Kuroda s Identities. We find that Kuroda s Identities can be very useful in making the implementation of Richard s transformations more practicable.

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1 4//7 Kurodas Idetities 1/7 Kuroda s Idetities We fid that Kuroda s Idetities ca be very useful i makig the implemetatio of Richard s trasformatios more practicable. Kuroda s Idetities essetially provide a list of equivalet two port etworks. By equivalet, we mea that they have precisely the same scatterig/impedace/admittace/trasmissio matrices. I other words, we ca replace oe two-port etwork with its equivalet i a circuit, ad the behavior ad characteristics (e.g., its scatterig matrix) of the circuit will ot chage! Q: Why would we wat to do this? A: Because oe of the equivalet may be more practical to implemet! For example, we ca use Kuroda s Idetities to: 1) Physically separate trasmissio lie stubs. ) Trasform series stubs ito shut stubs. 3) Chage impractical characteristic impedaces ito more realizable oes.

2 4//7 Kurodas Idetities /7 Four Kuroda s idetities are provided i a very ambiguous ad cofusig table (Table.7) i your book. We will fid the first two idetities to be the most useful. Cosider the followig two-port etwork, costructed with a legth of trasmissio lie, ad a ope-circuit shut stub: Note that the legth of the stub ad the trasmissio lie are idetical, but the characteristic impedace of each are differet. The first Kuroda idetity states that the two-port etwork above is precisely the same two-port etwork as this oe:

3 4//7 Kurodas Idetities 3/7 = 1 + Thus, we ca replace the first structure i some circuit with the oe above, ad the behavior that circuit will ot chage i the least! Note this equivalet circuit uses a short-circuited series stub. The secod of Kuroda s Idetities states that this two port etwork:

4 4//7 Kurodas Idetities 4/7 Is precisely idetical to this two-port etwork: = 1 +

5 4//7 Kurodas Idetities 5/7 With regard to Richard s Trasformatio, these idetities are useful whe we replace the series iductors with shorted stubs. To see why this is useful whe implemetig a lowpass filter with distributed elemets, cosider this third order filter example, realized usig Richard s Trasformatios: 1 3 C ω c 1 ω c 3 1 ω cc

6 4//7 Kurodas Idetities 6/7 Note that we have a few problems i terms of implemetig this desig! First of all the stubs are ideally ifiitely close to each other how do we build that? We could physically separate them, but this would itroduce some trasmissio lie legth betwee them that would mess up our filter respose! Secodly, series stubs are difficult to costruct i microstrip/striplie we like shut stubs much better! To solve these problems, we first add a short legth of trasmissio lie ( ad = ) to the begiig ad ed of the filter: ω c 1 ω c 3 1 ω cc

7 4//7 Kurodas Idetities 7/7 Note addig these legths oly results i a phase shift i the filter respose the trasmissio ad reflectio fuctios will remai uchaged. Now we ca use the secod of Kuroda s Idetities to replace the series stubs with shuts: ω 1 c 1 ω 3 c ω cc 3 where: = ωc 1 = ωc 3 Now this is a realizable filter! Note the three stubs are separated, ad they are all shut stubs. Note that a specific umerical example (example.5) of this procedure is give o pp of your book.