Circuit Theorems Part I

1
Tutorial 4
Circuit Theorems Part I
2
Agenda

• Linearity
• Superposition
• Source Transformation

3
Linearity
1. Linearity is the property of an element
describing a linear relationship between cause
(input) and effect (output). It is a combination
of scaling and additivity property.
2. Scaling Property If the input is multiplied
by a constant, then the output is multiplied by
the same constant.
v iR
V kv kiR
4
Linearity
3. Additivity property The output of a sum of
inputs is the sum of the output to each input
applied separately.
V1 i1R V2 i2R
V i1R i2R V1 V2
5
Linearity
12O

10V
5O
8O
Vo
-
Assume Vo 1V and use linearity to calculate the
actual value of Vo.
Answer 4V
6
Linearity

• Assume Vo 1V
• I8ohm 1/8 0.125A
• ? Vsource (0.125)(128) 2.5V
• However, actual Vsource 10V, by linearity
  principle (scaling property)
• Vo actual 10/2.51 4V

7
Superposition

• Superposition principle states that for a circuit
  having multiple independent sources, the voltage
  across (or current through) an element is equal
  to the algebraic sum of all the individual
  voltages (or currents) due to each independent
  source acting alone.

• 2. Steps to apply superposition principle
• Turn off all independent sources except one
  source. Find the output (voltage or current) due
  to the active source using nodal or mesh
  analysis.
• Repeat step (a) for each of the other independent
  sources.
• Find the total contribution by adding
  algebraically all the contributions due to the
  independent sources.

8
Superposition
Answer 13.33A
Use superposition to find ix.
Note We treat current source as open circuit (I
0) and voltage source as short circuit (V 0).
Dependent sources cannot be eliminated and should
be treated as resistors only.
9
Superposition

• Step 1 Consider the 6A source only

By Mesh Analysis 2i1 8i2 4ix1 0 (1) i2
i1 6 (2) i1 -ix1 (3) Solving (1) (3),
ix1 8A
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Superposition

• Step 2 Consider the 4A source only

By Mesh Analysis 2i1 8i2 4ix2 0 (1) i2
i1 4 (2) i1 -ix2 (3) Solving (1) (3),
ix2 5.33A
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Superposition

• Step 3 Add the contribution from the sources
  together
• ix ix1 ix2
• 85.33
• 13.33A

12
Source Transformation

• A source transformation is the process of
  replacing a voltage source Vs in
• series with a resistor R by a current source
  Is in series with a resistor R,
• or vice versa.

a
a
R
Vs
R
Is
b
b
Vs IsR or Is Vs/R
Note These two circuits are equivalent to the
circuits that connected between node a and b
13
Source Transformation
2. Source transformation is also applicable to
dependent sources, provided that we
carefully handle the dependent variables. 3. The
arrow of a current source should be consistent
with the positive terminal of an equivalent
voltage source and vice versa. The source
transformation is not valid when R0.
14
Source Transformation

• Question Consider the transform pair

• What are IRL and VRL before and after the
  transformation?
• What are IRS and VRS before and after the source
  transformation?
• Be careful IRS and VRS changes after the
  transformation, but for the load
• connected to A and B, the transform pair is
  equivalent

15
Source Transformation

• Problem 4.25 Determine vx using source
  transformation

Answer 3.652V Note Be careful on the
definition of vx when applying source
transformation