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Transcript
```FUNDAMENTALS OF ELECTRICAL
ENGINEERING
[ ENT 163 ]
LECTURE #4
CIRCUIT THEOREMS
HASIMAH ALI
Programme of Mechatronics,
School of Mechatronics Engineering, UniMAP.
Email: [email protected]
CONTENTS







Introduction
Linearity Property
Superposition
Source Transformation
Thevenin’s Theorem
Northon’s Theorem
Maximum Power Transfer
INTRODUCTION
1. Disadvantage of Kirchhoff’s laws – tedious computation for
lage , complex circuit.
2. To handle complexity – Thevenin’s and Northon’s theorems
(applicable to linear circuits)
3. Related concept – superposition, source transformation and
maximum power transfer.
LINEARITY PROPERTY
A linear circuit is one whose output is linearly related (or directly
proportional) to its input.
1. Consist of linear elements, linear dependent sources and
independent sources.
2. Property of linearity:
a) Homogeneity property
•
If input/ excitation is multiplied by a constant, then the output/
response is multiplied by the same constant
•
E. g. from Ohm’s law
•
If multiply by a constant k
v  iR
kv  kiR
LINEARITY PROPERTY
• Response to a sum of inputs is the sum of the responses to each
input applied separately.
• E.g. Voltage-current relationship  v1  i1R and v2  i2 R
• Applying :
(i1  i2 )  v  (i1  i2 ) R  i1R  i2 R  v1  v2
Therefore we can say that a resistor is a linear element (voltage –current
relationship satisfies both properties).
LINEARITY PROPERTY
•
•
•
Linearity principle can be illustrated by considering circuit below. The
linear circuit has no independent sources inside it.
It is excited by a voltage source vs , which serve as the input. The circuit
is terminated by a load R.
Suppose vs= 10V and output i=2A. Linearity principle would gives i=0.2A
when vs=1V and i=1mA when vs=5mV.
LINEARITY PROPERTY
Example: For the circuit in Figure , find I when vs=12V and vs=24V.
SUPERPOSITION
1. Use to determine the value of a specific variable (voltage or current);
based on linearity.
The superposition principle states that the voltage across (or current
through )an element in a linear circuit is the algebraic sum of the voltage
across ( or current through) that element due to each independent source
acting alone.
1. Steps to apply superposition principle:
a) Turn off all independent sources except one source. Find the output
(voltage or current ) due to that active source using previous
techniques.
b) Repeat step(a) for each of the other independent sources.
c) Find the total contribution by adding algebraically all the contributions
due to the independent sources.
Disadvantage – may involve more work.
SUPERPOSITION
Keep in mind:
•
To consider only one independent source, every voltage sources are
replaced by 0 V (short circuit) and current sources by 0 A (open circuit).
•
Dependent sources are left intact because they are controlled by circuit
variables.
SUPERPOSITION
Example: Use the superposition theorem to find v in the circuit in Figure
shown.
Solution
SUPERPOSITION
SUPERPOSITION
Example: For the circuit shown, use the superposition theorem to find i.
Solution:
SUPERPOSITION
SUPERPOSITION
SOURCE TRANSFORMATION
•
Source transformation is another tool for simplifying circuits (like seriesparallel and wye – delta ).
A source transformation is the process of replacing a voltage source
vs in series with a resistor R by a current source is in parallel with a
resistor R, or vice verse.
•
Basic to these tools is the concept of equivalence.
vs  is R
vs
is 
R
SOURCE TRANSFORMATION
•
Source transformation also applies to dependent source.
Note:
1. The arrow of the current source is directed toward the positive terminal of
the voltage source.
2. Source transformation is not possible when R=0 (ideal voltage source) and
R=∞ (ideal current source)
SOURCE TRANSFORMATION
Example: Use source transformation to find vo in the circuit given.
Solution:
1. Transform the current and voltage sources.
SOURCE TRANSFORMATION
2. Combine the 4Ω and 2Ω resistors in series and transform the 12-V voltage
source.
3. Combine the 2-A and 4-A current sources to get 2-A source.
4. Use current division to get i.
THEVENIN’S THEOREM
•
Usage: to avoid analyzing entire circuit for every chnaes in variable
element .
•
Provides a technique by which the fixed part of the circuit is replaced by
an equivalent circuit.
Thevenin’s theorem states that a linear two – terminal circuit can be
replaced by an equivalent circuit consisting of a voltage source VTH in series
with a resistor RTH, VTH is the open circuit voltage at the terminals and RTH is
the input or equivalent resistance at the terminals when the independent
sources are turned off.
