Instruction FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 30, N o 2, June 2017, pp. 179 - 185 DOI: 10.2298/FUEE1702179D A NEW LUMPED ELEMENT BRIDGED-T ABSORPTIVE BAND- STOP FILTER  Suhash C. Dutta Roy Formerly at the Department of Electrical Engineering, Indian Institute of Technology, Delhi, New Delhi India Abstract. Following a brief review of previous work on bandstop filters, the inadequacy of a recent work to obtain a perfect notch or perfect absorption at the notch frequency ω0 is demonstrated. A simple and elegant alternative solution, based on purely analytical arguments, is then presented. The resulting network is shown to achieve perfect matching as well as perfect absorption at the notch frequency and has several other advantages. A comparison has also been made with the conventional bridged-T band-stop filter. Key words: bandstop filter, bridged-T network, circuit design. 1. INTRODUCTION Bandstop filters are circuits which reject, to within a specified tolerance, a band of frequencies around a centre frequency at which there is complete rejection. Such filters are known by various names, such as band rejection filters, notch filters, null networks etc. and are required in many situations in communication and instrumentation. Bandstop filters have fascinated a large number of researchers, including the present author, who has written papers on the analysis [1-7], design [8] and its limitations [9], and analysis and applications of dual input techniques to such filters [10-12]. All these contributions relate to analog circuits. Bandstop filters are also required in digital signal processing, and the author and his students have done extensive work on digital notch filters, using both FIR and IIR techniques [13-21]. Of these, [21] is a review of FIR notch filter design, which appeared in this journal. At low frequencies, passive RC networks are mostly used, except in situations where a selectivity, defined as (notch frequency)/(3 dB stop bandwidth), is required to be more than half. In the latter cases, either active RC filters or LC networks are to be used. For high frequencies, LC networks are easily designed and implemented. At microwave frequencies, distributed networks are preferred over lumped networks, although the latter  Received November 3, 2016 Corresponding author: Suhash C. Dutta Roy Department of Electrical Engineering, Indian Institute of Technology, 164, Hauz Khas Apartments, New Delhi 110016, India (E-mail: s.c.dutta.roy@gmail.com) 180 S. C. DUTTA ROY have the advantage of occupying less space, and as is well known, space is a premium in microwave integrated circuits. Examples of lumped element microwave bandstop filters can be found in [22-29], while bandstop filters with distributed elements can be found in [30,31]. 2. SCOPE AND ORGANIZATION OF THE PAPER This paper is concerned with the design of a band-stop filter which achieves a perfect notch and perfect absorption at some frequency ω0. In this context, we first demonstrate, in Section 3, the inadequacy of a recent solution proposed by Chieh and Rowland [32], by network theoretic arguments. In the next Section, we present a new, simple and elegant alternative design, based on purely analytical arguments. The resulting network is shown to achieve perfect matching as well as perfect absorption at the notch frequency, and has several other advantages. A normalized design is discussed in Section 5, and the simulation results are presented. A comparison of the new design with the conventional bridged-T band- stop filter is made in Section 6. Finally, Section 7 gives the concluding comments. 3. CHIEH AND ROWLAND'S DESIGN Chieh and Rowland [32] proposed the symmetrical network of Fig. 1 where Z1(jω)=1/(jωC), (1a) Z2(jω)=R1+jωL1+1/( jωC1) (1b) and Z3(jω)=R2+jωL2+1/(jωC2). (1c) and both Z2 and Z3 resonate at the same frequency ωo. For ready reference, we reproduce here the expressions for the z-parameters of the network and the scattering parameters, in slightly different forms: Fig. 1 The bridged-T network z11=z22=Z2+(Z1 2 +Z1Z3)/(2Z1+Z3), (2) z12=z21=Z2+Z1 2 /(2Z1+Z3), (3) S12=S21=2z21 Zo/[(z11+ Zo) 2 -z21 2 ], (4) A New Lumped Element Bridged-T Absorptive Band-stop Filter 181 and S11=S22=(z11 2 - z21 2 ]/[(z11+ Zo) 2 -z21 2 ]. (5) Note from (2) and (3) that z11=z21+Z1Z3/(2Z1+Z3). (6) From (4) and (5), we observe that for a perfect notch as well as perfect absorption at the frequency ωo, we require z21(jωo)=0 (7) and z11(jωo)= Zo. (8) From (1), we have Z1(jωo)=1/(jωoC), Z2(jωo)=R2 , and Z3(jωo)=R1. (9) Substituting these values in (3) gives, on simplification, z21(jωo)= R2+1/[ jωoC(2+ jωoC R1)], (10) which cannot be made zero. Also, under this condition, z11(jωo)= R2+(1+ jωoC R1)/[ jωoC(2+ jωoC R1)], (11) which cannot be equal to Zo if the latter is purely resistive, which is usually the case. Equation (8) can be satisfied only if Zo is a complex series RC impedance. Thus the network of Fig. 1 with the element values given by (1) can achieve neither perfect notch nor perfect absorption. 