Superposition, Thevenin /
Norton Equivalents,
Maximum Power Transfer
Circuits 1
Fall 2005
Harding University
Jonathan White
Outline – Ch. 4
 Superposition
 Method of analyzing a circuit by turning off all sources but 1 and
then finding their contributions individually. End by summing up
all the contributions.
 Thevenin Equivalent Circuits
 A circuit at a given 2 terminals can be replaced by a voltage
source with a resistor in series.
 Norton Equivalent Circuits
 A circuit at a given 2 terminals can also be replaced with a
current source and a parallel resistor.
 Maximum Power Transfer
 When you have a load, when does it receive the maximum
power? We’ve already answered this in lab.
Superposition
 Resistors are linear elements, meaning that the
output is linearly related to the input.
 Voltages around a loop can simply be added up – no
non linear math is required.
 Instead of analyzing circuits like we did in Ch. 2
and Ch. 3, we can analyze them using
Superposition.
 Definition: The voltage across (or current through) a
resistor is the algebraic sum of all the contributions
due to each source acting alone.
 So, another way to analyze a circuit is to find the
contribution of each source individually and them add
them up at the end to get the total.
Superposition 2
 We only consider 1 independent source at
a time when we use superposition. This
means that we:
 Replace voltage sources with a wire (0 V).
 Replace current sources with an open circuit
(no current can flow).
 Dependent sources are left intact since
they are controlled by circuit variables.
Superposition 3
 To solve a circuit using superposition:
 Turn off all independent sources but 1. Use
the techniques of Ch. 2 and Ch. 3 to solve for
the desired voltage or current.
 Repeat for each independent source.
 Find the total voltage or contribution by taking
the algebraic sum.
Superposition – Exp. 1
Find the voltage over the 2 Ohm resistor using superposition.
Superposition Exp. 2
Find the voltage over the 5 ohm resistor using superposition.
+V -
Equivalent Circuits
 A model of the real thing.
 Used to capture only the necessary details
of a potentially complex circuit.
 Examples of various models:
 Battery
 OSI network layer
 Function calls
 You (as a user), don’t really care how the function
operates, just that it does.
Thevenin Equivalent Circuits
 Consists of a voltage source and a resistor
in series.
 Used to provide a “black box” picture from the
view of a load. The load, looking back in to
the circuit, only wants to know the voltage and
current that is provided to it.
Finding a TEC
 Steps:
 Find the open circuit voltage – disconnect the load
from the circuit and calculate the voltage looking in to
the circuit.
 Find the open circuit equivalent resistance looking
back in to the circuit
 Remove all independent current sources
 Replace all independent voltage sources with wires.
 Rth is then that equivalent resistance and Vth is just
the voltage that you found.
TEC Example - 1
Find the Thevenin Equivalent Circuit:
a
b
TEC Example - 2
Find the Thevenin Equivalent Circuit:
a
b
Norton Equivalent Circuits
 Consists of a current source with a resistor




in parallel.
Electrically equivalent to the Thevenin
model
Rth is the same
In is equal to Vth / Rth
When finding Norton equivalents, I often
recommend just finding the Thevenin
equivalent and then just switch at the end.
Norton Example
Find the Norton Equiv. Circuit
Source Transformations
 Like the Wye-Delta transformation, we can transform a voltage
source with a resistor in series into a current source with a resistor in
parallel without changing the rest of the circuit and vice versa.
 Like superposition, however, this is often more work than just
using mesh currents to solve the problem.
Source Transformation Exp.
Find i0 and the voltage over the 3 ohm resistor
using source transformations.
i0
+
V
-
Maximum Power Transfer
 When does the load receive maximum
power? – see notes
 When RL = Rth
Maximum Power Example
Find the RL that achieves maximum power
transfer. Find the power it absorbs. Note:
You must find Vth to calculate the power.