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**Lecture 11 Thévenin’s Theorem Norton’s Theorem and examples**

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**

Kill independent sources in circuit A Get equivalent resistance seen at terminals a-b Resulting voltage across terminals: v1=RTH·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: v2 = voc

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**“Derivation” – continued**

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

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**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

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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 RTH

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Example 1 – Get Voc

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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**

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: i2 = -isc

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**“Derivation” – continued**

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

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**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

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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 RTH

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Example 1 – Get isc

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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 RL so that RL receives the maximum power For maximum power transfer, choose RL = RTH

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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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Discussion D2.5 Sections 2-9, 2-11

Discussion D2.5 Sections 2-9, 2-11

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