THEVENIN’S THEOREM
Let’s consider the figure:
•
We would like to replace the original
linear two-terminal circuit by its
equivalent Thevenin circuit.
•
The circuits are said to be equivalent
if they have same voltage-current
relation at their terminals.
•
If the terminals a-b are made opencircuited (by removing the load), no
current will flows, so that the terminals
a-b must be equal to the voltage
source VTH , since the two circuits
are equivalent, Therefore:
VTh  voc
THEVENIN’S THEOREM
•
With the load disconnected and terminals a-b open-circuited, we turn
off all independent sources.
•
The input resistance of the dead circuit at the terminals a-b must be
equal to RTH since the two circuits are equivalent,
RTh  Rin
THEVENIN’S THEOREM
In finding the Thevenin resistance RTh, we need to consider two cases:
Finding VTh and RTh
•
If the network has no dependent sources, we turn off all independent sources.
RTh is the input resistance of the network looking between terminals a and b.
•
If the network has dependent sources, we turn off all independent sources. We
apply a voltage source vo at the terminals a and b determine the resulting
current io.
THEVENIN’S THEOREM
•Then RTh= vo/io as shown below. Alternatively, we may insert a current
source io at terminals a-b and find the terminal voltage vo. Again RTh= vo/io
(either of the two approaches will give the same result.)
THEVENIN’S THEOREM
•
Important of Thevenin’s theorem- helps simplify a circuit (a large circuit
may be replaced by a single independent voltage source and a single
resistor)
•
The current IL through the load and the voltage VL across the load are
easily determined once the Thevenin equivalent of the circuit at the
V Th
IL 
RTh  RL
VL  RL I L 
RL
VTh
RTh  RL
Note: A negative RTh value shows that the
circuit is supplying power (circuit with
dependent source)
THEVENIN’S THEOREM
Example: Find the Thevenin equivalent circuit shown in Figure below, to the left
of the terminals a-b. Then find the current through RL=6, 16, and 36Ω.
Solution:
THEVENIN’S THEOREM
THEVENIN’S THEOREM
THEVENIN’S THEOREM
Exercise: Using Thevenin’s theorem, find the equivalent circuit to the left of
the terminals in the circuit shown. Then find I.
NORTHON’S THEOREM
Norton’s theorem states that a linear two-terminal circuit can be replaced
by an equivalent circuit consisting of a current source IN in parallel with a
resistor RN, where IN is the short circuit consisting of a current through the
terminals and RN is the input or equivalent resistance at the terminals
when the independent sources are turned off.
We find RN in the same way we find RTh
RN  RTh
NORTHON’S THEOREM
To find IN:
•
Determine the short-circuit current flowing from
terminal a to b in both circuits.
•
Since the two circuits are equivalent,
I N  isc
VTh
IN 
RTh
Source transformation
NORTHON’S THEOREM
To determine the Thevenin or Northon equivalent circuit, we must find:
•
The open – circuit voltage vc across terminals a and b, since Vth=voc.
•
The short – circuit current Isc at terminals a and b , since IN=iSC
•
The equivalent or input resistance Rin at terminals a and b when all
independent sources are turned off, yields:
voc
RTh 
 RN
isc
NORTHON’S THEOREM
Example: Find the Northon equivalent of the circuit below.
Solution:
NORTHON’S THEOREM
NORTHON’S THEOREM
NORTHON’S THEOREM
Exercise: Find the Northon equivalent circuit for the circuit shown.
MAXIMUM POWER TRANSFER
•
Practically, a circuit is designed to provide power to a load.
•
Certain application, e. g., communication – desire to maximize the power
•
Thevenin equivalent – useful in finding the maximum power a linear circuit
•
By assuming that the load resistance, RL can be adjusted, the power
2
 VTh 
 RL
p  i RL  
 RTh  RL 
2
Where VTh and RTh are fixed
MAXIMUM POWER TRANSFER
•
Relationship of power and load resistance:
Where maximum power occurs when RL=RTh.
MAXIMUM POWER TRANSFER
•
Maximum power theorem stated that the maximum power is
resistance as seen from the load (RL=RTh)
•
Therefore when RL=RTh, maximum power is equal to:
pmax
2
Th
V

4 RTh
MAXIMUM POWER TRANSFER
Example: Find the value of RL for maximum power transfer in the circuit
shown. Find the maximum power.
Solution:
MAXIMUM POWER TRANSFER