thevenin - norton equivalents and maximum power transfer
DESCRIPTION
Lec #Circuit analysisTRANSCRIPT
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Circuit Theorems:Thevenin and Norton Equivalents,
Maximum Power Transfer
Dr. Mustafa Kemal Uygurolu
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Thevenins Theorem
z 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.
z Thevenins theorem implies that we can replace arbitrarily complicated networks with simple networks for purposes of analysis.
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Independent Sources (Thevenin)
Circuit with independent sources
RTh
VTh
Thevenin equivalent circuit
+
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No Independent Sources
Circuit without independent sources
RTh
Thevenin equivalent circuit
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Thevenin Equivalent Circuit
z Basic steps to determining Theveninequivalent are Find vTh RTh
VTh+
i = 0 i = 0
+
Voc-
+
Voc-
Linear
Two-terminal circuit
Voc = VTh
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Thevenin Equivalent Circuit
- 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).
RThLinear circuit with all independent sources set equal to zero.
Rin = RTh
RinRin
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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
RThLinear circuit with dependent sources
Itest
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Vtest
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Linear circuit with dependent sources
Vtest+
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Itest
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Thevenin Equivalent Circuit
(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
+
Linear
Two-terminal circuit
i = isc i = iscVTh
isc = VTh / RTh
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Example
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Example cont.
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Example cont.
Ref.
5 3 5.52 2oc oc
ocV V V V+ = =KCL at Voc :
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Example cont.
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Example cont.
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z Problem: for the following circuit , determine the Thevenin equivalent circuit.
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z Solution:z Step 1: In this circuit, we have a dependent
source. Hence, we start by finding the open circuit voltage Voc = Vab.
z KCL at node C z (5 - Voc)/2 + Voc/4 = 0 z Voc = 10 V
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z Step 2: We obtain the short circuit current Isc by shorting nodes a-b and finding the current through it.
z
z 5 = 2 Isc + 3 Isc => Isc = 5/5 z Isc = 1 A
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z Step 3: Find the equivalent Thevenin Voltage and Resistance
z Vth = Voc = Vab = 10V z Rth = Voc/Isc => Rth = 10/1 z Vth = 10Vz Rth = 10
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Norton Equivalent Circuit
z Any Thevenin equivalent circuit is in turn equivalent to a current source in parallel with a resistor [source transformation].
z A current source in parallel with a resistor is called a Norton equivalent circuit.
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Norton Equivalent Circuit
z Finding a Norton equivalent circuit requires essentially the same process as finding a Thevenin equivalent circuit.
VTh
RTh
IN RN
Th N NV R I= ThNTh
VIR
=
+
Th NR R=
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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.
If there is not any independent source then both VOC=0 and ISC=0 [so skip step 2]
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Thevenin/Norton Analysis
3. Calculate RTh(RN) = Voc / Isc4. Thevenin: Replace circuit with VOC in series with RTh
Norton: Replace circuit with ISC in parallel with RTh
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!
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Maximum Power Transfer
I
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Maximum Power Transfer
Power delivered to the load as a function of RL.
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Maximum Power Transfer
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Maximum Power Transfer
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Example
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Example cont.
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Example
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Example cont.
17 6 5 172 2 33 2 6 3
6 32 64 12.85 3 5
Th ThTh
Th
V V V
V V
+ ++ = = = = =
KCL at the supernode:
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Example cont.
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Example cont.
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Example cont.
Circuit Theorems:Thevenin and Norton Equivalents, Maximum Power TransferThevenins TheoremIndependent Sources (Thevenin)No Independent SourcesThevenin Equivalent CircuitThevenin Equivalent CircuitThevenin Equivalent CircuitThevenin Equivalent CircuitExampleExample cont.Example cont.Example cont.Example cont.Norton Equivalent CircuitNorton Equivalent CircuitThevenin/Norton AnalysisThevenin/Norton AnalysisMaximum Power TransferMaximum Power TransferMaximum Power TransferMaximum Power TransferExampleExample cont.ExampleExample cont.Example cont.Example cont.Example cont.