Filters

1. Unit – VI
Filters & Attenuators
Dr.G.Radhika
Dept. of EEE

2. Filters and Attenuators
https://www.youtube.com/watch?v=LqEiIeN96c0
https://www.youtube.com/watch?v=NndixY_aw7k
• Classification of Filters,
• Classification of Pass band and Stop band,
• Characteristic Impedance in the Pass and Stop Bands,
• Constant-k and m derived filters
– Low Pass Filter and High Pass Filters, Band Pass filter and Band Elimination filters
(qualitative treatment only),
• Attenuators-Symmetrical and asymmetrical (qualitative treatment only).

11. Filters and Attenuators
https://www.youtube.com/watch?v=LqEiIeN96c0
https://www.youtube.com/watch?v=NndixY_aw7k
• Classification of Filters,
• Classification of Pass band and Stop band,
• Characteristic Impedance in the Pass and Stop Bands,
• Constant-k and m derived filters
– Low Pass Filter and High Pass Filters, Band Pass filter and Band Elimination filters
(qualitative treatment only),
• Attenuators-Symmetrical and asymmetrical (qualitative treatment only).

12. • Filter is a electrical circuit which can be designed to modify,
reshape or reject all the undesired frequencies of an electrical
signal and pass only the desired signals.
• It is usually a frequency selective network that passes a specified
band of frequencies and blocks signals of frequencies outside this
band.
• Filters are electronic circuits that remove any unwanted
components or features from a signal.
• Filters alter the amplitude and/or phase characteristics of a signal
with respect to frequency.
Introduction

13. • Filter is basically a linear circuit that helps to remove unwanted
components such as Noise, Interference and Distortion from the
input signal.
• Ideally filters should produce no attenuation in the desired band,
called the transmission band or pass band and should provide
total or infinite attenuation at all other frequencies, called
attenuation band or stop band.
• The frequency which separates the transmission band and the
attenuation band is defined as the cut-off frequency of the filters
and is designated by fc.

14. Function and Applications of Filters
• Eliminate background noise
• Radio tuning to a specific frequency
• Direct particular frequencies to different speakers
• Modify digital images
• Remove specific frequencies in data analysis
• Filter networks are widely used in communication systems to
separate various voice channels in carrier frequency telephone
circuits.
• Filters also find applications in instrumentation, tele-metering
equipment, etc. where it is necessary to transmit or attenuate a
limited range of frequencies.

15. Filters
• Filters are electronic circuits that remove any unwanted components or
features from a signal. In simple words, you can understand it as the circuit
rejects certain band of frequencies and allows others to pass through. They
are widely used in Instrumentation, Electronics and Communication
Systems especially in Signal and Image processing systems.

16. Applications of Filters
• Filter Circuits are used to eliminate background Noise.
• They are used in Radio tuning to a specific frequency.
• Used in Pre-amplification, Equalization, Tone Control in Audio
Systems.
• They are also used in Signal Processing Circuits and Data Conversion

17. Function and Applications of Filters
• Eliminate background noise
• Radio tuning to a specific frequency
• Direct particular frequencies to different speakers
• Modify digital images
• Remove specific frequencies in data analysis
• Filter networks are widely used in communication systems to
separate various voice channels in carrier frequency telephone
circuits.
• Filters also find applications in instrumentation, telemetering
equipment, etc. where it is necessary to transmit or attenuate a
limited range of frequencies.

18. Types of Filters
Basically filters are of two types: Active filters and Passive filters.
• Active filters are the filters having active elements like OP-AMP and
Transistor, in addition to resistor and capacitor. An active filter not only
passes or stops a particular band of frequency but also amplifies the signal
that passes through it.
• Passive filters are made up of only Passive components like inductor
and capacitor. Such filters cannot amplify the signal that passes through
them.

19. Classification of Passive Filters
On the basis of functions they perform, Passive filters are classified
as follows:
1. Low-Pass Filters (LPF)
2. High-Pass Filters (HPF)
3. Band-Pass Filters (BPF)
4. Band-Stop Filters (BSF)

20. (i) Low-Pass Filters
• These are the filters that pass all the frequencies lower than the selected
cut-off frequency fc and attenuate/stop/suppress the signals whose
frequency is greater than fc
• For the low-pass filter the pass-band is from 0 to fc and stop-band is from fc to ∞.
• This transmits currents of all frequencies from zero up to the cut-off frequency. The
band is called pass band or transmission band.
• Thus, the pass band for the Low Pass filter is the frequency range 0 to fc.
• The frequency range over which transmission does not take place is called the stop
band or attenuation band.
• The stop band for a Low Pass filter is the frequency range above fc.

