Richards’s transformation is a remarkable scheme that takes into account the actual properties of transmission lines, yielding broadband transmission line-based implementations of lumped-element filter prototypes [12, 13, 14, 15]. 2.12.1 Richard's Transformation and Transmission Lines Consider a section of transmission line of electrical length θ with ABCD parameters T=[cos(θ)ȷ/Z0sin(θ)ȷZ0sin(θ)cos(θ)](2.12.1) If this line is terminated in a load, ZL , then its input impedance is Zin(θ)=cos(θ)ZL+ȷZ0sin(θ)ȷ/Z0sin(θ)ZL+cos(θ)(2.12.2) Now examine two extreme conditions. As the load impedance increases, eventually becoming an open circuit, the input impedance of a line with electrical length θ is defined in terms of a cotangent of the electrical length: ZL→∞⇒Zin(θ)Yin(θ)=Z0ȷcot(θ)=ȷY0tan(θ)(2.12.3)(2.12.4) As the load impedance reduces to become a short circuit, the input impedance of a line with electrical length θ is defined in terms of a tangent of the electrical length: ZL→0⇒Zin(θ)=ȷZ0tan(θ)(2.12.5) These results lead to the Richards’s transformation, which replaces the Laplace variable, s , by Richards’s variable, S , where S=ȷαtan(θ) . This transformation is written s→S=ȷαtan(θ)(2.12.6) For now α and θ are constants that can be chosen as design variables. θ , of course, is the electrical length of the line. Also, α must have the units of impedance and it is the characteristic impedance of the transmission line. Applying the Richards’s transformation to a capacitor, the admittance of the element is transformed as follows: y=sC→Y=SC=ȷαCtan(θ)(2.12.7) so that the capacitor is transformed into an open-circuited stub with characteristic admittance Y0=αC(2.12.8) If a lumped-element capacitor with admittance y=sC is to be realized using a transmission line, the admittance Y=SC=ȷαCtan(θ) is instead realized. There are two parameters to select to realize this admittance. The first, α , is the characteristic admittance of the transmission line (and for any given transmission line topology there is a minimum and maximum characteristic admittance or impedance that can be realized), and θ is the electrical length of the line. Applying the transformation to an inductor, the impedance of the element is transformed as follows: Z=sL→Z=SL=ȷαLtan(θ)(2.12.9) clipboard_e459a77cb1196c9d7559c1b3c74a1364a.png Figure 2.12.1 : Equivalences resulting from Richards’s transformation. With fr=2f0 the transmission line stubs are one-eighth wavelength long at f0 . so that the inductor is transformed into a short-circuited stub with characteristic impedance Z0=αL(2.12.10) Thus the Richards’s transform converts an inductor into a short-circuited stub and a capacitor into an open-circuited stub. 2.12.2 Richard's Transformation and Stubs There is a duality between stubs and inductors and capacitors; they are coupled by Richards’s transformation. One of the important quantities used in the transformation is the commensurate frequency, , which most of the

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