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Module 9 - Thévenin and Norton
Equivalent Circuits
In this module, we’ll learn about an important property
of resistive circuits called Thévenin Equivalence.
M. Leon Thévenin (1857-1926), published
his famous theorem in 1883.
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Thévenin’s Theorem applies to circuits containing
resistors, voltage sources, and/or current sources
Thêvenin Equivalent Circuit
2
Thévenin’s Theorem: A resistive circuit can be represented
by one voltage source and one resistor:
RTh
VTh
Resistive Circuit
Thévenin Equivalent Circuit
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Definition of a “Port”
Port: Set of any two terminals
PORT
Resistive Circuit
PORT
4
Illustrate concept with a simple resistive circuit:
• Any two terminals can be designated as a port.
• Our objective: Find the equivalent circuit seen looking
into the port
iX
R1
+
Vo
R2
vX
_
Simple resistive circuit
Define port
variables vX and iX
ix flows to some load
(not shown)
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Find an equation that relates vx to ix
i1
Vo
R1
iX
R2
i2
KVL:
i1R1 + i2R2 = Vo
KCL:
i1 = i2 + iX
+
v_X
(Each resistor voltage expressed
using Ohm’s Law)
Also note: vX = i2R2
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Solve these equations for vX versus iX :
i1R1 + i2R2 = Vo
i1 = i X + i2
vX = i2R2
(iX + i2) R1 + i2R2 = Vo
(iX + vX/R2) R1 + vX = Vo
Rearrange the variables…
iX R1 + vX (R1 /R2 + 1) = Vo
or
Vo – iX R1
vX = –––––––––
1 + R1 /R2
vX = Vo
R2
––––––– –
R1 + R2
R1 R2
iX –––––––
R1 + R2
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Examine this last equation:
R1
Vo
iX
R2
vX = Vo
R2
––––––– –
R1 + R2
+
v_X
R1 R2
iX –––––––
R1 + R2
It has the form vX = VTh – iX RTh
R2
VTh = Vo –––––––
R1/ + R2
R1 R2
RTh = –––––––
R1 + R2
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Constructing the Thévenin Equivalent Circuit
RTh
+i R –
X Th
VTh
iX
+
v_X
Write down KVL for this circuit:
vX = VTh – iXRTh
“Output voltage = voltage source – voltage drop across RTh”
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Actual Circuit:
Model:
R1
Vo
+
R2
vX = Vo
RTh
iX
R2
––––––– –
R1 + R2
iX
+
v_X
VTh
R1 R2
iX –––––––
R1 + R2
v_X
vX = VTh – iXRTh
Choose model parameters VTh and RTh:
R2
VTh = Vo –––––––
R1 + R2
and
RTh
R1 R2
= ––––––= R1 || R2
R1 + R2
• From the point of view of vX and iX, the Thévenin circuit
models the actual circuit in every way.
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Actual Circuit:
iX
R1
+
Vo
R2
vX
_
PORT
Thévenin Equivalent:
R1 || R2
R2
Vo –––––––
R1 + R2
iX
+
vX
_
PORT
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Significance of RTh
R1
Vo
R2
Equivalent resistance
• Set all independent sources in the actual circuit to zero.
• For a voltage source, that means substituting a short circuit.
• Equivalent resistance RTh= R1||R2
R
is the equivalent resistance seen looking into the port
with all independent sources set to zero.
Th
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Setting a Voltage Source to Zero
Current determined by
what’s connected…
Voltage between
nodes fixed at Vo
V
o
13
Setting a Voltage Source to Zero
Voltage between
nodes fixed at 0 V
by short circuit
LOAD
14
Setting a Current Source to Zero
Current through
branch set to Io
x
Voltage between
nodes determined by Ioopen circuit
what’s connected
x
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Significance of VTh
R1
iX = 0
+
Vo
R2
_
Open Circuit Voltage
• Connect nothing to the port
• iX automatically set to zero.
• Port voltage is called the open circuit voltage.
RTh
iX = 0
+0V–
VTh
KVL
+
Open Circuit Voltage
_
• VTh represents the open circuit voltage of the actual circuit
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Example: Resistor Network
Balanced audio microphone system
50 k = Input resistance of typical audio amplifier.
What voltage is developed across a 50 k resistive load?
R1=100 k
Vmic
R3 = 10 k
+
R2 = 30 k
vLOAD
–
10 mV
50 k
R4 =10 k
Load
Microphone network
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Solution Method: Find the Thévenin Equivalent of the Microphone Network
• Disconnect the load.
• Find Thevenin Equivalent remaining circuit.
• Reconnect the load.
• Find vLOAD from simplified circuit.
R1=100 k
Vmic
R3 = 10 k
R2 = 30 k
10 mV
R4 =10 k
Find VTh
and RTh
Load
Load
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Step 1: Find the Equivalent Resistance
• Set the voltage source to zero. (Substitute a short circuit.)
