ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Regsitered Electrical & Mechanical Engineer Engineering 43 Chp 5.4 Maximum Power Transfer

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis ReCall Thévenin Equivalent  Thevenin Equivalent Circuit for PART A  v TH = Thevenin Equivalent VOLTAGE Source  R TH = Thevenin Equivalent SERIES RESISTANCE

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Recall Norton Equivalent  Norton Equivalent Circuit for PART A  i N = Norton Equivalent CURRENT Source  R N = Norton Equivalent PARALLEL RESISTANCE

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  Recognize Mixed sources Must Compute Open Circuit Voltage, V OC, and Short Circuit Current, I SC  The Open Ckt Voltage  Use V-Divider to Find V X  For V b Use KVL  Solve for V TH  The Short Ckt Current Note that Shorting a-to-b Results in a Single Large Node  Now V TH = V x - V b

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example cont  Need to Find V x  KCL at Single Node  Then R TH Single node  Solving For V x  KCL at Node-b for I SC  The Equivalent Circuit

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Analysis  Using Excel Spreadsheet SShort INDEPENDENT Sources to Find R TH AAnd V OC by 12V Source and V-Divider for  V across R X

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Analysis - Plot

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Analysis - Limits

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Plot using MATLAB Script File  ENGR43_Chp5_Rth_Voc_Analysis_MATLAB _0602.m % 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]')

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevenin Theorem – General View  Typical Interpretation  The General View Looks Like Series Resistance

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevenin General - Comments  V TH 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 v o  This Interpretation Applies Even When The Passive Elements Include INDUCTORS and CAPACITORS

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Amplifier Driving Speaker  Consider an Amplifier Circuit connected to a Speaker Driving Circuit a.k.a. the “SOURCE” Speaker a.k.a. the “LOAD”

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Circuit Simplification  Thévenin’s Equivalent Circuit Theorem Allows Tremendous Simplification of the Amp Ckt Thevenin +  R S V S

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Maximum Power Transfer  Consider The Amp-Speaker Matching Issue From PreAmp (voltage ) To speakers

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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 BASIC MODEL FOR THE ANALYSIS OF POWER TRANSFER

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Maximum Power Transfer  Consider Thevenin Equivalent Ckt with Load R L  Find Load Pwr by V-Divider  For every choice of R L we have a different power. How to find the MAXIMUM Power value?  Consider P L as a FUNCTION of R L and find the maximum of such a function  have at left! i.e., Take 1 st Derivative and Set to Zero

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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” (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 resistance of the circuit

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Power Quantified  By Calculus we Know R L for P L,max  Recall the Power Transfer Eqn  Sub R TH for R L  So Finally

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer Example  Determine R L for Maximum Power Transfer  Need to Find R TH Notice This Ckt Contains Only INDEPENDENT Sources  Thus R TH By Source Deactivation a b  This is Then the R L For Max Power Transfer  To Find the AMOUNT of Power Transferred Need the Thevenin Voltage  Then use R TH = 6kΩ along with V TH

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer Example cont  To Find V TH Use Meshes  The Eqns for Loops 1 & 2  Solving for I 2  Now Apply KVL for V OC  Recall  At Max: P L = P MX, R L = R TH

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer  Determine R L and Max Power Transferred  Find Thevenin Equiv. At This Terminal-Set  Recall for Max Pwr Xfer a b  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

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer cont  The Controlling Variable  Now Short Ckt Current The Added Wire Shorts the 2k Resistor  Remember now the partition points  Then R TH c d  The R TH for ckt at a-b = 2kΩ+4kΩ; So a b

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevenin & Norton Summary  Independent Sources Only R TH = R N by Source Deactivation V TH –= V OC or –= R N ·I SC I N –= I SC or –= V OC /R TH  Mixed INdep and Dep Srcs Must Keep Indep & dep Srcs Together in Driving Ckt V TH = V OC I N = I SC R TH = R N = V OC / I SC  DEPENDENT Sources Only Must Apply V or I PROBE –Pick One, say I P = 1.00 mA, then Calculate the other, say V P V TH = I N = 0 R TH = R N = V P / I P

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis WhiteBoard Work LLet’s Work Problem FFind P max for Load R L

ENGR-43_Lec-04-3b_Thevein-Norton_Part-b.ppt 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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.