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Module 5. Reciprocity and Maximum power transfer

Module 6. Star- Delta conversion solution of DC ci...

Module 7. Sinusoidal steady state response of circ...

Module 8. Instantaneous and average power, power f...

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Module 11. Series and parallel resonance

Module 12. Classification of filters

Module 13. Constant-k, m-derived, terminating half...

## LESSON 28. Classification of filters

**28.1. Classification of filters**

Wave filters were first invented by G.ACampbell and O.I. Lobel of the Bell Telephone Laboratories. A filter is a reactive network that freely passes the desired bands of frequencies while almost totally suppressing all other bands. A filter is constructed from purely reactive elements, for otherwise the attenuation would never become zero in the pass band of the filter network. Filters differ from simple resonant circuits in providing a substantially constant transmission over the band which they accept; this band may lie between any limits depending on the design. Ideally, filters should produce no attenuation in the desired band, called the transmission band or pass band, an 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 wave filters, and is designated by ƒc.

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.

A filter may, principle, have any number of pass bands separated by attenuation bands. However, they are classified into four common types, viz. low pass, high pass, band pass and band elimination.

**Decibel and Neper**

The attenuation of a wave filter can be expressed in decibels or nepers. Neper is defined as the natural logarithm of the ratio of input voltage (or current) to the output voltage (or current), provided that the network is properly terminated in its characteristic impedance Z_{0}.

From Fig. 28.1 (a) the number of nepers, \[N={\log _e}\left[ {{{{V_1}} \over {{V_2}}}} \right]\,or\,\,{\log _e}\left[ {{{{I_1}} \over {{I_2}}}} \right]\]

**Fig. 28.1 (a)**

A neper can also be expressed in terms of input power, P_{1} and the output power P_{2} as N = ½ log_{e} P_{1}/P_{2}.

A decibel is defined as en times the common logarithms of the ratio of the input power to the output power.

Decibel \[D=10\,{\log _{10}}{{{P_1}} \over {{P_2}}}\]

The decibel can be expressed in terms of the ratio of input voltage (or current) and the output voltage (or current).

\[D=10\,{\log _{10}}\left[ {{{{P_1}} \over {{P_2}}}} \right]\,\,=20\,\,{\log _{10}}\left[ {{{{I_1}} \over {{I_2}}}} \right]\]

One decibel is equal to 0.115 N.

**Low Pass Filter**

By definition, a low pass (LP) filter is one which passes without attenuation all frequencies up to the cut-off frequency f_{c}, and attenuates all other frequencies greater than ƒ_{c}. The attenuation characteristic of an ideal LP filter is shown in Fig.28.1 (b). 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 LP filter is the frequency range 0 to ƒ_{c}. The frequency range over which transmission does not take place is called the stop band or attenuation band. The stop band for a LP filter is the frequency range above ƒ_{c}.

**Fig.28.1 (b)**

**High Pass Filter **

A high pass (HP) filter attenuates all frequencies below a designated cut-off frequency, ƒ_{c}, and passes all frequencies above ƒ_{c}. Thus the pass band of this filter is the frequency range above ƒ_{c}, and the stop band is the frequency range below ƒ_{c}. The attenuation characteristic of a HP filter is shown in Fig.28.1 (b).

**Band Pass Filter**

A band pass filter passes frequencies between two designated cut-of frequencies and attenuates all other frequencies. It is abbreviated as BP filter. As shown in Fig.28.1 (b), a BP filter has two cut-off frequencies and will have the pass band _{ƒ2} – ƒ_{1}; _{ƒ1} is called the lower cut-off frequency, while ƒ_{2} is called the upper cut-off frequency.

**Band Elimination Filter**

A band elimination filter passes all frequencies lying outside a certain range, while it attenuates all frequencies between the two designated frequencies. It is also referred as band stop filter. The characteristic of an ideal band elimination filter is shown in fig.28.1 (b).

All frequencies between _{ƒ1} and ƒ_{2} will be attenuated while frequencies below _{ƒ1} and above ƒ_{2} will be passed.