21. (ii) High-Pass Filters
• It is a filter that passes all the signals whose frequency is higher than the
cut-off frequency fc and stops the signals whose frequency is less than fc .
Therefore, for a high-pass filter (HPF) the pass band is from fc to ∞ and
the stop-band is from 0 to fc.

22. (iii) Band-Pass Filters
• It has two cut-off frequencies: lower cut-off frequency (f1) and higher
cutoff frequency (f2).
• This filter passes all those signals whose frequency lies inside the band f1
to f2 and stops all other frequency signals.
• Pass band: f1 to f2
• Stop band: 0 to f1 and f2 to ∞.

23. (iv) Band-Stop Filters
• It is a filter that stops particular band of frequencies and passes all other
frequencies. It is just opposite to that of a band-pass filter (BPF).
• Therefore, for a BPF, the following are given:
• Pass band: 0 to f1 and f2 to ∞
• Stop band: f1 to f2 .

27. Filter Networks
• Ideally a filter should have zero attenuation in the pass band.
• This condition can only be satisfied if the elements of the filter are dissipation less,
which cannot be realized in practice.
• Filters are designed with an assumption that the elements of the filters are Purely
Reactive.
• Filters are made of Symmetrical T, or π sections.
• T and π sections can be considered as combinations of unsymmetrical L sections as
shown in Fig.1
Fig. 1. Filter Networks

28. Fig. 1. Filter Networks

29. • The ladder structure is one of the commonest forms of filter network. A cascade
connection of several T and π sections constitutes a ladder network.
• A common form of the ladder network is shown in Fig. 2.
• Fig. 2 (a) represents a T section ladder network, whereas
Fig. 2 (b) represents the π section ladder network.
• It can be observed that both networks are identical except at the ends.
Fig. 2. Ladder Networks

30. Fig. 2. Ladder Networks

31. Parameters of A Filter
• There are four important parameters that are necessary to analyze
the performance of a filter network.
1. Propagation constant γ = (α+jβ)
2. Attenuation constant α
3. Phase shift constant β
4. Characteristic impedance Z0

32. Propagation Constant γ
• For any two-port network terminated by characteristic impedance Z0, as
shown in Figure, we can write the following:
where γ is known as propagation constant. Propagation constant
determines the propagation performance of any two-port network.
where α is real part of γ and is known as attenuation constant of the
filter, and β is imaginary part of γ and is known as phase constant.

33. Attenuation Constant α
• Whenever a signal passes through a passive network/filter, it gets attenuated,
because passive components like capacitors and inductors consume some of the
signal energy. The attenuation constant determines the attenuation of the signal
when it passes through the filter.
• Attenuation can be expressed in decibels or nepers.
• Neper: It is defined as the natural log of the ratio of input current or voltage or
power to the output current or voltage or power.
Decibel: It is defined as the ten times the common log of the ratio of input
current/voltage/power and output current/voltage/power.
Decibel (D) can be written as follows:

34. Phase Shift Constant (β)
• When the signal passes through the filter, it gets some shift in phase. Phase
shift constant signifies the phase shift in the signal when it passes through
the filter.
• The unit of phase shift is radians or degrees. The relation between radians
and degrees can be written as follows:

35. Characteristics Impedance
FILTER
Consider filter as a two port network, where we will give alternating
input of multiple frequencies and at output we will get alternating
signal of selected frequency range only.

36. Image Parameters
if port 1-1’ of the network is
terminated in ZI1, the input
impedance of port 2-2’ is ZI2; and
if port 2-2’ is terminated in ZI2, the
input impedance at port 1-1’ is ZI1.
Then, ZI1 and ZI2 are called image
impedances of the two port
network
The reason for using image parameter here is they link us with
maximum power transfer theorem, that says when load impedance is
equal and conjugate of source impedance, maximum power will be
transferred.

37. Analysis of Filter Networks
• Filters are made of Symmetrical T- or π-Sections.
• In this section, we will explain how the basic parameters of
T- and π-networks are determined.

38. Symmetrical T Network
Characteristic Impedance (Z0)
• If a two-port network is symmetrical, the image impedance Zi1 at port 1-1′ is
equal to the image impedance Zi2 at port 2-2′ and that image impedance is
called the characteristic impedance Z0.
• When the network is terminated with Z0 (characteristic impedance), then
input impedance Zin =Z0
The input impedance of the T-network shown in figure is written as follows:

43. Propagation Constant (γ)
• As shown in Figure, I1, I2, V1 and V2 are input current, output current,
input voltage and output voltage, respectively.
• Applying KVL in mesh II, as shown in Figure, we get the following:
Now, by definition, we get the following form:
Substituting this value in eq.

