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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