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Note for Network Theory - NT by Suman Kumar Acharya

  • Network Theory - NT
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  • Biju Patnaik University of Technology Rourkela Odisha - BPUT
  • Electrical and Electronics Engineering
  • B.Tech
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Whites, EE 481/581 Lecture 30 Page 1 of 14 Lecture 30: Scaling of Low Pass Prototype Filters. Stepped Impedance Low Pass Filters. In the last lecture, we discussed the design of prototype low pass filters where Rs  RL  1  and c  1 rad/s. Of course, one generally is not going to implement the prototype filter. So what good is it? It is possible to scale and transform the low pass prototype filter to obtain a low pass, high pass, band pass, and band stop filters for any impedance “level” ( Rs  RL ) and cutoff frequency. Nice! The process of filter design has three basic steps as discussed in the last lecture: (1) collect the filter specifications, (2) design the low pass prototype filter, (3) scale and convert the prototype. The first two steps were performed in the previous lecture. We’ll now consider the last step, beginning with scaling. Scaling Low Pass Prototype Filters There are two types of scaling for low pass prototype circuits, impedance scaling and frequency scaling: © 2016 Keith W. Whites

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Whites, EE 481/581 Lecture 30 Page 2 of 14 1. Impedance Scaling. Since the filter is a linear circuit, we can multiply all the impedances (including the terminating resistances) by some factor without changing the transfer function of the filter. Of course, the input and output impedances will change. If the desired source and load impedances equal R0 , then (8.64a),(1)  X L  R0 X L    R0 L  . Therefore, L  R0 L . 1R  C  X C  R0 X C    0  . Therefore, C   . (8.64b),(2) R0  C   Rs  R0 1  R0 . (8.64c),(3)  RL  R0  RL  R0 RL . (8.64d),(4) The primed quantities are the scaled quantities while the unprimed are those from the low pass prototype circuit (i.e., the unscaled quantities). 2. Frequency Scaling. As defined for the prototype c  1 rad/s. To scale for a different low pass cutoff frequency, we substitute   c (8.65),(5) where c is the desired cutoff frequency of the low pass filter. Applying this to the inductive and capacitive reactances in the prototype filter we find

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Whites, EE 481/581 Lecture 30 Page 3 of 14  L L  X L   L        . Therefore, L  . (8.66a),(6) c c  c  C 1 1     c  . Therefore, C   . (8.66b),(7)  X C  C      C  c c For a one-step impedance and frequency scaling process, we can combine (1)-(4), (6), and (7) to obtain RL  Lk   0 k (8.67a),(8) c  Ck   Ck  c R0  Rs  R0 (8.67b),(9) (10)  RL  R0 RL (11) where k  1,, N as in Fig. 8.25. For example, in the circuit of Fig. 8.25a, C1  g1 , L2  g 2 , C3  g 3 , etc. Example N30.1. Design a 3-dB, equi-ripple low pass filter with a cutoff frequency of 2 GHz, 50- impedance level, and at least 15-dB insertion loss at 3 GHz. The first step is to determine the order of the filter needed to achieve the required IL at the specified frequency. From equation (7) in the previous lecture for   c

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Whites, EE 481/581 Lecture 30 k 2  2  PLR    4  c  Page 4 of 14 2N (12) (This is just an approximation here since  / c  1.5 .) What value do we use for k? From Fig. 8.21 1 PLR Equal ripple 1 1 1 k 2 Pass band c Stop band  we see that the passband ripple equals 1  k 2 . So, with A = ripple in dB, then 10log 1  k 2   A k  10 A /10  1 (13) so that Consequently, for A  3 dB then k  0.998  1. Therefore, at  c  1.5 , equation (12) becomes PLR  32 N / 4 so that N 1 3 5 10logPLR 3.5 dB 22.6 dB 41.7 dB Fig. 8.27b w/ |/c|-1=0.5 6 dB 19 dB 35 dB

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