キーワード: analog filter design, second order filters, highpass, high pass, lowpass, low pass, filters, notch, allpass, high order, filters, Butterworth, Chebychev, Bessel, elliptic, state variable, filter
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チュートリアル 733

要約: This comprehensive article covers all aspects of analog filters. It first addresses the basic types: first- and second-order filters, highpass and lowpass filters, notch and all-pass filters, and high-order filters. The tutorial then explains the characteristics of the different implementations, such as Butterworth filters, Chebychev filters, Bessel filters, elliptic filters, state-variable filters, and switched-capacitor filters.

Starting with a simple integrator, we first develop an intuitive approach to active filters in general. We then introduce practical realizations such as the state-variable filter and its implementation in switched-capacitor form. Specific integrated filters described here include Maxim's MAX7400 family of higher-order switched-capacitor filters.

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,

You can easily derive the transfer function as:

V_{OUT}/V_{IN} = X_{C}/R = (1/sC)/R = -1/(sCR) = -ω_{0}/s |
Eq. 1 |

Where s is the complex-frequency variable σ + jω and ω

V_{OUT}/V_{IN} = (1/sC)/(R + 1/sC) = 1/(1 + sCR) = ω_{0}/(s + ω_{0}) |
Eq. 2 |

When s = 0, the function reduces to ω

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 jωL for the impedance of an inductor and 1/jωC 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 jω 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 jω.

If we replace s by jω in any of the above equations, we have the circuit's response to an angular frequency, ω. In the complex plot in Figure 2b, σ = 0 and hence s = jω along the positive jω axis. Thus, the function's value along this axis is the frequency response of the filter. We have sliced the function along the jω axis and emphasized the RC lowpass filter's frequency-response curve by adding a heavy line for function values along the positive jω axis. The more familiar Bode plot (

While the complex frequency's imaginary part, jω, helps describe a response to AC signals, the real part, σ, helps describe a circuit's transient response. Looking at Figure 2b, we can therefore say something about the RC lowpass filter's response compared to that of the integrator. The lowpass filter's transient response is more stable, because its pole is in the negative real half of the complex plane. Restated, the lowpass filter makes a decaying-exponential response to a step-function input; the integrator makes an infinite response. For the lowpass filter, pole positions further down the -σ axis mean a higher ω

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 application needs, we must move to higher orders. From now on, we will describe the transfer function using f(s) rather than the cumbersome V

ƒ(s) = X_{C}/(R + X_{L} + X_{C}) = (1/sC)/[R + sL + (1/sC)] = 1/(LC_{S}² + RC_{S} + 1) |
Eq. 3 |

If we define:

ω_{0}² = 1/LC and Q = ω_{0}L/R = 1/(RCω_{0}) |
Eq. 4 |

Then:

ƒ(s) = 1/[(s/ω_{0})² + s/(ω_{0}Q) + 1] = ω_{0}²/[s² + s(ω_{0}/Q) + ω_{0}²]
| Eq. 5 |

Where ω

The poles occur at s values for which the denominator becomes zero; that is, when s² + sω

Eq. 6 |

In this case, a = 1, b = ω

The real part is therefore -b/2a, which is -ω

Varying ω

Now we should examine the second-order function's frequency response and see how it varies with Q. As before,

Increasing the Q moves the poles in a circular path toward the jω axis.

There is also an effect on the filter's transient response. Because the poles' negative-real part is smaller, an input step function will cause ringing at the filter output. Lower values of Q result in less ringing, because the damping is greater. If Q becomes infinite, the poles reach the jω axis, causing an infinite frequency response (instability and continuous oscillation) at s = ω

A second-order filter provides the variables ω

Having discussed first- and second-order lowpass filters, we now need to extend our concepts in two directions: we will discuss other filter configurations, such as highpass and bandpass sections, and then we will address higher-order filters.

ƒ(s) = ω_{0}²/[(ω_{0}^{4}/s²) + (ω_{0}³/Qs) + ω_{0}²] |
Eq. 7 |

If we multiply the numerator and the denominator by s²/ω

ƒ(s) = s²/[s² + (sω_{0}/Q) + ω_{0}²] |
Eq. 8 |

This form is the same as before, except that the numerator is s² instead of ω

The Bode plot offers another perspective on lowpass-to-highpass transformations.

We can use the same idea to generate a bandpass filter. Multiply the lowpass responses by s, which adds a +20dB/decade slope. The net response is then +20dB/decade below the cutoff and -20dB/decade above. This yields the bandpass response in

ƒ(s) = ω_{0}s/[s² + (sω_{0})/Q) + ω_{0}²] |
Eq. 9 |

Notice that the rate of cutoff in a second-order bandpass filter is half that of the other types. This is because the available 40dB/decade slope must be shared between the two skirts of the filter.

