SC resistor, summing integrator and inverting lossy integrator

Switched capacitor filter

A switched capacitor (SC) filter is an active filter in which every resistor of an active-RC filter is replaced by a capacitor and MOS switches driven by a two-phase non-overlapping clock. The switched capacitor behaves like a resistor $R = 1/(f_cC)$, so the filter needs only op-amps, capacitors and switches, all easily built on one MOS chip.

Resistor

Consider a capacitor $C_R$ with two MOS switches driven by non-overlapping clocks $\phi_1$ and $\phi_2$ of frequency $f_c = 1/T_c$:

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

1. Phase $\phi_1$ (left switch closed): $C_R$ charges to $V_1$; stored charge $q_1 = C_RV_1$.
2. Phase $\phi_2$ (right switch closed): $C_R$ charges/discharges to $V_2$; stored charge $q_2 = C_RV_2$.
3. Charge moved from node 1 to node 2 in one clock period:

$$\Delta q = q_1 - q_2 = C_R(V_1 - V_2)$$

4. If $f_c$ is much higher than the signal frequency, $V_1$ and $V_2$ are nearly constant over a period, so the average current is

$$\begin{aligned} I_{avg} &= \frac{\Delta q}{T_c} = C_Rf_c(V_1 - V_2) \\ R_{eq} &= \frac{V_1 - V_2}{I_{avg}} = \frac{1}{f_cC_R} = \frac{T_c}{C_R} \end{aligned}$$

Example: $C_R = 1$ pF at $f_c = 100$ kHz simulates 10 MΩ.

Summing integrator

Summing integrator. Active-RC: $V_o = -\frac{1}{sC_F}\left(\frac{V_1}{R_1} + \frac{V_2}{R_2}\right)$. With switched capacitors:

$$V_o = -\frac{f_c}{sC_F}\left(C_1V_1 + C_2V_2\right)$$

                    CF
              +-----||-----+
              |            |
 V1 -[C1-SC]--+   |\       |
              +---|-\      |
 V2 -[C2-SC]--+   |  >-----+---> Vo
               +--|+/
               |  |/
              GND

Each input has its own integrator gain $f_cC_i/C_F$.

Inverting lossy integrator

Active-RC form: input resistor $R_1$, feedback $R_2 \parallel C_F$:

$$T(s) = -\frac{1/(R_1C_F)}{s + 1/(R_2C_F)}$$

Replacing $R_1$ and $R_2$ by switched capacitors $C_1$ and $C_2$ ($1/R = f_cC$):

$$T(s) = -\frac{f_cC_1/C_F}{s + f_cC_2/C_F}$$

                    CF
              +-----||-----+
              |            |
              +--[C2-SC]---+
              |            |
              |   |\       |
 Vi -[C1-SC]--+---|-\      |
                  |  >-----+---> Vo
               +--|+/
               |  |/
              GND

DC gain $= -C_1/C_2$ and pole $\omega_p = f_cC_2/C_F$, both set by capacitor ratios. It is the first-order low-pass section used in SC ladders and biquads.