• Find the equivalent resistance RTh
• RTh = R3 + R1||R2 + R4 = 10 k + 23 k + 10 k = 43 k
R1=100 k
Vmic
R3 = 10 k
R2 = 30 k
RTh
RTh
43 k
R4 =10 k
Note: R1||R2 = (100 k)||(30 k) = 23 k
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Step 2: Find the Open Circuit Voltage
• Analyze the circuit under no-load conditions.
• Voltage across port terminals will be VTh
• From KVL around the inner loop*: v2 = VmicR2/(R1 + R2) = 2.3 mV
*basically, voltage division
R1=100 k
R3 = 10 k
+
Vmic
R2 = 30 k
VTh = 2.3 mV
–
10 mV
R4 =10 k
• Note that no current flows through R3 and R4.
 Voltage across these resistors is zero.
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Step 3: Reconnect the Load to the
Thévenin Equivalent Model
RTh = 43 k
+
VTh
RLOAD
2.3 mV
vLOAD
–
50 k
Thévenin equivalent of
microphone network
From simple voltage division:
vLOAD = VTh (RLOAD/(RLOAD + RTh)
= 2.3 mV  (50 k)/(93 k) = 0.54 mV
Answer
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More Examples:
The Norton Equivalent Circuit
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Short Circuit Current
Another important parameter of a circuit is its short circuit current
The short circuit current of a port is defined as the current that
will flow if:
 The load is disconnected
 A short circuit is connected instead
RTh
VTh
Isc = VTh /RTh
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Circuit Containing a Current Source
Consider the following simple circuit:
I1
R1
Port
What is the Thévenin equivalent circuit seen looking into
the port?
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Step 1: Find the open circuit voltage:
Current is zero

+
I1
–
R1
+
VTh
–
• Open circuit conditions  All of I1 flows through R1
• Voltage develops across R1 with polarity shown.
• From Ohm’s Law:
VTh = I1R1
(That part is simple…)
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Step 2: Find the equivalent resistance
• Set the current source to zero.
I1
R1
RTh
• Set the current source to zero  open circuit
• Find the resistance looking into the port.
• Trivially, by inspection: RTh = R1
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The Thévenin Equivalent Circuit:
R1
I1
I1R1
R1
Thévenin Equivalent
Actual Circuit:
RTh = R1
VTh = I1R1
Done!
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Norton Equivalent Circuit
RN
IN
RN
Norton Circuit.
INRN
Thévenin Circuit.
• The Norton and Thévenin equivalents of a circuit are interchangeable.
• The equivalent resistance is the same: RN = RTh
• The open circuit voltage is the same: VTh = INRN
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What about the short-circuit current from a Norton Circuit?
• Apply a short circuit:
+
IN
RN VN = 0
–
Isc = IN
• The voltage across the Norton resistance becomes zero.
• No current flows through the Norton resistance (I = V/R).
• All the current flows through the short circuit.
• The short circuit current is the source current IN.
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Norton Equivalent Circuit
IN
IN =
VTh/RTh
RN = RTh
Norton Circuit
VTh =
INRN
IN
RTh = RN
Thévenin Circuit
• The short circuit current is the same in each circuit:
IN = VTh/RTh
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Example: Resistive Network
Find the Norton Equivalent of the following circuit using the
short-circuit current method
R1=100 k
Vmic
R3 = 10 k
R2 = 30 k
RTh or RN
10 mV
R4 =10 k
Step 1: Find RTh (same as RN) by setting the source to zero.
By inspection,
RTh = R3 + R1||R2 + R4 = 10 k + 23 k + 10 k = 43 k
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Step 2: Apply a short circuit to the port and compute the
short-circuit current.
R1=100 k
Vmic
R3 = 10 k
R2 = 30 k
ISC = 0.54 A
10 mV
R4 =10 k
R1=100 k
Vmic
IP = Vmic/(R1 + RP) = 0.9
A
RP = R2 || (R3 + R4) = 12 k
10 mV
From current division:
R2
30 k
ISC = IP
= 0.9 A
= 0.54 A = ISC
50
k
[R2 + (R3 + R4)]
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Find the Norton Equivalent of the Circuit
ISC = 0.54 A
RN = 43 k
+
IN = 0.54 A
RN = 43 k vOC = 23 mV
–
“Open Circuit Voltage”
vOC = IN RN = (0.54 A)(43 k) = 23 mV
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Construct the Thévenin Equivalent of the Circuit
ISC = 0.54 A
RTh = 43 k
VTh = ISC RTh = (0.54 A )(43 k) = 23 mV
RTh = 43 k
VTh = 23 mV
This result is the same one obtained in the previous example!
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A circuit that can be represented by a Thévenin Equivalent
can also be represented by its corresponding Norton circuit
RTh
IN
RN
VTh
Norton Equivalent
Thévenin Equivalent
VTh= INRN
RTh = RN
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End of This Module
Do the Homework Exercises
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