Chapter 2
Properties of Resistive Circuits
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Resistive Circuits : circuits consisting entirely of sources and resistors
¾ their properties will be used in the next chapters for important practical
applications
¾ their concepts and techniques are foundation for all aspects of circuit analysis,
where the circuit also contains other types of elements
2.1 SERIES AND PARALLEL RESISTANCE
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network : the small unit into which a complete circuit is divided, for easier analysis or
design
¾ may include any number of elements, but must have at least two terminals
¾ for example, the load that is attached to a source is often a two-terminal network
(several resistors in series or in parallel)
Series Resistance and Potentiometers
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A simple circuit consisting of two resistors
Figure 2.1
¾
(1) from Ohm’s law, v1 = R1i, v2 = R2i
(2) from KVL, v = v1 + v2 = (R1 + R2 ) i
∴ the i-v relationship is
v = Rseri
Rser = R1 + R2
¾
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Rser : the series equivalent resistance, in other words, Rser can replace the two
resistors R1 plus R2
Two-terminal networks are equivalent if they have exactly the same i-v
characteristics at their terminals
1
¾
¾
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the conditions at the “terminals” of a complicated network can be more easily
calculated using a simpler equivalent network
however, the conditions “within” the network might be quite different
voltage divider : because the series resistors carry the same current, the voltage v is
“divided” among the series resistors
¾ for the two-resistor case,
R1
R2
v1 =
v, v 2 =
v
R1 + R2
R1 + R2
¾
for the N-resistor case,
Rser = R1 + R2 + + RN
the voltage across any resistor Rn is
R
Rn
vn = n v =
v
Rser
R1 + R2 + + RN
‹
the value of Rser will always be larger than the
largest individual resistance
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potentiometer (pot) : a three-terminal device
¾ W : movable contact point
RAW + RWB = RAB
¾
¾
adjustable resistance (in Figure (c)) : RWB ≤ RAB
R
adjustable voltage (in Figure (d)) : vW = AW vs (W must not carry any
RAB
current)
Figure 2.3
2
Example 2.1
Audio Volume Control
Figure 2.4
Parallel Resistance
Figure 2.5
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N resistors are in parallel, so voltage v appears across each one
¾
(Here, using conductance Gn = 1 Rn may be easier to analysis)
from Ohm’s law : i1 = G1v, i2 = G2v,
, iN = G N v
from KCL :
i = i1 + i2 +
+ iN
=G1v + G2v +
= (G1 + G2 +
∴ G par = G1 + G2 +
+ GN v
+ GN ) v = G parv
+ GN : parallel equivalent conductance
the corresponding parallel equivalent resistance is
3
Rpar =
∴
‹
z
1
G par
1
1
1
=
+
+
Rpar
R1 R2
+
1
RN
Rpar has the same i-v relationship as the N parallel resistors
current divider : parallel-connected resistors having the same voltage, the total
current i is “divided” into the small i1, i2, , iN , through the resistors R1, R2, , RN
¾ the current through any resistor Rn is given by
1 Rn
in =
Gn
Gn
i=
G par
G1 + G2 +
‹
special case : if all N resistors have the same conductance G = 1 R , then
i=
+ GN
1 R1 + 1 R2 +
+ 1 RN
i
in = i N ,G par = NG , Rpar = 1 G par = R N
¾
the value of Rpar will always be smaller than the smallest individual
resistance
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two parallel resistors :
−1
¾
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Rpar
⎛1
1⎞
= ⎜⎜ + ⎟⎟⎟
⎜⎝ R1 R2 ⎠⎟
i1 =
R2
R1
i, i2 =
i
R1 + R2
R1 + R2
=
R1R2
= R1 R2
R1 + R2
Application : in order to reduce the present resistance R1 to a specified smaller value
Rpar , you can parallel R1 with another resistance R2 to get
Rpar = R1 R2 ≤ R
∵
1
1
1
=
+
Rpar
R1 R2
∴ R2 =
1
1
1
−
Rpar R1
=
4
R1Rpar
R1 − Rpar
Example 2.2
Parallel Resistance Calculations
Figure 2.6
Example 2.3
Electric Grill Unit
Figure 2.7
Resistive Ladders
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A network consisting entirely of series and parallel resistors
¾ Given a resistive-ladder circuit, we may need to find some or all of the voltages
and currents. They can be determined by the series-parallel reduction method in
pages 48-49 at the textbook
5
Example 2.4
Ladder Calculations
Figure 2.8
6
2.2 DUALS
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Table 2.1
Series Network
Parallel Network
KVL loop equation:
v = v1 + v2 = R1i + R2i
KVL node equation:
i = i1 + i2 = G1v + G2v
Equivalent resistance:
Rser = R1 + R2
Equivalent conductance:
G par = G1 + G2
Voltage divider:
R1
v1 =
v
R1 + R2
Current divider:
G1
i1 =
i
G1 + G2
Open circuit:
v1 = v when R1 = ∞
short circuit:
i1 = i when G1 = ∞
duality :
Two different networks are duals when the i − v equations that describe one of them
have the same mathematical form as the i − v equations for the other with voltage an
current variables interchanged
¾ is not a technique for analyzing particular circuits
¾ is a conceptual aide that enhances intuition and underscores important
relationships
¾ for instance:
a series circuit: v = (R1 + R2 + + RN ) i
its dual, a parallel circuit i = (G1 + G2 + + GN ) v
Example 2.5
Constructing a Dual Circuit
7
2.3 CIRCUITS WITH CONTROLLED SOURCES
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Independent sources : the source voltage or current does not depend on any other
voltage or current
¾ For instance : the ideal sources introduced in Chapter 1
Controlled sources
