Filter for thought

Have you ever wondered how engineers designed active filters before the birth of software? They were able to do it using nomographs - but before we talk about what these are, let’s refresh our memory a little bit on active filters.

Magnitude is the amplitude of the output. Phase is the angle of the output. Every pole adds -90° at high frequencies. Theoretically, a fourth order filter would phase shift through 360°, though the magnitude would be very small.

Group delay, on the other hand, is the derivative of phase with respect to frequency. It’s a measure of the time delay (filter’s time) in frequency. A filter with a flat group delay means that the output signals come out with the same relative phase as the input signals. In essence, everything is delayed by the same amount.

Before you decide on your filter design consider the tradeoffs between the various implementations and types. Let’s review a few important ones:

Narrower transition regions require higher filter order filters. That is, you’ll need a higher component count - active and passive. If you can tolerate more ripple in the passband, you can get a smaller transition region for the desired attenuation. However, a monotonic passband  gives you a smoother phase response which yields a constant group delay in the passband, important in multi frequency communication.

Now that we touched on some of the tradeoffs, let’s take a look at some of the most common types of active filters.

• The Butterworth has a monotonic passband and stopband. It’s optimal with respect to passband ripple, also known as maximally flat but has a wide transition region. It’s often chosen for anti aliasing.
• The Chebyshev has an equal amount of ripple in the passband and a monotonic stopband, providing a fast transition region.
• The Elliptic, sometimes referred to as a Cauer filter, has equal-ripple in both the passband and the stopband and gives you the fastest transition band of any filter. It does, however, have a long tail of settling time and requires a more complex implementation including poles and zeros in the transfer function.
• The Bessel, also known as Thomson filter, offers the most constant group delay in the passband. It’s optimal with respect to group delay and is also called a linear phase filter, not to be confused with zero phase. The Bessel has a monotonic passband and stopband but has a wide transition region.
• The Inverse Chebyshev gives you a monotonic passband and equal ripple in stopband, with the same transition region as the Chebyshev .  

Take a look at figure 1 below to see the different filter bands described above.

Figure 1. Representation of different filter bands

So now what about those nomographs?

Suppose you want to design a low pass filter, assuming an active filter with -3dB at 300kHz and -60dB at 1.2MHz. Once you’ve decided which filter type you’re going to use, you’ll need to determine the minimum filter order.

Suppose you want to design an anti-aliasing low pass filter where your goal is to minimize ripple. You’ll need to look at the Butterworth nomograph and follow these steps:

• Determine Ω= Ω_S/Ω_P
• Extend a line to connect between  M_P and M_S
• Draw a horizontal line across graph to intersect Ω
• The higher number represents the minimum filter order

In this example, you’ll need a fifth order.

If you plug the same numbers into FilterPro, you get the same answer. See the below figure.

Suggesting a “good” op amp for an active filter is not a trivial task and depends greatly on the implementation as well as the application. If you have a low voltage design and are in need of good settling time and low ripple the OPA320 is a good candidate.  If your design uses higher voltages and requires good linearity consider the LMP8671 or OPA211.

For more info on how to ensure measurement set-up is properly calibrated and matched to avoid measurement errors due to ripples check out Habeeb Ur Rahman Mohammed’s “How to understand ripples in RF perfomances” Engineer It video.