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. 17.6 2 1 2ZZ2+ 0 Z0 = 2ZZ2 0 Z1 + + 2Z2 2ZZ2+ 0 2 2ZZ2 0 2ZZZZZZZ2( 2 1 2+ 0 1 + 2 0 2 ) ZZ0 1 + +2ZZ0 2 = 2ZZ2+ 0 ()2ZZ2+ 0 2 2 2 2 2ZZ0 1ZZ2+ Z 1 Z 0 +2 Z0 Z2 + 4 Z 2 Z0 + 2 Z 2 Z 0 2 2 =4ZZZZZZZ1 2 + 20 1 2 + 4 0 2 2 2 2 ZZ1 0 + 44ZZZZ2 0 = 4 1 2 2 2 ZZZZZ0 ()1+4 2 = 4 1 2 2 2 4ZZ1 2 Z0 = ZZ1+ 4 2 Rearranging the above equation leads to

ZZ1 2 Z0 = (17.8) 1+ ZZ1/ 4 2 Filters and Attenuators 817

which is the characteristic impedance of a symmetrical p-network,

ZZ1 2 Z0p = 2 ZZZ1 2+ 1 / 4

Z 2 From Eq. 17.1 Z =1 + ZZ 0T 4 1 2

ZZ1 2 ∴ Z0p = (17.9) Z0T

Z0p can be expressed in terms of the open circuit impedance Z0c and short circuit impedance Zsc of the p network shown in Fig. 17.6 exclusive of the load Z0. From Fig. 17.6, the input impedance at port 1-19when port 2-29 is open is given

2ZZZ2() 1+ 2 2 by Z0C = ZZ1+ 4 2 Similarly, the input impedance at port 1-19 when port 2-29 is short circuited is

2ZZ1 2 given by Zsc = 2ZZ2+ 1 2 4ZZ1 2 ZZ1 2 Hence ZZ0c× sc = = ZZ1+ 4 2 1+ ZZ1/ 4 2 Thus from Eq. 17.8

ZZZ0p = 0c sc (17.10)

Propagation Constant of p-Network The propagation constant of a symmetrical p-section is the same as that for a symmetrical T-section. Z i.e. cosh g =1 + 1 2Z2

17.4 CLASSIFICATION OF PASS BAND AND STOP BAND

It is possible to verify the characteristics of filters from the propagation constant 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 p-section, the expression for propagation constant g in terms of the hyperbolic functions is given by Eqs 17.5 g Z and 17.7 in Section 17.3. From Eq. 17.7, sin h = 1 . 2 4Z2 If Z1 and Z2 are both pure imaginary values, their ratio, and hence Z1/4Z2, will be a pure real number. Since Z1 and Z2 may be anywhere in the range from – j t0 1 j, Z1/4Z2 818 Circuits and Networks

g may also have any real value between the infinite limits. Then sinh= ZZ / 4 2 1 2 will also have infinite limits, but may be either real or imaginary depending upon whether Z1/4Z2 is positive or negative. We know that the propagation constant is a complex function g 5 a 1 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. From Eq. 17.7, we have ␥ ␣j  ␣  ␣  sinh= sinh + = sinh cos+ j cosh sin 2 2 2 2 2 2 2 Z = 1 (17.11) 4Z2 Case A Z1 If Z1 and Z2 are the same type of reactances, then is real and equal to say a + x. 4Z2 The imaginary part of the Eq. 17.11 must be zero. ␣  ∴ cosh sin = 0 (17.12) 2 2 ␣  sinh cos = x (17.13) 2 2 a and b must satisfy both the above equations. Equation 17.12 can be satisfied if b/2 5 0 or np, where n 5 0, 1, 2, ..., then Z cos b/2 = 1 and sinh a/2 = x = 1 4Z2 That x should be always positive implies that Z Z 1 >0and a = 2sinh−1 1 (17.14) 4Z2 4Z2 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 ZZ14/ 2 imaginary and equal to say Jx ∴ The real part of the Eq. 17.11 must be zero. ␣  sinh cos = 0 (17.15) 2 2 ␣  cosh sin = x (17.16) 2 2 Both the above equations must be satisfied simultaneously by a and b. Equation 17.15 may be satisfied when a 5 0, or when b 5 p. These conditions are considered separately hereunder. Filters and Attenuators 819

(i) When a 5 0; from Eq. 17.15, sinh a/2 5 0. And from Eq. 17.16 sinb / 2 =x = Z1/ 4 Z 2 . 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 Z −1 ≤1 ≤ 0 4Z2 Z b = 2sin−1 1 (17.17) 4Z2 (ii) When b 5 p, from Eq. 17.15, cos b/2 5 0. And from Eq. 17.16, sin b/2 5 1; cosh a/2 = x= Z14/ Z 2 .

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. Z −a ≤1 ≤ −1 4Z2 Z a = 2cosh−1 1 (17.18) 4Z2 It can be observed that there are three limits for case A and B. Knowing the values of Z1 and Z2, it is possible to determine the case to be applied to the filter. Z1 and Z2 are made of different types of reactances, or combinations of reactances, so that, as the frequency changes, a filter may pass from one case to another. Case A and (ii) in case B are attenuation bands, whereas (i) in case B is the transmission band. The frequency which separates the attenuation band from pass band or vice versa is called cut-off frequency. The cut-off frequency is denoted by fc, and is also termed as nominal frequency. Since Z0 is real in the pass band and imaginary in an attenuation band, fc is the frequency at which Z0 changes from being real to being imaginary. These frequencies occur at Z1 17.18(a) =0or Z1 = 0 4Z 2 Z 1 = −1or ZZ + 4 = 0 17.18(b) 1 2 4Z2 The above conditions can be represented graphically, as in Fig. 17.7.

Fig. 17.7 820 Circuits and Networks

17.5 CHARACTERISTIC IMPEDANCE IN THE PASS AND STOP BANDS

Referring to the characteristic impedance of a symmetrical T-network, from Eq. 17.1 we have

Z 2 Z Z =1 +ZZZZ =1 + 1 0T 1 2 1 2 4 4Z2

If Z1 and Z2 are purely reactive, let Z1 5 jx1 and Z2 5 jx2, then

x Z= − x x 1 + 1 (17.19) 0T 1 2 4x2

A pass band exists when x1 and x2 are of opposite reactances and x −1 <1 < 0 4x2

Substituting these conditions in Eq. 17.19, 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; then x1/4x2 . 0. Substituting these conditions in Eq. 17.19, 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 Eq. 17.19, 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 5 R0), the input impedance is R0 and the network transmits the power received from the source to the R0 without any attenuation. In a stop band Z0T is reactive. Therefore, if the network is terminated in a pure reactance (Z0 5 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 p-section can be expressed in terms of T. Thus, from Eq. 17.9, Z0p 5 Z1Z2/Z0T. Since Z1 and Z2 are purely reactive, Z0p is real if Z0T is real, and Z0x is imaginary if Z0T is imaginary. Thus the conditions developed for T-sections are valid for p sections.

17.6 CONSTANT—K LOW PASS FILTER

A network, either T or p, is said to be of the constant-k type if Z1 and Z2 of the network satisfy the relation

2 Z1Z2 5 k (17.20) where Z1 and Z2 are impedances in the T and p sections as shown in Fig. 17.8. Equation 17.20 states that Z1 and Z2 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 p type filter is also known as the prototype because other more complex networks can be derived from it. A prototype T and p-sections are shown in Filters and Attenuators 821

Fig. 17.8

L 2 Fig. 17.8 (a) and (b), where Z1 5 jvL and Z2 5 1/jvC. Hence ZZ1 2 = = k which is independent of frequency. C

