp issued by the USSR in 1983 to commemorate the 1200th anniversary of Muhammad al-Khowarizmi, after whom algorithms are named.