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Bruton Charisma: Make those inductors vanish using savvy scaling

October 04, 2010 | Kendall Castor-Perry | 222901145
Bruton Charisma: Make those inductors vanish using savvy scaling The Filter Wizard, Kendall Castor-Perry, continues to offer his insights into how to optimize spreadsheet techniques to plan and create active filters. In this article the Filter Wizard looks at impedance scaling and the Bruton Transform.
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I must confess that I just didnt realize how much mess the monkeys would leave in the Monkey House when I let them loose with a few innocent software tools during the development of Filter Design Using the Million Monkeys Method. Todays column is the third sequel to that article, which was Filter Wizard #13, and I am still flushing out the little promises that got made in that episode. Maybe those triskaidekaphobics are onto something...

Most recently, you were left hanging by a finger at the end of Filter Wizard #15 (Dualling Master: Swap Current and Voltage for Easier Filter Design). We set out to eliminate the inductors in our LC lowpass filter and ended up doubling the number! Well see shortly just how these can be made to disappear in one mathematical stroke. But first, lets talk about impedance scaling. In the text that follows, Ill use w to indicate w wed normally use a lower-case omega but the Web seems to have an aversion to Greek characters in text. Ill also use ^2 to mean squared just in case that wasnt obvious...

Our filter uses source and load resistances of 1 ohm. Now, if all the impedances in the network change by the same factor, the voltage transfer function a dimensionless value which is only ever determined by ratios of impedances in a network is unchanged. So its easy to scale the filter to make it suit source and load resistances of, for example, 1 kilohm. The impedance of an inductor, Z=jwL (I dont have to explain what j is, I hope) is proportional to its inductance, so we just make all the inductor values 1000 times higher. We reduce the capacitor values by 1000, because their impedance is inversely proportional to their value, Z=1/(jwC).

The scaled circuit is shown in Figure 1, ready for simulation. The cutoff frequency is unchanged from the original 1/(2*pi) Hz; one way of convincing yourself of that is that the product of any inductor and capacitor pair is unchanged by the impedance scaling we just did.

The values are even less practical than they were before. Weve taken inductor L`3, for instance, from its previous already-enormous 1.621 H to a rather bizarre 1.621 kH yes, thats kilo-Henry, not kilo-Hertz with a missing z. Ill forgive those inductor-fearing readers for feeling that things are just getting worse and worse! But just hold on for a moment longer.

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