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Chapter 08

Methods of Analysis

Source: Circuit Analysis: Theory and Practice Delmar Cengage Learning

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Source Conversion

Mesh Analysis

Nodal Analysis

Delta-Wye (-Y) Conversion

Bridge Networks

Outline

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Linear and Nonlinear V-I Curves

Ohm’ Law I = V / R R is fixed

I V / R R may be thermistor or photocell

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Constant Current Sources

Maintains same current in branch of circuit

Regardless of how components are connected external to the source

Direction of current source indicates direction of current flow in branch

For example:

Calculate the voltage Vs across current source I if the resistor is 100 Ω

Vs = I *R

= 2 * 100

= 200 V

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Example: Constant Current Sources

Determine VS

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Example: Constant Current Sources

Determine the voltage VS and currents I1 and I2

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Source Conversions

Ideal current source I

Infinite shunt (parallel) resistance Rs = ∞

Real current source I

Some shunt (parallel) resistance Rs

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Source Conversions

If internal resistance of a source is considered:

Voltage source may be converted to current source

Calculate current from E/RS , RS does not change, and place current source and resistor in parallel

Current source may be converted to voltage source

E = I RS and place voltage source in series with resistor

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Current and Voltage Sources Exchange

A load connected to a voltage source or its equivalent current Should have same voltage and current for either

source

Although sources are equivalent Currents and voltages within sources may differ

Sources are only equivalent external to terminals

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Determine IL

Voltage Source Current Source

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Determine IL

Current Source Voltage Source

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Current Sources in Parallel and Series

Current sources in parallel

Simply add together algebraically

Add magnitude currents in one direction

Subtract magnitude currents in opposite direction

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Noted: Current Sources in Parallel and Series

Current sources with different values

Never place in series and This violates KCL

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Example1: Current Sources in Parallel and Series

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Example2: Current Sources in Parallel and Series

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Branch Current Analysis

Used for circuits having more than one source

Use different methods of analysis

Begin by arbitrarily assigning current directions in each branch

Label polarities of the voltage drops across all resistors

Step0: Assume all the current I1, I2, …

Step1: Write KVL around all loops

Step2: Apply KCL at enough nodes

so all branches have been included

Step3: Solve resulting equations

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Example1: Branch Current Analysis

From KVL:

6 - 2I1 + 2I2 - 4 = 0

4 - 2I2 - 4I3 + 2 = 0

From KCL:

I3 = I1 + I2

Solve simultaneous equations

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Example2: Branch Current Analysis

Loop badb: - 2I2 + 3I3 - 8 = 0

Loop bacb:

- 2I2 + I4 - 6 = 0

Node a: I3 + I4 = 5 + I2

Solve simultaneous

equations

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Ex0 using Source Conversions

R1

2Ω

V1

6 V

R3

4Ω

R2

2Ω

V2

4 V

V3

2 V

U1

DC 10MOhm

3.6 V

+

-

R4

2Ω

R5

4Ω

R6

2ΩI1

3 A

I2

2 A

I3

0.5 A

U2

DC 10MOhm

3.6 V

+

-

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Mesh Analysis

Step0: Arbitrarily assign a clockwise current

to each interior closed loop (Mesh)

Step1: Indicate voltage polarities across

all resistors

Step2: Write KVL equations

Step3: Solve resulting simultaneous

equations

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Example1: Mesh Analysis

Assign loop currents and voltage polarities

Using KVL: 6 - 2I1 - 2I1 + 2I2 - 4 = 0

4 - 2I2 + 2I1 - 4I2 + 2 = 0

Simplify and solve equations

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Example2: Mesh Analysis

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Example3: Mesh Analysis

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Example4: Mesh Analysis

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Example5: Mesh Analysis

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Nodal Analysis

Step0: Assign a reference node within circuit and indicate node as ground

Convert voltage sources to current sources

Arbitrarily assign a current direction to each branch where there is no current source

Step1: Assign voltages V1, V2, etc. to

remaining nodes

Step2: Apply KCL to all nodes except

reference node

Rewrite each current in terms of voltage

Step3: Solve resulting equations for voltages

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Example0: Nodal Analysis

Assign voltage at node v1, then using KVL

(V1-6)/2 + (V1-4)/2 + (V1-(-2))/4= 0

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Example1: Nodal Analysis

Using KCL for nodes V1 and V2

200mA+50mA = I1+I2

200mA+I2 = 50mA+I3

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Example2: Nodal Analysis

Using KCL for nodes V1 and V2

I1+I2 = 2A

3A+I2 = I3+I4

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Example3: Nodal Analysis

Using KCL for nodes V1 and V2

V1/3+(V1-V2)/5+6 = 1

V2/4+(V2-V1)/5+2+1 = 0

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Example4: Nodal Analysis

Using KCL for nodes V1 and V2

V1/5K+V1/3K+(V1-V2)/4K+3mA = 2mA

V2/2K+(V2-V1)/4K = 2mA

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Example5: Nodal Analysis

Determine voltages V1 and V2

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Delta-Wye (-Y) Conversion

Resistors connected to a point of Y

Obtained by finding product of resistors connected to same point in Delta

Divided by sum of all Delta resistors

R1=(RC*RB) / (RA+RB+RC)

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Example: Y Conversion

Given a Delta circuit with resistors of 30, 60, and 90

Resulting Y circuit will have resistors of 10, 15, and 30

R1=(30*60) / (30+60+90) = 10

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Wye-Delta Conversions

A Delta resistor is found: Taking sum of all two-product combinations of Y

resistor values

Divided by resistance of Y directly opposite resistor being calculated

RA=(R1R2+R2R3+R1R3) /R1

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Example1: Y Conversions

For a Y circuit having resistances of 2.4, 3.6, and 4.8 K

Resulting Delta resistors will be 7.8, 10.4, and 15.6 K

RA=(3.6K*2.4K+2.4K*4.8K+4.8K*3.6K) /4.8K = 7.8K

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Example2: Y- Conversions

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Bridge Networks

Three same equivalent bridge networks

Balanced bridge: R1R4 = R2R3 and IR5=0

Unbalanced bridge: R1R4 R2R3 and IR50

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Examples: Bridge Networks

Balanced bridge:

30*240 = 60*120

R1R4 = R2R3 and IR5=0

Unbalanced bridge:

20*80 40*60

R1R4 R2R3 and IR50

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Example: Bridge Networks

Balanced bridge:

3*24 = 6*12

R1R4 = R2R3 and IR5=0

R5 can be replaced with an open circuit or

a short circuit.

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Example: Bridge Networks

Unbalanced bridge:

6*3 12*3 R1R4 R2R3 and IR50

Mathod1：Using mesh analysis with KVL

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Example: Bridge Networks

Unbalanced bridge:

6*3 12*3 R1R4 R2R3 and IR50

Mathod2： Using node analysis with KCL

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Example: Bridge Networks

Unbalanced bridge:

6*3 12*3

Mathod3： Using Y

conversion

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Kernel abilities

1. Can use Mesh Analysis for solving the unknown voltage and current of a circuit.

2. Can use Nodal Analysis for solving the unknown voltage and current of a circuit.

3. Can use Delta-Wye (-Y) Conversion for solving the unknown voltage and current of a circuit.

4. Can recognize a Bridge circuit whether is balance or unbalance and solve the unknown voltage and current.

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Problem 14

Determine the voltage Vab

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Problem 21

Determine the current I2

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Problem 47

Determine the current I