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Lecture 11: Thévenin's Theorem, Norton's Theorem, Source Transformations

This lecture covers Thévenin's Theorem, Norton's Theorem, and Source Transformations in circuit analysis. It includes explanations, derivations, examples, and applications of these concepts.

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Lecture 11: Thévenin's Theorem, Norton's Theorem, Source Transformations

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  1. Lecture 11 Thévenin’s Theorem Background and justification Examples Norton’s Theorem and examples Source Transformations Maximum Power Transfer Related educational modules: Sections 1.7.4, 1.7.5

  2. Thévenin’s Theorem • We want to replace a complicated circuit with a simple one without affecting the load • We can do this by taking advantage of superposition

  3. Thévenin’s Theorem • Lecture 10: Any linear circuit can be represented by an ideal voltage source in series with a resistance, without affecting any “load” connected to the circuit • Why?

  4. Thévenin’s Theorem – “Derivation” • Represent circuit “B” (load) as a current source, providing some voltage • Note that we haven’t changed the i-v characteristics at terminals!

  5. “Derivation” – continued • Kill independent sources in circuit A • Get equivalent resistance seen at terminals a-b • Resulting voltage across terminals: v1=RTH·i

  6. “Derivation” – continued 2. Replace sources in circuit A and kill current source representing circuit B • Get voltage seen at terminals a-b • Resulting voltage across terminals: v2 = voc

  7. “Derivation” – continued • 3. Superimpose v1 and v2 • Get expression for voltage at terminals of circuit A • Represent as a conceptual “circuit”

  8. Creating the Thévenin equivalent circuit • Identify the circuit for which the Thévenin equivalent circuit is desired • Kill sources and determine RTH of the circuit • Re-activate the sources and determine VOC • Place the Thévenin equivalent circuit into the original overall circuit and perform the desired analysis • Note: a slightly different process is necessary if the circuit contains dependent sources

  9. Thévenin’s Theorem – example 1 • Replace everything except the load resistor R with its Thévenin equivalent

  10. Example 1 – Get RTH

  11. Example 1 – Get Voc

  12. Example 1 – Thévenin circuit

  13. Norton’s Theorem • Norton’s Theorem: any linear circuit can be modeled as a current source in parallel with a resistor

  14. Norton’s Theorem – “Derivation” • Represent circuit “B” (load) as a voltage source, providing some current • Note that we still haven’t changed the i-v characteristics at terminals!

  15. “Derivation” – continued • Kill independent sources in circuit A • Get equivalent resistance seen at terminals a-b • Resulting voltage across terminals:

  16. “Derivation” – continued 2. Replace sources in circuit A and kill voltage source representing circuit B • Get current seen at terminals a-b • Resulting current: i2 = -isc

  17. “Derivation” – continued • 3. Superimpose i1 and i2 • Get expression for voltage at terminals of circuit A • Represent as a conceptual “circuit”

  18. Creating the Norton equivalent circuit • Identify the circuit for which the Norton equivalent circuit is desired • Kill sources and determine RTH of the circuit • Re-activate the sources, short the output terminals, and determine isc • Place the Norton equivalent circuit into the original overall circuit and perform the desired analysis • Note: a slightly different process is necessary if the circuit contains dependent sources

  19. Norton’s Theorem – example 1 • Replace everything except the load resistor R with its Norton equivalent

  20. Example 1 – Get RTH

  21. Example 1 – Get isc

  22. Example 1 – Norton circuit

  23. Source Transformations • The Thévenin and Norton equivalent circuits both represent the same circuit • They have the same voltage-current characteristics

  24. Source Transformations – continued • We can equate the two representations • Solving for i from the Thévenin equivalent • Equating this current with the Norton Equivalent circuit: • So that:

  25. Using Source Transformations in Circuit Analysis • Any voltage source in series with a resistance can be modeled as a current source in parallel with the same resistance and vice-versa

  26. Source Transformation – example • Use source transformations to determine the voltage v

  27. Maximum Power Transfer • We can use Thevenin’s Theorem to show how to transfer the maximum amount of power to a load • Problem: choose RL so that RL receives the maximum power • For maximum power transfer, choose RL = RTH

  28. Maximum Power Transfer – example • Choose R so that maximum power is delivered to the load • Previously found the loaded Thévenin equivalent circuit:

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