Lesson 10: Thevenin’s Theorem and Max Power Transfer

1. Lesson 10: Thevenin’s Theorem and Max Power Transfer

2. Learning Objectives • State and explain Thèvenin's theorem. • List the procedure for determining the Thèvenin equivalence of an actual circuit from the standpoint of two terminals. • Apply Thèvenin's Theorem to simplify a circuit for analysis. • Analyze complex series-parallel circuits using Thèvenin's theorem. • Apply the Maximum Power Transfer theorem to solve appropriate problems.

3. Thévenin’s theorem greatly simplifies analysis of complex circuits by allowing us to replace all of the elements with a combination of just one voltage source and one resistor. Thévenin’s theorem provides a simplified circuit that provides the same response (voltage and current) at the load terminals. This allows the response to be easily determined for various load values. Thévenin’s Theorem

4. Any complex two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh and a series resistor RTh. Thévenin’s Theorem Original Circuit ThéveninEquivalent Circuit • The Thévenin equivalent circuit provides an equivalence at the terminals only. • The internal construction and characteristics of the original network and the Thévenin equivalent are usually quite different.

5. ETh is the open circuit voltage at the terminals. RTh is the input or equivalent resistance at the terminals when the sources are turned off. Thévenin’s Theorem Original Circuit ThéveninEquivalent Circuit

6. Thévenin’s Theorem 5 Steps: • Remove the load and • Label the terminals a and b. • SolveforRTHbysettingallsources to zero. • SolveforVTHbyreturningallsources to their original position and findingthe open-circuitvoltagebetweena and b. • Draw the new equivalent circuit.

7. Thévenin’s TheoremSteps 1 & 2 Convert to a Thévenin circuit: • Identify and remove the load from the circuit. • Label the resulting open terminals.

8. Thévenin’s TheoremStep 3 3. Solve for RTH and isolate the resistance from the source. Set all sources to zero: • Replace voltage sources with shorts. • Replace current sources with opens.

9. 3. “Zeroing” a source means setting its value equal to zero. Zeroing Sources • Voltage sources – 0 V is equivalent to a short-circuit. • Current sources – 0 A is equivalent to a open-circuit. Voltage Sources BecomeShort-Circuits Current Sources BecomeOpen-Circuits

10. With the load disconnected, turn off all source. RThis the equivalent resistance looking into the “dead” circuit through terminals a-b. Thévenin’s TheoremStep 3 Rth

11. Thévenin’s TheoremStep 3 3. Set all sources to zero, and calculateRTh . Remember, calculate RTH from the a and b perspective!

12. Thévenin’s TheoremStep 4 4. Solve for VTH and then, as needed: • Calculate the voltage (VLD) across the RLD. • Calculate the current (ILD) through RLD.

13. 5. REDRAW the circuit showing the Thèvenin equivalents (VTH and RTH) with the load installed. Thévenin’s Theorem Step 5

14. Repetitive solutions for various load resistances now becomes easy with the transformed circuit. Applying Thévenin Equivalent

15. Example Problem 1 Find the Thévenin equivalent circuit external to RLD. Determine ILDand VLD whenRLD = 2.5 Ω. a 15Ω 9Ω b RTH = 18.6Ω = 2.5Ω 1/2. Remove the load label the terminals a and b. 3. Solve for RTH. 4. Solve for VTH. 5. Draw the new equivalent circuit.

16. Example Problem 2 Find the Thévenin equivalent circuit external to RLD and determine ILD. RTH = 70kΩ = 180kΩ ETH = 5V

17. Maximum Power Transfer In someapplications, thepurpose of a circuitis to providemaximumpower to a load. Someexamples: • Stereoamplifiers • Radio transmitters • Communicationsequipment Thequestionis: Ifyouhave a system, what load shouldyouconnect to thesystem in orderforthe load to receivethemaximumpowerthatthesystem can deliver?

18. How might we determine RLD such that PLD is maximized? Maximizing PLD

19. Maximum power is transferred to the load when the load resistance equals the Thévenin resistance as seen from the load (RLD = RTh). When RLD = RTh, the source and load are said to be matched. Maximum Power Transfer Theorem

20. As RLDincreases, a higherpercentage of the total powerisdissipated in the load resistor. Butsincethe total resistanceisincreasing, the total currentisdropping, and a pointisreachedwherethe total powerdissipatedbytheentirecircuitstartsdropping. Maximizing PLD

21. Maximum Power • The max power happens when RLD = RTh, but what is the level of power at this point? • Showing the derivation, we get: • BE CAREFUL!!! Note that this is not true if RLD RTh. • If RLD RThthen use:

22. Maximum Power Transfer Theorem • The total power delivered by a supply such as ETh is absorbed by both the Thévenin equivalent resistance and the load resistance. • Any power delivered by the source that does not get to the load is lost to the Thévenin resistance.

23. Example Problem 3 • Find the Thévenin equivalent circuit to the left of terminals a-b. • Calculate the maximum power transfer to the load if RLD = RTH. • Determine the power dissipated by RLD for load resistances of 2 and 6. a 1Ω 12Ω b 4Ω

24. Example Problem 4 A stereo is rated for max output power of 150W per channel when RLD = 8Ω • Sketch the Thévenin Equivalent circuit. • What would the output power be with two 8Ω speakers as the load and are connected in parallel to one of the channels? RTH = 8Ω RTH = 8Ω = 8Ω = 8Ω//8Ω =4Ω

25. When maximum power is delivered to RLD, the efficiency is a mere 50%. Remember that max power occurs whenRLD= RTH. Efficiency

26. Efficiency • Communication Circuits and Amplifiers: • Max Power Transfer Is More Desirable Than High Efficiency. • Power Transmission (115 VAC 60 Hz Power ): • High Efficiency Is More Desirable Than Max Power Transfer. • Load Resistance Kept Much Larger Than Internal Resistance Of Voltage Source.

27. QUESTIONS?