4. THE NEW DESIGN The problem to be solved can be restated as follows: Given ωo and Ro and the network topology of Fig.1, find Z1, Z2 and Z3 such that z21(jωo)=Z2(jωo)+[Z1(jωo)] 2 /[2Z1(jωo)+Z3(jωo)]=0, (12) and z11(jωo)=z21(jωo)+Z1(jωo)Z3(jωo)/[2Z1(jωo)+Z3(jωo)]=Ro. (13) where Zo has been assumed to be resistive, equal to Ro. In view of (12), (13) reduces to z11(jωo)=Z1(jωo)Z3(jωo)/[2Z1(jωo)+Z3(jωo)]=Ro. (14) From (14), Z3 is expressed in terms of Z1 as Z3(jωo)=2RoZ1(jωo)/[Z1(jωo)-Ro]. (15) Combining this with (12) and simplifying, we get Z2(jωo)=[Ro-Z1(jωo)]/2. (16) We can now choose a Z1. If we take Z1(jωo)=1/(jωoC), as in [1], then (15) gives, on simplification, Z3(jωo)=[2Ro/(1+ωo 2 C 2 Ro 2 )]+jωo[2CRo 2 /(1+ωo 2 C 2 Ro 2 )] (17) 182 S. C. DUTTA ROY which represents a series combination of an inductance L3 and a resistance R3, where R3=[2Ro/(1+ωo 2 C 2 Ro 2 )] and L3=2CRo 2 /(1+ωo 2 C 2 Ro 2 ). (18) Similarly, (16) gives Z2(jωo)=(Ro/2)+jωo/(2ωo 2 C), (19) which also represents a series combination of an inductance L2 and a resistance R2, where R2=(Ro/2) and L2=1/(2ωo 2 C). (20) In theory, C can be chosen to have any value, but as we shall see, it will be most convenient to choose C from the expression for R3 given in (18), which gives C=[(2Ro/R3)-1] 1/2 /(ωoRo) (21) Note that if we choose C=1/(ωoRo), (22) then R3 becomes equal to Ro. Also, under this condition, (17) and (18) give L3=Ro/ωo and L2=Ro/(2ωo). (23) This choice of C is advantageous because then Z3 can be obtained by a series combination of Z2 and Z2 and there is no spread in the element values of the network. Also note that lossy inductors can be used with ease because their losses can be absorbed in their series resistances. Finally, the element valus of the network are consolidated as C=1/(ωoRo), L3=2L2=Ro/ωo and R3=2R2= Ro. (24) Fig. 2 The normalized design of the absorptive bandstop filter 5. A NORMALIZED DESIGN It is always convenient to have a normalized design which can be denormalized by impedance and frequency scaling. Let Ro=1 ohm and ωo=1 rad/sec. Then (24) gives the element values as C=1F, L3=2L2=1H and R3=2R2=1 ohm. (25) A New Lumped Element Bridged-T Absorptive Band-stop Filter 183 The resulting network is shown in Fig. 2. This network has been simulated with MATLAB and the obtained plots of │S11(jω)│and│S21(jω)│are shown in Fig. 3. These plots exactly match the theoretical predictions. 6. COMPARISON WITH THE CONVENTIONAL BRIDGED-T BANDSTOP FILTER It may be noted that compared to network proposed in [32], the conventional bridged- T bandstop filter [3] performs better because it achieves a perfect notch but not perfect absorption. In this network, Z1(jωo)=1/(jωC), Z2(jωo)= r+jωL, and Z3(jω)=R. (26) The network then achieves a perfect notch at ω=[2/(LC)] 1/2 under the condition L=CRr, but it cannot achieve S11(jωo)=0 unless Zo is a parallel combination of a capacitor C and a resistor R/2, which is not the usual case. Also, if we choose r=R, then there is no spread in the component values. Further, as in the proposed alternative, a lossy inductor can be used here. In addition, in comparison with the networks of [32] and that proposed here, it uses the least number, viz. three of reactive elements, yielding a transfer function of order three. 10 -1 10 0 10 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 3 Performance of the normalized design. The upper curve is a plot of │S21(jω)│and the lower curve represents │S11(jω)│ 7. CONCLUDING COMMENTS It has been shown that the network proposed in [32] achieves neither a perfect notch nor perfect absorption. An alternative solution is proposed here purely by analytical, rather than physical or heuristic arguments, which achieves these two objectives simultaneously. The element values are obtained very simply, rather than by numerical and parametric methods as in [32]. Also, the new solution uses only two capacitors, instead of four, which 184 S. C. DUTTA ROY reduces the order of the transfer function by two. By an appropriate choice of the elements, there is no spread in the element values. A normalized design has been presented and the resulting characteristics of │S11(jω)│and│S21(jωo)│ have been plotted. A comparison of the two circuits has also been made with the conventional bridged-T bandstop filter. Acknowledgement: The author thanks Professor Y. V. 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