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12.10.6
Lc circuit as an Averaging Filter
Let us see how the low-pass filter in Example 12.10-1 has achieved this extent of attenuation of AC
component. We noted that the resonant frequency of the circuit is about 0.98 kHz. Thus, the ratio
between the lowest frequency present in the input, i.e., 10 kHz, and the resonant frequency of the
circuit is about 10. We now show that the gain of a LC low-pass filter for a sinusoidal input at f Hz is
∝
( f
n
/f )
2
, where f
n
is the resonant frequency in Hz unit and f >> f
n
. We show it in two ways.
Refer to Eqn. 12.9-3 that gives the frequency-response function for capacitor voltage in a standard
series RLC circuit. It is reproduced in the following.
V
j
V j
C
S
n
n
n
C
C
n
n
(
)
(
)
(
)
tan
w
w
w
w
w
x w w
f
f
xw w
w
=
−
+
∠
= −
−
2
2
2 2
2
2
2
1
4
2
where
22
2
−
w
rad
V
j
V j
x
x
x
C
S
n
n
n
(
)
(
)
(
)
(
)
w
w
w
w
w
x w w
x
w
=
−
+
=
−
+
=
2
2
2 2
2
2
2
2 2
2 2
4
1
1
4
where
w
w
x
w
w
n
n
n
f
f
x
x
x
x
f
f
=
≈
−
+
≈
−
=
=
=
1
4
1
1
2 2
2 2
2 2
2
2
2
(
)
(
)
n
for
x
>>
1
Fig. 12.10-10
Theripplecomponent
ofoutputvoltagein
Example:12.10-1
0.1
Volts
Time in ms
–0.1
0.05 0.1 0.15 0.2 0.25
12.40
SeriesandParallel
RLC
Circuits
The second way to appreciate the second-order gain characteristic of LC low-pass filter is by
considering the following qualitative argument based on phasor impedances. The purpose of the
averaging filter is to eliminate AC components in the output. Some intervening element has to absorb
the AC component in the input if it is not to appear at output. An element which can take large
AC voltage across it while keeping current low is inductance (since its impedance increases with
frequency) and a capacitor across the output will shunt out whatever AC current that tries to get
into the load resistance (because capacitor has low impedance at high frequency). Thus, the inductor
chokes the high frequency current while absorbing almost all the input AC voltage content and the
capacitor located across the output absorbs whatever AC current that appears even after the inductor
chokes it. This is how a LC filter does averaging. There is a two-fold action – a series element
that makes it more and more difficult for AC current to flow as frequency increases and a shunt
element which makes it more and more difficult for AC voltage to develop across it as frequency
increases. This explains the inverse square dependence of gain on frequency at high frequency
values.
At a sufficiently high frequency, any resistance or inductor that is connected in parallel with a
capacitor may be ignored for approximate calculation. Similarly, at a sufficiently high frequency
any resistance or current connected in series with an inductor may be ignored for approximate
calculations. For example, see the phasor equivalent of circuit in this example at 10 kHz in
Fig. 12.10-11 (a).
(a)
–
v
o
(
j
)
ω
0.1
Ω
2
Ω
j
5.026
Ω
–j
0.0482
Ω
+
(b)
j L
ω
j C
ω
1
Fig. 12.10-11
PhasorequivalentcircuitsforExample:12.10-1
Obviously, that 2
W
and 0.1
W
can be safely ignored. Therefore, the circuit can
be approximated by the circuit in (b) for high frequencies. Then the gain function is
=
+
=
−
=
−
≈
=
>
1
1
1
1
1
1
2
2
2
2
2
j C
j L
j C
LC
f
f
f
n
n
n
w
w
w
w
w w
w
w
/
for
>>
f
n
. Current in the circuit
(b) lags voltage by 90
°
at high frequency because the net reactance is inductive at
w
>
w
n
. And the
capacitance voltage lags behind circuit current by 90
°
. Therefore, the output voltage will be 180
°
out
of phase with respect to input at high frequencies.
We could have used this approximate gain expression to solve this example. The gain at 10 kHz
will be
=
(0.98/10)
2
=
0.0096, at 30 kHz
=
(0.98/30)
2
=
0.00106 etc. Phase of output will be 180
°
at all
these frequencies.
Note that a LC filter will offer superior performance in averaging applications compared to RC
circuit. This is due to the fact that the gain of an RC averaging circuit falls off in inverse proportion
to
w
for large values of
w
(large compared to 1/
t
), whereas the gain of an LC filter falls in inverse
proportion to
w
2
for large values of
w
(large compared to
w
n
). Moreover, averaging by an LC filter is
more efficient since an inductor does not dissipate power. Hence averaging at high power levels (few
10
’
s of watts and up) is usually done by LC filters. RC averaging is commonly used in low-power
signal processing applications.
ResonanceinSeries
RLC
Circuit
12.41
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