Lecture 11 slides

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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:

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