A Filter Primer (MXM AN733)
A Filter Primer (MXM AN733)
A Filter Primer (MXM AN733)
A Filter Primer
This comprehensive article covers all aspects of analog filters. It first addresses the basic types like first and second
order filters, highpass and lowpass filters, notch and allpass filters and high order filters. It then goes on to explain the
characteristics of the different implementations such as Butterworth filters, Chebychev filters, Bessel filters, Elliptic
filters, State-variable filters, and Switched-capacitor 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).
A Filter Primer
+j
and
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
0,
0/
0,
i.e., 1. When s tends to infinity, the function tends to zero, so this is a low-
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.
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 j L for impedance of an inductor and 1/j C 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 the complex
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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.
A Filter Primer
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 VOUT/VIN.
and if we define
= 1/LC and Q =
0L/R,
then
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where
is the filter's characteristic frequency and Q is the quality factor (lower R means higher Q).
0/Q
= 0. We can solve
In this case, a = 1, b =
0/Q,
and c =
2.
roots are real and lie on the negative-real axis. The circuit's behavior is much like that of two first-order RC filters in
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cascade. This case isn't very interesting, so we'll consider only the case where Q > 0.5, which means (b2 -4AC) is
negative and the roots are complex.
The real part is therefore -b/2a, which is -
0/2Q,
and common to both roots. The roots' imaginary parts will be equal
and opposite in signs. Calculating the position of the roots in the complex plane, we find that they lie at a distance of x0
from the origin, as shown in Figure 3b. (The associated mathematics, which are straightforward but tedious, will be left
as an exercise for the more masochistic readers.)
Varying 0 changes the poles' distance from the origin. Decreasing the Q moves the poles toward each other,
whereas increasing the Q moves the poles in a semicircle away from each other and toward the j axis. When Q = 0.5,
the poles meet at - 0 on the negative-real axis. In this case, the corresponding circuit is equivalent to two cascaded
first-order filters, as noted earlier.
Now let's examine the second-order function's frequency response and see how it varies with Q. As before, Figure 4a
shows the function as a curved surface, depicted in the three-dimensional space formed by the complex plane and a
vertical magnitude vector. Q = 0.707, and you can see immediately that the response is a low-pass filter.
A Filter Primer
plane. These poles must, however, occur as complex-conjugate pairs, in which the real parts are equal and the
imaginary parts have opposite signs. This flexibility in pole placement is a powerful tool and one that makes the secondorder stage a useful component in many switched-capacitor filters. As in the first-order case, the second-order low-pass
transfer function tends to zero as frequency tends to infinity. The second-order function decreases twice as fast,
however, because of the s2 factor in the denominator. The result is a double zero at infinity.
Having discussed first- and second-order low-pass filters, we now need to extend our concepts in two directions: We'll
discuss other filter configurations such as high-pass and bandpass sections, and then we'll address higher-order filters.
the old and new values of s are identical. The double zero that was at s = 1 moves to zero, and the finite
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2,
2.
pass function into a high-pass one by changing the numerator, leaving the denominator alone.
The Bode plot offers another perspective on low-pass to high-pass transformations. Figure 5a shows the Bode plot of a
second-order low-pass function: flat to the cutoff frequency, then decreasing at -40db/decade. Multiplying by s2 adds a
+40db/decade slope to this function. The additional slope provides a low-frequency rolloff below the cutoff frequency,
and above cutoff it gives a flat response (Figure 5b) by canceling the original -40db/decade slope.
We can use the same idea to generate a bandpass filter. Multiply the low-pass responses by s, which adds a +20db/
decade slope. The net response is then +20db/decade below the cutoff and -20db/decade above, yielding the bandpass
response in Figure 5c:
Notice that the rate of cutoff in a second-order bandpass filter is half that of the other types, because the available 40db/
decade slope must be shared between the two skirts of the filter.
In summary, second-order low-pass, bandpass, and high-pass functions in normalized form have the same
denomination, but they have numerators of 02, 0s, and s2, respectively.
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z,
2/
2,
the numerator becomes zero, f(s) becomes zero (a double zero, in fact, because of s2 in the numerator),
and we have the characteristic of a notch filter. The gain at frequencies above and below the notch will differ unless
=
In other words, the notch filter is based on the sum of a low-pass and a high-pass characteristic. We use this fact in
practical filter implementations to generate the notch response from existing high-pass and low-pass responses. It may
seem odd that we create a zero by adding two responses, but their phase relationships make it possible.
