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Module 4. Network theorems Thevenin’ s, Norton’ s,...

Module 5. Reciprocity and Maximum power transfer

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Module 11. Series and parallel resonance

Module 12. Classification of filters

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

## LESSON 30. Constant-k filters

**30.1. Constant – K Low Pass Filter**

A network, either *T* or \[\pi\], is said to be of the constant-*k* type if Z_{1} and Z_{2} of the network satisfy the relation

*Z _{1}Z_{2} = k^{2 }\[...................................................\left( {30.1} \right)\]*

where Z_{1} and Z_{2} are impedance in the T and \[\pi\] sections as shown in Fig.17.8. Equation 17.20 states that Z_{1} and Z_{2} are inverse if their product is a constant, independent of frequency. *k* is a real constant, that is the resistance. *k *is often termed as design impedance or nominal impedance of the constant *k*-filter.

The constant *k, T *or \[\pi\] type filter is also known as the prototype because other more complex networks can be derived from it. A prototype T and \[\pi\]-sections are shown in

**Fig. 30.1**

Fig.30.1 (a) and (b), where Z_{1} = jω_{L} and Z_{2} = 1/jω_{C}. Hence Z_{1}Z_{2}= \[{L \over C}={k^2}\] which is independent of frequency.

\[{Z_1}{Z_2}={k^2}={L \over C}\,\,\,or\,\,k=\sqrt {{L \over C}} ...................................................\left( {30.2} \right)\]

Since the product Z_{1} and Z_{2} is constant, the filter is a constant-k type. From Eq. 29.8 (a) the cut-off frequencies are Z_{1}/4Z_{2} = 0.

i.e. \[{{ - {\omega ^2}LC} \over 4}=0\]

i.e. \[f=0\,\,and\,\,{{{Z_1}} \over {4{Z_2}}}=-1\]

\[{{ - {\omega ^2}LC} \over 4}=-1\]

or \[{f_c}={1 \over {\pi \sqrt {LC} }}...................................................\left( {30.3} \right)\]

The pass band can be determined graphically. The reactances of Z_{1} and 4Z_{2} will vary with frequency as drawn in Fig.30.2. The cut-off frequency at the intersection of the curves Z_{1} and 4Z_{2} is indicated as ƒ_{c}. On the X-axis as Z_{1} = -4Z_{2} at cut-off frequency, the pass band lies between the frequencies at which Z_{1} = 0, and Z_{1}=-4Z_{2}.

**Fig.30.2**

All the frequencies above ƒ_{c} lie in a stop or attenuation band. Thus, the network is called a low-pass filter. We also have from Eq.28.7 (previous chapter) that

\[\sinh {\gamma\over 2}=\sqrt {{{{Z_1}} \over {4{Z_2}}}}=\sqrt {{{ - {\omega ^2}LC} \over 4}}={{J\omega \sqrt {LC} } \over 2}\]

From Eq. 30.3 \[\sqrt {LC}={1 \over {{f_c}\pi }}\]

\[\sinh {\gamma\over 2}={{j2\pi f} \over {2\pi {f_c}}}=j{f \over {{f_c}}}\]

We also know that in the pass band

\[-1<{{{Z_1}} \over {4{Z_2}}}< 0\]

\[-1<{{ - {\omega ^2}LC} \over 4}< 0\]

\[-1<-\left( {{f \over {{f_c}}}} \right)< 0\]

or \[{f \over {{f_c}}} < 1\]

and \[\beta=2{\sin ^{ - 1}}\left( {{f \over {{f_c}}}} \right);\,\,\alpha=0\]

In the attenuation band,

\[{{{Z_1}} \over {4{Z_2}}}<-1,\,\,i.e.{f \over {{f_c}}} < 1\]

\[\alpha=2{\cosh ^{ - 1}}\left[ {{{{Z_1}} \over {4{Z_2}}}} \right]=2{\cosh ^{ - 1}}\left[ {{f \over {{f_c}}}} \right];\,\beta=\pi\]

The plots of \[\alpha\] and β for pass and stop bands are shown in Fig.30.3.

Thus, from Fig.30.3, a = 0, b = 2 sinh^{-1} \[\left( {{f \over {{f_c}}}} \right)for\,\,f < {f_c}\]

\[\alpha=2{\cosh ^{ - 1}}\left( {{f \over {{f_c}}}} \right);\,\beta=\pi \,\,for\,\,f > fc\,\,f < {f_c}\]

**Fig.30.3**

The characteristic impedance can be calculated as follows.

\[{Z_{0T}}=\sqrt {{Z_1}{Z_2}\left( {1 + {{{Z_1}} \over {4{Z_2}}}} \right)}\]

\[=\sqrt {{L \over C}\left( {1 + {{{\omega ^2}LC} \over 4}} \right)}\]

\[{Z_{0T}}=k{\sqrt {1 - \left( {{f \over {{f_c}}}} \right)} ^2}...................................................\left( {30.4} \right)\]

From Eq.30.4, Z_{0T} is real when ƒ < ƒ_{c} i.e. in the pass band at ƒ = ƒ_{c}, Z_{0T} = 0; and for ƒ > ƒ_{c}, Z_{0T} is imaginary in the attenuation band, rising to infinite reactance at infinite frequency. The variation of Z_{0T} with frequency is shown in Fig.30.4.

