BEF 12403 - Week 4 - Resistive Circuit Elements.ppt

Week 4
RESISTIVE CIRCUIT
ELEMENTS

2
Objectives
In this topic we will investigate the behaviours of several common types of
circuit element:
• Resistors
• Independent voltage and current sources
• Open-circuit and short circuits
• Ideal Diode

3
Circuit Model
Circuit Model
A model is a description of the properties of a device that we think are
important. Frequently the model will consist of an equation relating the
element voltage and current. Though the model is different from the
electric device, the model can be used in calculations that will predict
how a circuit comprised of actual devices will operate.
Idealised circuit models are called circuit elements.
The behaviour of an electric circuit depends on the behaviours of the individual
circuit elements that comprise the circuit.

4
The Ideal Basic Circuit
Element
The ideal basic circuit element has three attributes:
(1) it has only two terminals, which are points of connections to
other circuit components;
(2) It is described mathematically in terms of current and/or
voltage; and
(3) It cannot be subdivided into other elements.
We use the word ideal to imply that a basic circuit element does not exist
as a realisable physical component. However, ideal elements can be
connected in order to model actual devices and systems.
We use the word basic to imply that the circuit element cannot be further
reduced or subdivided into other elements.

5
Measuring the i-v
Characteristic
Voltmeter-Ammeter Method
The full i-v characteristic can be obtained using the voltmeter-ammeter
method shown in Figure x. In this method a variable source is used to supply
the positive and negative voltages to the circuit element under test and the
ammeter measures the current that flows into circuit for each value of applied
voltage measured by the voltmeter. The advantage of this method is that it
allows four-quadrant measurement of the v-i characteristics since it allows
the application of both positive and negative voltages to the circuit element,
V
A
B
Unknown
circuit
element
Variable
source
i

6
A resistor is a passive circuit element
whose terminal voltage is some
function of the terminal current.
Mathematically, a resistor is a circuit
element that satisfies the equation
Definition
v = f(i)
v
i
a
b
Figure x. Measured volt-ampere
characteristics of physical resistors:
(a) linear; (b) nonlinear
Figure x shows the volt-ampere
characteristics of two physical
resistors; curve a represents a linear
resistor and curve b represents a
nonlinear resistor.
v
Circuit
element
i
Resistor

7
Figure x. v-i characteristics of a linear
resistor.
A linear resistor is a circuit element whose
terminal voltage is directly proportional to
the terminal current. For the reference
directions shown in Figure x(a), we can
write an equation of the form
v = iR
where R is a constant of proportionality
called the resistance of the circuit element.
This proportionality relationship is referred
to as Ohm’s law.
v
Circuit
element
i
R has the dimension of volt per ampere,
or ohm (Ω).
i
v
Δi
Δv
i
v
R



Linear Resistor

8
The schematic symbol for a resistor is shown in Figure x.
Linear Resistance Symbol
R
A
B
i
v
Figure x. Resistance symbol.

9
Ohm’s Law
There two ways of expressing Ohm’s law,
depending on the sign convention used.
Ohm’s law according to the passive sign
convention
The passive sign convention is said to be
used when the ammeter and voltmeter are
connected in the way shown in Figure x. The
measured volt-ampere characteristic shown
in Figure x(b) can then be expressed in the
form
Circuit
element
under
test
v
i
v
i
Δv
Δi
i
v
R



v = iR

10
Ohm’s law according to the active sign
convention
The active sign convention is said to be
used when the ammeter and voltmeter are
connected in the way shown in Figure x. The
measured volt-ampere characteristic shown
in Figure x(b) can be expressed as
v = - iR
v
Circuit
element
i
v
i
Δv
Δi
i
v
R




11
Exercise
The volt-ampere measurements of a device are shown in Figure x, What value of
resistance would have this terminal equation?

12
A nonlinear resistance is a resistance
whose ohmic value does not remain
constant.
Nonlinear Resistor
The dynamic resistance of a nonlinear
resistor is defined as the reciprocal of
the slope of the i-v curve at the point in
question. Thus,










v
i
1
slope
1
R
i
v
Δi
Δv
Figure x. v-i curve for a nonlinear
resistance.

13
Bilateral and Unilateral Resistances
A bilateral resistance is a device that
exhibits the same current-voltage
characteristics regardless of the
direction of current through the device.
Figure x. i-v curves: (a) bilateral resistor ; (b) unilateral resistor.
i
v
Δ
i
Δ
v
A unilateral resistance is one whose
resistance or current-voltage curve
changes markedly with the direction
of the current through the device.
i
v
(a) (b)

14
Short Circuit
A circuit element with resistance
approaching zero is called a short
circuit. Formally, a short circuit is
defined as a circuit element across
which the voltage is zero, regardless
of the current flowing through it. Figure
z depicts the circuit symbol for an ideal
short circuit.
Circuit
element
under
test
v
i
v i
Mathematically, v = 0 for all values of i.
i
v
0
v = 0 for all i

15
Open Circuit
An open circuit is defined as a circuit
element through which the current flow
is zero, regardless of the voltage
across it. Figure z depicts the circuit
symbol for an ideal open circuit.
Circuit
element
under
test
v
i
Mathematically, i = 0 for all values of v.
v
i = 0
i
v
0
i = 0 for all v

16
Ideal Diode
An ideal diode is a circuit element that
allows current to flow through it in one
direction only.
Mathematically, i = 0 for v < 0
v = 0 for i > 0
Circuit
element
under
test
v
i
A
B
v
i
A
B
An ideal diode behaves as a short-circuit
when operating in the 1st quadrant and as
an open-circuit when operating in the 3rd
quadrant.
(a)
(b)
Figure x. (a) Symbol of an ideal
diode; (b) i-v characteristics of an
ideal diode
i
v
0
v = 0 for i > 0
i = 0 for v < 0

17
Ideal Current Source
A n ideal voltage source is a circuit element that maintains a prescribed
voltage across its terminals regardless of the current flowing in those
terminals.
Ideal Voltage Source
An ideal curernt source is a circuit element that maintains a prescribed
current through its terminals regardless of the voltage across those
terminals.
Ideal Sources

18
Ideal voltage and current sources can be further described as either
independent sources or dependent sources.
An independent source establishes a voltage or a current in a circuit
without relying on voltages or current elsewhere in the circuit. The value of
the voltage or current supplied is specified by the value of the
independent source alone.
A dependent source establishes a voltage or current whose value
depends on the value of a voltage or current elsewhere in the circuit.
Dependent sources are sometimes called controlled sources.
Ideal Sources
Classification of Ideal Sources
Independent Source
Dependent Source

19
Time-Invariant and Time-Varying Sources
Independent and dependent sources can be of the the time-invariant type
or the time-varying type. Time-invariant sources are called constant voltage
sources or constant-current sources.
Figure x shows the symbol of a voltage source. To complete specify an ideal
independent voltage source in a circuit, you must include the value of the
supplied voltage and the reference polarity.
Figure x
v(t)

20
The ideal constant voltage source is a
two-terminal element which supplies its
specified current to the circuit it is placed
independently of the value and direction of
the voltage appearing across its terminals.
Figure xa shows the symbol of a voltage
source and Figure xb shows the i-v
characteristics.
(b)
i
v
0
i2
i1
V
Constant Voltage Source
The constitutive equation of a constant
voltage source is of the form
v(t) = V
where V is a constant, independent of
terminal current and time.
(a)
v(t) = V

21
i
v
R



where Δv denotes the change in source voltage
with a given change in current, Δi = i2 – i1. From
Figure xa, we note that the change in voltage
associated with the change in current is zero.
Hence,
(a)
(b)
i
v
0
i2
i1
V
The internal resistance of the voltage source is
0
i
-
i
0
i
v
R
1
2





v(t) = V
Hence, the internal resistance of voltage source
is zero.

22
Example
Using the definition of the ideal independent voltage source, state which
connections in Figure x are permissible and which violate the constraints
imposed by the ideal voltage source.
10 V
10 V 5 V
10 V
Figure x

23
Solution
Connection (a) is valid. Each source supplies voltage across
the same pair of terminals, marked a, b. This requires that
each source supply the same voltage with the same polarity,
which they do.
10 V
10 V

24
Solution
Connection (b) is not permissible. Each source supplies
voltage across the same pair of terminals, marked a,b. This
requires that each source supplies the same voltage with the
same polarity, which they do. valid. Each source supplies
voltage across the same pair of terminals, marked a, b. This
requires that each source supply the same voltage with the
same polarity, which they do not.
5 V
10 V

25
The reference polarity and the source function assigned an unknown
voltage source will depend on the voltmeter connections used in the
measurement. Because one is free to connect a voltmeter in any arbitrary
way across the terminals of the unknown source, it is possible to have two
complementary representations of the same source as demonstrated by
the following example.
V2 V1
A
B
Unknown
circuit
element
Variable
source
i
Schematic symbol of voltage source
Figure x

26
i-v characteristics of unknown circuit
element as measured by voltmeter v1.
i
v1
0 V
Symbolic representation of
unknown circuit element based on
voltmeter v1 connections and
readings.
V
A
B
Schematic symbol of voltage source (continued)

27
Symbolic representation of unknown
circuit element based on voltmeter v2
connections and readings.
- V
A
B
i
V2
0
-V
As a consequence, it should be quite clear that the ideal voltage sources in
Figure x and Figure y are equivalent.
Schematic symbol of voltage source (continued)

28
An AC voltage source is a circuit element whose terminal voltage is
alternating between positive and negative values with respect to time.
V2 V1
A
B
Unknown
circuit
element
Variable
source
i
Time-Varying Voltage Source

29
Figure x. Symbolic representation
of unknown circuit element based
on voltmeter v1 connections and
readings.
AC Voltage Source (continued)
v1(t)
t
0 t1
t2 t3 t4
Vm
i
v1
0 Vm
-Vm
t = 0,2 t = 1
t = 3
v(t) = Vmsin ωt
A
B

30
on voltmeter v2 connections and
readings.
AC Voltage Source (continued)
v2(t)
t
0 t1
t2 t3 t4
Vm
i
v2
0 Vm
-Vm
t = 0,2 t = 3
t = 1 v(t) = -Vmsin
ωt
A
B

31
i(t) = I
The ideal constant current source is a
two-terminal element which supplies its
specified current to the circuit it is
placed independently of the value and
direction of the voltage appearing
across its terminals.
Figure x. Symbol of a constant current
source.
To complete specify an ideal constant
current source you must include the
value of the supplied current and its
reference direction, as shown in Figure
x.
Constant Current Source
The constitutive equation of a constant
current source is of the form
i(t) = I
where I is a constant.

32
Constant Current Source (continued)
i(t) = I
The i-v characteristic of an ideal constant current source is a horizontal line
on the i-v plane.
v
i
0
I
v2 v1
Figure x. i-v characteristics of a current source.

33
Constant Current Source (continued)
i(t) = I
The internal resistance of the current
source is
v
i
0
I
v2 v1






0
v
-
v
I
V
R 1
2
Accordingly, when one “looks” into the
terminals of a current source, one
“sees” an open circuit.
(a)
(b)

34
The reference polarity and the
source function assigned an
unknown current source will
depend on the ammeter
connections used in the
measurement. Because one is
free to connect an ammeter any
direction in the measuring circuit,
it is possible to have to
complementary representations of
the same current source as
demonstrated by the following
example.
Schematic symbol of current source
V
A
B
Unknown
circuit
element
Variable
source
i1 i2

35
element as measured by ammeter i1.
Figure x. Symbolic representation of
ammeter i1 connections and readings.
Schematic symbol of current source (continued)
v
i1
0
I i(t) = I
A
B

36
element as measured by ammeter i2.
ammeter i2 connections and readings.
Schematic symbol of current source (continued)
v
i2
0
-I
i(t) = I
A
B

37
10 V
2 A
Example
In the circuit of Figure x, determine which element is generating
power and which element is consuming power, and how much?
Figure x

38
Solution
The current reference direction of the
voltage source is defined by that of the
current source, which in this case follows
the passive sign convention.
Figure x
10 V
2 A
Hence, power consumed by the source is
W
20
(10)(2)
vi
p 


Since p is positive, the voltage source IS
consuming power.

39
Solution
The voltage reference direction of the
current source is defined by that of the
voltage source, which in this case follows
the active sign convention.
Figure x
Hence, power generated by the source is
W
20
(10)(2)
vi
p 


Since p is positive, the current source IS
generating power.
10 V
2 A

40
An AC current source is a circuit element whose terminal current is alternating
between positive and negative values with respect to time.
V
A
B
Unknown
circuit
element
Variable
source
i1 i2
Time-Varying Current Source

41
element as measured by ammeter i1 and
voltmeter v.
ammeter i1 and voltmeter v
AC Current Source (continued)
v
i1
0
Im
-Im
t = 0,2
t = 1
t = 3
i1(t)
t
0 t1
t2 t3 t4
Im
i1(t) = Imsinωt
A
B
v(t)

42
element as measured by ammeter i1 and
voltmeter v.
on ammeter i1 and voltmeter v
AC Current Source (continued)
i1(t) = Imsinωt
A
B
v(t)
v
i2
0
Im
-Im
t = 0,2
t = 3
t = 1
i2(t)
t
0 t1
t2 t3 t4
Im

BEF 12403 - Week 4 - Resistive Circuit Elements.ppt

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