ECA1212Introduction to Electrical &
Electronics EngineeringChapter 2: Circuit Analysis Techniques
by Muhazam Mustapha, September 2011
Learning Outcome
• Understand and perform calculation on circuits with mesh and nodal analysis techniques
• Be able to transform circuits based on Thevenin’s or Norton’s Theorem as necessary
By the end of this chapter students are expected to:
Chapter Content
• Mesh Analysis
• Nodal Analysis
• Source Conversion
• Thevenin’s Theorem
• Norton’s Theorem
Mash AnalysisMesh
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Mesh Analysis
• Assign a distinct current in clockwise direction to each independent closed loop of network.
• Indicate the polarities of the resistors depending on individual loop.
• [*] If there is any current source in the loop path, replace it with open circuit – apply KVL in the next step to the resulting bigger loop. Use back the current source when solving for current.
Steps:
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Mesh Analysis
• Apply KVL on each loop:– Current will be the total of all directions– Polarity of the sources is maintained
• Solve the simultaneous equations.
Steps: (cont)
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Mesh AnalysisExample: [Boylestad 10th Ed. E.g. 8.11 - modified]
R1
Ia Ib2V
2Ω
R2
1Ω
6V
R3 4Ω
a bI1
I3
I2
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Mesh Analysis
Example: (cont)
Loop a: 2 = 2Ia+4(Ia−Ib) = 6Ia−4Ib
Loop b: −6 = 4(Ib−Ia)+Ib = −4Ia+5Ib
After solving: Ia = −1A, Ib = −2A
Hence: I1 = 1A, I2 = −2A, I3 = 1A
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Noodle AnalysisNodal
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Nodal Analysis
• Determine the number of nodes.• Pick a reference node then label the rest with
subscripts.• [*] If there is any voltage source in the branch,
replace it with short circuit – apply KCL in the next step to the resulting bigger node.
• Apply KCL on each node except the reference.• Solve the simultaneous equations.
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Nodal AnalysisExample: [Boylestad 10th Ed. E.g. 8.21 - modified]
R14A 2Ω
R2
6Ω 2AR3
12ΩI1
I2
I3
a b
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Nodal Analysis
Node a:
Node b:
After solving: Va = 6V, Vb = − 6A
Hence: I1 = 3A, I2 = 1A, I3 = −1A
Example: (cont)
487122
4
babaa VV
VVV
243126
2
babab VV
VVV
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Mesh vs Nodal Analysis
• Mesh: Start with KVL, get a system of simultaneous equations in term of current.
• Nodal: Start with KCL, get a system of simultaneous equations on term of voltage.
• Mesh: KVL is applied based on a fixed loop current.
• Nodal: KCL is applied based on a fixed node voltage.
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Mesh vs Nodal Analysis
• Mesh: Current source is an open circuit and it merges loops.
• Nodal: Voltage source is a short circuit and it merges nodes.
• Mesh: More popular as voltage sources do exist physically.
• Nodal: Less popular as current sources do not exist physically except in models of electronics circuits.
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Thevenin’s Theorem
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Thevenin’s TheoremStatement:
Network behind any two terminals of linear DC circuit can be replaced by an
equivalent voltage source and an equivalent series resistor
• Can be used to reduce a complicated network to a combination of voltage source and a series resistor
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• Calculate the Thevenin’s resistance, RTh, by switching off all power sources and finding the resulting resistance through the two terminals:– Voltage source: remove it and replace with short
circuit– Current source: remove it and replace with open
circuit
• Calculate the Thevenin’s voltage, VTh, by switching back on all powers and calculate the open circuit voltage between the terminals.
Thevenin’s Theorem
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Thevenin’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω9V
Convert the following network into its Thevenin’s equivalent:
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Thevenin’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω
RTh calculation:
263ThR
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Thevenin’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω9V
VTh calculation:
V6963
6
ThV
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Thevenin’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
2Ω
6V
Thevenin’s equivalence:
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Norton’s Theorem
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Norton’s TheoremStatement:
Network behind any two terminals of linear DC circuit can be replaced by an
equivalent current source and an equivalent parallel resistor
• Can be used to reduce a complicated network to a combination of current source and a parallel resistor
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• Calculate the Norton’s resistance, RN, by switching off all power sources and finding the resulting resistance through the two terminals:– Voltage source: remove it and replace with short
circuit– Current source: remove it and replace with open
circuit
• Calculate the Norton’s voltage, IN, by switching back on all powers and calculate the short circuit current between the terminals.
Norton’s Theorem
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Norton’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω9V
Convert the following network into its Norton’s equivalent:
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Norton’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω
RN calculation:
ThN RR 263
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Norton’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
3Ω
6Ω9V
IN calculation:
A33
9NI
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Norton’s TheoremExample: [Boylestad 10th Ed. E.g. 9.6 - modified]
2Ω3A
Norton’s equivalence:
OR,
We can just take the Thevenin’s equivalent and calculate the short circuit current.
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Maximum Power Consumption
An element is consuming the maximum power out of a network if its resistance is equal to the
Thevenin’s or Norton’s resistance.
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Source ConversionUse the relationship between Thevenin’s and Norton’s source to convert between voltage and current sources.
2Ω3A
2Ω
6V
V = IR
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