Circuit Theorems:
Thevenin and Norton Equivalents,
Maximum Power Transfer
Dr. Mustafa Kemal Uyguroğlu
Independent Sources (Thevenin)
Thevenin’s Theorem
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Any circuit with sources (dependent and/or
independent) and resistors can be replaced
by an equivalent circuit containing a single
voltage source and a single resistor.
Thevenin’s theorem implies that we can
replace arbitrarily complicated networks with
simple networks for purposes of analysis.
No Independent Sources
RTh
VTh
Circuit with
independent sources
+
–
Thevenin equivalent
circuit
RTh
Circuit without
independent sources
Thevenin equivalent
circuit
Thevenin Equivalent Circuit
Thevenin Equivalent Circuit
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Basic steps to determining Thevenin
equivalent are
–
Find vTh
i=0
RTh
+
Linear
Voc
Two-terminal
circuit
VTh
-
- Compute the Thevenin equivalent resistance, RTh
(a) If there are only independent sources, then short
circuit all the voltage sources and open circuit the current
sources (just like superposition).
i=0
RTh
+
+
–
Voc
-
Linear circuit with
all independent
sources set equal
to zero.
Rin
Rin
Voc = VTh
Rin = RTh
Thevenin Equivalent Circuit
Thevenin Equivalent Circuit
(b) If there are only dependent sources, then must use a test
voltage or current source in order to calculate
RTh = VTest/Itest
(c) If there are both independent and dependent sources, then
compute
(i) RTh = VTest/Itest (all independent sources set equal to zero)
(ii) compute RTh from VOC/ISC.
RTh
RTh
+
Linear circuit with
dependent sources
Itest
Vtest
Two-terminal
circuit
-
Itest
Linear circuit with
dependent sources
Linear
+
-
Vtest
i = isc
VTh
+
–
i = isc
isc = VTh / RTh
Example
Example cont.
Example cont.
Example cont.
Ref.
KCL at Voc :
Voc Voc − 5
+
= 3 ⇒ Voc = 5.5V
2
2
Example cont.
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Solution:
Step 1: In this circuit, we have a dependent
source. Hence, we start by finding the open
circuit voltage Voc = Vab.
KCL at node C
(5 - Voc)/2 + Voc/4 = 0
Voc = 10 V
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Problem: for the following circuit , determine the
Thevenin equivalent circuit.
Step 2: We obtain the short circuit current Isc by
shorting nodes a-b and finding the current through
it.
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5 = 2 Isc + 3 Isc
Isc = 1 A
=>
Isc = 5/5
Norton Equivalent Circuit
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Step 3: Find the equivalent Thevenin Voltage
and Resistance
Vth = Voc = Vab = 10V
Rth = Voc/Isc
=>
Rth = 10/1 Ω
Vth = 10V
Rth = 10 Ω
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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.
Thevenin/Norton Analysis
Norton Equivalent Circuit
RTh
VTh
+
–
IN
VTh = RN I N
z
IN =
VTh
RTh
RN
RTh = RN
Finding a Norton equivalent circuit requires essentially the
same process as finding a Thevenin equivalent circuit.
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.
If there is not any independent source then both VOC=0 and
ISC=0 [so skip step 2]
Thevenin/Norton Analysis
Maximum Power Transfer
3. Calculate RTh(RN) = Voc / Isc
4. Thevenin: Replace circuit with VOC in series with RTh
Norton: Replace circuit with ISC in parallel with RTh
I
Note: for circuits containing no independent sources the
equivalent network is merely RTh , that is, no voltage (or
current) source.
Only steps 2 & 4 differ from Thevenin & Norton!
Maximum Power Transfer
Power delivered to the load as a function of RL.
Maximum Power Transfer
Maximum Power Transfer
Example
Example cont.
Example
Example cont.
KCL at the supernode:
Example cont.
Example cont.
VTh + 17 VTh + 6
5
17
+
= −2 ⇒ VTh = −2 − − 3
3
2
6
3
6  −32 
64
VTh = 
= −12.8V
=−
5 3 
5
Example cont.