richard transformation method its used to so many things.ppt

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 capacitortan 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/C0
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 identities2 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