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Controlled sources : Sources that depend on some other voltage or current
¾ is also called dependent sources
¾ example :
if vin = 10 V, then vout = 100vin = 1000 V
if vin = 0.2 cos ωt V, then vout = 100vin = 20 cos ωt V
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Classification
:
8
¾
VCVS : voltage-controlled voltage source
‹ vc is proportional to some other voltage vx , regardless of the current i
vc = µvx
vx : control variable
µ : voltage gain, dimensionless
‹
¾
CCCS : current-controlled current source
‹ the dual of VCVS
‹ ic = βix , independent of v
ix : control variable
β : current gain, dimensionless
‹
¾
CCCS often represents the current-amplifying property of a bipolar junction
transistor
VCCS : voltage-controlled current source
‹ ic = gmvx , independent of v
gm is called the transconductance,
‹
¾
VCVS may be any linear voltage amplifier or voltage-amplifying device
because
it
represents
a
voltage-to-current transfer effect and has the same units as conductance
example : the current through a field-effect transistor depends on a
controlling voltage
CCVS : current-controlled voltage source
‹ vc = rmix , independent of i
rm is called the transresistance, because it represents a current-to-voltage
transfer effect and has the same units as resistance
¾
These controlled sources mentioned above are linear elements, because the
current or voltage is directly proportional to the control variable
¾
a controlled source produces voltage or current only when an independent
source activate the control variable
‹
¾
if there is no independent source, then all v and i will be zero
In the analysis of circuits with controlled source, an essential step is relating the
control voltage or current to other quantities of interest
9
Example 2.6
Amplifier with a Field-Effect Transistor
Figure 2.15
Example 2.7
Analysis with a VCVS
Figure 2.16
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Generalized Equivalent Resistance
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recall that a resistive ladder network can be reduced to a single resistance
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load network :
any two-terminal network that contains no independent sources. If controlled sources
are included, then the control variable must be within the same network
¾ controlled source are allowed since they remain “off” until an independent source
activates the control variables
Figure 2.18
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equivalent resistance theorem :
When a load network consists entirely of resistances or resistances and controlled
sources, the terminal voltage and current are related by v = Reqi , where Req is a
constant
¾
Note : the reference polarities of v and i
Example 2.8
Equivalent Resistance with a VCCS
Figure 2.19
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2.4 LINEARITY AND SUPERPOSITION
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For linear circuits, two analysis techniques can be applied
¾ Proportionality Principle
¾ Superposition Theorem
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Linear elements and circuits
¾ A resistor is a linear element, ∵ v = Ri
¾ A controlled source is a linear element, ∵ y = kx
¾
Linear circuit :
A circuit is linear when it contains entirely of linear elements and
independent sources
¾
Let the excitation (source) be x , and the response (any branch variable) be y ,
then y = f (x ) , where x is the cause and y is the effect
For a linear circuit, f (x ) must be a linear function having the the
‹
properties of proportionality and superposition.
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Proportionality Property :
if f (x ) = y, then f (Kx ) = Kf (x ) = Ky
K : constant
for any value of x
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Superposition Property :
if f (xa ) = ya , f (xb ) = yb , then f (xa + xb ) = f (xa ) + f (xb ) = ya +yb
for any values of xa and xb
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Linearity :
The property of Linearity combines Proportionality Property and Superposition
Property. That is
if f (xa ) = ya , f (xb ) = yb , then f (Kaxa + Kbxb ) = Ka f (xa ) + Kb f (xb ) = Kaya +Kbyb
for any values of xa and xb , and Ka and Kb are arbitrary constants
¾
For example, an amplifier, vout = 100vin , that is f (x ) = 100x
the input might come from an audio mixer combining the voltage v1 from a
keyboard with the voltage v2 from a microphone to produce vin = 0.5v1 + 3v2 ,
then
vout = f (vin ) = f (0.5v1 + 3v2 ) = 0.5 f (v1 ) + 3 f (v2 )
f (v1 ) = 100v1, f (v2 ) = 100v2, ∴ vout = 50v1 + 300v2
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¾
Testing if an element is linear :
for an ideal resistor,
v = Ri = f (i )
Let i = Kaia + Kbib
f (ia ) = Ria , f (ib ) = Rib
f (i ) = f (Kaia + Kbib ) = R (Kaia + Kbib ) = Ka (Ria ) + Kb (Rib )
= Ka f (ia ) + Kb f (ib )
∴ a resistor is linear
power dissipation by a resistor is not a linear function of the current i
∵ p = Ri 2
2
R (Kaia + Kbib ) ≠ KaRia 2 + KbRib 2
Proportionality Principle
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Consider a linear circuit driven by a single independent source x (which may be
time-varying) that produces the response y = f (x )
if x̂ = Kx
then from proportionality property,
yˆ = f (xˆ) = f (Kx ) = Kf (x ) = Ky
∴ any branch variable in a linear circuit varies in direct proportion to the source
→ proportionality principle : x̂ = Kx then ŷ = Ky
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From proportionality principle, it is easy to find all branch variables and the source
