5.2.2 Switched-Capacitor Resistors

One important class of CMOS analog integrated circuits are switched-capacitor (SC) circuits, where high-precision resistors with large resistance values are implemented by switches and capacitors [28]. Figure 5.9 shows implementations of a switched-capacitor resistor and the equivalent circuit. Ideally, the equivalent resistance is

$$R = {\displaystyle\frac{1}{f C}} .$$ (5.2)

Figure 5.9: Switched-capacitor resistors

[Equivalent circuit]

[Simple SC resistor]

[Fringe insensitive SC resistor]

However, the fringing and switch capacitances in Fig. 5.9(b) would cause a large error and prevent the implementation of large resistances. Therefore, the circuit in Fig. 5.9(c), which is largely insensitive to fringing capacitances, is preferred in practice. It should be noted, however, that these spurious effects still cause parasitic components as shown in Fig. 5.9(a). But the effect of these components can be circumvented when the SC resistor is connected only to input/output nodes (e.g. OPAMP outputs) or to ``cold'' nodes (i.e. virtually grounded nodes).

Simulation results of an SC resistor are shown in Figs. 5.10-5.13. The circuit in Fig. 5.9(c) was implemented using the transmission-gate switches analyzed in Section 5.2.1 with $\ensuremath{W_{\mathit{p}}}\xspace =\rm 1\mu m$, $\ensuremath{W_{\mathit{p}}}\xspace =\rm 2.5\mu m$. To determine the error and distortion of the equivalent resistor a fifth-order polynomial was fitted to the IV data. The signal used for distortion analysis was a sine signal with an amplitude of $0.75 \ensuremath{V_{\mathit{DD}}}\xspace$.

Figure 5.10: Resistance and distortion vs. capacitance ( $\ensuremath{V_{\mathit{DD}}}\xspace =\rm0.4V$, $f=\rm 100MHz$)

Figure 5.11: Resistance and distortion vs. clock frequency ( $\ensuremath{V_{\mathit{DD}}}\xspace =\rm0.4V$, $C=\rm 1fF$)

Figure 5.12: Resistance and distortion vs. fringing capacitance ( $\ensuremath{V_{\mathit{DD}}}\xspace =\rm0.4V$, $C=\rm 1fF$, $f=\rm 100MHz$)

Figure 5.13: Resistance and distortion vs. supply voltage ( $\ensuremath{V_{\mathit{DD}}}\xspace =\rm0.4V$, $C=\rm 1fF$, $f=\rm 100MHz$)

The error seems to be acceptable over wide parameter ranges. The distortion, however, is generally in the order of 1% which is equivalent to -40dB. As the spurious switch capacitances are in the range of 10fF the results in Figs. 5.12 and 5.13 suggest that the distortions are primarily caused by the non-linear MOSFET capacitances. Although, practically, the switches work only well when the transistors can be operated in strong inversion, the results in Fig. 5.13 show that, apart from a larger error, the SC resistor does not fail completely when $\ensuremath{V_{\mathit{DD}}}\xspace /2<\ensuremath{V_{\mathit{T}}}\xspace = \rm 90 \ldots 100 mV$.