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A basic filter circuit consists of a combination of a resistor and a capacitor. This combination of R and C has a time constant which defines the cutoff frequency of the filter; but the actual mode of working also depends on the configuration of the two components (see Fig. 1.) With R in series and C across the circuit, frequencies lower than the cutoff frequency are passed without attenuation. Frequencies at above the cutoff frequency are then sharply attenuated. This is called a lowpass filter. With C in series and R across the circuit, frequencies above the cutoff frequency are passed without attenuations. Frequencies be low the cutoff frequency are then sharply attenuated. This is called a highpass filter. Practical circuits for these two types of filter are shown in Fig. 2. The amount of attenuation provided by a filter is expressed by the ratio volts out/volts in, or voltage ratio. This is quoted in decibels (dB)  a 3dB drop being equivalent to a voltage ratio drop from 1.0 to 0.707, or a power loss of 50 percent. Opamps can be used as practical filters associated with an external capacitor, with the advantage that the more sophisticated circuitry involved can provide superior performance to straightforward RC combinations.
Two filter circuits based on the CA301 opamp are shown in Fig. 2. In the case of the lowpass filter component values are calculated from these formulas: C1 R1 + R2 1.414R1R2f, 1.414 C2 (R1 + R2) fc ... where fc is the effective cutoff point. In the case of the highpass filter circuit: C1 R1 + R2 1.732R1R2 fc 1.732 C2 = (R1 + R2) fc Bandpass filters or bandwidth filters can be produced by combining a lowpass filter in series with a highpass filter. If the bandwidth is from fL to fH, then the cutoff frequency for the lowpass filter is made fH and that of the highpass filter fLFig. 3 (top). This filter combination will pass frequencies from 1L to 1H, i.e., in the desired band. To produce a bandreject filter, a lowpass filter is used in parallel with a highpass filter, as in the second diagram. This combi nation will reject all frequencies within the band fL to fH. Bandpass and bandreject filters are essentially functional mirror images of each other, as illustrated in the frequency response graphs of Fig. 4. For convenience, we will confine our discussion to the bandpass filter. This type of filter has two cutoff frequenciesthe upper cut off and the lower cutoff. The action of the filter can not be de scribed with a single specification as the lowpass and highpass filters can. Usually the cutoff frequencies are not identified specifically in describing bandpass filters. Instead, we identify the midpoint between the two cutoff frequencies. This is called the center frequency, for obvious reasons.
But we still need another specification to indicate how far apart the cutoff frequencies are. This specification is generally given in one of two forms. The most direct is the bandwidth specification. This is simply the frequency distance from the lower cutoff to the upper cutoff. This can all be made clearer with an example. Let's say the lower cutoff frequency is 3000 Hz, and the upper cutoff frequency is 5000 Hz. The center frequency is 4000 Hz, and the bandwidth is the difference between the cutoffs (5000 3000 = 2000 Hz). More commonly, the bandwidth specification is given in terms of Q, or the Quality factor. The lower the Q, the wider the bandwidth. Q is defined as the center frequency divided by the bandwidth: Q = Fc / BW or, for our example: Q = 4000 / 2000 = 2 A typical bandpass filter circuit is shown in Fig. 5. Using the component values specified, the center frequency will be about 1000 Hz. Some practical bandreject (or notch) filter circuits are shown in Figs. 6 and 7. There is one final type of filter of interest which we should look at before moving on. The state variable filter has multiple outputs for the various filter types described in the last few pages. A fairly typical state variable filter circuit is shown in Fig. 8. For the component values shown the Q is 3.4, and the cutoff frequencies for the lowpass and highpass sections is 3000 Hz. This is also the center frequency for the bandpass section. The notch frequency is 9,500 Hz. Fig. 9 shows another state variable filter circuit. In this one the highpass and lowpass cutoff frequencies are adjustable from 300 Hz to 3000 Hz. The potentiometers should have a reverse log type taper.

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