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Gee, I see! The ins and outs of generalized impedance converters

December 07, 2010 | Kendall Castor-Perry | 222901381
Gee, I see! The ins and outs of generalized impedance converters The Filter Wizard, Kendall Castor-Perry, returns and takes a look at how to practically realize a new element by attaching an initial component to some kind of 'impedance scaler'.
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OK, settle down at the back of the room, gentlemen and ladies! I know it made a nice change for you all that Jorge ran the last class while I was off doing battle with some hot transistors. Let him be an example to you all of what you can achieve when you actually do the darned homework. But Im back, and theres more work to do to actually make some filters.

Filter Wizard #16,Bruton Charisma, showed how we could apply an arbitrary scaling factor to each impedance in an LRC filter network, to create another network that has the same transfer function but is assembled from a different set of components, with familiar Rs and Cs and also a new guy, the D-element. But we didnt get as far as figuring out how we would actually make up such a thing, a component whose impedance equals -1/(Dw^2).

Such a component doesnt exist in passive form. How might we systematically develop a way of creating such a circuit configuration? Do we have to randomly juggle around a bunch of Rs, Cs and op-amps until we encounter a circuit that has an impedance formula that does the job? Well, thats another Million Monkeys project in its own right; interesting work has been done on topology development by guesswork {cough} sorry, I mean optimization. The subject of a future article, perhaps, but not disciplined enough for my purpose here.

Here well address the central problem straight on. Weve shown that we can
conceptually create a new element by multiplying a components value by a factor K thats a function of frequency. Can we now practically realize such an element by attaching our initial component to some kind of impedance scaler? With such a scaler, indicated in Figure 1(a), we could hang a certain component of impedance Z on one port, and when we look in on the other port, wed see an impedance of Z*K. Now, if K were a real constant, we already know how to do that, its just a transformer with a turns ratio of sqrt(K). But that doesnt help us when K=1/(jw), or any other function of jw for that matter.

Well, it turns out that we can perform such impedance scaling on actual components; lets look at how we deploy key properties of the humble op-amp to do this. In all of this work, well investigate only the creation of scaled elements with one terminal connected to ground. Remember that in Dualling Master I gave advance warning that this would be much easier, or at least much more economical of parts, than generating truly floating scaled components.

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