Filters

Ease of use makes integrated, switched-capacitor filters attractive for many new applications. This article helps you prepare for such designs by describing the filter products and explaining the concepts that govern their operation.
Starting with a simple integrator, we first develop an intuitive approach to active filters in general and then introduce practical realizations such as the state-variable filter and its implementation in switched-capacitor form. Specific integrated filters mentioned include Maxim's MAX7400 family of higher-order switched-capacitor filters.

First-Order Filters

An integrator (Figure 1a) is the simplest filter mathematically, and it forms the building block for most modern integrated filters. Consider what we know intuitively about an integrator. If you apply a DC signal at the input (i.e., zero frequency), the output will describe a linear ramp that grows in amplitude until limited by the power supplies. Ignoring that limitation, the response of an integrator at zero frequency is infinite, which means that it has a pole at zero frequency. (A pole exists at any frequency for which the transfer function's value becomes infinite.)
We also know that the integrator's gain diminishes with increasing frequency and that at high frequencies the output voltage becomes virtually zero. Gain is inversely proportional to frequency, so it has a slope of -1 when plotted on log/log coordinates (i.e., -20db/decade on a Bode plot, Figure 1b).

Figure 1a. A simple RC integrator

Figure 1b. A Bode plot of a simple integrator

You can easily derive the transfer function as

where s is the complex-frequency variable + j and ₀ is 1/RC. If we think of s as frequency, this formula confirms the intuitive feeling that gain is inversely proportional to frequency. We will return to integrators later, in discussing the implementation of actual filters.
The next most complex filter is the simple low-pass RC type (Figure 2a). Its characteristic (transfer function) is

When s = 0, the function reduces to ₀/ ₀, i.e., 1. When s tends to infinity, the function tends to zero, so this is a low-pass filter. When s = - ₀, the denominator is zero and the function's value is infinite, indicating a pole in the complex frequency plane. The magnitude of the transfer function is plotted against s in Figure 2b, where the real component of s () is toward us and the positive imaginary part (j) is toward the right. The pole at - ₀ is evident. Amplitude is shown logarithmically to emphasize the function's form. For both the integrator and the RC low-pass filter, frequency response tends to zero at infinite frequency; that is, there is a zero at s = . This single zero surrounds the complex plane.

Figure 2a. A simple RC low-pass filter

Figure 2b. The complex function of an RC low-pass filter

But how does the complex function in s relate to the circuit's response to actual frequencies? When analyzing the response of a circuit to AC signals, we use the expression jL for impedance of an inductor and 1/jC for that of a capacitor. When analyzing transient response using Laplace transforms, we use sL and 1/sC for the impedance of these elements. The similarity is apparent immediately. The j in AC analysis is in fact the imaginary part of s, which, as mentioned earlier, is composed of a real part s and an imaginary part j.
If we replace s by j in any equation so far, we have the circuit's response to an angular frequency . In the complex plot in Figure 2b, = 0 and hence s = j along the positive j axis. Thus, the function's value along this axis is the frequency response of the filter. We have sliced the function along the j axis and emphasized the RC low-pass filter's frequency-response curve by adding a heavy line for function values along the positive j axis. The more familiar Bode plot (Figure 2c) looks different in form only because the frequency is expressed logarithmically.

Figure 2c. A Bode plot of a low-pass filter

While the complex frequency's imaginary part (j) helps describe a response to AC signals, the real part () helps describe a circuit's transient response. Looking at Figure 2b, we can therefore say something about the RC low-pass filter's response as compared to that of the integrator. The low-pass filter's transient response is more stable, because its pole is in the negative-real half of the complex plane. That is, the low-pass filter makes a decaying-exponential response to a step-function input; the integrator makes an infinite response. For the low-pass filter, pole positions further down the - axis mean a higher ₀, a shorter time constant, and therefore a quicker transient response. Conversely, a pole closer to the j axis causes a longer transient response.
So far, we have related the mathematical transfer functions of some simple circuits to their associated poles and zeroes in the complex-frequency plane. From these functions, we have derived the circuit's frequency response (and hence its Bode plot) and also its transient response. Because both the integrator and the RC filter have only one s in the denominator of their transfer functions, they each have only one pole. That is, they are first-order filters.
However, as we can see from Figure 1b, the first-order filter does not provide a very selective frequency response. To tailor a filter more closely to our needs, we must move on to higher orders. From now on, we will describe the transfer function using f(s) rather than the cumbersome V_OUT/V_IN.