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Norton's Theorem (4.3, 7.8)
Dr. S. M. Goodnick
October 22, 2003
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Introduction
• Any Thevenin equivalent circuit is in turn
equivalent to a current source in parallel
with a resistor [source transformation].
• A current source in parallel with a resistor is
called a Norton equivalent circuit.
• Finding a Norton equivalent circuit requires
essentially the same process as finding a
Thevenin equivalent circuit.
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Independent Sources
Isc
Circuit with one or
more independent
sources
RTh
Norton equivalent
circuit
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No Independent Sources
RTh
Circuit without
independent sources
Norton equivalent
circuit
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Finding the Norton Equivalent
• Circuits with independent sources:
– Find Voc and Isc
– Compute RTh
• Circuits without independent sources:
– Apply a test voltage (current) source
– Find resulting current (voltage)
– Compute RTh
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Example: Strain Gauge
• Strain is the amount of deformation of a
body due to an applied force-it is defined as
the fractional change in length.
• Strain can be positive (tensile) or negative
(compressive).
• One type of strain gauge is made of a foil
grid on a thin backing.
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A Strain Gauge
Backing
Foil
• The strain gauge’s resistance varies as a
function of the strain:
DR = GF e R
• e is the strain, R is the nominal resistance,
GF is the Gauge Factor
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Typical values
• Measured strain values are typically fairly
small-usually less than 10-3.
• GF is usually close to 2.
• Typical values for R are 120W, 350W, and
1000W.
• A typical change in resistance is
DR = 2•10-3•120W = 0.24W
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Measuring Small Changes in R
• To measure such small changes in
resistance, the strain gauge is placed in a
Wheatstone bridge circuit.
• The bridge circuit uses an excitation voltage
source and produces a voltage that depends
on DR.
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The Bridge Circuit
R+DR
R
Vex
–V
+
–
+
out
R
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Norton Equivalent for Any e
Vex 2e
I sc 
R 4  6e
R 4  6e
RTh 
4 1  2e
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Thevenin/Norton Analysis
1. Pick a good breaking point in the circuit (cannot split a
dependent source and its control variable).
2. Thevenin: Compute the open circuit voltage, VOC.
Norton: Compute the short circuit current, ISC.
For case 3(b) both VOC=0 and ISC=0 [so skip step 2]
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Thevenin/Norton Analysis
3. Compute the Thevenin equivalent resistance, RTh (or
impedance, ZTh).
(a) If there are only independent sources, then short
circuit all the voltage sources and open circuit the current
sources (just like superposition).
(b) If there are only dependent sources, then must use a
test voltage or current source in order to calculate
RTh (or ZTh) = VTest/Itest
(c) If there are both independent and dependent sources,
then compute RTh (or ZTh) from VOC/ISC.
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Thevenin/Norton Analysis
4. Thevenin: Replace circuit with VOC in series with RTh, ZTh.
Norton: Replace circuit with ISC in parallel with RTh, ZTh.
Note: for 3(b) the equivalent network is merely RTh (or ZTh),
that is, no voltage (or current) source.
Only steps 2 & 4 differ from Thevenin & Norton!
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Class Examples
• Learning Extension E4.4
• Learning Extension E7.15(c)
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