ELEctromagnetic DIAgnostics Lab.DIT Università di TrentoDIT ‐ Università di Trento

Via Sommarive 14, I‐38050 Trento ItaliaE‐mail: massimo.donelli@disi.unitn.it@

Richards and Kurodatransformations

Master Master DegreeDegree Electronic and TelecommunicationElectronic and TelecommunicationA.A. A.A. 20122012--20132013

Filter Implementation

The lumped element filter design discussed in theprevious lessons generally work wel at lowprevious lessons generally work wel at lowfrequencies, but two problems arise at microwavefrequencies. First of all commercial capacitors andq pinductors are generally available only for limited rangeof values and are difficult to implements at microwavef i b t th t b i l t d ithfrequencies, but they must be implemented withdistributed elements. Second at microwave frequenciesthe dimension and the distances between filterthe dimension and the distances between filtercomponents are not negligible. Richard’stransformations are used to convert lumped circuits totransmission line sections, while Kuroda’s identitiescan be used to separate filter elements by usingt i i li titransmission line sections.

Richard’s TransformationThe following transformation was introduced by Richard tosynthesize an LC network using open- short-circuited transmissionlineslines.

Thus if we replace the frequency ω with Ω the reactance of an inductor can be written as:

And the susceptance of a capacitor can be written as

Richard’s TransformationsThe above results indicate that an inductor can be replaced with a short circuitedstub of length βl and characteristic impedance L while a capacitor C can bereplaced with an open circuit stub of length βl and characteristic impedance 1/C Areplaced with an open circuit stub of length βl and characteristic impedance 1/C. Aunity filter impedance has been assumed.

Cutoff occours at unity frequency for a low pass filter prototype: to obtain the sameCutoff occours at unity frequency for a low pass filter prototype: to obtain the samecutoff frequency for the Richard’s transformed filter we must impose that:

This impose a stub length l=λ/8 where λ is the wavelength of the line at the cutoffp g gfrequency ωc . At a frequency two times the cutoff the line lenght will be a quarter-wavelength.

At frequencies far away from the cutoff the impedance of the stubs will no longermatch the original lumped element impedances, and the filter response will differfrom the desired prototype response. However the response will be periodic infrequency with a repetition time of 4ωc

Richard’s TransformationsThen the inductor and capacitors of a lumped-element filter designcan be replaced with short circuited and open circuited stubs asf llfollows:

Since the inductors and capacitors can be replaced with stubs ofth l th λ/8 th li ll d t lithe same length λ/8 these lines are called commensurate lines

Kuroda’s IdentitiesThe four Kuroda identities are used to achieve a more practical microwave filter design by performing one of the following 

ioperation:

1) Physically separate transmission line stubs1) Physically separate transmission line stubs2) Transform series stubs into shunt stubs and vice versa.3) Change impractical impedances into more realizable ones.g p p

The Kuroda’s identities use additional transmission line sectionll d i l f l h λ/8 h ff Th icalled unit elements of length λ/8 at the cutoff. The unit

elements are thus commensurate with the stubs used toimplement the inductors and capacitors of the prototypeimplement the inductors and capacitors of the prototypedesign.

Kuroda’s Identities

Where n2= 1 + Z2/Z1

Kuroda’s IdentitiesLet us consider the first Kuroda identity. It tells us that the following networks areequivalent.

Kuroda’s IdentitiesWe are able to analyze the behavior of the network considering theABCD matrix of the unit element is given by:

Which is nothing else that the ABCD matrix of a transmisison line oflength l and impedance Z1

Where

Kuroda’s IdentitiesThe open circuited stub that represent the capacitor has an impedance of 

and the related ABCD matrix of the entire circuit is given by:

Kuroda’s IdentitiesThe impedance of the short circuited series stub has an impedanceof:

The ABCD matrix of the second section of the Kurodatransformation is the following:

Impedance and Admittance InvertersAs we have seen it is often desiderable to use only series, oronly shunt elements when implementing a filter with aparticular type of transmission line. The Kuroda identities can beused for conversions of this form, nut another possibility is touse impedance (K) or admittance (J) inverters Such inverters areuse impedance (K) or admittance (J) inverters. Such inverters areespecially useful for bandpass or bandstop filters with narrow(<10%) bandwidth. The conceptual operation of impedance and( 10%) bandwidth. The conceptual operation of impedance andadmittance inverters is reported in the following slides.Fundamentally the inverters can be made with quarterwavelength transformers of given length.

Impedance and Admittance Inverters

Equivalent circuits for short transmission line sectionsThe impedance The approximate equivalent circuits for a shortlength of transmission line having a very large or very smallh d h f h k d f lcharacteristic impedance. The Z‐parameters for such kind of line aregiven by

A d id i h i l T k h i lAnd considering the equivalent T network the series elements aregiven by:

While the shunt elements are given by Z12

Equivalent circuits for short transmission line sections

Equivalent circuits for short transmission line sectionsSo if βl<π/2 we have the first approximation and the serieselements are inductors while the shunt element is a capacitor oflvalue:

Equivalent circuits for short transmission line sectionsIf we assume βl<π/4 and large characteristic impedance thecapacitor could be neglected and we have only a serie inductor oflvalue:

Equivalent circuits for short transmission line sectionsIf we assume βl<π/4 and short characteristic impedance theinductor can be neglected and we have only a shunt capacitor oflvalue: