A Beginner's Guide to Filter Topologies - AN1762

This article is an excellent introduction to analog filters. It covers the basic first and second order filter types as well as the advantages and disadvantages of passive and active filters. It derives and analyzes the basic biquad implementation using several op amp implementation examples. The world of filter design is often thought of as black magic because of the myriad of configurations, unique terminology, and complex equations. This application note will look at different filter types, shed some light on terminology and provide a foundation to base the selection and allow the design of a filter to be a little more science and a little less magic. We will focus primarily on low-pass filters, although both band-pass and high-pass filters could be analyzed and designed with the same approach. One of the first questions that arises is: Can I use a simple passive filter (i.e., only passive components such as resistors, capacitors and inductors—no operational amplifiers) or is it best if I use an active filter. The advantages of a passive filter are that it is quite simple to design and implement. It also provides a simple single pole or two pole filter whose electrical response can be easily calculated. For a single pole low-pass filter, fc = 1/(2 × π × RC) the filter roll-off is 6dB per octave or 20dB/decade. However, this filter does have some noticeable drawbacks: 1. It is very sensitive to the component value tolerances. 2. For low frequencies, the values of R and C can be quite large, leading to physically large components. 3. A first or second-order filter may not give adequate roll-off 4. If gain is required in the circuit, it cannot be added to the filter itself. 5. The filter can potentially have a high output impedance. Since the resistor value is typically large, to keep the capacitors a reasonable value, the next stage device can see a significant source impedance. An op amp can be added at the output, but why add the op amp here when it can be used to improve the filter's performance in addition to lowering the output impedance? If the passive filter doesn't meet your needs you can opt for an active filter. By adding an op amp, we can easily implement a 2nd order filter. Second order filters are often the building blocks of higher order filters since they can be easily cascaded to obtain the higher orders. The general form of a filter can be written in equation form as follows: H(s) = K (s + z1)(S + z2)/[(s + p1)(s + p2)] This equation is called a biquadratic equation, or a biquad for short. The zn terms in the numerator represent zeros and the pn terms in the denominator represent poles. Looking at the equation, the first thing that becomes clear is that filters that can be written in this form are biquads. This means Sallen-Key filters, state-variable variable filters, multiple feedback filters and other types are all biquads. There also is a "biquad" topology to help further confuse things. Thus, the real filter names are biquad Sallen-Key, biquad state variable, and biquad (which will all be explained a little later). Using low pass filters as our example, a low pass filter can be written in a general equation form as: H(s) = K/(as2 + bs + 1), where a = R1R2C1C2 and b = R1C1 + R2C1 This can be simplified by making R1 = R2 and C1 = C2, resulting in: H(s) = K/(R2C2s2 + 2RCs + 1) The block diagram of a low-pass 2nd order Sallen-Key filter is shown in Figure 1. This filter is also referred to as a positive feedback filter since the output feeds back into the positive terminal of the op amp. This topology is popular because it requires only a single op amp, thus making it relatively inexpensive.