Filter is the circuit that capable of passing signal from input to output that has frequency within a specified band and attenuating all others outside the band. This is the property of selectivity. They are four basic types of filters. They are lowpass, high-pass, band-pass, and band-stop.
The basic filter is achieved with various combinations of resistors, capacitors, and sometimes inductors. These are named as passive filters. Active filters besides using passive element, it also uses active element such as transistors or operational amplifiers to provide desired voltage gains or impedance characteristics.
Each type of filter response can be tailored by circuit component values that have Butterworth, Chebyshev, or Bessel characteristics. Each of these characteristics is identified by the shape of its response curve and each has an advantage in certain application.
In this chapter, we shall explore the basic concepts of designing passive and active filters namely the low-pass, high-pass, band-pass, and band-stop filters.
The characteristics of each filter type in terms of it bandwidth, critical frequency, and gain would be discussed in details. We shall not be discussing the methods to design filter that has Butterworth, Chebyshev, or Bessel characteristics.
This part of the lecture shall be left to be discussed in the analogue circuit design subject.
Ideally, filter should have the characteristics as shown in Fig. 13.1. In practice, such characteristics are not possible to be achieved. In practice, the attenuation of the signal after the critical frequency is either exponentially increased or decreased. It is not abruptly decreased or increased as shown in Fig. 13.1.
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13 Filter
Figure 13.1: Ideal filter characteristic – low-pass, high-pass, band-pass, and band-stop
Four basic passive filters namely the low-pass, high-pass, band-pass, and bandstop are be discussed in this section.
A basic passive low-pass circuit is shown in Fig. 13.2.
Figure 13.2: Passive low pass filter
The transfer function H(s) of the filter is
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13 Filter
H(s) =
V
0
V i
=
R
1 /
+
1 j
ω
C
/ j
ω
C
(13.1)
The magnitude of the function shall be
1
1
+
(
ω
RC )
2
. The critical frequency f
C
of the filter shall be f
C
=
1
2
π
RC
(13.2)
It can determined either by setting function by setting
ω
RC = 1 and the – 3dB
1 frequency is determined by setting
1
= .
1
+
(
ω
RC )
2 2
The phase of the filter shall be
φ
= - tan
-1
(
ω
RC).
Based on equation (13.1), as frequency increases, the function H(s) is approaching zero. As frequency decreases, the function H(s) is approaching one.
The characteristic of the filter is shown in Fig. 13.3.
Figure 13.3: The characteristics of a passive low pass filter
A basic passive high-pass filter circuit is shown in Fig. 13.4.
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13 Filter
Figure 13.4: Passive high-pass filter
The transfer function of the filter H(s) is
H(s) =
V
0
V i
=
R
R
+
1 / j
ω
C
The magnitude of the function shall be
ω
RC
1
+
(
ω
RC )
2
(13.3)
. The critical frequency f
C
of the filter shall be f
C
=
1
2
π
RC
(13.4)
It can determined either by setting function
ω
RC = 1 and the -3dB frequency can be determined by setting
1
+
ω
(
RC
ω
RC )
2
=
1
2
.
The phase of the filter shall be
φ
= 90
0
- tan
-1
(
ω
RC).
Based on equation (13.3), as frequency increases, the function H(s) is approaching one. As frequency decreases, the function H(s) approaches zero. The characteristic of the filter is shown in Fig. 13.5.
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13 Filter
Figure 13.5: The characteristics of a passive high pass filter
A basic passive band-pass filter circuit is shown in Fig. 13.6.
(13.5)
Figure 13.6: Passive band-pass filter
The transfer function of the filter H(s) is
H(s) =
V
0
V i
=
R
+
R j (
ω
L
−
1 /
ω
C )
The magnitude of the function shall be
R
R
2 +
(
ω
L
−
1 /
ω
RC )
2
. The resonant frequency f o
of the filter shall be f o
=
2
π
1
LC
(13.6)
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13 Filter
It can be determined by setting the imaginary part of the impedance to zero. The –
R
3dB frequencies are determined by solving expression =
1
.
R
2 +
(
ω
L
−
1 /
ω
RC )
2 2
Based on equation (13.5), the characteristic of the filter is shown in Fig. 13.7.
Figure 13.7: The characteristics of a passive band pass filter
A basic passive band-stop filter circuit is shown in Fig. 13.8.
Figure 13.8: Passive band pass filter
The transfer function of the filter H(s) is
H(s) =
V
0
V i
=
R j
+
(
ω j
L
(
ω
−
L
1 /
−
ω
1 /
C )
ω
C )
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(13.7)