44. Substituting this value in the equation, we get the following form:
Squaring both sides, the equation can be written as follows:

48. p-Network
Consider asymmetrical p-section shown in Fig. 1 When the network is
terminated in Z0 at port 2-2’, its input impedance is given by :
Fig. 1

49. which is the characteristic impedance of a symmetrical p-network

52. Attenuation Constant (α) and Phase Shift
Constant (β)
• Propagation constant is a complex function represented as γ = α + jβ.
The real part α is a measure of change in the magnitude of the current or
the voltage in the network and the imaginary part β is a measure of
difference in the phase between the input and output currents or
voltages.
Substituting γ = α+jβ, in the equation, we get the following:

55. Summary of Parameters of Filter Networks

56. Classification of Pass Band and Stop Band
• It is possible to verify the Filter characteristics from the Propagation Constant
(g) of the network.
• The propagation constant g , being a function of frequency, the pass band, stop
band and the cut-off point, i.e. the point of separation between the two bands, can
be identified.
• For symmetrical T or п- section, expression for propagation constant g in terms of
the hyperbolic functions is given by
• If Z1 and Z2 are both pure imaginary values, and hence Z1/4Z2, their ratio will be
a pure real number.
• Since Z1 and Z2 may be anywhere in the range from –j to +j ; Z1/4Z2 may also
have any real value between the infinite limits.
• Then sinh g /2 will also have infinite limits, but may be either real or imaginary
depending upon whether Z1/4Z2 is positive or negative.

57. • We know that the Propagation constant is a complex function,
g = a + jb.
• The real part of the complex propagation constant a, is a measure of the change
in magnitude of the current or voltage in the network, known as the
Attenuation constant.
• b is a measure of the difference in phase between the input and output currents
or voltages, known as Phase shift constant.
• Therefore a and b take on different values depending upon the range of Z1/4Z2.
(1)

58. Case A :
• If Z1 and Z2 are the same type of reactances, then is real
and equal to say x.
• The imaginary part of the Eqn. (1) must be zero.
• a and b must satisfy both the above equations.
• Eqn. (2) can be satisfied if b/2 = 0 or np, where n = 0, 1, 2, ...,
then cosb/2 = 1 and sinha/2 = x =
(2)
(3)
(1)

59. • That x should be always positive implies that
• Since a ≠ 0, it indicates that the attenuation exists.
Case B:
• Consider the case of Z1 and Z2 being opposite type of reactances,
i.e. Z1/4Z2 is negative, making imaginary and equal to say jx.
∴ The real part of the Eqn. (1) must be zero.
• Both the above equations must be satisfied simultaneously by a
and b. Eqn. (4) may be satisfied when a = 0, or when b = p. These
conditions are considered separately.
(4)
(5)

60. (i) When a =0; from Eqn. (4), sinh a/2 = 0 and from Eqn. (5),
sin b/2 =x= . But the sine can have a maximum value of 1.
• Therefore, the above solution is valid only for negative Z1/4Z2, and having
maximum value of unity.
• It indicates the condition of Pass band with zero attenuation and follows the
condition as
(6)

61. (ii) When b=p, from Eqn. (4), cos b/2=0 and from Eqn.(5), sinb/2= ±1; cosh a/2 = x
= .
• Since cosh a/2 ≥1, this solution is valid for negative Z1/4Z2, and having
magnitude greater than, or equal to unity. It indicates the condition of Stop
band since a ≠ 0.
(7)

63. Characteristic Impedance In The Pass and Stop Bands
• Referring to the Characteristic impedance of a Symmetrical T-
network,
we have
• If Z1 and Z2 are purely reactive, let Z1 = jx1 and Z2 = jx2, then
• A Pass band exists when x1 and x2 are of opposite reactances
and
(1)

64. • Substituting these conditions in Eqn. (1), we find that Z0T is positive and
real.
• Now consider the Stop band. A Stop band exists when x1 and x2 are of the
same type of reactances; and x1/4x2 > 0.
• Substituting these conditions in Eqn.(1), we find that Z0T is purely
imaginary in this attenuation region.
• Another stop band exists when x1 and x2 are of the same type of
reactances, but with x1/4x2 < -1.
• Then from Eqn. (1), Z0T is again purely imaginary in the attenuation
region.
• Thus, in a Pass band if a network is terminated in a pure resistance R0 (Z0T
= R0), the input impedance is R0 and the network transmits the power
received from the source to the R0 without any attenuation.

