E1
APPENDIX E
SINGLETIMECONSTANT CIRCUITS
Introduction
Singletimeconstant (STC) circuits are those circuits that are composed of or can be reduced
to one reactive component (inductance or capacitance) and one resistance. An STC circuit
formed of an inductance
L
and a resistance
R
has a time constant
.
The time constant
τ
of an STC circuit composed of a capacitance
C
and a resistance
R
is given by
=
CR.
Although STC circuits are quite simple, they play an important role in the design and analysis
of linear and digital circuits. For instance, the analysis of an amplifier circuit can usually be
reduced to the analysis of one or more STC circuits. For this reason, we will review in this appen
dix the process of evaluating the response of STC circuits to sinusoidal and other input signals
such as step and pulse waveforms. The latter signal waveforms are encountered in some amplifier
applications but are more important in switching circuits, including digital circuits.
E.1 Evaluating the Time Constant
The first step in the analysis of an STC circuit is to evaluate its time constant
.
E.1.1 Rapid Evaluation of
In many instances, it will be important to be able to evaluate rapidly the time constant
of a
given STC circuit. A simple method for accomplishing this goal consists first of reducing the
excitation to zero; that is, if the excitation is by a voltage source, short it, and if by a current
=
LR
⁄
Example E.1
Reduce the circuit in Fig. E.1(a) to an STC circuit, and find its time constant.
Solution
The reduction process is illustrated in Fig. E.1 and consists of repeated applications of Thévenin’s theo
rem. From the final circuit (Fig. E.1c), we obtain the time constant as


=
CR
4
{
[
R
3
R
1
(
+
R
2
)
]
}
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Appendix E
SingleTimeConstant Circuits
source, open it. Then, if the circuit has one reactive component and a number of resistances,
“grab hold” of the two terminals of the reactive component (capacitance or inductance) and
find the equivalent resistance
R
eq
seen by the component. The time constant is then either
or
CR
eq
.
As an example, in the circuit of Fig. E.1(a), we find that the capacitor
C
“sees”
a resistance
R
4
in parallel with the series combination of
R
3
and
R
2
in parallel with
R
1
. Thus


and the time constant is
CR
eq
.
In some cases it may be found that the circuit has one resistance and a number of capacitances
or inductances. In such a case, the procedure should be inverted; that is, “grab hold” of the resis
tance terminals and find the equivalent capacitance
C
eq
, or equivalent inductance
L
eq
, seen by
this resistance. The time constant is then found as
C
eq
R
or
L
eq
/
R.
This is illustrated in Exam
ple E.2.
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 Spring '10
 kal
 Electronic amplifier, Lowpass filter, Thévenin's theorem, Electronics terms, STC Circuits

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