•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 i B
(Load)

“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

“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

“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 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

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

Example 1 – Get R
TH

Example 1 – Get V oc

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
v B
(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
Circuit
A i sc
+ v
2
= 0
-
• Resulting current: i
2
= -i sc

“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 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

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

Example 1 – Get R
TH

Example 1 – Get i sc

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 R
L so that R
L receives the maximum power
• For maximum power transfer, choose R
L
= R
TH

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