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 (“An Appendix to Filter Wizard #14 (“Match Point”)”) while I was away doing battle with a tangle of hot transistors. Let him be an example to you all of what you can achieve when you actually do the darned homework. But I’m back, and there’s more work to do to actually make some filters.

Filter Wizard #16, “Bruton Charisma: Make Those Inductors Vanish Using Savvy Scaling”, 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 didn’t 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 doesn’t 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, that’s another Million Monkeys (Filter Design using the “Million Monkeys Method” project in its own right; interesting work has been done on topology development by guesswork *coughs* sorry, I mean optimization. The subject of a future article, perhaps, but not disciplined enough for my purpose here.

Here we’ll address the central problem straight on. We’ve shown that we can **conceptually** create a new element by multiplying a component’s value by a factor K that’s 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, we’d ‘see’ an impedance of Z*K. Now, if K were a real constant, we already know how to do that, it’s just a transformer with a turns ratio of sqrt(K). But that doesn’t 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; let’s look at how we deploy key properties of the humble op-amp to do this. In all of this work, we’ll investigate only the creation of scaled elements with one terminal connected to ground. Remember that in “Dualling Master: Swap Current and Voltage for Easier Filter Design” I gave advance warning that this would be much easier, or at least more robust and economical of parts, than generating truly floating scaled components.

**Figure 1**

**An ‘ideal’ impedance scaler, and an op-amp circuit to explore.**

Two key attributes of a good, high-gain op-amp when it’s running linearly in a stable, closed-loop circuit are that (1) there’s essentially no voltage between the two input pins and (2) there’s essentially no current flowing into either input pin. So, consider the circuit shown in figure 1(b) (and yes, I know the input polarity symbols are missing from the amplifier, I’ll get to that). Because of property (1) we can see that V2=V1=V, say, and because of property (2) we can write down by inspection

This looks promising, because of course our Zs don’t have to be resistive; they can have some frequency dependence. But before you get too excited I need to tell you that this **isn’t** our ‘Holy Grail’ circuit yet, for a couple of good reasons. Firstly, there’s **no** combination of Rs and Cs we can use for Z1 through Z3 that quite leads to the correct impedance function for our D-element.

It gets worse. Remember I left out the polarity signs for the op-amp? That was to emphasize that we can write down the circuit equations that lead to eqn.1 without knowing the polarity! But, remember the caveat for the initial conditions: this has to be a **stable** circuit. Clearly this circuit has both positive and negative feedback, in amounts that depend not only on Z1 through Z3 but also on the impedance of whatever is connected to terminal 1 (and also, obviously, on which way round we connect the amplifier). Any combination of these impedances and choice of feedback phase that cannot deliver stability cannot be used as a practical circuit. One usage configuration that does crop up occasionally is Figure 2, which realizes a negative capacitance – note that this is only usable where such a component wouldn’t itself destabilize your circuit!

**Figure 2**

**A simple negative impedance converter realizing a negative capacitance.**

This circuit is a simple example of the class of circuits called Impedance Converters (it’s a **negative** impedance converter, by virtue of the sign in eqn.1). The next step in our journey is to investigate a more elaborate two-op-amp configuration whose flexibility has earned it the term *Generalized Impedance Converter*, or GIC for short. The circuit we’ll look at has become the most popular of the many theoretical ways to achieve this function, and is usually referred to as the Antoniou GIC [Ref.1].