SC summer, inverting and non-inverting integrators

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 basic SC resistor (charge $C_1(V_1 - V_2)$ moved each clock period) gives $R_{eq} = 1/(C_1f_c)$. The building blocks below use the parasitic-insensitive four-switch branch: capacitor $C_1$ with two switches on each plate.

        S1        C1        S3
 Vi o---/ ---+---| |---+---/ ---o to (-) of op-amp
             |         |        (virtual ground)
             / S2      / S4
             |         |
            GND       GND

Inverting integrator

Switching: $\phi_1$: S2 and S4 closed (C1 discharged). $\phi_2$: S1 and S3 closed: $C_1$ charges to $V_i$ through the virtual ground, so charge $C_1V_i$ is pushed onto the feedback capacitor $C_2$ with negative sign.

$$H(z) = -\frac{C_1}{C_2}\cdot\frac{1}{1 - z^{-1}} \qquad\approx\qquad H(s) = -\frac{C_1 f_c}{C_2}\cdot\frac{1}{s}$$

Non-inverting integrator

Change the phases of the left switches: $\phi_1$: S1 and S4 closed ($C_1$ charges to $V_i$ with its right plate grounded). $\phi_2$: S2 and S3 closed (left plate grounded, right plate to the virtual ground). The charge is now delivered with the opposite polarity, one half-period later:

$$H(z) = +\frac{C_1}{C_2}\cdot\frac{z^{-1}}{1 - z^{-1}} \qquad\approx\qquad H(s) = +\frac{C_1 f_c}{C_2}\cdot\frac{1}{s}$$

Circuit for both integrators (the same box, with the switch phases of the table above):

                    +-----| C2 |-----+
                    |                |
 Vi o--[SC branch C1]--+---(-)       |
                       |      A -----+--o Vo
                 GND--(+)

Summer (and summing integrator)

Connect several SC branches $C_1, C_2, \dots$ (one per input) to the same virtual-ground node. The charges add:

              +------------| CF |------------+
              |        (reset switch across CF|
              |         for the summer)       |
              |            |\                 |
 V1 o--[SC C1]+------------|-\                |
              |            |  >---------------+--o Vo
 V2 o--[SC C2]+       +----|+/
                      |    |/
                     GND

Each box [SC Ci] is the four-switch branch shown above.

• With a feedback capacitor $C_F$ only: summing integrator $V_o = -\dfrac{f_c}{s}\left(\dfrac{C_1}{C_F}V_1 + \dfrac{C_2}{C_F}V_2\right)$.
• With $C_F$ reset each period (a switch across $C_F$), or with an SC resistor in feedback: summer (amplifier) $V_o = -\left(\dfrac{C_1}{C_F}V_1 + \dfrac{C_2}{C_F}V_2\right)$.

A branch wired with non-inverting phasing gives a + sign for that input, so sums and differences are both possible with one op-amp. All gains are capacitor ratios, hence accurate.