**28.2. 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 p sections. T and p sections can be considered as combinations of unsymmetrical L sections as shown in Fig.28.2.

**Fig.28.2**

The ladder structure is one of the commonest forms of filter network. A cascade connection of several T and \[\pi\] sections constitutes a ladder network. A common form of the ladder network is shown in Fig.28.3.

Figure 28.3 (a) represents a T section ladder network, whereas Fig.28.3 (b) represents the \[\pi\] section ladder network. It can be observed that both networks are identical except at the ends.

**Fig. 28.3**

**28.3. Equations of Filter Networks**

The study of the behavior of any filter requires the calculation of its propagation constant \[\gamma\] , attenuation \[\alpha\] , phase shift β and its characteristic impedance Z_{0}.

**T-Network**

Consider a symmetrical T-network as shown in Fig.28.4

**Fig.28.4**

If the image impedances at port 1-1’ and port 2-2’ are equal to each other, the image impedance is then called the characteristic, or the iterative impedance, Z_{0}. Thus, if the network in Fig.28.4 is terminated in Z_{0}, its input impedance will also be Z_{0}. The value of input impedance fro the T-network when it is terminated in Z_{0} is given by

The characteristic impedance of a symmetrical T-section is

\[{Z_{0T}}=\sqrt {{{Z_1^2} \over 4} + Z_1^{}{Z_2}} ...................................................\left( {28.1} \right)\]

Z_{0T} can also be expressed in terms of open circuit impedance Z_{OC} and short circuit impedance Z_{SC} of the T-netowrk. From Fig.17.4, the open circuit impedance \[Z_{OC}^{}={{{Z_1}} \over 2} + {Z_2}\,\,and\]

**Propagation Constant of T-Network**

By definition the propagation constant g of the network in Fig.28.5 is given by

\[\gamma\] = log_{e} I_{1}/I_{2}

Writing the mesh equation for the 2^{nd} mesh, we get

**Fig.28.5**

\[{{{Z_1}} \over 2} + {Z_2} + {Z_0}={Z_2}{e^\gamma }\]

\[{Z_0}={Z_2}\left( {{e^\gamma } - 1} \right) - {{{Z_1}} \over 2}...................................................\left( {28.3} \right)\]

The characteristic impedance of a T-network is given by

\[{Z_{OT}}=\sqrt {{{Z_1^2} \over 4} + {Z_1}{Z_2}} ...................................................\left( {28.4} \right)\]

Squaring Eqs. 28.3 and 28.4 and subtracting Eq.28.4 from Eq.28.3, we get

\[Z_2^2{\left( {{e^\gamma } - 1} \right)^2} + {{Z_1^2} \over 4} - {Z_1}{Z_2}\left( {{e^\gamma } - 1} \right) - {{Z_1^2} \over 4} - {Z_1}{Z_2}=0\]

\[Z_2^2{\left( {{e^\gamma } - 1} \right)^2} - {Z_1}{Z_2}\left( {1 + {e^\gamma } - 1} \right)=0\]

\[Z_2^2{\left( {{e^\gamma } - 1} \right)^2} - {Z_1}{Z_2}{e^\gamma }=0\]

\[{Z_2}{\left( {{e^\gamma } - 1} \right)^2} - {Z_1}{e^\gamma }=0\]

\[{\left( {{e^\gamma } - 1} \right)^2}={{{Z_1}{e^\gamma }} \over {{Z_2}}}\]

\[{e^\gamma } + 1 - 2{e^\gamma }={{{Z_1}} \over {{Z_2}{e^{ - \gamma }}}}\]

Rearranging the above equation, we have

\[{e^{ - \gamma }}\left( {{e^{2\gamma }} + 1 - 2{e^\gamma }} \right) = {{{Z_1}} \over {{Z_2}}}\]

\[\left( {{e^\gamma } + {e^{ - \gamma }} - 2} \right)={{{Z_1}} \over {{Z_2}}}\]

Dividing both sides by 2, we have

\[{{{e^\gamma } + {e^{ - \gamma }}} \over 2}=1 + {{{Z_1}} \over {2{Z_2}}}\]

