1. Analog Filters: Singly-Terminated LC Ladders Franco Maloberti

2. Analog Filters: Singly-Terminated LC Ladders2 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

3. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders3 LC Ladder with Current source Consider a singly-terminated filter with a current source and a normalized resistive load

4. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders4 LC Ladder with Current source (ii) How to realize an LC network for a given Z 21 (s)? Properties of z 21 and z 22 z 21 = (even poly)/(odd poly) or vice versa Zeros of Z 21 (transmission zeros) are zeros of z 21 z 22 is a lossless function P 22 /Q 22 = (even poly)/(odd poly) or vice versa

5. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders5 Implementing Z 21 (s) Consider the transfer function If P(s) is an even polynomial

6. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders6 Implementing Z 21 (s) (ii) If P(s) is an odd polynomial For a given Z 21 (s) we have to design an LC network that realizes z 21 (s) and z 22 (s) simultaneously Proceed from left to right instead of going from right to left

7. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders7 Transmission zeros at 0 or infinite Use of Cauer’s realizations to remove zeros at the origin or infinite completely. Use an intuitive view. Example Three non-dissipative elements Input is a parallel element Series L zero @ ∞ Series C Zero @ 0 Shunt C Zero @ ∞ Shunt L Zero @ 0

8. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders8 Intuitive view (Example Q 3 (s))

9. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders9 Example 1

10. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders10 Example 1 (ii) Three transmission zeros at s = ∞

11. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders11 Example 1 (iii) Network realization Evaluation of k

12. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders12 Example 1 (iv) Use of Matlab for removing transmission zeros 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) ex6_1 c1 = 0.5000 0 l = 1.3333 0 c2 =1.5000 0 r3 = 0 0

13. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders13 Example 2 High-pass Butterworth filter One zero at 0 and two zero at ∞

14. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders14 Example 2 (ii) Remove the pole of z 22 at infinite 1/2 H 3/4 F 3/2 H I1I1 ILIL

15. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders15 Zeros at finite frequency Finite zeros are zeros of P(s) (or z 21 ) We need to create the finite zeros of Z 12 (s) while realizing z 22. z 22 does not have the zeros of Z 12 ! Partial (and complete) removal of poles shifts the zeros

16. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders16 Zero Shifting The partial removal of poles from a function DOES affect the zeros of the remainder  Zero shifting is necessary Consider When part of the pole at  (left) and at the origin (right) is removed

17. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders17 Zero Shifting (ii) A partial pole removal shits all the finite (and non- zero) 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(  ) 

18. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders18 Example 6.4 Zeros of z 22 is at Poles of z 22 are at We can move the zeros only in the range Work with y 22 !!

19. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders19 Example 6.4 (ii) Zeros of y 22 is at Poles of y 22 are at We can move the zeros only in the range

20. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders20 Example 6.4 (iii)

21. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders21 Example 6.4 (iv) Another option: Remove completely the pole of z 22 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

22. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders22 LC Ladder with Voltage Source Consider a singly-terminated filter with a voltage source and a normalized resistive load

23. Franco MalobertiAnalog Filters: Singly-Terminated LC Ladders23 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)