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Lecture 08
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1
Lecture 8, Slide 1EECS40, Fall 2003 Prof. King
Lecture #8ANNOUNCEMENTS
• HW#2 solutions, HW#3 are posted online• Change in Farhana’s O.H. : Th 5-6 instead of Mo 3-4• Prof. King will be away next Monday & Wednesday
– Guest lecturer: Prof. Neureuther– Prof. King’s office hour on Wed. 9/17 cancelled
OUTLINE• Thévenin and Norton equivalent circuits• Maximum power transfer• Superposition
ReadingChapter 4.10-4.13
Lecture 8, Slide 2EECS40, Fall 2003 Prof. King
Thévenin Equivalent Circuit• Any network of voltage sources, current sources,
and 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.
networkof
sourcesand
resistors
≡ –+
VTh
RTh
RL
iL+
vL
–
a
b
RL
iL+
vL
–
a
b
Thévenin equivalent circuit
“load” resistor
2
Lecture 8, Slide 3EECS40, Fall 2003 Prof. King
I-V Characteristic of Thévenin Equivalent• The I-V characteristic for the series combination of
elements is obtained by adding their voltage drops:
–+
VTh
RTh a
b
i
i
+
vab
–
v = VTh+ iR
I-V characteristic of resistor: v = iRI-V characteristic of voltage source: v = VTh
For a given current i, the voltage drop vab is equal to the sum of the voltages dropped across the source (VTh)and the across the resistor (iRTh)
v
Lecture 8, Slide 4EECS40, Fall 2003 Prof. King
Finding VTh and RTh
Only two points are needed to define a line. Choose two convenient points:1. Open circuit across terminals a,b
i = 0, vab ≡ voc
2. Short circuit across terminals a,bvab = 0, i ≡ -isc = -VTh/RTh
–+
VTh
RTh
+
voc = VTh
–
–+
VTh
RTh
+
voc = VTh
–
i
v = VTh+ iR
vab-isc
voc
–+
VTh
RTh
isci
i
3
Lecture 8, Slide 5EECS40, Fall 2003 Prof. King
Calculating a Thévenin Equivalent1. Calculate the open-circuit voltage, voc
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 !
networkof
sourcesand
resistors
a
b
+
voc
–
networkof
sourcesand
resistors
a
b
isc
sc
ocTh
ocTh
ivR
vV
=
=
Lecture 8, Slide 6EECS40, Fall 2003 Prof. King
Thévenin Equivalent ExampleFind the Thevenin equivalent with respect to the terminals a,b:
4
Lecture 8, Slide 7EECS40, Fall 2003 Prof. King
Alternative Method of Calculating RTh
For a network containing only independent sources and resistors:
1. Set all independent sources to zerovoltage source short circuitcurrent source open circuit
2. Find equivalent resistance Req between the terminals
For a network containing dependent sources:1. Set all independent sources to zero2. Apply a test voltage source VTEST
3. Calculate ITEST
eqRR =Th
TEST
TESTTh I
VR =
network ofindependentsources and
resistors, witheach sourceset to zero
Req
network ofindependentsources and
resistors, witheach sourceset to zero
ITEST
–+
VTEST
Lecture 8, Slide 8EECS40, Fall 2003 Prof. King
RTh Calculation Example #1
Set all independent sources to 0:
5
Lecture 8, Slide 9EECS40, Fall 2003 Prof. King
Comments on Dependent SourcesA 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.
(imaginary device, used to model behavior of transistors & amplifiers)
To specify a dependent source, we must identify:1. the controlling voltage or current (must be calculated, in general)2. the relationship between the controlling voltage or current
and the supplied voltage or current3. the reference direction for the supplied voltage or current
The relationship between the dependent sourceand its reference cannot be broken!
– Dependent sources cannot be turned off for various purposes (e.g. to find the Thévenin resistance).
Lecture 8, Slide 10EECS40, Fall 2003 Prof. King
RTh Calculation Example #2Find the Thevenin equivalent with respect to the terminals a,b:
6
Lecture 8, Slide 11EECS40, Fall 2003 Prof. King
Networks Containing Time-Varying SourcesCare must be taken in summing time-varying sources!
Example:
20 cos (100t)
1 kΩ
1 kΩ
10 sin (100t)
+
–
– +
[ ]
Ω=ΩΩ=
+=+Ω+Ω
Ω=
500k1 k1
)90100sin(210)100sin(10)100cos(20k1k1
k1
Th
oTh
R
tttV
Lecture 8, Slide 12EECS40, Fall 2003 Prof. King
Norton equivalent circuit
Norton Equivalent Circuit• Any network of voltage sources, current sources,
and 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.
networkof
sourcesand
resistors
≡RL
iL+
vL
–
a
b
RL
iL+
vL
–iN
a
b
RN
7
Lecture 8, Slide 13EECS40, Fall 2003 Prof. King
Finding IN and RN
• We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation:
LTh
ThL RR
vi+
=
RLRN
iL
iN
+
vL
–
a
b
–+
RL
iL+
vL
–vTh
RTh
NLN
NL i
RRRi+
=
scTh
ThN
sc
ocThN ; i
Rvi
ivRR ====
a
b
Lecture 8, Slide 14EECS40, Fall 2003 Prof. King
Maximum Power Transfer Theorem
A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit.
L
2
LTh
ThL
2L R
RRVRip
+
==
–+
VTh
RTh
RL
iL+
vL
–
Thévenin equivalent circuit
( ) ( )( )
( ) ( )LTh
LThL2
LTh
4LTh
LThL2
LTh2Th
02
02
RRRRRRR
RRRRRRRV
dRdpL
=⇒=+×−+⇒
=
++×−+=
To find the value of RL for which p is maximum, set to 0:
Power absorbed by load resistor:
LdRdp
8
Lecture 8, Slide 15EECS40, Fall 2003 Prof. King
SuperpositionA linear circuit is constructed only of linear elements
– can be described by a linear differential equation
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.
Note: Superposition cannot be used to find power!– This principle is useful for analysis of op-amp circuits.
Procedure:1. Determine contribution due to an independent source
• Set all other sources to 02. Repeat for each independent source3. Sum individual contributions to obtain desired
voltage or current
Lecture 8, Slide 16EECS40, Fall 2003 Prof. King
Superposition Example• Find Vo
–+
24 V
2 Ω
4 Ω4 A
4 V+ – +
Vo
–
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