Module 2

Network Analysis and Synthesis

Classical Filter Design Techniques

constant- k filters,
m-derived filters
composite filters
image parameter technique
CONSTANT-k LOW PASS FILTER
A T or π network is said to be of the constant k type if Z1 and
Z2 are opposite types of reactance satisfying the relation
Z1 Z2 = k 2 where k is a constant, independent of frequency. k is
often referred to as design impedance or nominal impedance of
the constant-k filter. The constant-k, T or π-type filter is also
known as the prototype filter because other complex networks
can be derived from it. Figure 1 shows constant-k, T and
π-section filters.

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Con...

Figure 1: Constant–k Filter

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Hence, the pass band starts at f = 0 and continues up to the

cut-off frequency fc. All the frequencies above fc are in the
attenuation or stop band.

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The characteristic impedance ZOT is real when f <fc . At f =

fc , ZOT = 0. For f >fc , ZOT is imaginary in the stop band,
rising to infinite reactance at infinite frequency. ZOπ is real
when f <fc . At f = fc , ZOπ is infinite and for f >fc , ZOπ is
imaginary.
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Example 1
1.A low-pass filter is composed of a symmetrical π section.
Each series branch is a 0.02 H inductor and shunt branch is a 2
F capacitor. Find (a) cut-off frequency, (b) nominal impedance,
(c) characteristic impedance at 200 Hz and 2000 Hz, (d)
attenuation at 200 Hz and 2000 Hz, and (e) phase shift constant
at 200 Hz and 2000 Hz.

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Cont...

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CONSTANT-k HIGH-PASS FILTER

A constant-k high-pass filter is obtained by changing the
positions of series and shunt arm of the constant-k low-pass
filter, Figure 2 shows a constant-k, T and π section,
high-pass filter.

Figure 2: Constant-k high pass filter

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Cont...

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Example2

1.Find the characteristic impedance, cut-off frequency and pass

band for the network shown below.

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Cont...

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Example 3

3. A high-pass filter section is constructed from two capacitors

of 1 F each and a 15 mH inductance. Find (a) cut-off frequency,
(b) infinite frequency characteristic impedance, (c)
characteristic impedance at 200 Hz and 2000 Hz, (d)
attenuation at 200 Hz and 2000 Hz, and (e) phase constant at
200 Hz and 2000 Hz.

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Cont...

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Cont...

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BAND-PASS FILTER
A band-pass filter attenuates all the frequencies below a lower
cutoff frequency and above an upper cut-off frequency. It passes
a band of frequencies without attenuation. A band pass filter is
obtained by using a low pass filter followed by a high-pass filter.
Figure 3 shows a band pass filter. The series arm is a series
resonant circuit comprising L1 and C1 while its shunt arm is
formed by a parallel resonant circuit L2 andC2 . The resonant
frequency of series arm and shunt arm are made equal.

Figure 3: Band-pass fi lter

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Example 4
4.In a constant-k band-pass filter, the ratio of capacitance in
the shunt and series arms is 100:1. The resonant frequency of
both arms is 1000 Hz. Find the bandwidth,

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BAND-STOP FILTER
A band-stop filter attenuates a specified band of frequencies
and allows all frequencies below and above this band. A
band-stop filter is realized by connecting a low-pass filter in
parallel with a high-pass filter. Figure 4 shows a band-stop
filter. As in the band-pass filter, the series and shunt arms are
chosen to resonate at same frequencyw0 .

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Figure 4: Band-stop filter

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Design of Filter If the load is terminated in load resistance R =

k then at lower cut-off frequency,

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Example5
6. Design a constant-k band-stop filter having cut-off
frequencies at 2000 Hz and 5000 Hz and characteristic
resistance of 600 W.

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m-DERIVED FILTERS
There are two disadvantages of constant-k filters.
(i) The attenuation is not sharp in the stop band.
(ii) The characteristic impedance varies widely in the pass band.
If the constant-k filter is regarded as prototype, it is possible to
design a filter to have rapid attenuation in the stop band and
the same characteristic impedance as the prototype throughout
the pass band and stop band. Such a filter is called an m-
derived filter. Constant-k and m-derived T sections are shown
in Fig 7. The constant-k T-network is identical to that of the
m-derived T-network except that the series arm is multiplier by
m.

