The document discusses Richard's transformation method for lumped-element filter design, emphasizing its application at low and microwave frequencies. It introduces Kuroda's identities for separating filter elements with transmission line sections and outlines specific filter design problems, including low-pass filters for various cut-off frequencies and impedances. The content covers key concepts in filter design, practical realizations, and transformation techniques for effective circuit implementation.

1. Richards Transformation
method
Class 29 & 30

2. Lumped-element filter design
 Works well at low frequencies
 At microwave frequencies
 inductors and capacitors are generally available only
for a limited range of values
 Distances between filter components is not negligible
 Richard’s transformation is used to convert lumped
elements to transmission line sections
 LC network using open and short circuited
transmission lines
 Kuroda’s identities can be used to separate filter
elements using transmission line sections

3. Richard’s transformation

4. Richard’s transformation cont..
 The reactance of an inductor
 The susceptance of a capacitor
tan tan
2
= maps the plane to plane,
which repeats with a period of =
p
p
l
l
v
l
v

 


  
tan (1)
(if we map to )
L
jX j L j L jL l
 

   

tan (2)
c
jB j C j C jC l
 
   

5. Richard’s transformation cont..
 WKT input impedance for a short circuit stub is
 Comparing this with equation (1)
 An inductor can be replaced by
 a short circuited stub of length l and
 Characteristic impedance L
 Similarly from equation (2) a capacitor can be replaced
with
 An open circuited stub of length l and
 Characteristic impedance 1/C
0 tan
in
Z jZ l



6. Richard’s transformation cont..
 Cut-off frequency of the low pass filter
prototype is
 c=1
 For Richard’s transform filter Cut-off
frequency is
  =1
 tanl=1
 ie. L=/8

7. Kuroda’s identities
2 2
1
1
Z
n
Z
 

8. Kuroda’s identities cont..
 Physically separate transmission line stubs
 Using unit elements
 an additional transmission line section of length
/8 at c
 Transform series stubs into shunt stubs
 Change impractical characteristic
impedances into more realizable ones

9. Equivalent circuit illustrating Kuroda’s
identity

10. Problem
 Design a low pass filter for fabrication using
microstrip lines. The specifications are: cut-
off frequency of 4GHz, third order,impedance
of 50, and a 3dB equal ripple characteristic

11. Element values for equal-ripple low-
pass filter prototype

12. Solution

13. Solution cont..

14. Problems
 Design a three-element maximally flat low-
pass filter with its cut-off frequency as 1GHz.
It is to be used between a 50 load and a
generator with its internal impedance at 50
 Design a low pass fourth order maximally flat
filter using only shunt stubs. The cut-off
frequency is 8GHz and the impedance is 50