Analog Filters: Singly-Terminated LC Ladders Franco Maloberti Introduction The purpose of this part is to design a LC ladder network that: Is a two-port network It contains inductors and capacitors Has a resistive termination at the output The source is a voltage or a current generator Or a voltage source E1 z11I1 z12I2 I2 I1 LC Ladder 1 E2 E 2 z21I1 z22I2 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 2 LC Ladder with Current source Consider a singly-terminated filter with a current source and a normalized resistive load I2 I1 LC Ladder 1 E2 E2 z21 I1 z22 I 2 z21 I1 z22 E2 E2 z21 Z 21 ( s ) I1 1 z22 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 3 LC Ladder with Current source (ii) Z21(s) E2 z 21 I1 1 z22 How to realize an LC network for a given Z21(s)? Properties of z21 and z22 z21 = (even poly)/(odd poly) or vice versa Zeros of Z21 (transmission zeros) are zeros of z21 z22 is a lossless function P22/Q22 = (even poly)/(odd poly) or vice versa Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 4 Implementing Z21(s) Consider the transfer function P( s ) KN1 ( s) KM 1 ( s) Z 21 ( s) or Q( s ) M 2 ( s ) N 2 ( s ) M 2 ( s) N 2 ( s) M(s) even terms N(s) odd terms Q(s) Hurwitz polynomial If P(s) is an even polynomial KM1 (s) N 2 (s) Z 21(s) M (s) 1 2 N 2 (s) z22 M 2 (s) and N 2 (s) z21 KM1 (s) N 2 (s) Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 5 Implementing Z21(s) (ii) If P(s) is an odd polynomial KN1 (s) M 2 (s) Z 21(s) N (s) 1 2 M 2 (s) z22 N 2 (s) and M 2 (s) z21 KN1 (s) M 2 (s) For a given Z21(s) we have to design an LC network that realizes z21(s) and z22(s) simultaneously Proceed from left to right instead of going from right to left Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 6 Transmission zeros at 0 or infinite Use of Cauer’s realizations to remove zeros at the origin or infinite completely. Use an intuitive view. Series L P(s) Z21(s) Q3 (s) Example Three non-dissipative elements Input is a parallel element Z2 Z1 Z3 Franco Maloberti Z2 Z1 Z3 Analog Filters: Singly-Terminated LC Ladders zero @ ∞ Series C Zero @ 0 Shunt C Zero @ ∞ Shunt L Zero @ 0 7 Intuitive view (Example Q3(s)) Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 8 Example 1 Z 21 ( s ) K even numerator 3 2 s 2s 2s 1 K 3 s 2s Z 21 ( s ) 2s 2 1 1 3 s 2s three transmission zeros at s 2s2 1 K z22 3 ; z21 3 s 2s s 2s Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 9 Example 1 (ii) Three transmission zeros at s = ∞ Applying the long division to get circuit parameters 1 3 s 3 2 s (2 s 2 1) s s 2 2 3 4 2s 2 1 s s 1 2 3 3 3 s 1 s 0 2 2 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 10 Example 1 (iii) Network realization 4 H 3 I1 3 F 2 1 F 2 1 E2 Evaluation of k E2 1 Z 21 (0) K I1 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 11 Example 1 (iv) Use of Matlab for removing transmission zeros 2s2 1 z22 3 s 2s clear all num=[1 0 2 0]; den=[2 0 1]; [c1,r1]=deconv(num,den); c1 r1=r1(3:4); [l,r2]=deconv(den,r1); l r2=r2(3); [c2,r3]=deconv(r1,r2) Franco Maloberti ex6_1 c1 = 0.5000 l = 1.3333 c2 =1.5000 r3 = 0 Analog Filters: Singly-Terminated LC Ladders 0 0 0 0 12 Example 2 High-pass Butterworth filter s3 K 2 Ks3 2s 1 Z 21(s) 3 ; Z (s) 21 s3 2s s 2s2 2s 1 1 2 2s 1 s3 2s Ks3 z22 2 ; z21 2 2s 1 2s 1 One zero at 0 and two zero at ∞ Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 13 Example 2 (ii) Remove the pole of z22 at infinite s3 2s 1 3s z1 2 s; z1 2 2s 1 2 4s 2 Ks3 z21 2 Ks 2s 1 1/2 H I1 3/2 H IL 3/4 F 3 s IL 3 2 z21 s Ks I1 1 s 1 2 2 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 14 Zeros at finite frequency Finite zeros are zeros of P(s) (or z21) P(s) P(s) Z 21(s) Q(s) M 2 (s) N 2 (s) We need to create the finite zeros of Z12(s) while realizing z22. z22 does not have the zeros of Z12! Partial (and complete) removal of poles shifts the zeros Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 15 Zero Shifting The partial removal of poles from a function DOES affect the zeros of the remainder Zero shifting is necessary ( s 1)( s 9) Consider Z (s) s(s 4) 2 2 2 X ( ) Im[Z ( j )] X ( ) Im[Z ( j )] k' 0 1 2 3 1 2 3 0 k ' / When part of the pole at (left) and at the origin (right) is removed Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 16 Zero Shifting (ii) A partial pole removal shits all the finite (and nonzero) zeros toward the affected pole The larger part is removed the more the zeros are shifted toward the pole Zeros cannot be shifted beyond adjacent poles Shifting a zero in a given desired position is not always possible X() Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 17 Example 6.4 K(s2 4) Z 21(s) 3 s 2s2 2s 1 2s 2 1 z 22 (s) 3 s 2s 1 Zeros of z22 is at 2 Poles of z22 are at 0; y !! Work with 22 Franco Maloberti We can move the zeros only in the range 2 0; 2 s3 2s y 22 (s) 2 2s 1 Analog Filters: Singly-Terminated LC Ladders 18 Example 6.4 (ii) 2 Zeros of y22 is at Poles of y22 are at 2 Franco Maloberti 1/ 2; y 22 C1s 0 s j 2 1 2 1 ; 2 We can move the zeros only in the range 2 s3 2s y 22 (s) C1s s j 2 2 C1s 2s 1 j4 2 j2C1 0 C1 7 7 Analog Filters: Singly-Terminated LC Ladders 19 Example 6.4 (iii) 2 3s(s2 4) y 22 (s) s 7 7(2s2 1) 7(2s2 1) k1s z1 z 2 3s(s2 4) s2 4 7(2s2 1) s2 4 49 k1 2 s s2 4 12 3s(s 4) 49 s 7 12 z2 z1 2 s 4 12s Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 20 Example 6.4 (iv) Another option: Remove completely the pole of z22 at infinite This produces a zero at infinite Produce the required pair of zeros by partially removing the zero at infinite The partial removal leave the two zeros only Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 21 LC Ladder with Voltage Source Consider a singly-terminated filter with a voltage source and a normalized resistive load I2 y 21E1 y 22 E 2 y 21E1 y 22I2 I2 y 21 Y21(s) E1 1 y 22 Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 22 LC Ladder with Voltage Source (ii) The y parameters and the z parameters of a lossless two-port have the same properties. Use the same procedure studied for current source Transmission poles (of y) at the origin and infinite Non-zero transmission poles (of y) Franco Maloberti Analog Filters: Singly-Terminated LC Ladders 23
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