CHAPTER 17 FILTER NETWORKS
Exercise 99, Page 277
1. Determine the cut-off frequency and the nominal impedance of each of the low-pass filter
sections shown below.
(a) Comparing the low-pass T section with circuit (i) below shows that
L
= 0.5 H, i.e. L = 1 H
2
and C = 0.04 106 F
Cut-off frequency, f C 
1
1

= 1592 Hz
 LC  1  0.04 106  


Nominal impedance, R O 
(i)
L
1


= 5 k
 
6 
C
 0.04 10 
(ii)
(b) Comparing the low-pass  section with circuit (ii) above shows that
C
= 27.8 nF,
2
i.e. C = 2  27.8 = 55.6 109 F and L = 20 103 H
Cut-off frequency, f C 
1
1

= 9545 Hz
 LC   20 103  55.6 109  


Nominal impedance, R O 
 20 103 
L
= 600 
 
9 
C
 55.6 10 
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2. A filter section is to have a characteristic impedance at zero frequency of 500  and a cut-off
frequency of 1 kHz. Design (a) a low-pass T section filter, and (b) a low-pass  section filter to
meet these requirements.
With R O  500  and f C  1kHz ,
capacitance, C =
1
1
= 636.6 nF or 0.6366 F

R O f C   500 1000 
and inductance, L =
RO
500
= 159.2 mH

f c  1000 
(a) A low-pass T section filter is shown in (i) below, where the series arm inductances are
L 159.2

= 79.60 mH, and shunt arm capacitance is 0.6366 F
2
2
(b) A low-pass  section filter is shown in (ii) below, where the series arm inductance is
159.2 mH, and the shunt arm capacitances are
(i)
C 0.6366

= 0.3183 F
2
2
(ii)
3. Determine the value of capacitance required in the shunt arm of a low-pass T-section if the
inductance in each of the series arms is 40 mH and the cut-off frequency of the filter is 2.5 kHz.
For a low-pass T-section filter series arm inductance =
Cut-off frequency, f C 
1
from which,
 LC
L
= 40 mH, hence, L = 80 mH
2
 fC 
1
LC
and
  fC 
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2

1
LC
225
Hence, capacitance, C =
1
L   fC 
2

1
80 103   2.5 103 
2
= 203 nF or 0.203 F
4. The nominal impedance of a low-pass -section filter is 600 . If the capacitance in each of the
shunt arms is 0.1 F determine the inductance in the series arm.
For a low-pass -section filter series arm inductance =
Nominal impedance, R O 
L
C
from which,

R0 

2
C
= 0.1 F, hence, C = 0.2 F
2

L
C
from which, inductance, L = C  R 0   0.2 106  600  = 72 mH
2
2
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226
Exercise 100, Page 280
1. Determine the cut-off frequency and the nominal impedance for each of the high-pass filter
sections shown below
(a) Comparing circuit (a) with circuit (a) below gives: 2C = 500 pF, i.e. C = 250 pF
and L = 50 mH
Cut-off frequency, f C 
1

4 LC 4
Nominal impedance, R O 
1
50 10  250 10 
3
12
= 22.51 kHz
 50 103 
L
= 14.14 k
 
12 
C
 250 10 
(a)
(b)
(b) Comparing circuit (b) with circuit (b) below gives: 2L = 800 mH, i.e. L = 400 mH
and C = 0.2 F
Cut-off frequency, f C 
1

4 LC 4
Nominal impedance, R O 

1

 400 103  0.2 106 
= 281.3 Hz
 400  103 
L
= 1414 
 
6 
C
0.2

10


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2. A filter is required to pass all frequencies above 4 kHz and to have a nominal impedance of
750 . Design (a) an appropriate T section filter, and (b) an appropriate  section filter to meet
these requirements.
f C  4 103 Hz
and R O  750 
Capacitance, C =
1
1

