Switched-Capacitor Filters:Second-Order SC Circuit

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Second-Order SC Circuit

The general expression of a second-order (biquadratic) z-transfer function can be written in the form

Switched-Capacitor Filters-0179

The denominator coefficients (a and b) fix the pole frequency and quality factor, while the numerator coefficients (g, e, and d) define the types of filter frequency response. Several SC biquadratic cells have been proposed in the literature to implement the above t.f. (transfer function). In the following two of them are presented: the Fleischer&Laker biquadratic cell and another one useful for high sampling frequency.

The Fleischer&Laker Biquad

A popular biquadratic cell, proposed by Fleischer and Laker [5], is shown in Figure 62.14 in its most general form. The cell is composed of:

• an input branch (capacitors G, H, I, and J), which allows to select the zero positions, and therefore the kind of frequency response; and

• a resonating loop (capacitors A, B, C, and D in conjunction with the damping capacitors E and F), which sets the pole frequency and the pole quality factor. The two damping capacitors are not usually adopted together. In general, E-capacitor is used for high-Q filters, while F-capacitor is preferred for low-Q circuits.

The transfer function of the Fleischer&Laker biquad cell can be written as

Switched-Capacitor Filters-0180

This cell allows to synthesize any kind of transfer function by using parasitic-insensitive structures, which ensures performance accuracy. The key observation related to this biquad cell is that the first opamp operates in cascade to the second one; therefore its settling is longer than that of the second one.

Design Methodology

At the first order, the SC circuits can be derived from continuous-time circuits implementing the desired frequency response, by a proper substitution of each resistor with the equivalent SC structures, as shown for the first-order cell. An alternative approach is to optimize the transfer function in the z-domain, and

Switched-Capacitor Filters-0181

to numerically fit the desired transfer function with the transfer function implemented by the second- order cell. A third possibility is to adapt the s-domain transfer function to the z-domain signal processing of the SC structures. This procedure will be used in the following.

Let us consider the case of a given transfer function in s-domain, for instance when an approximation table (Butterworth, Chebichev, Bessel, etc.) is used. The s-domain transfer function to be implemented is written as:

Switched-Capacitor Filters-0182

as it is indicated in Figure 62.15 for a band-pass-type response.

The characteristic frequency can be the -3 dB frequency for a low-pass filter. The H¢(s) that will satisfy the “prewarped” filter mask, will be automatically transformed by Eq. (62.17) into a z-domain transfer function whose frequency response satisfies the desired filter mask in the w domain. Obviously, if wi Ts << 1, no predistortion is needed, being Wi » wi. Assuming that wiTs << 1, H¢(s) » H(s) and Eq. (62.17)

Switched-Capacitor Filters-0183

Switched-Capacitor Filters-0184Switched-Capacitor Filters-0185
No frequency has been prewarped, since, applying Eq. (62.18) to fo, the prewarped pole frequency should result 19.765 kHz, with a negligible deviation of about 0.1%.

For the bandpass response, using the bilinear s-to-z mapping, the zero positions are at {z = 1, z = –1}. The frequency response is shown in Figure 62.16 with line I. The normalized capacitance value, obtained equating the transfer function of Eq. (62.23) with the transfer function of the biquadratic cell of Eq. (62.15), are given in Table 62.1, for the E- and F-type structures, in column I. A very large capacitor spread (>78) is needed. This results in large die area, and large power consumption. The capacitor spread could be reduced with a slight modification of the transfer function to the one given in the following:

With respect to the bilinear transformation of Eq. (62.23), in this case, the zero at DC is maintained, while the zero at Nyquist frequency (at Fs/2, i.e., at z = –1) is eliminated. The normalized capacitor values are indicated again in Table 62.1, in Column II. It can be seen that a large reduction of the capacitor spread is obtained (from 80 to 8, for the E-type). The obtained frequency response is reported in Figure 62.16 with line II. In the passband no significant changes occurs; in contrast in the stopband, the maximum signal attenuation is about –35 dB. In some applications, this solution is acceptable, also in consideration of the considerable capacitor spread reduction. For this reason, if not strictly necessary, the zero at Fs/2 can be eliminated. However reducing the factor fo/Fs results in reducing the stopband attenuation. For instance, for fo = 200 kHz (i.e., fo/Fs = 0.2), the frequency responses with and without the Nyquist zero are reported in Figure 62.16 with lines III and IV, respectively. In this case the stopband attenuation is reduced to –22 dB and therefore the Nyquist zero could be strongly needed. The relative normalized capacitor values are indicated in Table 62.1 in Column III (with zeros at {z = 1, z = –1}) and in Column IV (with zeros at z = 1).

A Biquadratic Cell for High Sampling Frequency

In the previous biquadratic cell, the two opamps operate in cascade during the same clock phase. This requires that the second opamp in cascade waits for the complete settling of the first opamp to complete its settling. This, of course, reduces the maximum achievable sampling frequency, or, alternatively, for a given sampling frequency increases the required power consumption since opamp with larger bandwidth

Switched-Capacitor Filters-0186

are needed. To avoid this aspect, the biquad shown in Figure 62.17 can be used. In this scheme the two opamps settle in different clock phase and thus they have the full clock phase time slot to settle.

The transfer function of the biquadratic cell is given in Eq. (62.25). As seen a limitation occurs in the possible transfer function, since the term in z -2 is not present.

Switched-Capacitor Filters-0187

High-Order Filters

The previous first- and second-order cells can be used to build up high-order filters. The main architectures are taken from the theory for the active-RC filters. Some of the most significant ones are: ladder [6] (with good amplitude response robustness with respect to component spread), cascade of first- and second-order cells (with good phase response robustness with respect to component spread), follow-the-leader feedback (for low-noise systems, such as reconstruction filters in oversam- pled DAC).

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