Announcements
New topics:
Mesh (loop) method of circuit analysis
Superposition method of circuit analysis
Equivalent circuit idea (Thevenin, Norton)
Maximum power transfer from a circuit to a load
To stop blowing fuses in the lab, note how the
breadboards are wired …
EECS 42, Spring 2005
Week 3a
1
Top view of board
EECS 42, Spring 2005
Week 3a
2
Bottom view of board – note which way
the wires go
EECS 42, Spring 2005
Week 3a
3
Primary Formal Circuit Analysis Methods
NODAL ANALYSIS
(“Node-Voltage Method”)
0) Choose a reference node
1) Define unknown node voltages
2) Apply KCL to each unknown
node, expressing current in
terms of the node voltages
=> N equations for
N unknown node voltages
3) Solve for node voltages
=> determine branch currents
MESH ANALYSIS
(“Mesh-Current Method”)
1) Select M independent mesh
currents such that at least one
mesh current passes through each
branch*
M = #branches - #nodes + 1
2) Apply KVL to each mesh,
expressing voltages in terms of
mesh currents
=> M equations for
M unknown mesh currents
3) Solve for mesh currents
=> determine node voltages
*Simple method for planar circuits
A EECS
mesh
current
identified with a branch current.
42, Spring
2005 is not necessarily
Week 3a
4
Mesh Analysis: Example #1
1. Select M mesh currents.
2. Apply KVL to each mesh.
3. Solve for mesh currents.
EECS 42, Spring 2005
Week 3a
5
Mesh Analysis with a Current Source
ia
ib
Problem: We cannot write KVL for meshes a and b
because there is no way to express the voltage drop
across the current source in terms of the mesh currents.
Solution: Define a “supermesh” – a mesh which avoids the
branch containing the current source. Apply KVL for this
supermesh.
EECS 42, Spring 2005
Week 3a
6
Mesh Analysis: Example #2
ia
ib
Eq’n 1: KVL for supermesh
Eq’n 2: Constraint due to current source:
EECS 42, Spring 2005
Week 3a
7
Mesh Analysis with Dependent Sources
• Exactly analogous to Node Analysis
• Dependent Voltage Source: (1) Formulate and
write KVL mesh eqns. (2) Include and express
dependency constraint in terms of mesh
currents
• Dependent Current Source: (1) Use supermesh.
(2) Include and express dependency constraint
in terms of mesh currents
EECS 42, Spring 2005
Week 3a
8
Superposition Method (Linear Circuits
Only)
A linear circuit is constructed only of linear elements (linear
resistors, linear dependent sources) and independent sources.
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.
Procedure:
1. Determine contribution due to an independent source
•
Set all other sources to zero (voltage source  short circuit;
current source  open circuit)
2. Repeat for each independent source
3. Sum individual contributions to obtain desired voltage or
current
EECS 42, Spring 2005
Week 3a
9
Superposition Example
• Find Vo
–
+
24 V
2W
4V
+–
+
4 W Vo
4A
–
EECS 42, Spring 2005
Week 3a
10
EECS 42, Spring 2005
Week 3a
11
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)
EECS 42, Spring 2005
EECS40, Spring 2004
Week 3a
12
Lecture 6, Slide 1
Prof. Sanders
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
EECS 42, Spring 2005
i2
≡
Week 3a
i1+i2
13
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
b
EECS 42, Spring 2005
a
+
iL
vL
RL
–
b
Week 3a“load”
resistor
14
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 the across the resistor (iRTh)
RTh
a
v = VTh+ iR
v
i +
–
+
VTh
i
vab
–
b
I-V characteristic of resistor: v = iR
I-V characteristic of voltage source: v = VTh
EECS 42, Spring 2005
Week 3a
15
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 = 0, vab ≡ voc
–
+
VTh
i +
2. Short circuit across terminals a,b
vab = 0, i ≡ -isc = -VTh/RTh
voc = VTh
–
i
voc
RTh
vab
i
EECS 42, Spring 2005
–
+
VTh
isc
-isc
v = VTh+ iR
Week 3a
16
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
EECS 42, Spring 2005
Week 3a
isc
VTh  voc
RTh
b
voc

isc
17
Thévenin Equivalent Example
Find the Thevenin equivalent with respect to the terminals a,b:
EECS 42, Spring 2005
Week 3a
18
EECS 42, Spring 2005
Week 3a
19
Alternative Method of Calculating RTh
For a network containing only independent sources
network of
and linear resistors:
1. Set all independent sources to zero
voltage source  short circuit
current source  open circuit
independent
sources and
resistors, with
each source
set to zero
Req
2. Find equivalent resistance Req between the terminals by
inspection
RTh  Req
Or, set all independent sources to zero
RTh
EECS 42, Spring 2005
VTEST

I 3a
WeekTEST
network of
independent
sources and
resistors, with
each source
set to zero
VTEST
20
–
+
1. Apply a test voltage source VTEST
2. Calculate ITEST
ITEST
RTh Calculation Example #1
Set all independent sources to zero:
EECS 42, Spring 2005
Week 3a
21
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
EECS 42,using
Spring 2005
22
Superposition). Week 3a
RTh Calculation Example #2
Find the Thevenin equivalent with respect to the terminals a,b:
EECS 42, Spring 2005
Week 3a
23
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
–
EECS 42, Spring 2005
Week 3a
24
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
≡
RN
–
+
iL
vL
RL
–
b
EECS 42, Spring 2005
iN
a
b
Week 3a
25
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
EECS 42, Spring 2005
Week 3a
26
Finding IN and RN = RTh
Analogous to calculation of Thevenin Eq. Ckt:
1) Find open-circuit voltage and short-circuit current
IN ≡ isc = VTh/RTh
2) Or, find short-circuit current and Norton
(Thevenin) resistance
EECS 42, Spring 2005
Week 3a
27
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
EECS 42, Spring 2005
Week 3a
28
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
EECS 42, Spring 2005
A resistive load receives maximum power from a circuit if the
load resistance equals the Thévenin resistance of the circuit.
Week 3a
29
EECS 42, Spring 2005
Week 3a
30