2 L L Z1 Z 2 = k =or k = (17.21) C C

Since the product Z1 and Z2 is constant, the filter is a constant-k type. From Eq. 17.18(a) the cut-off frequencies are Z1/4Z2 5 0, −v2 LC i.e. = 0 4 Z i.e. f = 0 and 1 = −1 4Z2 −v2 LC = −1 4 1 or fc = (17.22) p LC

The pass band can be determined graphically. The reactances of Z1 and 4Z2 will vary with frequency as drawn in Fig. 17.9. The cut-off frequency at the intersection of the curves Z1 and – 4Z2 is indicated as fc. On the X-axis as Z1 5 –4Z2 at cut‑off frequency, the pass band lies between the frequencies at which Z1 5 0, and Z1 5 – 4Z2.

Fig. 17.9 822 Circuits and Networks

All the frequencies above fc lie in a stop or attenuation band. Thus, the network is called a low-pass filter. We also have from Eq. 17.7 that

␥ Z −2 LCJLC sinh =1 = = 2 4Z2 4 2 1 From Eq. 17.22 LC = fcp ␥j2 f f ∴ sinh = = j 2 2fcf c We also know that in the pass band Z −1 <1 < 0 4Z2 −v2 LC −1 < < 0 4 2 f −1 < − < 0 fc f or <1 fc f and  = 2sin−1 ; ␣ = 0 fc In the attenuation band, Z f 1 1, .i.e 1 < − < 4Z2 fc Z f ␣ = 2cosh−1 1 = 2cosh−1 ; = 4Z2 fc The plots of a and b for pass and stop bands are shown in Fig. 17.10. f Thus, from Fig. 17.10, ␣=0,  = 2 sinh−1 for f< f c fc −1 f ␣ = 2cosh ; = for f > f c fc The characteristic impedance can be calculated as follows

Z =1 + 1 ZZZ0T 1 2 4Z2 2 L v LC =1 − C 4

2 f Z= k 1 − (17.23) 0T Fig. 17.10 fc Filters and Attenuators 823

From Eq. 17.23, Z0T is real when f , fc, i.e. in the pass band at f 5 fc, Z0T 5 0; and for f . fc, Z0T is imaginary in the attenuation band, rising to infinite reactance at infinite frequency. The variation of Z0T with frequency is shown in Fig. 17.11.

Fig. 17.11 Similarly, the characteristic impedance of a p-network is given by

ZZ1 2 k Z0p = = (17.24) Z 2 0T f 1− fc

The variation of Z0p with frequency is shown in Fig. 17.11. For f , fc, Z0p is real; at f 5 fc, Z0p is infinite, and for f . fc, Z0p is imaginary. A low pass filter can be designed from the specifications of cut-off frequency and load resistance.

At cut-off frequency, Z1 5 – 4Z2 −4 jvc L = jvc C 2 2 p fc LC 5 1 Also we know that k= L/ C is called the design impedance or the load resistance

2 L ∴ k = C 2 2 2 2 p fc k C 5 1 1 C = gives the value of the shunt capacitance pfc k k and L= k2 C = gives the value of the series inductance. pfc Example 17.1 Design a low pass filter (both p and T-sections) having a cut-off frequency of 2 kHz to operate with a terminated load resistance of 500 V.

L Solution It is given that k = 5 500 V, and fc 5 2000 Hz C 824 Circuits and Networks

k 500 We know that L = = = 79. 6 mH pfc 3. 14× 2000 1 1 C = = = 0.F 318 3. 14 2000 500 fc k ⋅ ⋅ The T and p-sections of this filter are shown in Fig. 17.12 (a) and (b) respectively.

Fig. 17.12

17.7 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. 17.8. The prototype high pass filters are shown in Fig. 17.13, where Z1 5 – j/vC and Z2 5 jvL.

