SC inverting lossy, inverting and non-inverting integrators

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.

SC resistor. A capacitor $C$ switched between nodes 1 and 2 by non-overlapping clocks transfers $\Delta q = C(V_1 - V_2)$ each period $T_c$, so $I = f_cC(V_1 - V_2)$ and $R_{eq} = 1/(f_cC)$.

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

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.

Integrator (inverting)

The active-RC integrator has $T(s) = -\frac{1}{sRC_F}$. Replacing $R$ by a switched capacitor $C_1$:

$$T(s) = \frac{V_o}{V_i} = -\frac{f_cC_1}{sC_F} = -\frac{1}{s}\cdot\frac{C_1}{C_F}f_c$$

                    CF
              +-----||-----+
       phi1   |  phi2      |
 Vi ---o/o--+-o/o-+  |\    |
            |     +--|-\   |
           ===C1     |  >--+---> Vo
            |     +--|+/
           GND    |  |/
                 GND

The integrator time constant $C_F/(f_cC_1)$ is set by a capacitor ratio. (In the z-domain, the stray-insensitive version gives $H(z) = -\frac{C_1}{C_F}\cdot\frac{1}{1 - z^{-1}}$.)

Non-inverting integrator

Here the switched capacitor is connected so that its charge is reversed before it reaches the op-amp: during $\phi_1$, $C_1$ is charged to $V_i$ (top plate to $V_i$, bottom plate to ground); during $\phi_2$, the top plate is grounded and the bottom plate is connected to the virtual ground. Charge $-C_1V_i$ is pushed into $C_F$, so the switched capacitor acts as a negative resistor $R = -1/(f_cC_1)$:

$$T(s) = \frac{V_o}{V_i} = +\frac{f_cC_1}{sC_F}$$

       phi1   C1   phi2           CF
 Vi ---o/o--+-||-+--o/o--+    +---||---+
            |    |       |    |        |
  GND -o/o--+    +-o/o-  +----+  |\    |
       phi2        | phi1     +--|-\   |
                  GND            |  >--+--> Vo
                              +--|+/
                              |  |/
                             GND

(Left plate: $V_i$ on $\phi_1$, ground on $\phi_2$; right plate: ground on $\phi_1$, op-amp input on $\phi_2$.) In the z-domain: $H(z) = +\frac{C_1}{C_F}\cdot\frac{z^{-1}}{1 - z^{-1}}$. A non-inverting integrator removes the need for an extra inverter in biquads, which is a big advantage of SC over active-RC.