Figure 7: Constant-k and m-derived T sections 30 / 36

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Cont...

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Cont...

Figure 8: m-derived T-section filter 32 / 36

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Cont...
Similarly, the constant-k π-section filter can be modified to
m-derived π section filter. A constant-k and m-derived p
sections are shown in Fig 9

Figure 9: (a) Constant-k section (b) m-derived p section

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cont...

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Figure 11: m-derived π-section filter

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Active and passive Filters

m-DERIVED LOW-PASS FILTER

Figure 6 shows an m-derived low-pass T and π filter. At a

particular frequency, the shunt arm of the T section or series
arm of the p section will be in resonance giving a short circuit
to the input for the T section and open circuit for the p section
causing infinite attenuation

Figure 6: m-derived low-pass T and p filter

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Z1 = jωL
1
Z2 = jωC
For sharp cut-off, f∞ should be near to fc . For smaller value of
m, f∞ comes close to fc .
At resonant frequency

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Similarly, for m-derived π-section, the inductance and

capacitance in the series arm constitute a resonant circuit.

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An m-derived filter gives a sharp cut-off.

The decrease in attenuation for frequencies higher than F∞ can
be avoided by connecting an m-derived filter and a constant-k
filter in cascade.

Figure 7: The m-derived π section low-pass filter

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Example 1
1) Design an m-derived T-section low-pass filter having cut-off
frequency of 800 Hz, design impedance of 500ohm and
frequency of attenuation of 1000 Hz.

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m-DERIVED HIGH-PASS FILTER

Figure 8 shows m-derived high-pass T and p filter

Figure 8: m-derived high-pass filter

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The m-derived T-section high-pass filter is shown in Fig 9.

Figure 9: m-derived T-section high-pass filter

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Similarly, for an m-derived π section, the resonant circuit is

formed by the series arm inductance and capacitance.

Figure 10: m-derived π-section high-pass filter

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Example 1

1) Design an m-derived T-section high-pass filter with a cutoff

frequency of 2 kHz, design impedance of 700ohm and m = 0.6.

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COMPOSITE FILTER
A constant-k filter does not give a sharp cut-off.
An m-derived filter section has a sharp cut off, but its
attenuation decreases for frequencies beyond f∞ . Moreover, two
terminating half sections, one at each end,are needed for
impedance matching.
A filter composed of all the above sections is known as a
composite filter. Figure shows a composite low-pass filter.

Figure 11: Composite low-pass filter

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COMPOSITE FILTER

A composite filter consists of the following components

(i) One or more constant-k sections to produce cut-off
frequency, i.e. transition from pass band to the stop band at a
specified frequency fc.
(ii) One or more m-derived sections to give infinite attenuation
at a frequency f∞ near to fc .
(iii) Two terminating m-derived half sections with m = 0.6 to
provide matching with the source and the load.
example 1) Design a composite low-pass filter to have a cut-off
frequency of 1000 Hz and a characteristic impedance of
600ohm. Use one constant-k T section, one m-derived T section
and two terminating half sections with m = 0.6. The frequency
of infinite attenuation is 1050 Hz.

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The composite low-pass filter is shown in Fig 12.

Figure 12: composite low-pass filter

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IMAGE PARAMETERS
If the driving-point impedance at port 1, with impedance
Zi2 connected across port 2, is Zi1 and driving-point
impedance at port 2, with impedance Zi1 connected across
the port 1, is Zi2 then Zi1 and Zi2 are known as image
impedance of the network.
These are also known as image parameters.
The image parameters can be expressed in terms of ABCD
parameters. Figure 13 shows a two-port network
terminated in Zi2 Port 2.

Figure 13: Terminated two-port network

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Cont...

Similarly, if the two-port network is terminated in Zi1 at port 1

as shown in Fig 14.

Figure 14: Terminated two-port network

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Cont...

then solving zi1 andzi2

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Example

1.The ABCD parameters of a two-port network are given

as,A= 65 ,B= 17 1 7
5 , C= 5 ,D= 5 ,Find the image parameters.

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