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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

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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

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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?

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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!

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“Derivation” – continued 1.Kill independent sources in circuit A Get equivalent resistance seen at terminals a-b Resulting voltage across terminals: v 1 =R TH ·i

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“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: v 2 = v oc

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“Derivation” – continued 3.Superimpose v 1 and v 2 Get expression for voltage at terminals of circuit A Represent as a conceptual “circuit”

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Creating the Thévenin equivalent circuit 1.Identify the circuit for which the Thévenin equivalent circuit is desired 2.Kill sources and determine R TH of the circuit 3.Re-activate the sources and determine V OC 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

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

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Example 1 – Get R TH

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Example 1 – Get V oc

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Example 1 – Thévenin circuit

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

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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!

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“Derivation” – continued 1.Kill independent sources in circuit A Get equivalent resistance seen at terminals a-b Resulting voltage across terminals:

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

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“Derivation” – continued 3.Superimpose i 1 and i 2 Get expression for voltage at terminals of circuit A Represent as a conceptual “circuit”

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Creating the Norton equivalent circuit 1.Identify the circuit for which the Norton equivalent circuit is desired 2.Kill sources and determine R TH of the circuit 3.Re-activate the sources, short the output terminals, and determine i sc 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

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

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Example 1 – Get R TH

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Example 1 – Get i sc

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Example 1 – Norton circuit

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

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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:

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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

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

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Maximum Power Transfer We can use Thevenin’s Theorem to show how to transfer the maximum amount of power to a load Problem: choose R L so that R L receives the maximum power For maximum power transfer, choose R L = R TH

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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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