Editor’s note: This article was written by Jorge Garcia, EAGLE Product Support Specialist and guest Filter Wizard columnist. Kendall Castor-Perry is always open to guest spots here in his blog.
Hello! Filter Fans, I’m sure you are all familiar with the classical approach to filter design. First you get your specifications, and then you go through a few simple calculations to determine the required order and response type for the filter (Butterworth, Chebyshev, Cauer, etc.). From there you choose a suitable prototype circuit (Sallen-Key, LC ladder, etc) with a 1 radian/second center frequency. Then you open a volume of charts and select the normalized component values. With the end in sight you scale the prototype circuit to the desired center frequency. It’s very methodical and very tedious to design this way (though if you can afford fancy software, that can make things easier).
This was the only method I knew to design filters, so imagine my surprise when I read Kendall’s article, “An Excelent fit, Sir!, where he describes designing a filter using a spreadsheet. My first reaction was amazement. No sophisticated charts, no pole zero relationships, determination of a response type, none of those overly complex equations I learned in school.
His method seemed far more intuitive and practical. The ability to analyze circuits on a spreadsheet, to use Excel Solver to optimize circuit values, this was a technique that in my mind was essential for not only filter design, but as a powerful tool in my toolbox. So if you’ve been following along with his articles, besides answering many questions about filter design, I’m sure a few questions have popped up that need some small clarification and that’s what I intend to do.
Throughout his various articles Kendall has posted some small “homework assignments”. Being the inquisitive engineer that I am, I decided to explore those interesting assignments. This column is the result of those efforts. Specifically I’ll be showing you the answer to the question he posed in the “Filter Design using the “Million Monkeys Method”” article: “how could I modify the calculation to make use of all that gap between the upper and lower passband tolerance limits?” Specifically, how to allow the optimization to use gain values greater than the DC value, as well as less than it. Once that one is answered then the next question naturally follows: “Can the design still be optimized for minimum sensitivity?” I’ll be walking you through the answers to those questions, so let’s go!
It’s straightforward to adjust the calculations so that the Solver can use the total of 0.2 dB ripple allotted in “Million Monkeys”. The first method is to include a 0.1 dB “fudge factor” in such a way that the allowed ripple is between 0 dB and -0.2 dB referenced to the DC gain (instead of +0.1 dB and -0.1 dB) while keeping RL=RS. We still meet the condition of maximum power transfer at DC; there is no solution that will allow for a gain above the DC gain of -6.02 dB gain, so the solver ends up converging on a solution that pulls the pass band gain between -6.02 dB and -6.22 dB. So we still don’t really use the allowed tolerance of 0.1 dB either side of the DC gain.
A smarter method is to change the load impedance to shift the maximum power point to the top of the desired ripple band, making it different from the DC gain. This is the method we’re going to focus on here. Those who are used to designing filters using charts will recognize that when designing some responses, an adjustment factor is applied to the load impedance. In this column you’ll find out why. The key to understanding all of this is in the diagrams in Figure 1 below; I want you to take a good hard look at them, and engrave them in your memory. Got it? Good.
The key to this whole discussion is to recognize that in some applications the LC network can behave as a transformer (at least at some frequencies). So we can use the transformer relationship to determine the effective turns ratio of the equivalent transformer that gives maximum power into RL, given RL and RS. We then set this turns ratio to provide just that extra gain we need to push the passband ripple to our upper limit, making this the ‘maximum power’ gain.
In Figure 1(b) I’ve replaced the LC Network with a transformer in our filter circuit. If we specify the transformer turns ratio correctly then we can obtain the maximum power transfer case (c) and everything to the right of the first RS can be replaced with a single resistor whose value is also RS.
I’ll begin the derivation by first reviewing the basic transformer relations which will then be used to evaluate the maximum power voltage gain. A simple transformer is shown in Figure 2.
We have, from the standard transformer relationships:
From the transformer derivation we can see that the voltage gain from the matching transformer is given by the square root of the ratio between the load resistance and the source resistance. With this information in tow we’ll compare it with the DC gain. That’s trivial since we know that at DC, inductors behave like shorts and capacitors like open circuits. The DC gain therefore is the voltage divider formed by RL and RS. By setting the ’maximum power’ voltage gain equal to the DC gain times the ripple gain k (in V/V, not dB), we will be able to solve for the value(s) of the ratio of load to source resistor, a=RL/RS, that will yield a minimum sensitivity design. The resulting expression is derived below.
Using a little algebra (yes, I said it, mwah hah hah) we arrive at a quadratic equation, as shown below: