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Lecture 11 •Thévenin’s Theorem •Background and justification •Examples •Norton’s Theorem and examples •Source Transformations •Maximum Power Transfer •Related educational materials: –Chapter 4.5, 4.6 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 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? 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! Circuit iB (Load) “Derivation” – continued 1. Kill independent sources in circuit A • Get equivalent resistance seen at terminals a-b • Resulting voltage across terminals: v1=RTH·i “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 “Derivation” – continued • 3. Superimpose v1 and v2 • Get expression for voltage at terminals of circuit A • Represent as a conceptual “circuit” Creating the Thévenin equivalent circuit 1. Identify the circuit for which the Thévenin equivalent circuit is desired 2. Kill sources and determine RTH of the circuit 3. Re-activate the sources and determine VOC 4. 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 Thévenin’s Theorem – example 1 • Replace everything except the load resistor R with its Thévenin equivalent Example 1 – Get RTH Example 1 – Get Voc Example 1 – Thévenin circuit Norton’s Theorem • Norton’s Theorem: any linear circuit can be modeled as a current source in parallel with a resistor 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! Circuit + vB (Load) “Derivation” – continued 1. Kill independent sources in circuit A • Get equivalent resistance seen at terminals a-b • Resulting voltage across terminals: “Derivation” – continued 2. Replace sources in circuit A and kill voltage source representing circuit B • Get current seen at terminals a-b isc Circuit A • Resulting current: i2 = -isc + v2 = 0 - “Derivation” – continued • 3. Superimpose i1 and i2 • Get expression for voltage at terminals of circuit A • Represent as a conceptual “circuit” Creating the Norton equivalent circuit 1. Identify the circuit for which the Norton equivalent circuit is desired 2. Kill sources and determine RTH of the circuit 3. Re-activate the sources, short the output terminals, and determine isc 4. 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 Norton’s Theorem – example 1 • Replace everything except the load resistor R with its Norton equivalent Example 1 – Get RTH Example 1 – Get isc Example 1 – Norton circuit Source Transformations • The Thévenin and Norton equivalent circuits both represent the same circuit • They have the same voltage-current characteristics 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: 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 Source Transformation – example • Use source transformations to determine the voltage v 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 Maximum Power Transfer – example • Choose R so that maximum power is delivered to the load • Previously found the loaded Thévenin equivalent circuit: