ClassificationofTwo-TerminalElements
1.37
1.7.1
lumped and distributed Elements
Electromagnetic effects propagate within the circuit in the form of waves with a finite velocity. Hence,
the time-variation of electromotive force taking place within electrical sources will be felt at different
points in the circuit with different time delays. Therefore, the description of electrical phenomena
in circuit elements, in general, will involve time and space variables. A circuit element can not be
described by a unique voltage and current variable pair in that case.
However, if the circuit dimensions and element dimensions are
such that the time taken by
electromagnetic waves to propagate over the largest dimension in the circuit is small compared to
the
characteristic time of variation of the electromotive forces acting in the circuit, then, the retardation
effect due to finite velocity of electromagnetic waves can be ignored and a simple circuit model for
elements can be used.
Assume that the circuit contains many sources of sinusoidal nature and the maximum angular
frequency of such source functions is
w
o
rad/sec. That is, there is some voltage or current variable
of the form
X sin(
w
o
t) present in one of the sources in the circuit. Then, this variable will complete
one cycle of oscillation in 2
p
/
w
o
seconds. The
characteristic time of variation in this circuit is then
2
p
/
w
o
seconds. That is, this is a measure of the minimum time-interval over which significant changes
in circuit variables will take place. Now let us assume that the largest dimension of any element in the
circuit (including connecting wires) is
d meters. Then, electromagnetic waves will take
d/
c seconds
to cover this distance where
c is the velocity of light in free space. If
d/
c is much less than 2
p
/
w
o
, we
may ignore the travel time of electromagnetic disturbances and model all the elements in the circuit
by terminal voltage-current relationships. Note that this conclusion is valid only for operation at
≤
w
o
rad/sec.
The ‘characteristic time of variation’ of a circuit depends on the waveshape of source functions
present in the circuit. The source functions need not be sinusoidal always. However, it is possible to
expand arbitrary time-functions in terms of sinusoidal functions under certain conditions. The highest
frequency that appears in such expansions will have to be used to decide
whether the circuit can
be modeled by ignoring retardation effect. There is one kind of source function,
which if present
in a circuit, will not permit us to ignore retardation effect. That is a source function that contains
sudden, instantaneous changes in values – that is, a function that has step discontinuities. Obviously,
the characteristic time of variation of such a function is zero.
An element is classified as a
lumped element if the net effect of electrical phenomena taking place
within that element can be described in terms of only its terminal voltage and current variables,
irrespective of its internal details and geometry. This amounts to neglecting the retardation effect in
the element. If the electrical description of an element calls for voltage and current variables that are
functions of space variables over the element (in addition to time variable), the element is called a
distributed element.
An electrical device can be modeled by a
lumped model only for a range of frequencies in the
source functions in the circuit. The same electrical device may call for a
distributed model if the source
functions in the circuit vary rapidly enough to make retardation time within the device significant.
For instance, consider a solenoid coil of length 5 cm and diameter 1 cm with 100 turns of wire. One
may be tempted to assume that the largest dimension of the coil is its length-
i.e., 5 cm. It is not. The
largest dimension that we need here is the length of the wire and that is about 314 cm. The retardation
time over this length
=
3.14/3
×
10
8
≈
10 ns. If
w
o
is the highest frequency of sinusoidal components
present in the sources within the circuit, then, the ‘characteristic time of variation’ is 2
p
/
w
o
sec. If this
time is 10ns then
w
o
is 628 Mrad/sec. The corresponding cyclic frequency will be 100 MHz. Thus, this
coil can be modeled as a two-terminal lumped inductance with good accuracy if the circuit contains
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1.38
CircuitVariablesandCircuitElements
source sinusoidal components at 1 MHz or below. However, it will call for a distributed model if the
sources contain
>
10 MHz sinusoidal components.
Consider a power transmission line of length 300 km. The retardation time over the length of the
line is 1 ms. 50 Hz sinusoidal source functions have a waveform period of 20 ms. Hence, a lumped
parameter model for this line amounts to ignoring 1 ms in comparison with 20 ms. But 20 ms is the
time required for one full oscillation of source function. Significant change in function value takes
place within a quarter cycle –
i.e., in 5 ms. Obviously, this power line requires a distributed model
even at 50 Hz.
A 1 nF ceramic capacitor typically has two leads of 1.5 cm each. The retardation time over 3 cm
is 100 ps (1 ps
=
10
-
12
sec). This corresponds to a frequency value of 10 GHz. Therefore, a lumped
parameter model will be satisfactory for frequencies below 10 MHz. A
distributed model will be
necessary for frequencies
>
50 MHz.
All circuit elements of arbitrary dimensions can be modeled by lumped elements if all the sources
are DC sources. But, no element, of any dimension whatsoever, can be modeled by lumped parameter
model to obtain
detailed circuit solution at and around the instants at which such DC sources are either
switched into the circuit or switched out of the circuit. Such switching operations represent very rapid
changes in circuit variables and retardation time can not be ignored in comparison with infinitesimal
intervals.
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