65. • In a stop band Z0T is reactive. Therefore, if the network is
terminated in a pure reactance (Z0 = pure reactance), the input
impedance is reactive, and cannot receive or transmit power.
• However, the network transmits voltage and current with 90° phase
difference and with attenuation.
• It has already been shown that the characteristic impedance of a
symmetrical п-section can be expressed in terms of Z0T.
• Thus, Z0п = Z1Z2 / Z0T
• Since Z1 and Z2 are purely reactive, Z0п is real, if Z0T is real, and
Z0п is imaginary, if Z0T is imaginary.
• Thus the conditions developed for T-sections are valid for
п sections.

66. Classification of Filters

83. Constant K
• Design T and π section low pass filter which has series
inductance 80 mH and shunt capacitance 0.022 μf. Find the
cutoff frequency and design impedance.

86. 1. Design a Constant k high pass filter to cut-off at 5Khz with a design
impedance of 600Ω. Draw the π and T- sections of the high pass filter and
show the circuit elements in each configuration.
L = 9.454mH
C = 0.0265 uF
2. Design a Constant-k low pass filter to have a cut-off frequency at 796 Hz,
when terminated in a 500Ω resistance in T configuration.
L = 199.8 mH
C = 0.799 uF

87. • A Constant k, T-section high pass filter has a cutoff
frequency of 10 KHz. The design impedance is 600 Ω.
Determine the value of L.
Given: K = 600 ohm fc= 10 KHz
L = K / 4πfc
= (600/4 x πx 10x 103) = 0.004771 = 4.77mH
• Design a low pass filter having a cut-off frequency of 2 KHz
to operate with a design impedance of 500Ω.

90. Problems with k-constant filter
• The attenuation is not sharp in the stop band for k-type filters.
• The characteristic impedance, Z0 is a function of frequency and
varies widely in the transmission band.
• Attenuation can be increased in the stop band by using ladder
section, i.e. by connecting two or more identical sections.
• However, cascading is not a proper solution from a practical point
of view. This is because practical elements have a certain
resistance, which gives rise to attenuation in the pass band also.
• If the constant k section is regarded as the Prototype, it is possible
to design a filter to have rapid attenuation in the stop band, and
the same characteristic impedance as the prototype at all
frequencies. Such a filter is called m-derived filter.

91. Figure (a) is a constant K filter T-prototype, and figure (b) is prototype
derived from constant k filter by adding constant m to series branch and
changing shunt branch such that characteristic impedance remains same.
is characteristic impedance of modified (m-derived) T-network.

93. M-derive Π prototype
Again by modifying series and shunt branches, such that characteristics
impedance remans same:
Where Z’0Π is characteristic impedance of modified (m-derived) Π network.

94. • Squaring and cross multiplying the last equation
• Thus Z1’ can be a parallel combination of two impedances mZ1 and

95. M-Derived Low Pass Filter
• The shunt arm is to be chosen so that it is resonant at some frequency fr
above cut-off frequency fc.
• If the shunt arm is series resonant, its impedance will be minimum or zero.
• Therefore, the output is zero and will correspond to infinite attenuation at
this particular frequency. Thus at resonance

96. Thus we have
Since cut off frequency of low pass filter is
Thus for sharp attenuation at or close to cutoff frequency fc should be close
to fr, or m should be close to 0 or very small.

97. Π Network at Resonance
• Similarly, for m-derived pi section, the inductance and capacitance in the
series arm constitute a resonant circuit. Thus at resonance frequency we
will have
• The expression is same as for T network, Thus for sharp attenuation at or
close to cutoff frequency fc should be close to fr, or m should be close to
0 or very small.

98. Attenuation Constant for M-derive Low
Pass Filter
• We have expression for attenuation constant as

100. Phase Constant for m-derived Low Pass Filter
• We know the expression for phase constant β as
• Substituting value of Z1/4Z2, we will get
• The variation of attenuation
constant and phase constant
with frequency is shown
in the diagram:

101. Characteristic impedance of m-derive Low
Pass Filters

102. • Similarly for a Π network

105. M-derive High Pass Filter
Again for a T network shunt branch forms a resonant circuit, thus at
frequency fr it offers minimum impedance in stop band and making output
zero. At resonant frequency we will have

106. Since cut off frequency is given by
Similarly, for the m-derived pi-section, the resonant circuit is constituted by the series arm
inductance and capacitance. Thus, at resonant frequency:

107. Attenuation Constant and Phase Constant
Since,

110. Summary
• M-derive filter can be derived from constant k type filter by:
1. if it is a T prototype, multiply series impedance by “m” and divide shunt
impedance by “m” and add opposite impedance in series with shunt branch by
multiplying it with value (1-m2)/4m.
2. If it is a pi prototype, divide shunt impedance by “m”, multiply series
impedance by “m” and add extra opposite impedance in parallel with series
branch by multiplying with 4m/(1-m2).