value in a ladder network if you know the value of a branch variable “farthest” from
the source
¾ The procedure for analyzing a ladder network is in p.61 of the text
Example 2.9 Analysis of a Ladder Network
Figure 2.20
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Superposition Theorem
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When a linear circuit contains two or more independent sources, the value of any
branch variable equals the algebraic sum of the individual contributions from each and
every independent source with all other independent sources set to zero
¾ Very useful fro circuits with multiple sources
Illustration :
yn : the value of any branch variable
x 1, x 2 : the values of two independent sources
then yn = f (x 1, x 2 ) : a linear function of both x 1 and x 2
the individual contributions from each source are
yn −1 = f (x 1, 0) , yn −2 = f (0, x 2 )
then from the superposition theorem :
yn = f (x 1, x 2 ) = yn −1 + yn −2 = f (x 1, 0) + f (0, x 2 )
¾
In order to set one independent source to zero (suppress a source)
‹ v = 0 : "independent" voltage source ⇒ short circuit
i = 0 : "independent" current source ⇒ open circuit
‹
Because controlled sources affect the individual contribution of each
independent source, they cannot be suppressed during analysis by
superposition
Example 2.10 Superposition Calculations
Figure 2.21
14
Example 2.11 Superposition with a Controlled Source
Figure 2.22
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2.5 THÉVENIN AND NORTON NETWORK
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Two equivalent “source networks” are to be discussed based on Thévenin’s and
Norton’s Theorems
¾ Leads to handy source conversions that simplify many analysis and design
problems
Thévenin’s and Norton’s Theorems
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These two theorems are related to the “terminal” behavior of networks containing
independent sources
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source network:
¾ a two-terminal network
¾ consists entirely of linear elements and sources
¾ if controlled sources are present, then the control variables
must be within the same network
¾
Note : the reference polarities for v and i show that the source network
“supplies” power
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terminology :
¾
open-circuit voltage
voc
¾
i =0
short-circuit current
isc
¾
v
i
v =0
Thévenin resistance
v
Rt = oc , the slope of the v − i line is −Rt
isc
16
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Thévenin’s theorem :
Any linear resistive source network acts at its terminals like an ideal voltage
source of value voc in series with a resistor having resistance Rt
¾
Thévenin equivalent network:
v = voc − Rti
if i = 0, then v = voc
if v = 0, then i = voc Rt = isc
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Norton’s Theorem :
Any linear resistive source network acts at its terminals like an ideal current
source of value isc in parallel with a resistor having resistance Rt
¾
Norton equivalent network:
i = isc − v Rt
if i = 0, then v = Rtisc = voc
if v = 0, then i = isc
¾
voc , isc , Rt : Thévenin parameters
¾
Note : the equivalence holds only at the terminals of the source network, not
within the network
‹ e.g. for Thévenin network, p = 0 when i = 0
for Norton Network, p = isc2 Rt when i = 0
Example 2.12 Thévenin Parameters from a v-i Curve
17
Figure 2.27
Example 2.13 Equivalent Source Network
Figure 2.28
18
Thévenin Resistance
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It has been known that voc and isc have clear physical interpretations. Here, we are
v
to discuss the physical interpretations of Rt = oc .
isc
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In Thévenin’s and Norton’s equivalent networks,
¾
If voc = 0 or isc = 0 , then the networks reduce to a source-free network with
equivalent resistance Rt . Thus,
The Thévenin resistance of a source network equals the equivalent
resistance of that network when all independent sources have been
suppressed.
‹ v = 0 : "independent" voltage source ⇒ short circuit
i = 0 : "independent" current source ⇒ open circuit
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Calculation of Rt :
¾
¾
If no controlled sources are contained, the ladder structure of a source network
can be reduced to Rt by series-parallel reduction.
If controlled sources are contained, then Rt can be found using the method
below. The “dead” source network is connected to an external test source ( vt or
it )
v
Rt = t
it
After finding Rt , then
given isc , then voc = Rtisc , or
given voc , then isc = voc Rt
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Source Conversions
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Since both Thévenin’s network and Norton’s network are equivalent to a source
network, they are also equivalent to each other
=
z
=
voltage sources ( vs ) in series with resistance ( Rs )
= current sources ( is ) in parallel with resistance ( Rs )
where vs = Rtis , is = vs Rt
¾
¾
Note : the arrow for the current source points in the direction of the higher
potential end of the voltage source
Source conversions may be applied to portions of a circuit or network to simplify
intermediate calculations
‹ Note : when controlled sources are present, source conversions must not
obliterate the identity of any control variables
Example 2.15 Circuit Reduction by Source Conversions
20
Figure 2.34
Example 2.16 Thévenin Network via Source Conversions
Figure 2.35
21