Lecture 11 Thvenins Theorem Background and justification Examples

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

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!

“Derivation” – continued 1. Kill independent sources in circuit A • Get equivalent resistance seen at terminals a-b • Resulting voltage across terminals: v 1=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: v 2 = voc

“Derivation” – continued • 3. Superimpose v 1 and v 2 • 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!

“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 • Resulting current: i 2 = -isc

“Derivation” – continued • 3. Superimpose i 1 and i 2 • 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 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: