The active filters (filters using active devices) are classified as Butterworth, Bessel and Chebyshev type filters. These filters can be selected according to the purposes of applications.

The Butterworth type active filter focuses on the flatness of a passband. But characteristics of attenuation and response are inferior to those of the Bessel and Chebyshev type active filters.

For applications where severe flatness of a passband is not required as is the case with LPF used for voice synthesis, the Chebyshev type active filter is recommended where by allowing ripples, abrupt attenuation characteristics can be attained and active filter can be composed by smaller number of parts.

The Chebyshev type active filter can be designed by selecting appropriate ripple amplitude and attenuation characteristics.

If the frequency characteristics of a speaker itself does not reach the desired cut-off frequency, no filter is required.

Configuration

An LPF, consisting of one RC stage, containing no active device is as shown in Figure 10.7, and its transfer characteristics are given by the formula below.

Figure 10.8 plots F (jw).

As shown in Figure 10.8, frequency characteristics shows attenuation at –6 dB/oct above wo. At wo, a value of –3 dB is observed.

Figure 10.9 illustrates the circuit configuration of a second-order Chebyshev type filter.

The following gives transfer characteristics of the circuit.

The gain is one because the voltage follower consisting of an operational amplifier is used. Assuming that R1 and R2 are equal to R, the following expressions are obtained:

Design of high-order Chebyshev type filter

Such even-order filters as the fourth- and sixth-ones can be resolved into second-order elements. Such odd-order filters as third- and fifth-ones can be resolved into second- and first-order (passive filter consisting of one RC stage) elements.

For example, a fourth-order filter can be resolved into two second-order elements as shown in Figure 10.10. Determining fn and qn allows a fourth-order filter to be readily built.

The Chebyshev type filter incurs a ripple in the passband. Attenuation characteristics vary with permissible ripples, as fn and qn settings in each stage is changed.

Table 10.4 lists fn and qn values for the Chebyshev type LPF.

Example of design

The following gives an example of designing the fifth-order Chebyshev type LPF. A permissible ripple is 0.5 dB. Figure 10.11 provides the circuit.

The fn and qn values are covered in Table 10.4. Data for the fifth order and a ripple of 0.5 dB provides the following fn and qn values.

For the second-order filter in the first stage fn = 1.0177347 and qn = 4.5449633 For the second-order filter in the second stage fn = 0.6904832 and qn = 1.1778056 For the first-order filter in the third stage fn = 0.3623196 and qn = 0.5

The values above can be used to calculate constants.

When the cut-off frequency is 2.8 kHz, the constants are determined as follows.

The RF value can be changed to reflect the actual capacitor value. If C1 and C2 values have been multiplied by 1.5, the RF1 value should be divided by 1.5.

Selecting appropriate capacitor values leads to the final determination of the filter constants. See Figure 10.12.

Then, filter characteristics in Figure 10.12 are plotted.

Transfer characteristics for the second-order filters in the first and second stages are expressed by the formula below.

Transfer characteristics of the first-order filter in the third stage are expressed by the formula below.

For example, if w is equal to wo for the filter in the first stage, we get the following expression.

The following gives the absolute dB value of the I/O voltage ratio.

            20 log 4.545 = 13.15 (dB)

The absolute value is 13.15 dB for a frequency of 2850 Hz.

Figure10.13provides the plotted dashed curve. The alternate dot-dashline and the alternate two dot-dash line cover the second and third stages, respectively. The solid line provides total characteristics.

As the capacitor values have been approximated, total characteristics indicate that the maximum ripple is 0.6 dB and the cut-off frequency is about 2.9 kHz.

Figures 10.14 and 10.15 provide constants and frequency characteristics of a designed third-order Chebyshev type LPF.

The solid line in Figure 10.15 provides total characteristics.

As the capacitor values have been approximated, the characteristics indicate that the maximum ripple is 3 dB and the cut-off frequency is 2.56 kHz.