Fig. 17.13

Again, it can be observed that the product of Z1 and Z2 is independent of frequency, and the filter design obtained will be of the constant k type. Thus, Z1Z2 are given by

− j L 2 ZZ1 2 = jv L = = k vC C L k = C

The cut-off frequencies are given by Z1 5 0 and Z1 5 – 4Z2. j Z 5 0 indicates = 0 , or v → 1 vC

From Z1 5 – 4Z2 Filters and Attenuators 825

− j = −4 jv L vC 1 v2 LC = 4 1 or fc = (17.25) 4p LC

The reactances of Z1 and Z2 are sketched as functions of frequency as shown in Fig. 17.14. As seen from Fig. 17.14, the filter

transmits all frequencies between f 5 fc and f 5 . The point fc from the graph is a point at which Z1 5 – 4Z2. From Eq. 17.7,

␥ Z −1 sinh =1 = 2 4Z 42 LC 2 1 From Eq. 17.25, fc = 4p LC Fig. 17.14 1 ∴ LC = 4pfc

␥ −()()4 2f 2 f ∴ sinh = c = j c 2 42 f Z f In the pass band,−1 <1 < 0 , a 5 0 or the region in which c <1 is a pass band 4Z2 f −1 f b = 2sin c f Z f In the attenuation band 1 < −1, .i.e c > 1 4Z f 2 Z ␣ = 2 cosh−1 1 4Z2 −1 fc = 2 cos ; = − f The plots of a and b for pass and stop bands of a high pass filter network are shown in Fig. 17.15. 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 Fig. 17.15 R= k = L/ C 826 Circuits and Networks

1

fc = 4p LC/ k 1 f = = c 4pL4 p Ck L Since C = , k k 1 L = and C = 4pfc 4pfc k The characteristic impedance can be calculated using the relation

Z L 1 ZZZ=1 + 1 =1 − 0T 1 2 2 4Z2 C 4v LC

2 fc Z0 = k 1 − T f Similarly, the characteristic impedance of a p-network is given by

2 ZZ1 2 k Z0p = = Z0TTZ0 k = (17.26) 2 f 1− c f

The plot of characteristic impedances with respect to frequency Fig. 17.16 is shown in Fig. 17.16.

Example 17.2 Design a high pass filter having a cut-off frequency of 1 kHz with a load resistance of 600 .

Solution It is given that RL 5 K 5 600 V and fc 5 1000 Hz

K 600 ∴ L = = = 47. 74 mH 4pfc 4× p × 1000

1 1 C = = = 0.F 133 44 × 600 × 1000 kfc

The T and p-sections of the filter are shown in Fig. 17.17. Filters and Attenuators 827

Fig. 17.17

17.8 m-DERIVED T-SECTION

It is clear from Figs 17.10 and 17.15 that 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. In order to join the filter sections, it would be necessary that their characteristic impedances be equal to each other at all frequencies. If their characteristic impedances match at all frequencies, they would also have the same pass band. 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. Therefore, any attempt to increase attenuation in stop band by cascading also results in an increase of ‘a’ in the pass band. 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. Suppose a prototype T-network shown in Fig. 17.18 (a) has the series arm modified as shown in Fig. 17.18 (b), where m is a constant. Equating the characteristic impedance of the networks in Fig. 17.18, we have

Fig. 17.18

Z0T 5 Z0T where Z0T is the characteristic impedance of the modified (m-derived) T-network.

2 2 2 Z1 m Z1 +ZZ = + mZ Z ′ 4 1 2 4 1 2 828 Circuits and Networks

Z 2 m2 Z 2 1 +ZZ =1 + mZ Z ′ 4 1 2 4 1 2 Z 2 mZ Z ′ =1 ()1 −m2 + Z Z 1 2 4 1 2

Z1 2 Z2 Z2′ =(1 −m )) + (17.27) 4m m

It appears that the shunt arm Z2′ consists of two impedances in series as shown in Fig. 17.19.

