Filters and Attenuators 17.1 CLASSIFICATION OF FILTERS Wave filters were first invented by G.A Campbell 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 becomes 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 CHAPTER 17 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 wave filters, and is designated by fc. 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, in 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 Z0. From Fig. 17.1 (a) the number of V1 I1 nepers, N = loge or loge . V2 I2 A neper can also be expressed in terms of input power, P1 and the output Fig. 17.1 (a) power P2 as N 5 1/2 loge P1/P2. Filters and Attenuators 811 A decibel is defined as ten times the common logarithms of the ratio of the input power to the output power. P1 ∴ Decibel D =10 log10 P2 The decibel can be expressed in terms of the ratio of input voltage (or current) and the output voltage (or current.) V1 I1 D = 20 log10 = 20 log10 V2 I2 ∴ 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 fc, and attenuates all other frequencies greater than fc. The attenuation characteristic of an ideal LP filter is shown in Fig. 17.1 (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 fc. 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 fc. Fig. 17.1 (b) High Pass Filter A high pass (HP) filter attenuates all frequencies below a designated cut-off frequency, fc, and passes all frequencies above fc. Thus the pass band of this filter is the frequency range above fc, and the stop band is the frequency range below fc. The attenuation characteristic of a HP filter is shown in Fig. 17.1 (b). 812 Circuits and Networks Band Pass Filter A band pass filter passes frequencies between two designated cut-off frequencies and attenuates all other frequencies. It is abbreviated as BP filter. As shown in Fig. 17.1 (b), a BP filter has two cut-off frequencies and will have the pass band f2 – f1; f1 is called the lower cut-off frequency, while f2 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. 17.1 (b). All frequencies between f1 and f2 will be attenuated while frequencies below f1 and above f2 will be passed. 17.2 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 dissipationless, 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. 17.2. Fig. 17.2 The ladder structure is one of the commonest forms of filter network. A cascade connection of several T and p sections constitutes a ladder network. A common form of the ladder network is shown in Fig. 17.3. Figure 17.3 (a) represents a T section ladder network, whereas Fig. 17.3 (b) represents the p section ladder network. It can be observed that both networks are identical except at the ends. Filters and Attenuators 813 Fig. 17.3 17.3 EQUATIONS OF FILTER NETWORKS The study of the behaviour of any filter requires the calculation of its propagation constant g, attenuation a, phase shift b and its characteristic impedance Z0. T-Network Consider a symmetrical T-network as shown in Fig. 17.4. As has already been mentioned in Section 16.13, if the image impedances at port 1-19 and port 2-29 are equal to each other, the image impedance is then called the characteristic, or the iterative impedance, Z0. Thus, if the network in Fig. 17.4 is terminated in Z0, its input impedance will also be Z0. The value Fig. 17.4 of input impedance for the T-network when it is terminated in Z0 is given by Z Z 1 + Z Z 2 2 0 Z =1 + in 2 Z 1 +ZZ + 2 2 0 also Zin= Z 0 Z1 2Z2 + Z0 Z1 22 ∴ Z0 = + 2 ZZZ1+2 2 + 2 0 Z1 (ZZZZ1 2+ 2 2 0 ) Z0 = + 2 ZZZ1+2 2 + 2 0 814 Circuits and Networks 2 ZZZZZZZZZ1 +21 2 + 2 1 0 + 2 1 2 + 4 0 2 Z0 = 2(ZZZ1+ 2 2 + 2 0 ) 2 2 4ZZZZ0 =1 + 4 1 2 Z 2 Z 2 =1 + ZZ 00 4 1 2 The characteristic impedance of a symmetrical T-section is Z 2 Z =1 + ZZ (17.1) 0T 4 1 2 Z0T can also be expressed in terms of open circuit impedance Z0c and short circuit impedance Zsc of the T-network. From Fig. 17.4, the open circuit impedance Z Z =1 + Z and 0c 2 2 Z 1 Z Z × 2 Z =1 + 2 sc 2 Z 1 + Z 2 2 2 ZZZ1 + 4 1 2 Zsc = 2ZZ1+ 4 2 Z 2 ZZZZ× = + 1 0c sc 1 2 4 2 = ZZ0T or Z0T= Z 0 c Z sc (17.2) Propagation Constant of T-Network By definition the propagation constant g of the network in Fig. 17.5 is given by g 5 loge I1/I2 Writing the mesh equation for the 2nd mesh, we get Z IZI= 1 +ZZ + 1 2 2 2 2 0 Z 1 ZZ I +2 + 0 1 = 2 = eg I2 Z2 Fig. 17.5 Z ∴ 1 +Z + Z = Z eg 2 2 0 2 Z Z= Z() eg −1 − 1 (17.3) 0 2 2 Filters and Attenuators 815 The characteristic impedance of a T-network is given by Z 2 Z =1 + ZZ (17.4) 0T 4 1 2 Squaring Eqs 17.3 and 17.4 and subtracting Eq. 17.4 from Eq. 17.3, we get Z 2 Z 2 Z2() eg −1 2 +1 −Z Z() eg −1 −1 −ZZ = 0 2 4 1 2 4 1 2 2g 2 g Z2 ()( e−1 − Z1 Z 2 1 + e −−1) = 0 2g 2 g Z2 () e−1 − Z1 Z 2 e = 0 g 2 g Z2 () e−1 − Z1 e = 0 Z eg ()eg −1 2 = 1 Z2 Z e2g+1 − 2 e g = 1 −g Z2 e Rearranging the above equation, we have −g2 g g Z1 e( e+1 − 2 e ) = Z2 g− g Z1 (e+ e −2) = Z2 Dividing both sides by 2, we have eg+ e− g Z =1 + 1 2 2Z2 Z cosh g =1 + 1 (17.5) 2Z2 Still another expression may be obtained for the complex propagation constant in terms of the hyperbolic tangent rather than hyperbolic cosine. 2 sinhg= cos h g −1 Z 2 Z Z 2 =1 + 1 −1 =1 + 1 2Z2 Z1 2Z2 2 1 Z1 Z0T sinh g =ZZ1 2 + = (17.6) Z2 4 Z2 Dividing Eq. 17.6 by Eq. 17.5, we get Z tanh g = 0T Z Z + 1 2 2 816 Circuits and Networks Z But Z +1 = Z 2 2 0c Also from Eq. 17.2, ZZZ0T= 0 c sc Z tanh g = sc Z 0c g 1 Also sinh= (coshg −1 ) 2 2 Where coshg =1 + (ZZ1/ 2 2 ) Z = 1 (17.7) 4Z2 p-Network Consider asymmetrical p-section shown in Fig. 17.6. When the network is terminated in Z0 at port 2-29, its input impedance is given by 2ZZ 2ZZ + 2 0 2 1 2ZZ2+ 0 Zin = 2ZZ2 0 Z1 + + 2Z2 2ZZ2+ 0 By definition of characteristic impedance, Zin 5 Z0 2ZZ 2ZZ + 2 0 Fig.