Cosh \[\gamma =1 + {{{Z_1}} \over {2{Z_2}}}...................................................\left( {28.5} \right)\]

Still another expression may be obtained for the complex propagation constant in terms of the hyperbolic tangent rather than hyperbolic cosine.

\[\sinh \,\gamma=\sqrt {\cos \,{h^2}\gamma-1}\]

\[=\sqrt {{{\left( {1 + {{{Z_1}} \over {2{Z_2}}}} \right)}^2}}-1=\sqrt {{{{Z_1}} \over {{Z_2}}} + {{\left( {{{{Z_1}} \over {2{Z_2}}}} \right)}^2}} \]

\[Sinh\,\,\gamma=\,{1 \over {{Z_2}}}\sqrt {{Z_1}{Z_2} + {{Z_1^2} \over 4}}={{{Z_{0T}}} \over {{Z_2}}}...................................................\left( {28.6} \right)\]

Dividing by Eq.28.6 by Eq.28.5, we get

But \[{Z_2} + {{{Z_1}} \over {{Z_2}}}={Z_{0C}}\]

Also from Eq. 28.2, \[\tanh \,\,\gamma=\sqrt {{{{Z_{sc}}} \over {{Z_{0c}}}}}\]

\[\tanh \,\,\gamma=\sqrt {{{{Z_{sc}}} \over {{Z_{oc}}}}}\]

Also \[\sinh \,{\gamma\over 2}=\sqrt {{1 \over 2}\left( {\cosh \,\,\gamma-1} \right)}\]

Where \[\cosh \,\,\gamma=1 + \left( {{Z_1}/2{Z_2}} \right)\]

\[=\sqrt {{{{Z_1}} \over {4{Z_2}}}} ...................................................\left( {28.7} \right)\]

**\[\pi\] -Network**

Consider asymmetrical p-section shown in Fig.28.6. When the network is terminated in Z_{0} at port 2-2’, it input impedance is given by

By definition of characteristic impedance, Z_{in} = Z_{0}

**Fig.28.6**

Rearranging the above equation leads to

\[{Z_0}=\sqrt {{{{Z_1}{Z_2}} \over {1 + {Z_1}/4{Z_2}}}} ...................................................\left( {28.8} \right)\]

which is the characteristic impedance of a symmetrical pnetwork,

\[{Z_{0\pi }}=\sqrt {{{{Z_1}{Z_2}} \over {{Z_1}{Z_2} + Z_1^2/4}}\]

From Eq. 28.1 \[{Z_{OT}}=\sqrt {{{Z_1^2} \over 4} + {Z_1}{Z_2}}\]

\[{Z_{0\pi }}={{{Z_1}{Z_2}} \over {{Z_{OT}}}}...................................................\left( {28.9} \right)\]

Z_{0}_{p} can be expressed in terms of the open circuit impedance Z_{0c} and short circuit impedance Z_{SC} of the p network shown in Fig.28.6 exclusive of the lead Z_{0}.

\[by\,\,\,\,{Z_{0C}}={{2{Z_2}\left( {{Z_1} + 2{Z_2}} \right)} \over {{Z_1} + 4{Z_2}}}\]

Similarly, the input impedance at port 1-1’ when port 2-2’ is short circuited is given by

\[{Z_{sc}}={{2{Z_1}{Z_2}} \over {2{Z_2} + {Z_1}}}\]

Hence \[{Z_{oc}} \times {Z_{sc}}={{4{Z_1}Z_2^2} \over {{Z_1} + 4{Z_2}}} = {{{Z_1}{Z_2}} \over {1 + {Z_1}/4{Z_2}}}\]

Thus from Eq. 28.8

\[{Z_{0\pi }}=\sqrt {{Z_{oc}}{Z_{sc}}} ...................................................\left( {28.10} \right)\]

**Propagation Constant of ****p****-Network**

The propagation constant of a symmetrical p-section is the same as that for a symmetrical T-section.

i.e. \[\cosh \,\,\gamma=1 + {{{Z_1}} \over {2{Z_2}}}\]