DA & Control

# The Five-Minute Filter University, July Session

July 1, 2006 By: Ed Ramsden Sensors

Back in the late 1970s comedian Don Novello (a.k.a. Father Guido Sarducci) had a routine called the "Five-Minute University," which was supposed to impart to you, in the span of only five minutes, all the knowledge you would retain five years after graduating from a regular university. So, in the same spirit, I offer "Dr. Ed's Five-Minute Analog Filter Design University."

 Ed ramsden

## What Filters Do

The purpose of a filter is to pass along signals you are interested in, while blocking or attenuating (reducing) the ones you aren't. Traditional filter design focuses on discriminating between wanted and unwanted signals on the basis of their frequencies. Although there are potentially lots of ways to sort signals by frequency, the two most common are to pass along only those signals lower than some arbitrary cut-off frequency (low-pass filters), or pass along signals higher than the cutoff frequency (high-pass filters), as shown in Figure 1. Corner frequency is also referred to as the filter's –3 dB frequency because the response drops –3 dB (a factor of ~0.707) here.

 Figure 1. Low-pass filters pass along signals lower than some arbitrary cutoff frequency; high-pass filters pass signals higher than the cutoff frequency

For a real-world example of the difference between a low-pass and a high-pass filter, consider the bass and treble knobs on your stereo. The bass knob attenuates or enhances low-frequency sounds like drums or tubas, while the treble knob does the same for high-frequency sounds like flutes. The two main defining characteristics of a filter (among many) are the cutoff or corner frequency, and how well it discriminates against signals on the wrong side of that corner.

## Now for Some Circuits

Two minutes in and we have already covered theory and applications! Now it's time to see some circuits. Many simple passive filters can be derived by starting from a resistive voltage divider (Figure 2). If you don't know how one of these works, I suggest you enroll in the "Five-Minute Circuit Theory" course and come back next semester. If you already took this course (and remember only P = IV) then I can tell you that a voltage divider's gain (ratio of output voltage to input voltage) is R2/(R1+R2).

 Figure 2. You can derive a passive filter by starting from a resistive voltage divider

To turn a voltage divider into a filter, you substitute a capacitor or inductor for one of the resistors. This results in a voltage divider where the division ratio becomes frequency dependent. This is because the "resistance" of an inductor can be defined as 2πfLj and that of a capacitor can be defined as 1/2πfCj. If this were a real filter design course we would now combine these expressions with the voltage divider equation to derive frequency-dependent expressions for various filters. Note that these expressions have imaginary values, however, which means complex math. Since this is the five-minute course (and we are three minutes in), let's just skip the math part and take a look at some rules of thumb to describe qualitative behavior.

## Rules of Thumb—and Where It Gets Hard

• 1. At DC (frequency = 0), capacitors are open circuit (R = infinity) and inductors are short circuits (R = 0)
• 2. At infinitely high frequency, capacitors are short circuits (R = 0) and inductors are open circuits (R = infinity)
• 3. Resistors are resistors at all frequencies

`1 2 `

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