1

Chapter 2

Properties of Resistive Circuits

Resistive Circuits : circuits consisting entirely of sources and resistors their properties will be used in the next chapters for important practical

applications 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

network : the small unit into which a complete circuit is divided, for easier analysis or design

may include any number of elements, but must have at least two terminals 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

A simple circuit consisting of two resistors

Figure 2.1

(1) from Ohm’s law, 1 1 2 2, v R i v R i= = (2) from KVL, ( )1 2 1 2v v v R R i= + = + ∴ the i-v relationship is serv R i= 1 2serR R R= +

serR : the series equivalent resistance, in other words, serR can replace the two resistors 1R plus 2R

Two-terminal networks are equivalent if they have exactly the same i-v characteristics at their terminals

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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

for the two-resistor case, 1 2

1 21 2 1 2

, R Rv v v vR R R R

= =+ +

for the N-resistor case,

1 2ser NR R R R= + + + the voltage across any resistor nR is

1 2

n nn

ser N

R Rv v vR R R R

= =+ + +

the value of serR will always be larger than the largest individual resistance

potentiometer (pot) : a three-terminal device W : movable contact point

AW WB ABR R R+ = adjustable resistance (in Figure (c)) : WB ABR R≤

adjustable voltage (in Figure (d)) : AWW s

AB

Rv vR

= (W must not carry any

current)

Figure 2.3

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Example 2.1 Audio Volume Control

Figure 2.4

Parallel Resistance

Figure 2.5

N resistors are in parallel, so voltage v appears across each one

(Here, using conductance 1n nG R= may be easier to analysis)

from Ohm’s law : 1 1 2 2, , , N Ni G v i G v i G v= = = from KCL :

( )

1 2

1 2

1 2

=

=

N

N

N par

i i i i

G v G v G v

G G G v G v

= + + +

+ + +

+ + + =

1 2par NG G G G∴ = + + + : parallel equivalent conductance

the corresponding parallel equivalent resistance is

4

1 2

1

1 1 1 1

parpar

par N

RG

R R R R

=

∴ = + + +

parR 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 1 2, , , Ni i i , through the resistors 1 2, , , NR R R

the current through any resistor nR is given by

1 2 1 2

1

1 1 1nn n

npar N N

RG Gi i i iG G G G R R R

= = =+ + + + + +

special case : if all N resistors have the same conductance 1G R= , then

, , 1n par par pari i N G NG R G R N= = = =

the value of parR will always be smaller than the smallest individual

resistance

two parallel resistors :

1

1 21 2

1 2 1 2

1 1par

R RR R RR R R R

−⎛ ⎞⎟⎜ ⎟= + = =⎜ ⎟⎜ ⎟⎜ +⎝ ⎠

1 21 2 1 2

, R Ri i i iR R R R

= =+ +2 1

Application : in order to reduce the present resistance 1R to a specified smaller value

parR , you can parallel 1R with another resistance 2R to get

1 2parR R R R= ≤

1 2

12

1

1

1 1 1

11 1

par

par

par

par

R R RRR

RR R

R R

= +

∴ = =−−

∵

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Example 2.2 Parallel Resistance Calculations

Figure 2.6 Example 2.3 Electric Grill Unit

Figure 2.7

Resistive Ladders

A network consisting entirely of series and parallel resistors 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

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Example 2.4 Ladder Calculations

Figure 2.8

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2.2 DUALS

Table 2.1 Series Network Parallel Network

KVL loop equation:

1 2 1 2v v v R i R i= + = + KVL node equation:

1 2 1 2i i i G v G v= + = +

Equivalent resistance:

1 2serR R R= + Equivalent conductance:

1 2parG G G= +

Voltage divider:

11

1 2

Rv vR R

=+

Current divider:

11

1 2

Gi iG G

=+

Open circuit:

1 1 when v v R= =∞ short circuit:

1 1 when i i G= =∞

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

is not a technique for analyzing particular circuits is a conceptual aide that enhances intuition and underscores important

relationships for instance:

a series circuit: ( )1 2 Nv R R R i= + + + its dual, a parallel circuit ( )1 2 Ni G G G v= + + +

Example 2.5 Constructing a Dual Circuit

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2.3 CIRCUITS WITH CONTROLLED SOURCES

Independent sources : the source voltage or current does not depend on any other voltage or current

For instance : the ideal sources introduced in Chapter 1

Controlled sources

Controlled sources : Sources that depend on some other voltage or current is also called dependent sources example :

if 10inv = V, then 100 1000out inv v= = V if 0.2 cosinv tω= V, then 100 20 cosout inv v tω= = V

Classification :

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VCVS : voltage-controlled voltage source cv is proportional to some other voltage xv , regardless of the current i

c xv vµ= xv : control variable µ : voltage gain, dimensionless

VCVS may be any linear voltage amplifier or voltage-amplifying device

CCCS : current-controlled current source the dual of VCVS c xi iβ= , independent of v

xi : control variable β : current gain, dimensionless

CCCS often represents the current-amplifying property of a bipolar junction transistor

VCCS : voltage-controlled current source c m xi g v= , independent of v

mg is called the transconductance, 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 c m xv r i= , independent of i

mr is called the transresistance, because it represents a current-to-voltage transfer effect and has the same units as resistance

These controlled sources mentioned above are linear elements, because the current or voltage is directly proportional to the control variable

a controlled source produces voltage or current only when an independent source activate the control variable

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

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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

recall that a resistive ladder network can be reduced to a single resistance 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

controlled source are allowed since they remain “off” until an independent source activates the control variables

Figure 2.18

equivalent resistance theorem : When a load network consists entirely of resistances or resistances and controlled

sources, the terminal voltage and current are related by eqv R i= , where eqR is a

constant

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

For linear circuits, two analysis techniques can be applied Proportionality Principle Superposition Theorem

Linear elements and circuits

A resistor is a linear element, v Ri=∵ A controlled source is a linear element, y kx=∵ Linear circuit :

A circuit is linear when it contains entirely of linear elements and independent sources

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.

Proportionality Property :

( ) ( ) ( ) , :

if f x y then f Kx Kf x Ky K constant

for any value of x

= = =

Superposition Property :

( ) ( ) ( ) ( ) ( ) , , +

a b b a b a b a ba

a b

if f y f x y then f x x f x f x y yx

for any values of x and x

= = + = + =

Linearity : The property of Linearity combines Proportionality Property and Superposition Property. That is

( ) ( ) ( ) ( ) ( ) , , +

, and a b b a a b b a a b b a a b ba

a b a b

if f y f x y then f K x K x K f x K f x K y K yx

for any values of x and x K and K are arbitrary constants

= = + = + =

For example, an amplifier, ( )100 , that is 100out inv v f x x= = the input might come from an audio mixer combining the voltage 1v from a keyboard with the voltage 2v from a microphone to produce 1 20.5 3inv v v= + , then

( ) ( ) ( ) ( )( ) ( )

1 2 1 2

1 1 2 2 1 2

0.5 3 0.5 3

100 , 100 , 50 300out in

out

v f v f v v f v f v

f v v f v v v v v

= = + = +

= = ∴ = +

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Testing if an element is linear : for an ideal resistor,

( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

,

a resistor is linear

a a b b

a a b b

a a b b a a b b a a b b

a a b b

v Ri f i

Let i K i K i

f i Ri f i Ri

f i f K i K i R K i K i K Ri K Ri

K f i K f i

= =

= +

= =

= + = + = +

= +

∴

power dissipation by a resistor is not a linear function of the current i

( )

2

2 2 2a a b b a a b b

p Ri

R K i K i K Ri K Ri

=

+ ≠ +

∵

Proportionality Principle

Consider a linear circuit driven by a single independent source x (which may be time-varying) that produces the response ( )y f x= if x Kx= then from proportionality property, ( ) ( ) ( )ˆ ˆy f x f Kx Kf x Ky= = = = ∴ any branch variable in a linear circuit varies in direct proportion to the source → proportionality principle : x Kx= then y Ky=

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

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

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

Very useful fro circuits with multiple sources Illustration :

( )

( )

1 2

1 2 1 2

1 1 2

: the value of any branch variable

, : the values of two independent sources

then , : a linear function of both and

the individual contributions from each source are

, 0 , 0,

n

n

n n

y

x x

y f x x x x

y f x y f− −

=

= = ( )

( ) ( ) ( )

2

1 2 1 2 1 2

then from the superposition theorem :

, , 0 0,n n n

x

y f x x y y f x f x− −= = + = +

In order to set one independent source to zero (suppress a source) 0 : "independent" voltage source short circuitv = ⇒

0 : "independent" current source open circuiti = ⇒ 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

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Example 2.11 Superposition with a Controlled Source

Figure 2.22

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2.5 THÉVENIN AND NORTON NETWORK

Two equivalent “source networks” are to be discussed based on Thévenin’s and Norton’s Theorems

Leads to handy source conversions that simplify many analysis and design problems

Thévenin’s and Norton’s Theorems

These two theorems are related to the “terminal” behavior of networks containing

independent sources source network:

a two-terminal network consists entirely of linear elements and sources if controlled sources are present, then the control variables

must be within the same network Note : the reference polarities for v and i show that the source network

“supplies” power

terminology : open-circuit voltage

0oc iv v =

short-circuit current

0sc vi i =

Thévenin resistance oc

tsc

vRi

= , the slope of the v i− line is tR−

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Thévenin’s theorem : Any linear resistive source network acts at its terminals like an ideal voltage source of value ocv in series with a resistor having resistance tR

Thévenin equivalent network:

0,

0,

oc t

oc

oc t sc

v v R i

if i then v v

if v then i v R i

= −

= =

= = =

Norton’s Theorem : Any linear resistive source network acts at its terminals like an ideal current source of value sci in parallel with a resistor having resistance tR

Norton equivalent network:

0,

0,

sc t

t sc oc

sc

i i v R

if i then v R i v

if v then i i

= −

= = =

= =

é, , : Th venin parametersoc sc tv i R Note : the equivalence holds only at the terminals of the source network, not

within the network e.g. for Thévenin network, 0 when 0p i= =

for Norton Network, 2 when 0sc tp i R i= = Example 2.12 Thévenin Parameters from a v-i Curve

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Figure 2.27

Example 2.13 Equivalent Source Network

Figure 2.28

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Thévenin Resistance

It has been known that ocv and sci have clear physical interpretations. Here, we are

to discuss the physical interpretations of oct

sc

vRi

= .

In Thévenin’s and Norton’s equivalent networks,

If 0ocv = or 0sci = , then the networks reduce to a source-free network with

equivalent resistance tR . Thus, The Thévenin resistance of a source network equals the equivalent

resistance of that network when all independent sources have been

suppressed. 0 : "independent" voltage source short circuitv = ⇒

0 : "independent" current source open circuiti = ⇒

Calculation of tR : If no controlled sources are contained, the ladder structure of a source network

can be reduced to tR by series-parallel reduction. If controlled sources are contained, then tR can be found using the method

below. The “dead” source network is connected to an external test source ( tv or

ti )

tt

t

vRi

=

After finding tR , then given sci , then oc t scv R i= , or

given ocv , then sc oc ti v R=

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Source Conversions

Since both Thévenin’s network and Norton’s network are equivalent to a source network, they are also equivalent to each other

voltage sources ( sv ) in series with resistance ( sR ) = current sources ( si ) in parallel with resistance ( sR )

where , s t s s s tv R i i v R= =

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

= =

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Figure 2.34 Example 2.16 Thévenin Network via Source Conversions

Figure 2.35