1− 2 From Eq. 17.27, m should be positive 4m to realize the impedance Z2′ physically, i.e. 0 , m , 1. Thus m-derived section can be obtained from the prototype by modifying its series and shunt arms. The same technique can be applied to p section network. Suppose a prototype p-network shown in Fig. 17.20 (a) has the shunt arm modified as shown in Fig. 17.20 (b). The characteristic impedances of the prototype and its modified sections have to be equal for matching. Fig. 17.19

Fig. 17.20

′ ZZ0p= 0 p where Z90p is the characteristic impedance of the modified (m-derived) p-network.

Z Z ′ 2 ZZ 1 ∴ 1 2 = m Z Z ′ 1+ 1 1+ 1 4Z2 4⋅ Z2 / m Filters and Attenuators 829

Squaring and cross multiplying the above equation results as under.

4ZZZZ1′ 2+ 1 1′ (4Z1 Z 2+ mZ 1′ Z 1) = m Z14Z 2 Z1′ + −mZ1 = 4 Z 1 Z22 m m ZZ or Z ′ = 1 2 1 Z Z mZ 1+ 2 − 1 4m m 4 ZZ = 1 2 Z Z 2+ 1 ()1 −m2 m 4m 2 4m Z2 4 m ZZ1 2 2 mZ1 2 ()1−m ()1−m (17.28) Z1′ = = 4 2 Z4 m Z22 m + 2 + Z1 m mZ1 2 m()1− m2 ()1−m It appears that the series arm of the m-derived p section is a parallel combination of 2 mZ1 and 4mZ2 / 1 – m . The derived m section is shown in Fig. 17.21. m-Derived Low Pass Filter In Fig. 17.22, both m-derived low pass T and p filter sections are shown. For the T-section shown in Fig. 17.22 (a), the shunt arm is to be chosen so that it is resonant at some

frequency f 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 f 2 1 1−m v L, where v is the = r r mvr C 4m Fig. 17.21 resonant frequency

Fig. 17.22 830 Circuits and Networks

2 4 r = ()1−m2 LC 1 fr = = f∝ ()1− 2 LC m 1 Since the cut-off frequency for the low pass filter is fc = p LC fc (17.29) f = 1−m2

2 f or m =1 − c (17.30) f

If a sharp cut-off is desired, f should be near to fc. From Eq. 17.29, it is clear that for the smaller the value of m, f comes close to fc. Equation 17.30 shows that if fc and f are specified, the necessary value of m may then be calculated. Similarly, for m-derived p section, the inductance and capacitance in the series arm constitute a resonant circuit. Thus, at f a frequency corresponds to infinite attenuation, i.e. at f 1 mr L = 1 2 −m rC 4m

2 4 r = LC()1− m2 1 fr = LC()1− m2 1 Since, fc = p LC

fc fr = = f∝ (17.31) 1−m2 Thus for both m-derived low pass networks for a positive value of m (0 , m , 1), f . fc. Equations 17.30 or 17.31 can be used to choose the value of m, knowing fc and fr. After the value of m is evaluated, the elements of the T or p-networks can be found from Fig. 17.22. The variation of attenuation for a low pass m-derived −1 section can be verified from a = 2coshZZ1 / 4 2 for fc , f , f. For Z1 5 jvL and Z2 5 – j/vC for the prototype. f m f ∴ a = 2cosh−1 c 2 f 1− f∝ Filters and Attenuators 831

f m Z f and b = 2sin−1 1 = 2sin−1 c 4 2 Z1 f 1− () 1−m 2 fc

Figure 17.23 shows the variation of a, b and Z0 with respect to frequency for an m-derived low pass filter.

Fig. 17.23

Example 17.3 Design a m-derived low pass filter having cut-off frequency of 1 kHz, design impedance of 400 V, and the resonant frequency 1100 Hz.

