k
k
k
k
sin(tan
)
1
2
1
∴ =
+
+
=
+
−
(
)
+
+
−
−
A
R
k
k
k
v t
R
k
t
k
k
k
e
o
t
1
1
1
1
2
2
2
1
2
and
V
( )
sin
tan
w
t
for
t
≥
+
0
We normalise the time variable using the circuit time constant as the base and the output voltage
by using the value of R as the base value and obtain the following expression for normalised voltage
v
on
(t) as a function of normalised time t
n
.
v
t
k
kt
k
k
k
e
t
n
t
n
n
on
V for
wher
( )
sin
tan
=
+
−
(
)
+
+
≥
−
−
+
1
1
1
0
2
1
2
ee
and
on
o
v
t
v t
R
t
t
n
( )
( )
=
=
t
(11.3-2)
Zero-StateResponseof
RC
CircuitsforVariousInputs
11.15
This waveform for a case with k
=
4 is shown in Fig. 11.3-13. We had noted under a similar context
(Section 10.8 in Chapter 10 on RL Circuits) that the number k can be interpreted as a comparison
between the characteristic time, i.e., the period of the applied input and the characteristic time of the
circuit, i.e., its time constant. k can be expressed as 2
p
(
t
/T), where T is the period of input. The value
of T is indicative of the rate of change involved inthe waveform, i.e., the speed of the waveform and
time-constant is a measure of inertia in the system. Therefore, an input sinusoid is too fast for a circuit
to follow, if its T is smaller than the time constant
t
of the circuit. Similarly, if input sinusoid has a T
value much larger than time constant of the circuit, the circuit will perceive it as a very slow waveform
and will respond almost the same way it does to DC input. These aspects are clearly brought out in the
expression in Eqn. 11.3-2.
Total voltage
Circuit voltage
Applied current
Transient part
Forced response part
(b)
(a)
0.3
1
1
2
3
4
0.5
–0.5
–1
0.2
0.1
1
2
3
4
t/
τ
t/
τ
–0.1
–0.2
–0.3
Fig. 11.3-13
Unitsinusoidalresponse(normalised)ofaparallel
RC
circuitwith
k
=
4
We make the following observations on the sinusoidal steady-state response of Parallel RC Circuit
with current source excitation from Eqn. 11.3-2:
• The circuit voltage under sinusoidal steady-state response is a sinusoid at the same angular
frequency
w
rad/s as that of input current sinusoid.
• The circuit voltage initially is a mixture of an exponentially decaying unidirectional transient
component along with the steady-state sinusoidal component. This unidirectional transient imparts
an offset to the circuit voltage during the initial period.
• The circuit voltage at its first peak can go close to twice its steady-state amplitude in the case of
circuits with
w
t
>> 1 due to this offset.
• The amplitude of sinusoidal steady-state response is always less than the corresponding amplitude
when DC input is applied. This is due to the capacitive inertia of the circuit. When input is a
current, capacitor in a circuit behaves as electrical inertia, and, when input is a voltage, inductance
in a circuit behaves as electrical inertia. The amplitude depends on the product
wt
and decreases
monotonically with the
wt
product for fixed input amplitude.
• The response sinusoid (circuit voltage) lags behind the input sinusoid (applied current) under
steady-state conditions by a phase angle that increases monotonically with the product
wt
.
• The frequency at which the circuit gain becomes 1/
√
2 times that of DC gain is termed as cut-off
frequency and since this takes place as we go up in frequency it is called upper cut-off frequency.
Upper cut-off frequency of Parallel RC Circuit is at
w
=
1/
t
rad/s. The phase at this frequency
will be – 45
°
.
• Circuit voltage amplitude becomes very small at high frequencies (
w
t
>> 1) and the voltage lags
the input current by
≈
90
°
at such frequencies.
11.16
First-Order
RC
Circuits
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