Analogue Electronics (Aero) 1
1.1
Analogue Electronics (Aero) 1
§ Filter Circuits
1.2
At high enough frequencies, |ZC| << |ZR|, so
the circuit is approximately,
Low Pass Filter :
and Vout = 0. The gain curve of Vout/Vin from
low to high frequency looks like,
Before carrying out a detailed mathematical
analysis, how would we expect this to respond
as frequency is varied?
At low frequencies:
By going to low enough frequency we can
always ensure that |ZC| >> |ZR|. ZC dominates
the circuit, which becomes approximately,
!
i.e. Vout = Vin.
1
Simple analysis is a good guide to the
‘appearance’ of the frequency response, but to
get quantitative information we need to resort
to a proper mathematical analysis.
Vout
ZC
" j #C
=
=
Vin ZC + Z R " j #C + R
1
1
1
1
=
=
=
=
R
#CR 1+ j#CR
#
1+
1+
1+ j
" j #C
"j
#0
2
!
Analogue Electronics (Aero) 1
1.3
Analogue Electronics (Aero) 1
where !0 = 1/RC is referred to as the
characteristic frequency of the system.
1.4
Notice that if we injected an input signal
which was not a pure sine wave, but a mixture
(spectrum) of frequencies, the effect of this
circuit would be to pass signals with
frequency less than !0 unaltered (since in this
regime Vout/Vin ! 1), and to attenuate signals
with a frequency greater than !0. The
reduction is greater at higher frequencies.
Hence the circuit is called a low pass filter.
When ! << !0 then Vout/Vin ! 1
1
When ! >> !0 then Vout/Vin ! j ""
0
Magnitude of the output voltage decreases as
! is increased – if we make ! ten times
bigger, Vout becomes ten
! times smaller,
At the characteristic frequency ! = !0
High Pass Filter :
Vout
1
V
1
1
=
so out = 2 2 =
Vin 1+ j.1
Vin
2
1 +1
The Bode plot for this system displays the
amplitude ratio in dB where ‘ratio in dB’, R,
!
"V %
1
R = 20log10 $ out ' = 20log10
= 20 ( ).150 = )3
V
2
# in &
At low frequency, ZC = -j/!C so |ZC| >> |ZR|
and Vout ! 0. At high frequency |ZC| << |ZR| and
so Vout ! Vin. The frequency response, drawn
on a Bode plot, is expected to be,
!
3
4
Analogue Electronics (Aero) 1
1.5
Analogue Electronics (Aero) 1
This time the circuit passes high frequency
signals and attenuates low frequency signals,
acting as a high pass filter.
In the high pass and low pass filters we use the
fact that the impedance of an RC circuit can
be written as R–j/!C or R+jX where the
reactive term, X, becomes infinite as ! " !
and tends to zero as ! " 0. However if the
reactive part of the circuit arises from a
combination of inductors and capacitors we
can produce more complex variations of X
with frequency and can design circuits to our
own specification – in particular we can make
X become zero at a specific frequency of
interest to us.
Vout
R
1
=
=
1
1
Vin R +
1+
j"C
j"CR
When ! << !0 then
Vout
1
1
=
#
= j"CR
Vin 1+ j"CR 0 + j"CR
1
When ! >> !0 then Vout/Vin ! 1+ j0 ! 1
!
1.6
Series Resonant Circuits :
!
!
The total impedance of the circuit is given by
Z = R + j"L # j
5
6
!
Analogue Electronics (Aero) 1
1.7
Given particular values of L and C we can
make the reactive term, X, vanish if we choose
the frequency ! to be a particular !O such
that,
"O L #
1
1
1
= 0 $ "O L =
$ "O2 =
"O C
"O C
LC
fO =
!
1
1
=
2"# 2" LC
!
At this resonant frequency, the impedance of
the circuit is Z = R because X = 0. Since the
magnitude of Z is
!
Z = R 2 + ("L # "1C )
at any other frequency the magnitude of Z will
be greater than R. This means that the
magnitude of any current flows are at a
maximum at the resonant frequency.
2
7
$
1
1 '
= R + j&"L #
) = R + jX
%
"C
"C (