Announcement:

1
Lecture 6

• Announcement
• Lecture Notes Competition
• Next Tuesday 240 Cory Homework Box or
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  exchange the book coupons for prizes (TBD) in the
  last lecture.
• OUTLINE
• Superposition
• Thévenin and Norton equivalent circuits
• Maximum Power Transfer
• Reading
• Chapter 2

2
Circuit Analysis Approaches

• The Node-Voltage method can always be used to
  solve a circuit, but techniques for simplifying
  circuits (using equivalent circuits) are
  useful
• series and parallel combination reductions
• D-Y and Y-D conversions
• source transformations
• Thevenin and Norton equivalent circuits
• (to be covered in Lecture 8)

3
A Note of Caution

• These two resistive circuits are equivalent for
  voltages and currents external to the Y and D
  circuits. Internally, the voltages and currents
  are different.

RbRc Ra Rb Rc
RaRc Ra Rb Rc
RaRb Ra Rb Rc
R1
R2
R3
4
Circuit Simplification Example

• Find the equivalent resistance Rab

2W
2W
a
a
18W
12W
6W

4W
9W
b
4W
9W
b
5
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.

6
Superposition

• Procedure
• 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
• Repeat for each independent source
• Sum individual contributions to obtain desired
  voltage
• or current

7
Superposition Example

• Find Vo

4 V
2 W

Vo

24 V
4 W
4 A
8
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
iB
network A of sources and resistors
network B of sources and resistors
vA _
vB _

iA(vA) iB(vB)
9
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

v1v2


v2

i1i2
i1
i2
10
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
vL
iL
iL


VTh
RL
RL
b
b
load resistor
11
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)
i
RTh
a
v
i
vab
vab VTh iR

VTh
b
I-V characteristic of resistor v iR
I-V characteristic of voltage source v VTh
12
Thévenin Equivalent Example

• Find the Thevenin equivalent with respect to the
  terminals a,b

13
RTh Calculation Example 1

• Set all independent sources to 0

14
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).

15
RTh Calculation Example 2

• Find the Thevenin equivalent with respect to the
  terminals a,b

16
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
17
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
network of sources and resistors
vL
iL

RL
b
18
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
i
a
i -IN Gv
vab
v
iN
RN
b
I-V characteristic of resistor iGv
I-V characteristic of current source i -IN
19
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
20
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
vL
iL
iL

iN
vTh
RL
RN
RL
b
b
21
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.