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Lecture #6
OUTLINE
• Complete Mesh Analysis Example(s)
• Superposition
• Thévenin and Norton equivalent circuits
• Maximum Power Transfer
Reading
Chapter 2
EECS40, Fall 2004
Lecture 6, Slide 1
Prof. White
Superposition
A linear circuit is one constructed only of linear
elements (linear resistors, and linear capacitors and
inductors, linear dependent sources) and
independent sources. Linear
means I-V charcteristic of elements/sources are
straight lines when plotted
Principle of Superposition:
•
In any linear circuit containing multiple
independent sources, the current or voltage at
any point in the network may be calculated as
the algebraic sum of the individual contributions
of each source acting alone.
EECS40, Fall 2004
Lecture 6, Slide 2
Prof. White
Superposition
Procedure:
1. Determine contribution due to one independent source
•
Set all other sources to 0: Replace independent voltage
source by short circuit, independent current source by open
circuit
2. Repeat for each independent source
3. Sum individual contributions to obtain desired voltage
or current
EECS40, Fall 2004
Lecture 6, Slide 3
Prof. White
Superposition Example
• Find Vo
–
+
24 V
2W
4V
+–
4A
+
4 W Vo
–
EECS40, Fall 2004
Lecture 6, Slide 4
Prof. White
Equivalent Circuit Concept
• A network of voltage sources, current sources,
and resistors can be replaced by an
equivalent circuit which has identical terminal
properties (I-V characteristics) without
affecting the operation of the rest of the circuit.
iA
network A
of
sources
and
resistors
iB
+
vA
_
≡
network B
of
sources
and
resistors
+
vB
_
iA(vA) = iB(vB)
EECS40, Fall 2004
Lecture 6, Slide 5
Prof. White
Source Combinations
• Voltage sources in series can be replaced by an
equivalent voltage source:
≡
v1+v2
–
+
–
+
v2
–
+
v1
• Current sources in parallel can be replaced by
an equivalent current source:
i1
EECS40, Fall 2004
i2
≡
i1+i2
Lecture 6, Slide 6
Prof. White
Thévenin Equivalent Circuit
• Any* linear 2-terminal (1-port) network of indep. voltage
sources, indep. current sources, and linear resistors can
be replaced by an equivalent circuit consisting of an
independent voltage source in series with a resistor
without affecting the operation of the rest of the circuit.
Thévenin equivalent circuit
RTh
a
+
iL
vL
RL
≡
VTh
–
–
+
network
of
sources
and
resistors
a
b
+
iL
vL
RL
–
b
“load” resistor
EECS40, Fall 2004
Lecture 6, Slide 7
Prof. White
I-V Characteristic of Thévenin Equivalent
• The I-V characteristic for the series combination of
elements is obtained by adding their voltage drops:
For a given current i, the voltage drop
vab is equal to the sum of the voltages
dropped across the source (VTh)
and across the resistor (iRTh)
RTh
a
v
i +
–
+
VTh
i
vab = VTh+ iR
vab
–
b
I-V characteristic of resistor: v = iR
I-V characteristic of voltage source: v = VTh
EECS40, Fall 2004
Lecture 6, Slide 8
Prof. White
Finding VTh and RTh
Only two points are needed to define a line.
Choose two convenient points:
RTh
1. Open circuit across terminals a,b
i
i = 0, vab ≡ voc
–
+
VTh
2. Short circuit across terminals a,b
vab = 0, i ≡ -isc = -VTh/RTh
+
voc = VTh
–
i
voc
RTh
vab
i
EECS40, Fall 2004
–
+
VTh
isc
-isc
v = VTh+ iR
Lecture 6, Slide 9
Prof. White
Calculating a Thévenin Equivalent
1. Calculate the open-circuit voltage, voc
a
network
of
sources
and
resistors
+
voc
–
b
2. Calculate the short-circuit current, isc
•
Note that isc is in the direction of the open-circuit
voltage drop across the terminals a,b !
a
network
of
sources
and
resistors
isc
b
EECS40, Fall 2004
Lecture 6, Slide 10
VTh  voc
voc
RTh 
isc
Prof. White
Thévenin Equivalent Example
Find the Thevenin equivalent with respect to the terminals a,b:
EECS40, Fall 2004
Lecture 6, Slide 11
Prof. White
Alternative Method of Calculating RTh
For a network containing only independent sources
network of
and linear resistors:
independent
sources and
resistors, with
each source
set to zero
1. Set all independent sources to zero
voltage source  short circuit
current source  open circuit
Req
2. Find equivalent resistance Req between the terminals
by inspection
RTh  Req
Or, set all independent sources to zero
RTh
EECS40, Fall 2004
VTEST