In summary, second-order lowpass, bandpass, and highpass functions in normalized form have the same denomination, but they have numerators of ω

ƒ(s) = (s² + ω_{Z}²)/s² + (sω_{0}/Q) + ω_{0}² |
Eq. 10 |

Consider the limit cases. When s = 0, f(s) reduces to ω

ƒ(s) = s² + (ω_{Z}²)/s² + (sω_{0}/Q) + ω_{0}² = [s²/s² + (sω _{0}/Q) + ω_{0}²] + [ω_{Z}²/s² + (sω_{0}/Q) + ω_{0}²] |
Eq. 11 |

This can be stated simply. The notch filter is based on the sum of a lowpass and a highpass characteristic. We use this fact in practical filter implementations to generate the notch response from existing highpass and lowpass 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:

ƒ(s) = [s² - (sω_{0}/Q) + ω_{0}²]/[s² + (sω_{0}/Q) + ω_{0}²] |
Eq. 12 |

This response has poles and zeros placed symmetrically on either side of the jω axis, as shown in

ƒ(s) = 1/[s^{5} + a_{4}s^{4} + a_{3}s^{3} + a_{2}s^{2} + a_{1}s + a_{0}] |
Eq. 13 |

Where all the a

ƒ(s) = 1/[(s² + sω_{1}/Q_{1} + ω_{1}²)(s² + sω_{2}/Q_{2} + ω_{2}²)(s + ω_{3})] |
Eq. 14 |

Which is the same as:

ƒ(s) = [1/(s² + sω_{1}/Q_{1} + ω_{1}²)] × [1/(s² + sω_{2}/Q_{2} + ω_{2}²)] × [1/(s + ω_{3})] |
Eq. 15 |

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 complex-conjugate 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 second-order sections will be something other than lowpass 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.

The poles in Figure 7a have different Q values, but they all have the same ω

You can build Butterworth versions of highpass, bandpass, and other filter types, but the poles of these filters will not be arranged in a simple semicircle. In most cases, you begin by designing a lowpass filter and then applying transformations to generate the other types (such as the s 1/s that we used earlier to change a lowpass into a highpass filter).

You derive a Chebychev filter from a Butterworth by moving each pole closer to the jω axis in the same proportion, so that the poles lie on an ellipse (

The Bessel filter represents a trade-off in the opposite direction from the Butterworth. The Bessel's poles lie on a locus further from the jω axis (

Note that all the filters described have the same number of zeros as poles. (This must be the case, or the transfer function would not be a dimensionless expression.) Elliptic filters, for example, space their zeroes along the jω axis in the stopband. In the case of Bessel, Butterworth, and Chebychev, all the zeros are on top of each other at infinity. Because there are no zeros explicit in the numerator, these filter types are sometimes called all-pole filters.

We have now extended our concepts to cover not only first- and second-order filters, but also filters of higher order, including some particularly useful cases. Now it is time to shift from abstract theory to discuss practical circuits.

The state-variable filter is a convenient realization for the second-order section. It uses two cascaded integrators and a summing junction, as shown in

We know that the characteristic of an integrator is simply ω

L = B/s or B = sL | Eq. 16 |

B = H/s or H = sB = s²L | Eq. 17 |

The equation for the summing junction in Figure 11 is simply:

H = I - B - L | Eq. 18 |

If we substitute for H and B using the integrator equations, we get:

s²L = I - sL - L | Eq. 19 |

Or:

s²L + sL + L = I | Eq. 20 |

In which case:

L(s² + s + 1) = I | Eq. 21 |

Or:

L/I = 1/(s² + s + 1) | Eq. 22 |

Equation 22 is the classic, normalized, lowpass response. Because B = sL and H = s²L, then:

B/I = s/(s² + s + 1) and H/I = s²/(s² + s + 1) | Eq. 23 |

Equation 23 shows, respectively, the classic bandpass and highpass responses.

Thus, one filter provides simultaneous lowpass, bandpass, and highpass outputs. We can create actual filters with real values of ω

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 ω

In the switched-capacitor integrator shown in

The switch S

Q = C_{1}V_{IN} |
Eq. 24 |

Thus the average current transferred to the summing junction is:

I = Qf_{C} = C_{1}V_{IN} × f_{CLK} |
Eq. 25 |

Notice that the current is proportional to V

R = V_{IN}/I = 1/(C_{1}f_{CLK}) |
Eq. 26 |

The integrator's ω

ω_{0} = 1/RC_{2} = C_{1}f_{CLK}/C_{2} |
Eq. 27 |

Because ω

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.