111. Constant k LOW PASS FILTER
M-derived LOW PASS FILTER
T –section
Multiply series impedance by m:
In k type it is Z1/2, so now it will be
mZ1/2 since Z1=jωL, so Lm/2 is our
inductor value.
D2. divide shunt impedance by m, so
now it will be Z2/m
Z2=1/j ωC, so new capacitor will be
mC.

117. • Design a m-derived low pass filter having cutoff frequency of 1kHz, design
impedance of 400Ω and resonant frequency of 1100Hz ?

119. Attenuators-Symmetrical and Asymmetrical
• A Passive Attenuator is a special type of electrical or electronic bidirectional
circuit made up of entirely resistive elements.
Attenuator Summary
• An attenuator is a four terminal device that reduces the amplitude or power of a
signal without distorting the signal waveform, an attenuator introduces a certain
amount of loss.
• The attenuator network is inserted between a source and a load circuit to reduce
the source signal’s magnitude by a known amount suitable for the load.
• Attenuators can be fixed, fully variable or variable in known steps of attenuation, -
0.5dB, -1dB, -10dB, etc.
• An attenuator can be symmetrical or asymmetrical in form and either balanced or
unbalanced.
• Fixed attenuators also known as a “pad” are used to “match” unequal impedances.
• An attenuator is effectively the opposite of an amplifier. An amplifier provides
gain while an attenuator provides loss, or gain less than 1 (unity).
• Attenuators are usually passive devices made to from simple voltage divider
networks. The switching between different resistances produces adjustable stepped
attenuators and continuously adjustable ones using potentiometers.

120. • The attenuator is highly used after signal generator circuits.
• It helps in attenuating or reducing the strength of high-level signals before applying
them to the Antenna circuits.
• An attenuator is a two-port electronic device It is designed using resistors to
weaken or attenuate a signal. Attenuators are passive circuits, they work without
any power supply.
• Applications of Attenuators
• Attenuators are used as volume control equipment in broadcasting stations.
• For testing purposes in laboratories, to obtain smaller voltage signals, attenuators
are used.
• Fixed attenuators are used to improve the impedance matching in circuits.
• These are used to protect the circuits from damages caused by high voltage values.

122. T-TYPE ATTENUATOR
• Basically, there are four types of attenuators, T, pi, lattice, bridged & L-type. The
basic design principles are discussed in the following Sections. Figure 17.34 shows

124. Numerical Problem
• Exp: Design a T-pad attenuator to give an attenuation of 60 dB and to work
in a line of 500 ohm impedance.

126. Numerical Problem
• Design a p-type attenuator to give 20 dB attenuation and to have a
characteristic impedance of 100 ohm.

127. LATTICE ATTENUATOR

130. Numerical Problem
• Design a symmetrical lattice attenuator to have characteristic impedance of
800 ohm and attenuation of 20 dB.

131. BRIDGED-T ATTENUATOR

133. Numerical Problem
• Design a symmetrical bridged T - attenuator with an attenuation of 20 dB
and terminated into a load of 500 ohm

135. Numerical Problems
• Design a L-type attenuator to operate into a load resistance of 600 ohm
with an attenuation of 20 dB.

136. Thank You

137. BEE Exam Time:60M
Answer any THREE our of FIVE Dt:30/8/2021
1.) (a) Design a constant –k high pass filter to cut-off at 5Khz with a characteristic
impedance of 600Ω. Draw the π and T- sections of the high pass filter and show the
circuit elements in each configuration.
(b) Design M derived Low pass Filter
2.) Explain the following in detail
• High Pass Filter.
• Band Pass Filter
• Bridged T-type attenuator.
3) a) Design constant-k low pass filter.
b) Design a m-derived of low pass filter(π-section) to cut-off at 2Khz with a
characteristic impedance of 600Ω and f∞=2.1Khz.
4) Explain about the characteristic impedance in the Pass and Stop Band filters.
5) Discuss Classification of Pass and Stop Band