I TEST
Lecture 6, Slide 12
network of
independent
sources and
resistors, with
each source
set to zero
VTEST
–
+
1. Apply a test voltage source VTEST
2. Calculate ITEST
ITEST
Prof. White
RTh Calculation Example #1
Set all independent sources to 0:
EECS40, Fall 2004
Lecture 6, Slide 13
Prof. White
Comments on Dependent Sources
A dependent source establishes a voltage or current
whose value depends on the value of a voltage or
current at a specified location in the circuit.
(device model, used to model behavior of transistors & amplifiers)
To specify a dependent source, we must identify:
1.
2.
3.
the controlling voltage or current (must be calculated, in general)
the relationship between the controlling voltage or current
and the supplied voltage or current
the reference direction for the supplied voltage or current
The relationship between the dependent source
and its reference cannot be broken!
–
Dependent sources cannot be turned off for various
purposes (e.g. to find the Thévenin resistance, or in
analysis using Superposition).
EECS40, Fall 2004
Lecture 6, Slide 14
Prof. White
RTh Calculation Example #2
Find the Thevenin equivalent with respect to the terminals a,b:
EECS40, Fall 2004
Lecture 6, Slide 15
Prof. White
Networks Containing Time-Varying Sources
Care must be taken in summing time-varying sources!
Example:
10 sin (100t)
1 kW
–
+
+
20 cos (100t)
1 kW
–
EECS40, Fall 2004
Lecture 6, Slide 16
Prof. White
EECS40, Fall 2004
Lecture 6, Slide 17
Prof. White
Norton Equivalent Circuit
• Any* linear 2-terminal (1-port) network of indep. voltage
sources, indep. current sources, and linear resistors can
be replaced by an equivalent circuit consisting of an
independent current source in parallel with a resistor
without affecting the operation of the rest of the circuit.
Norton equivalent circuit
a
network
of
sources
and
resistors
+
iL
vL
RL
≡
iN
RN
–
+
iL
vL
RL
–
b
EECS40, Fall 2004
a
b
Lecture 6, Slide 18
Prof. White
I-V Characteristic of Norton Equivalent
• The I-V characteristic for the parallel combination of
elements is obtained by adding their currents:
For a given voltage vab, the current i is
equal to the sum of the currents in
each of the two branches:
a i
+
iN
RN
i
i = -IN+ Gv
v
vab
–
b
I-V characteristic of resistor: i=Gv
I-V characteristic of current source: i = -IN
EECS40, Fall 2004
Lecture 6, Slide 19
Prof. White
Finding IN and RN = RTh
Analogous to calculation of Thevenin Eq. Ckt:
1) Find o.c voltage and s.c. current
IN ≡ isc = VTh/RTh
2) Or, find s.c. current and Norton (Thev) resistance
EECS40, Fall 2004
Lecture 6, Slide 20
Prof. White
Finding IN and RN
• We can derive the Norton equivalent circuit from
a Thévenin equivalent circuit simply by making a
source transformation:
RTh
–
+
vTh
a
a
+
iL
vL
RL
iN
RN
–
+
iL
vL
RL
–
b
b
voc
vTh
RN  RTh 
; iN 
 isc
isc
RTh
EECS40, Fall 2004
Lecture 6, Slide 21
Prof. White
Maximum Power Transfer Theorem
Thévenin equivalent circuit
Power absorbed by load resistor:
RTh
2
–
+
VTh
+
iL
vL
RL
 VTh 
 RL
p  i RL  
 RTh  RL 
2
L
–
dp
To find the value of RL for which p is maximum, set
to 0:
dRL
2



dp
R

R
 RL  2RTh  RL 
2
Th
L
 VTh 
0
4
dRL
RTh  RL 


 RTh  RL   RL  2RTh  RL   0
2
 RTh  RL
EECS40, Fall 2004
A resistive load receives maximum power from a circuit if the
load resistance equals the Thévenin resistance of the circuit.
Lecture 6, Slide 22
Prof. White