**Fig.30.4**

Similarly, the characteristic impedance of a \[\pi\]-network is given by

The variation of Z_{0}_{\[\pi\]} with frequency is shown in fig. 30.4 for ƒ< ƒ_{c}, Z_{0}_{\[\pi\]} is real; at ƒ = ƒ_{c}, Z_{0}_{\[\pi\]} is infinite, and for ƒ > ƒ_{c}, Z_{0T} is imaginary. A low pass filter can be designed from the specifications of cut-off frequency and load resistance.

All cut-off frequency, Z_{1}= -4Z_{2}

\[j{\omega _c}L={{ - 4} \over {j{\omega _c}C}}\]

\[{\pi ^2}f_c^2LC=1\]

Also we know that \[k=\sqrt {L/C}\] is called the design impedance or the load resistance

\[{k_2}={L \over C}\]

\[{\pi ^2}f_c^2{k^2}{C^2}=1\]

\[C={1 \over {\pi \,{f_c}k}}\,gives\,\,the\,\,value\,\,of\,\,the\,\,shunt\,capaci\tan ce\]

\[and\,L={k^2}C={k \over {\pi {f_c}}}\,gives\,the\,value\,of\,the\,series\,induc\tan ce.\]

**30.2. Constant K-High Pass Filter**

Constant K-high pass filter can be obtained by changing the positions of series and shunt arms of the networks shown in Fig.30.1. The prototype high pass filters are shown in Fig.30.5, where Z_{1} =-j/ω_{C} and Z_{2} = jωL.

**Fig.30.5**

Again, it can be observed that the product of Z_{1} and Z_{2} is independent of frequency, and the filter design obtained will be of the constant k type. Thus, Z_{1}Z_{2} are given by

\[{Z_1}{Z_2}={{ - j} \over {\omega C}}j\omega L={L \over C}={k^2}\]

The cut-off frequencies are given by Z_{1}= 0 and Z_{1} =-4Z_{2}.

\[{Z_1}\,=\,0\,indicates\,{j \over {\omega C}}=0,\,\,or\,\,\omega\to \alpha\]

From Z_{1} =-4Z_{2}

\[{{ - j} \over {\omega C}}=-4j\omega L\]

\[{\omega ^2}LC={1 \over 4}\]

or \[{f_c}={1 \over {4\pi \sqrt {LC} }}...................................................\left( {30.6} \right)\]

The reactance of Z_{1} and Z_{2} are sketched as functions of frequency as shown in Fig.30.6.

As seen from Fig.30.6, the filter transmits all frequencies between ƒ = ƒ_{c} and ƒ = \[\propto\] . The point ƒ_{c} from the graph is a point at which Z_{1} = -4Z_{2}.

**Fig.30.6**

\[\sinh {\gamma\over 2}=\sqrt {{{{Z_1}} \over {4{Z_2}}}}=\sqrt {{{ - 1} \over {4{\omega ^2}LC}}}\]

\[From\,Eq.17.25,{f_c}={1 \over {4\pi \sqrt {LC} }}\]

\[\sqrt {LC}={1 \over {4\pi {f_c}}}\]

\[\sinh {\gamma\over 2}=\sqrt {{{ - {{\left( {4\pi } \right)}^2}{{\left( {{f_c}} \right)}^2}} \over {4{\omega ^2}}}}=j{{{f_c}} \over f}\]

In the pass band, \[- 1 < {{{Z_1}} \over {4{Z_2}}} < 0,\alpha=0\,\,or\,the\,region\,which\,{{{f_c}} \over f} < 1\,is\,a\,pass\,band\]

\[\beta=2{\sin ^{ - 1}}\left( {{{{f_c}} \over f}} \right)\]

In the attenuation band \[{{{Z_1}} \over {4{Z_2}}}<-1,i.e.{{{f_c}} \over f} > 1\]

\[\alpha=2{\cosh ^{ - 1}}\left( {{{{Z_1}} \over {4{Z_2}}}} \right)\]

\[=2{\cos ^{ - 1}}\left( {{{{f_c}} \over f}} \right);\,\,\beta=-\pi\]

The plots of a and b for pass and stop bands of a high pass filter network are shown in Fig.30.7

**Fig.30.7**

A high pass filter may be designed similar to the low pass filter by choosing a resistive load r equal to the constant k, such that \[R=k=\sqrt {L/C}\]

\[{f_c}={1 \over {4\pi \sqrt {L/C} }}\]

\[{f_c}={k \over {4\pi L}}={1 \over {4\pi Ck}}\]

Since \[\sqrt C={L \over k},\]

\[L={k \over {4\pi {f_c}}}and\,C={1 \over {4\pi {f_c}k}}\]

The characteristic impedance can be calculated using the relation

\[{Z_{0T}}=\sqrt {{Z_1}{Z_2}\left( {1 + {{{Z_1}} \over {4{Z_2}}}} \right)}=\sqrt {{L \over C}\left( {1 - {1 \over {4{\omega ^2}LC}}} \right)}\]

\[{Z_{0T}}=k\sqrt {1 - {{\left( {{{{f_c}} \over f}} \right)}^2}}\]

Similarly, the characteristic impedance of a p-network is given by

\[{Z_{0\pi }}={{{Z_1}{Z_2}} \over {{Z_{0T}}}}={{{k_2}} \over {{Z_{0T}}}}\]

The plot of characteristic impedance with respect to frequency is shown in Fig.30.8

**Fig.30.8**