Finally, there is the all-pass filter, which has the form
This response has poles and zeros placed symmetrically on either side of the j axis, as shown in Figure 6. The effects
of these poles and zeroes cancel exactly to give a level and uniform frequency response. It might seem that a piece of
wire could provide this effect more cheaply; however, unlike a wire, the all-pass filter offers a useful variation of phase
response with frequency.
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Higher-Order Filters
We are fortunate in not having to treat the higher-order filters separately, because a polynomial in s of any length can
be factored into a series of quadratic terms (plus a single first-order term if the polynomial is odd). A fifth-order low-pass
filter, for instance, might have the transfer function
where all the a0 are constants. We can factor the denominator as follows:
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The last equation represents a filter that we can realize physically as two second-order sections and one first-order
section, all in cascade. This configuration simplifies the design by making it easier to visualize the response in terms of
poles and zeroes in the complex-frequency plane. We know that each second-order term contributes one complexconjugate pole pair, and that the first-order term contributes one pole on the negative-real axis. If the transfer function
has a higher-order polynomial in the numerator, that polynomial can be factored as well, which means that the secondorder sections will be something other than low-pass sections.
Using the synthesis principles described above, we can build a great variety of filters simply by placing poles and
zeroes at different positions in the complex-frequency plane. Most applications require only a restricted number of these
possibilities, however. For them, many earlier experimenters such as Butterworth and Chebychev have already worked
out the details.
from the origin. The three-dimensional surface corresponding to this filter (Figure 7b) illustrates how, as the effect of the
lowest-Q pole starts to wear off, the next pole takes over, and the next, until you run out of poles and the response falls
off at -80db/decade.
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0/s.
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1.
In theory you can create higher-order filters by cascading more than two integrators. Some integrated-circuit filters use
this approach, but it has drawbacks. To program these filters, you must calculate coefficient values for the higher-order
polynomial. Also, a long string of integrators introduces stability problems. By limiting ourselves to second-order
sections, we have the advantage of working directly with the 0 and Q variables associated with each pole.
Switched-Capacitor Filters
The characteristics of all active filters, regardless of architecture, depend on the accuracy of their RC time constants.
Because the typical precision achieved for integrated resistors and capacitors is approximately 30%, a designer is
handicapped when attempting to use absolute values for the components in an integrated filter circuit. The ratio of
capacitor values on a chip can be accurately controlled, however, to about one part in 2000. Switched-capacitor filters
use these capacitor ratios to achieve precision without the need for precise external components.
In the switched-capacitor integrator shown in Figure 12, the combination of C1 and the switch simulates a resistor.
A Filter Primer
Notice that the current is proportional to VIN, so we have the same effect as a resistor of value
The integrator's
Because
is therefore
is proportional to the ratio of the two capacitors, its value can be controlled with great accuracy. Moreover,
the value is proportional to the clock frequency, so you can vary the filter characteristics by changing fCLK, if desired.
But the switched capacitor is a sampled-data system and therefore not completely equivalent to the time-continuous RC
integrator. The differences, in fact, pose three issues for a designer.
First, the signal passing through a switched capacitor is modulated by the clock frequency. If the input signal contains
frequencies near the clock frequency, they can intermodulate and cause spurious output frequencies within the system
bandwidth. For many applications, this is not a problem, because the input bandwidth has already been limited to less
than half the clock frequency. If not, the switched-capacitor filter must be preceded by an anti-aliasing filter that removes
any components of input frequency above half of the clock frequency.
Second, the integrator output (Figure 12) is not a linear ramp, but a series of steps at the clock frequency. There may
be small spikes at the step transitions caused by charge injected by the switches. These aberrations may not be a
problem if the system bandwidth following the filter is much lower than the clock frequency. Otherwise, you must again
add another filter at the output of the switch-capacitor filter to remove the clock ripple.
Third, the behavior of the switched-capacitor filter differs from the ideal, time-continuous model, because the input
signal is sampled only once per clock cycle. The filter output deviates from the ideal as the filter's pole frequency
approaches the clock frequency, particularly for low values of Q. You can, however, calculate these effects and allow for
them during the design process.
Considering the above, it is best to keep the ratio of clock-to-center frequency as large as possible. Typical ratios for
switched-capacitor filters range from approximately 28:1 to 200:1. The MAX262, for example, allows a maximum clock
frequency of 4MHz, so using the minimum ratio of 28:1 gives a maximum center frequency of 140kHz. At the low end,
switched-capacitor filters have the advantage that they can handle low frequencies without using uncomfortably large
values of R and C. You simply lower the clock frequency.
Conclusion
The purpose of this article was to introduce the concepts and terminology associated with switched-capacitor active
filters. If you have grasped the material presented here, you should be able to understand most filter data sheets.
A Filter Primer
More Information
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