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| SEC. 21-1 IDEAL RESPONSES There are five basic types of responses: low-pass, high-pass, bandpass, bandstop, and all-pass. The first four have a passband and a stopband. Ideally, the attenuation should be zero in the passband and infinite in the stopband with a brick wall transition. SEC. 21-2 APPROXIMATE RESPONSES The passband is identified by its low attenuation and its edge frequency. The stopband is identified by its high attenuation and edge frequency. The order of a filter is the number of reactive components. With active filters, it is usually the number of capacitors. The five approximations are the Butterworth (maximally flat passband), the Chebyshev (rippled passband), the inverse Chebyshev (flat passband and rippled stopband), the elliptic (rippled passband and stopband), and the Bessel (maximally flat time delay). SEC. 21-3 PASSIVE FILTERS A low-pass LC filter has a resonant frequency f0 and a Q. The response is maximally flat when Q = 0.707. As Q increases, a peak appears in the response, centered on the resonant frequency. The Chebyshev response occurs with Q greater than 0.707, and the Bessel with Q = 0.577. The higher the Q, the faster the roll-off in the transition region. SEC. 21-4 FIRST-ORDER STAGES First-order stages have a single capacitor and one or more resistors. All first-order stages produce a Butterworth response because peaking is possible only in second-order stages. A first-order stage can produce either a low-pass or a highpass response. SEC. 21-5 VCVS UNITY-GAIN SECOND-ORDER LOW-PASS FILTERS Second-order stages are the most common stage because they are easy to implement and analyze. The Q of the stage produces different K values. The pole frequency of a low-pass stage can be multiplied by its K values to get the resonant frequency if there is a peak, a cutoff frequency, and a 3-dB frequency. SEC. 21-6 HIGHER-ORDER FILTERS Higher-order filters are usually made by cascading second-order stages or a firstorder stage when the total order is odd. When filter stages are cascaded, we add the decibel gains of the stages to get the total decibel gain. To get the Butterworth response for a higher-order filter, we have to stagger the Qs of the stages. To get the Chebyshev and other responses, we have to stagger the pole frequencies and the Qs. SEC. 21-7 VCVS EQUALCOMPONENT LOW-PASS FILTERS The Sallen-Key equal-component filters control the Q by setting the voltage gain. The voltage gain must be less than 3 to avoid oscillations. Higher Qs are difficult to get with this circuit because the component tolerance becomes very important in determining the voltage gain and Q. SEC. 21-8 VCVS HIGH-PASS FILTERS VCVS high-pass filters have the same configuration as low-pass filters, except that the resistors and capacitors are interchanged. Again, the Q values determine the K values. We have to divide pole frequency by the K values to get the resonant frequency, cutoff frequency, and 3-dB frequency. SEC. 21-9 MFB BANDPASS FILTERS Low-pass and high-pass filters may be cascaded to get a bandpass filter, provided that Q is less than 1. When Q is greater than 1, we have a narrowband filter rather than a wideband filter. SEC. 21-10 BANDSTOP FILTERS Bandstop filters can be used to notch out a specific frequency such as the 60-Hz hum induced in circuits by ac power lines. With a Sallen-Key notch filter, the voltage gain controls the Q of the circuit. The voltage gain must be less than 2 to avoid oscillations. SEC. 21-11 THE ALL-PASS FILTER Somewhat of a misnomer, the all-pass filter does more than pass all frequencies with no attenuation. This type of filter is designed to control the phase of the output signal. Especially important is the use of an all-pass filter as a phase or time-delay equalizer. With one of the other filters producing the desired frequency response and an all-pass filter producing the desired phase response, the overall filter has a linear phase response, equivalent to a maximally flat time delay. SEC. 21-12 BIQUADRATIC AND STATE-VARIABLE FILTERS The biquadratic or TT filters use three or four op amps. Although more complex, the biquadratic filter offers lower component sensitivity and easier tuning. This type of filter also has simultaneous low-pass and bandpass outputs, or highpass and bandstop outputs. The statevariable or KHN filters also use three or more op amps. When a fourth op amp is used, it offers easy tuning because voltage gain, center frequency, and Q are all independently tunable. |