13 Filter
The magnitude of the function shall be
R
2
(
ω
L
−
1 /
ω
RC )
+
(
ω
L
−
1 /
ω
RC )
2
. The resonant frequency f o
of the filter shall be f o
=
2
π
1
LC
(13.8)
It can be determined by setting the imaginary part of the impedance to zero. The –
3dB frequencies are determined by solving
R
2
(
ω
L
+
−
(
ω
L
1 /
−
ω
RC
1 /
)
ω
RC )
2
=
1
2
.
Based on equation (13.7), the characteristic of the filter is shown in Fig. 13.9.
Figure 13.9: The characteristics of a passive band-pass filter
Four basic active filters employing operational amplifier namely the low pass, high pass, band pass, and band reject are be discussed in this section.
A basic first-order low-pass filter is shown in Fig. 13.10.
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13 Filter
Figure 13.10: An active first-order low pass filter
The transfer function, which the voltage gain function H(s) is
H(s) = −
R f
R i
⋅
1
+
1 j
ω
C f
R f
The magnitude of the gain function is
R
R i f ⋅
1
1
+
(
ω
C f
R f
)
2
(13.9)
. The critical frequency is
ω
C
=
1
R f
C f
, while the phase is – tan
-1
(
ω
C f
R f
).
A basic first order low pass filter is shown in Fig. 13.11. The transfer function, which the voltage gain function H(s) is
H(s) = −
R f
R i
⋅
1
+ j
ω
C f j
ω
C
R f f
R f
(13.10)
The magnitude of the gain function is
R
R i f ⋅
ω
C f
R f
1
+
(
ω
C f
R f
)
2
. The critical frequency is
ω
C
=
1
R f
C f
, while the phase
φ
is
φ
= 90
0
– tan
-1
(
ω
C f
R f
).
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13 Filter
Figure 13.11: An active first-order high pass filter
The basic circuit of an active band-pass filter is shown in Fig. 13.12. The circuit comprises of a low-pass filter stage, a high-pass filter stage, and an inverting stage that provides voltage gain.
Figure 13.12: An active band-pass filter
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13 Filter
The basic circuit of an active band-pass filter is shown in Fig. 13.13. The circuit comprises of a low-pass filter stage, a high-pass filter stage, and summing amplifier stage that provides voltage gain and band-reject.
Figure 13.13: An active band-stop filter
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13 Filter
13.1.
Show that circuit below is a low-pass filter. Calculate the corner frequency f c
if L = 2mH, C= 10
µ
F, and R = 10 k
Ω
.
13.2.
Find the transfer function V o
/V s
of the circuit in the figure. Show that the circuit is a low-pass filter.
13.3.
Determine the type of filter for the circuit shown in the filter and find its corner frequency.
13.4.
Design an RL low-pass filter that uses a 50 mH coil and has a cutoff frequency of 5.0 kHz.
13.5.
Design a series RLC type band-pass filter with cutoff frequencies of 10 kHz and 11 kHz. Assuming that C = 80.0 pF, find R, L, and Q.
13.6.
Find the transfer function for each of the active filters.
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13 Filter
13.7.
Obtain the transfer function of the active filter in the figure and determine the type.
13.8.
A high-pass filter is shown in the figure, show that the transfer function is
H(
ω
) =

 1

+
R f
R i


 1
+ j
ω
RC j
ω
RC
.
1.
Charles K. Alexander and Matthew N.O Sadiku, “Electric Circuits”, 2 nd edition, McGraw-Hill 2004.
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