Solution k 5 400 V, fc 5 1000 Hz; f 5 1100 Hz From Eq. 17.30 2 2 fc 1000 m =1 − =1 − = 0. 416 f 1100 Let us design the values of L and C for a low pass, K-type filter (prototype filter). Thus, k 400 L = = =127. 32 mH fc ×1000 1 1 C = = = 0. 795 F ×400 × 1000 kfc The elements of m-derived low pass sections can be obtained with reference to Fig. 17.22. Thus the T-section elements are

mL 0.. 416× 127 32 × 10−3 = = 26. 48 mH 2 2 mC 5 0.416 3 0.795 3 10–6 5 0.33 mF 832 Circuits and Networks

1−m2 1−(.) 0 416 2 L = ×127. 32 × 10−3 = 63. 27 mH 4m 4⋅ 0. 416 The p-section elements are

mC 0.. 416× 0 795 × 10−6 = = 0. 165 mF 2 2 1−m2 1−(.) 0 416 2 ×C = ×0. 795 × 10−6 = 0. 395 mF 4m 4× 0. 416 mL 5 0.416 3 127.32 3 10–3 5 52.965 mH The m-derived LP filter sections are shown in Fig. 17.24.

Fig. 17.24 m-derived High Pass Filter In Fig. 17.25 both m-derived high pass T and p‑sections are shown. If the shunt arm in T-section is series resonant, it offers minimum or zero impedance. Therefore, the output is zero and, thus, at resonance frequency, or the frequency corresponds to infinite attenuation. L 1 v = r m 4m vr C 1−m2

Fig. 17.25 Filters and Attenuators 833

1 1−m2 v2= v 2 = = r ∝ L 4m 4 C LC m 1−m2

2 2 1−m 1−m ∝ = or f∝ = 2 LC 4 LC

From Eq. 17.25, the cut-off frequency fc of a high pass prototype filter is given by

1 fc = 4p LC

2 f∝ = fc 1 − m (17.32)

2 f m =1 − ∝ (17.33) fc

Similarly, for the m-derived p-section, the resonant circuit is constituted by the series arm inductance and capacitance. Thus, at f

4m 1 vr L = 1 2 v −m r C m 2 2 2 1−m vr = v∝ = 4LC

2 1−m2 1−m v∝ = or fϰ = 2 LC 4 LC

Thus, the frequency corresponding to infinite attenuation is the same for both sections. Equation 17.33 may be used to

determine m for a given f and fc. The elements of the m-derived high pass T or p-sections can be found from Fig. 17.25. The variation of a,

b and Z0 with frequency is shown in Fig. 17.26 Fig. 17.26. 834 Circuits and Networks

Fig. 17.26 Example 17.4 Design a m-derived highpass filter with a cut-off frequency of 10 kHz; design impedance of 5 V and m 5 0.4.

Solution For the prototype high pass filter,

k 500 L = = = 3. 978 mH 4p4 p 10000 fc × × 1 1 C = = = 0.F 0159 44 × 500 × 10000 kfc The elements of m-derived high pass sections can be obtained with reference to Fig. 17.25. Thus, the T-section elements are

2C 2× 0. 0159 × 10−6 = = 0. 0795 mF m 0. 4

L 3. 978× 10−3 = = 9. 945 mH m 0. 4 4m 4× 0. 4 = ×0. 0159 × 10−6 = 0. 0302 mF 2 C 2 1−m 1−(.) 0 4 The p-section elements are

2L 2× 0. 0159 × 10−3 = =19. 89 mH m 0. 4 4m 4× 0. 4 ×L = ×3. 978 × 10−3 = 7. 577 mH 1 2 1(.) 0 4 2 −m − C 0. 0159 = ×10−6 = 0. 0397 mF m 0. 4 Filters and Attenuators 835

T and p sections of the m-derived highpass filter are shown in Fig. 17.27.

Fig. 17.27

17.9 BAND PASS FILTER

As already explained in Section 17.1, a band pass filter is one which attenuates all frequencies below a lower cut-off frequency f1 and above an upper cut-off frequency f2. Frequencies lying between f1 and f2 comprise the pass band, and are transmitted with zero attenuation. A band pass filter may be obtained by using a low pass filter followed by a high pass filter in which the cut-off frequency of the LP filter is