SC filter design: 20 dB between 10² and 10³ rad/s

Switched-capacitor filter

A switched-capacitor (SC) filter is an active filter in which the resistors of an active-RC filter are replaced by capacitors switched by MOS transistors driven by a clock. Each switched capacitor behaves like a resistor $R = 1/(Cf_c)$, so the time constants depend only on capacitor ratios and the clock frequency.

Applications

• Voice-band filters in telephone PCM codecs and modems.
• Anti-aliasing and reconstruction filtering in data converters; sigma-delta modulators.
• Audio equalisers and tone controls; DTMF (touch-tone) receivers.
• Programmable/tunable filters (cut-off set by the clock) and filter ICs (e.g. MF10 type).
• Biomedical and instrumentation front ends in mixed-signal ICs.

Transfer function from the plot

Slope changes: +20 dB/dec starts at ω = 10 (zero), flat from 100 (pole), −20 dB/dec from 1000 (pole), flat again from 10⁴ (zero). Check: rise $20\log(100/10)$ = 20 dB.

$$T(s) = \frac{(s+10)(s+10^4)}{(s+100)(s+1000)} = \frac{s+10}{s+100}\cdot\frac{s+10^4}{s+1000}$$

DC gain $= 10 \times 10^4/(100 \times 1000) = 1$ and gain at high frequency $= 1$ (0 dB).

Building block

A first-order section with an op-amp, input admittance $Y_1 = sC_1 + 1/R_1$ and feedback admittance $Y_2 = sC_2 + 1/R_2$:

$$T(s) = -\frac{Y_1}{Y_2} = -\frac{C_1}{C_2}\cdot\frac{s + 1/(R_1C_1)}{s + 1/(R_2C_2)}$$

With $C_1 = C_2 = C$ the zero is at $a = 1/(R_1C)$ and the pole at $b = 1/(R_2C)$. Each resistor is then replaced by a switched capacitor $C_R = 1/(Rf_c)$, i.e.

$$C_{R1} = \frac{aC}{f_c}, \qquad C_{R2} = \frac{bC}{f_c}$$

            +-------| C2 |-------+
            +----[SC: CR2]-------+
            |                    |
 Vi o--+--| C1 |--+              |
       |          |              |
       +-[SC: CR1]+-----(-)      |
                          A -----+----o Vo
                 GND-----(+)

Choices: $f_c$ = 20 kHz (more than 10 times the highest corner, 10⁴ rad/s ≈ 1.6 kHz). Because the corners are spread over three decades, a different $C$ is used in each section to keep the switched capacitors near 1–10 pF.

Answer: $f_c$ = 20 kHz; section 1: $C_1 = C_2$ = 2 nF, $C_{R1}$ = 1 pF, $C_{R2}$ = 10 pF; section 2: $C_1 = C_2$ = 20 pF, $C_{R1}$ = 10 pF, $C_{R2}$ = 1 pF.