Linearity & Superposition
Source Transformations
Sanity Check
Thevenin & Norton Equivalents
Maximum Power Transfer

Linearity

Superposition

Superposition: Voltage Sources

Superposition: Current Sources


For the circuit, use superposition to determine the unknown branch current i x
.


In the circuit, use the superposition principle to determine the value of i x
.

Example

Practical Voltage Sources

Practical Voltage Sources

Practical Voltage Sources

Practical Voltage Sources
The large, finite resistance of the current source acts to limit the voltage produced by the source.

Practical Voltage Sources
The small, nonzero resistance of the voltage source acts to limit the current produced by the source.

Equivalent Practical Sources

Solution by Source Transformation

Example 5.4

Compute the current through the 4.7 k resistor after transforming the 9 mA source into an equivalent voltage source.


Calculate the current through the 2 ohm resistor by making use of source transformations to first simplify the circuit.

Thevenin & Norton Equivalents
L. C. Thevenin -- French engineer; published his theorem in 1883
E. L. Norton -- scientist with Bell Telephone Laboratories
Any linear circuit network at two terminals may be replaced with a Thevenin equivalent ( V
TH
, R
TH
) or a Norton equivalent (I behave the same as the original network (v
L terminals.
, i
L
N
, R
N
). The equivalent will
) with respect to those two

Thevenin & Norton Equivalents
The Thevenin equivalent is more commonly seen in practice.
•   allows us to replace a large, complicated circuit with a much simpler 2element series/parallel circuit
•the simpler circuit allows for rapid calculations of V, I, P that the original circuit can deliver to a load
•helps us to choose the best value of load resistance to maximize the power delivered (e.g. from an amplifier, to a speaker)

Thevenin Equivalent, Method 1
How to determine V
TH
and R
TH
with respect to two terminals:
(1) Use repeated source transformations to arrive at a single voltage source in series with a single series resistance.

Thevenin Equivalent, Method 2
How to determine V
TH
and R
TH
with respect to two terminals:
(2) Open the load and determine the open-circuit voltage (V
OC load and determine the short-circuit current (I
SC
):
), then short the

Thevenin Equivalent, Method 3
How to determine V
TH
and R
TH
with respect to two terminals:
(3) Open the load and determine the open-circuit voltage (V
OC
), then deactivate all independent sources (short-circuit the V sources and opencircuit the I sources) and find the equivalent resistance (R eq
):

Thevenin Equivalent, Method 4
How to determine VTH and RTH with respect to two terminals:
(4) Open the load and determine the open-circuit voltage (VOC), then deactivate all independent sources and apply a test source:

Example 5.6
Consider the circuit shown below. Determine the Thévenin equivalent of network A , and compute the power delivered to the load resistor R
L
.

Example 5.7
Use Thévenin’s theorem to determine the Thévenin equivalent for that part of the circuit to the left of R
L
.

Thevenin Equiv: Dependent Sources
The only solution method for finding VTH and RTH (of the 4 presented in the prior slides) that is guaranteed to work when the circuit includes dependent sources is the test-source method.
(4) Open the load and determine the open-circuit voltage (VOC), then deactivate all independent sources and apply a test source:


Find the Thévenin and Norton equivalent circuits for the network faced by the 1 k resistor

Example 5.9
Determine the Thevenin equivalent of this network at the open-circuit terminals:

Example 5.10
Find the Thévenin equivalent of the circuit

Power from a Practical Source
The power delivered to a load from a practical voltage source is

Power from a Practical Source
The maximum value of pL vs. RL occurs when

Chapter 5 Summary & Review
• Superposition, Source Transformations, & Thevenin Equivalents
•useful for when we do not need to solve for all voltages or all
currents in a particular circuit
•allow us to modify a complicated circuit to a simpler equivalent in
order to study the complicated circuit’s behavior at a particular set
of terminals
• Superposition
•The response of a linear circuit is equal to the sum of responses
produced by individual independent sources (each acting alone).
• Source Transformations
•allow us to convert a practical voltage source (model) to a practical
current source (model), and vice versa & enable us to combine
sources & resistors using series/parallel equivalence to simplify the
circuit-under-analysis
• Thevenin Equivalent
•resistor in series with an independent voltage source
•useful for determining when a load is matched for maximum power
transfer
•may be obtained using source transformations, VOC & ISC, or
VOC and Vtest/Itest

(a)   Π network consisting of three resistors and three unique connections.
(b)   Same network drawn as a Δ network.
(c)   A T network consisting of three resistors.
(d)   Same network drawn as a Y network.

( a ) Labeled Π network; ( b ) labeled T network.
convert from a Π network to a Y network convert from a Y network to a Π network

Review: Nodal & Mesh Analysis
Set up a complete, valid set of simultaneous equations to solve for the power absorbed by the 5-
Ω
resistor. You are not required to solve these equations.

Review: Nodal & Mesh Analysis
Set up a complete, valid set of simultaneous equations to solve for the power absorbed by the 3Ω resistor. You are not required to solve these equations.