Now let’s play with this formula a little. Let’s say we’re designing a Butterworth (maximally-flat) response. What ratio of RL/RS will yield a minimum sensitivity design? Since we have no ripple, k=1, and using the quadratic formula we find that a=1 is the only solution.
So, if we are designing a maximally flat filter the impedance ratio for minimum sensitivity is 1 to 1 and this is perfectly inline with the conclusions stated in the “Match Point: Why Maximum Power means Minimum Sensitivity” article, just proven more rigorously. Now let’s repeat this calculation for a maximum upper ripple of 0.1 dB, which translates to k=1.011579. The two values of a are now distinct: a=1.3566 and a=0.7371.
So, we have two impedance ratios that should be able to generate a minimum sensitivity design using the Excel Solver method (and look how different from unity they are even for this small ripple). At first glance we might be tempted to drop the second solution since we don’t want to lose any more DC gain then necessary. However if you need to use the dual form of the filter then the lower value will be preferred since it will generate a larger output impedance hence preserving the dc gain of the dual form. See the “Dualling Master: Swap Current and Voltage for Easier Filter Design” and “Bruton Charisma: Make Those Inductors Vanish Using Savvy Scaling” Filter Wizard articles for why this is useful.
Next we run the spreadsheet to get the values that will potentially yield minimum sensitivity designs with these load ratios. The passband response of a converged a=1.3566 solution is shown in Figure 3. Note that it ripples both ways from the DC gain now.
Let’s look at some Monte Carlo simulations. I have centered the graphs for easy comparison and I assumed a 15% tolerance on the caps and inductors and 5% for the resistors since I run into these tolerances quite often. The basic schematic for the simulation is shown in Figure 4; I have also included a response obtained from a non-minimum sensitivity design for comparison.
As can be seen from figures 6 and 7 above, both of the impedance ratios we calculated yield low sensitivity solutions whose response doesn’t exceed the top of the ripple band.
The Monte Carlo simulations assume that the tolerance distribution is Gaussian or Normal. Many real-world processes follow a Gaussian distribution; for an easy experiment grab a bag of resistors (at least 30, the more, the better) and measure the values. You will notice that many of them have a value that is much closer than the specified tolerance would lead you to believe. The Gaussian distribution has the following interesting properties:
-approx. 68% of the components will have a value within one standard deviation of the mean
-approx. 95% of the components will have a value within two standard deviations of the mean
-approx. 99.7% of the components will have a value within three standard deviations of the mean
What does all this mumbo-jumbo mean? Manufacturers specify tolerances to make sure none of their components exceed the specified tolerance, and they base the specification on 3 standard deviations from the mean, which is at the specified component value. So if a component has a specified tolerance of 15% then 68% of the components will have a tolerance of 5% or less. A similar trend is observed in the Monte Carlo sims where most of the runs seem to stay within a certain window while only a few yield extreme responses.
So what have we learned? We can use a simple quadratic equation to define the impedance ratios that will allow for minimum sensitivity designs that use the entire ripple around the DC gain, regardless of the specified ripple. I hope to have also shed some light on the subject of component tolerances and how they are specified.
Thank you for your time, and special thanks to Kendall for the opportunity and insight. If anyone has any comments or suggestions feel free to e-mail me at Jorge.Garcia@autodesk.com. Till next time (that is, if I’m ever allowed to do this again)!
A Note From the Author:
I'm a product support specialist for EAGLE, here at Autodesk. I have been working with EAGLE for close to 10 years. I got my Bachelor's in Electrical Engineering from Florida International University in 2008. I love electronics and building projects focusing on power and control applications. I have a special spot in my heart for embedded systems and programming even though I still consider myself a novice in both. I love spending time with my wife Orlinda, in our little home here in Florida.