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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