Chapter 08
Methods of Analysis
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Tsai
Source:
Circuit Analysis: Theory and Practice Delmar Cengage Learning
Outline

Source Conversion

Mesh Analysis

Nodal Analysis

Delta-Wye (-Y) Conversion

Bridge Networks
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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


Although sources are equivalent


Should have same voltage and current for either
source
Currents and voltages within sources may differ
Sources are only equivalent external to
terminals
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Voltage Source  Current Source
Determine IL
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Current Source  Voltage Source
Determine IL
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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
R3
2Ω
4Ω
R2
2Ω
V1
6V
I1
3A
R4
2Ω
V3
2V
U1
+
-
3.6
V2
4V
I2
2A
V
DC 10MOhm
R6
2Ω
I3
0.5 A
R5
4Ω
U2
+
-
3.6
V
DC 10MOhm
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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 IR50
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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 IR50
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Example: Bridge Networks
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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 IR50
Mathod1：Using mesh analysis with KVL

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Example: Bridge Networks
Unbalanced bridge:
6*3  12*3  R1R4  R2R3 and IR50
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
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