SC filter applications and resistor simulation

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. The capacitors, MOS switches and op-amps are all made on one MOS chip.

Applications

• Voice-band filters in PCM telephone codecs, modems and fax.
• Anti-aliasing/reconstruction filters and sigma-delta modulators in data converters.
• Audio equalisers, tone decoders, DTMF receivers.
• Clock-tunable universal filter ICs and programmable filters.
• Filters inside mixed-signal ICs (biomedical, sensor interfaces).

Simulating a resistor with a switched capacitor

         phi1        phi2
  V1 o---/ ----+---- / ---o V2
               |
              === C
               |
              GND

Two non-overlapping clock phases $\phi_1$ and $\phi_2$ at frequency $f_c = 1/T$:

• During $\phi_1$ the capacitor connects to $V_1$: charge $q_1 = CV_1$.
• During $\phi_2$ it connects to $V_2$: charge $q_2 = CV_2$.

Charge moved from $V_1$ to $V_2$ in one clock period: $\Delta q = C(V_1 - V_2)$. Average current:

$$I = \frac{\Delta q}{T} = C f_c (V_1 - V_2) \quad\Rightarrow\quad R_{eq} = \frac{V_1 - V_2}{I} = \frac{1}{C f_c} = \frac{T}{C}$$

This holds when $f_c$ is much higher than the signal frequencies.

Example: $C$ = 1 pF switched at $f_c$ = 100 kHz gives $R_{eq} = 1/(10^{-12} \times 10^5)$ = 10 MΩ, which would need a huge area as a diffused resistor.

Use in an integrator: replacing the input resistor of an RC integrator by $C_1$ gives

$$\frac{V_o}{V_i} = -\frac{1}{sR_{eq}C_2} = -\frac{C_1 f_c}{C_2}\cdot\frac{1}{s}$$

so the time constant $C_2/(C_1f_c)$ depends only on a capacitor ratio and the clock, which is very accurate.