= 26.53 nF
4R O f C 4  750   4 103 
and inductance, L =
RO
750

= 14.92 mH
4f C 4  4 103 
(a) A high-pass T section filter is shown in circuit (i) below where the series arm capacitances are
each 2C, i.e. 2  26.53 = 53.06 nF and the shunt arm is 14.92 mH.
(b) A high-pass  section filter is shown in circuit (ii) below where the shunt arm inductances are
each 2L, i.e. 2  14.92 = 29.84 mH and the series arm is a 26.53 nF capacitor.
(i)
(ii)
3. The inductance in each of the shunt arms of a high-pass -section filter is 50 mH. If the nominal
impedance of the section is 600 , determine the value of the capacitance in the series arm.
50 mH is equivalent to 2L, hence, L = 25 mH
From equation (8), L =
R0
4 f C
from which,
the cut-off frequency, f C 
R0
600

= 1909.86 Hz
4 L 4(25 10 3 )
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From equation (7), capacitance, C =
1
1

 69.44 109 F = 69.44 nF
4 R 0 f c 4 (600)(1909.86)
4. Determine the value of inductance required in the shunt arm of a high-pass T section filter if in
each series arm it contains a 0.5 F capacitor. The cut-off frequency of the filter section is
1500 Hz.
A high-pass T section is shown in circuit (a) of question 1 above, where 2C = 0.5 F,
thus C = 0.25F. fC  1500 Hz .
Capacitance, C =
1
4R O f C
and inductance, L =
from which, R O 
1
1

= 212.2 
4Cf C 4  0.25 106  1500 
RO
212.2
= 11.26 mH

4f C 4 1500 
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229
Exercise 101, Page 282
1. A low-pass T section filter having a cut-off frequency of 20 kHz is connected in series with a
high-pass T section filter having a cut-off frequency of 8 kHz. The terminating impedance of the
filter is 600 . Determine the values of the components comprising the composite filter.
For a low-pass T section filter, fCL  20 103 Hz
Capacitance, C =
1
1

= 26.53 nF
R O f C   600   20 103 
and inductance, L =
RO
600

= 9.549 H
f C   20 103 
A low-pass T section filter is shown in circuit (i) below, where the series arm inductances are
each
L
9.549
, i.e.
= 4.77 mH and the shunt capacitance is 26.53 nF
2
2
(i)
(ii)
For a high-pass T section filter, fCH  8 103 Hz
Capacitance, C =
1
1

= 16.58 nF
4R O f C 4  600   8 103 
and inductance, L =
RO
600

= 5.97 mH
4f C 4  8 103 
A high-pass T section filter is shown in circuit (ii) above, where the series arm capacitances are
each 2C, i.e. 2  16.58 = 33.16 nF and the shunt inductance is 5.97 mH.
The band-pass filter is shown in the circuit below.
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2. A band-pass filter is comprised of a low-pass -section filter having a cut-off frequency of
50 kHz, connected in series with a high-pass -section filter having a cut-off frequency of
40 kHz. The terminating impedance of the filter is 620 . Determine the values of the
components comprising the composite filter.
For a low-pass -section filter:
With R O  620  and f CL  50 kHz ,
capacitance, C =
1
R O f CL
and inductance, L =

1
  620  50000 
= 10.268 nF
RO
620

= 3.95 mH
f CL   50000 
A low-pass  section filter is shown below, where the series arm inductance is 3.95 mH, and
each shunt arm capacitance is
C 10.268

= 5.13 nF
2
2
For a high-pass -section filter:
With R O  620  and f CH  40 kHz ,
capacitance, C =
1
1

= 3.21 nF
4R O f CH 4  620  40000 
and inductance, L =
RO
620

= 1.233 mH
4f CH 4  40000 
A high-pass  section filter is shown in circuit below where the shunt arm inductances are each 2L,
i.e. 2  1.233 = 2.47 mH and the series arm is a 3.21 nF capacitor.
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The band –pass filter section is shown below.
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