1. 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

2. Top view of board
EECS 42, Spring 2005
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3. Top view of board
EECS 42, Spring 2005
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4. Bottom view of board – note which way the wires go
EECS 42, Spring 2005
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5. Primary Formal Circuit Analysis Methods
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
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
*Simple method for planar circuits
A mesh current is not necessarily identified with a branch current.
EECS 42, Spring 2005
Week 3a 5

6. Mesh Analysis: Example #1
Select M mesh currents.
Apply KVL to each mesh.
Solve for mesh currents.
EECS 42, Spring 2005
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7. Mesh Analysis with a Current Sourceia 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 7

8. Mesh Analysis: Example #2ia ib
Eq’n 1: KVL for supermesh
Eq’n 2: Constraint due to current source:
EECS 42, Spring 2005
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9. 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 9

10. 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:
Determine contribution due to an independent source
Set all other sources to zero (voltage source  short circuit; current source  open circuit)
Repeat for each independent source
Sum individual contributions to obtain desired voltage or current
EECS 42, Spring 2005
Week 3a 10

11. Superposition Example
Find Vo
4 V
2 
+ – + Vo –
24 V + –
4 A
4 
EECS 42, Spring 2005
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12. EECS 42, Spring 2005
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13. EECS 42, Spring 2005
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14. Source Combinations ≡ ≡
Voltage sources in series can be replaced by an equivalent voltage source:
Current sources in parallel can be replaced by an equivalent current source: v1 + –
v1+v2 + – ≡ + – v2 ≡
i1+i2 i1 i2
EECS 42, Spring 2005
Week 3a 14

15. 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 a
network of
sources
and
resistors + vL – iL + vL – iL ≡ + – RL
VTh RL b b
“load” resistor
EECS 42, Spring 2005
Week 3a 15

16. 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) i
RTh a
v = VTh+ iR i v +
vab – + –
VTh b
I-V characteristic of resistor: v = iR
I-V characteristic of voltage source: v = VTh
EECS 42, Spring 2005
Week 3a 16

17. 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,b
vab = 0, i ≡ -isc = -VTh/RTh i i
voc R
vab Th i
-isc i sc + V Th –
v = VTh+ iR
EECS 42, Spring 2005
Week 3a 17

18. Calculating a Thévenin Equivalent
Calculate the open-circuit voltage, voc
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 +
voc – b a
network of
sources
and
resistors
isc
EECS 42, Spring 2005
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19. Thévenin Equivalent Example
Find the Thevenin equivalent with respect to the terminals a,b:
EECS 42, Spring 2005
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20. EECS 42, Spring 2005
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21. Alternative Method of Calculating RTh
For a network containing only independent sources
and linear resistors:
Set all independent sources to zero
voltage source  short circuit
current source  open circuit
Find equivalent resistance Req between the terminals by inspection
Or, set all independent sources to zero
Apply a test voltage source VTEST
Calculate ITEST
network of
independent
sources and
resistors, with
each source
set to zero
Req
ITEST
network of
independent
sources and
resistors, with
each source
set to zero + –
VTEST
EECS 42, Spring 2005
Week 3a 21

22. RTh Calculation Example #1
Set all independent sources to zero:
EECS 42, Spring 2005
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23. 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:
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).
EECS 42, Spring 2005
Week 3a 23

24. RTh Calculation Example #2
Find the Thevenin equivalent with respect to the terminals a,b:
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25. Networks Containing Time-Varying Sources
Care must be taken in summing time-varying sources!
Example:
10 sin (100t)
1 k – + –
20 cos (100t)
1 k
EECS 42, Spring 2005
Week 3a 25

26. 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 a RL iL + vL – iN b RN
network of
sources
and
resistors + vL – iL ≡ RL b
EECS 42, Spring 2005
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27. 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: i a i +
vab –
i = -IN+ Gv v iN RN b
I-V characteristic of resistor: i=Gv
I-V characteristic of current source: i = -IN
EECS 42, Spring 2005
Week 3a 27

28. 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 28

29. Finding IN and RN
We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation:
RTh a a + vL – iL + vL – iL
vTh + – iN RL RN RL b b
EECS 42, Spring 2005
Week 3a 29

30. Maximum Power Transfer Theorem
Thévenin equivalent circuit
Power absorbed by load resistor:
RTh + vL – iL + –
VTh RL
To find the value of RL for which p is maximum, set to 0:
A resistive load receives maximum power from a circuit if the
load resistance equals the Thévenin resistance of the circuit.
EECS 42, Spring 2005
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31. EECS 42, Spring 2005
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