A switched capacitor (SC) is an electronic circuit that implements a function by moving charges into and out of capacitors when electronic switches are opened and closed. Usually, non-overlapping clock signals are used to control the switches, so that not all switches are closed simultaneously. Filters implemented with these elements are termed switched-capacitor filters, which depend only on the ratios between capacitances and the switching frequency, and not on precise resistors. This makes them much more suitable for use within integrated circuits, where accurately specified resistors and capacitors are not economical to construct, but accurate clocks and accurate relative ratios of capacitances are economical. SC circuits are typically implemented using metal–oxide–semiconductor (MOS) technology, with MOS capacitors and MOS field-effect transistor (MOSFET) switches, and they are commonly fabricated using the complementary MOS (CMOS) process. Common applications of MOS SC circuits include mixed-signal integrated circuits, digital-to-analog converter (DAC) chips, analog-to-digital converter (ADC) chips, pulse-code modulation (PCM) codec-filters, and PCM digital telephony.
Parallel resistor simulation using a switched-capacitor
The simplest switched-capacitor (SC) circuit is made of one capacitor C S {\displaystyle C_{S}} and two switches S1 and S2 which alternatively connect the capacitor to either in or out at a switching frequency of f {\displaystyle f} . Recall that Ohm's law can express the relationship between voltage, current, and resistance as:
R = V I . {\displaystyle R={V \over I}.\ }
The following equivalent resistance calculation will show how during each switching cycle, this switched-capacitor circuit transfers an amount of charge from in to out such that it behaves according to a similar linear current–voltage relationship with R equivalent = 1 / ( C S f ) . {\displaystyle R_{\text{equivalent}}=1/(C_{S}f).}
Equivalent resistance calculation By definition, the charge q {\displaystyle q} on any capacitor C {\displaystyle C} with a voltage V {\displaystyle V} between its plates is:
q = C V . {\displaystyle q=CV.\ }
Therefore, when S1 is closed while S2 is open, the charge stored in the capacitor C S {\displaystyle C_{S}} will be:
q in = C S V in {\displaystyle q_{\text{in}}=C_{S}V_{\text{in}}}
assuming V in {\displaystyle V_{\text{in}}} is an ideal voltage source. When S2 is closed (S1 is open - they are never both closed at the same time), some of that charge is transferred out of the capacitor. Exactly how much charge gets transferred can't be determined without knowing what load is attached to the output. However, by definition, the charge remaining on capacitor C S {\displaystyle C_{S}} can be expressed in terms of the unknown variable V out {\displaystyle V_{\text{out}}} :
q out = C S V out . {\displaystyle q_{\text{out}}=C_{S}V_{\text{out}}.\ }
Thus, the charge transferred from in to out during one switching cycle is:
q in-out = q in − q out = C S ( V in − V out ) . {\displaystyle q_{\text{in-out}}=q_{\text{in}}-q_{\text{out}}=C_{S}(V_{\text{in}}-V_{\text{out}}).\ }
This charge is transferred at a rate of f {\displaystyle f} . So the average electric current (rate of transfer of charge per unit time) from in to out is:
I in-out = q in-out f = C S ( V in − V out ) f . {\displaystyle I_{\text{in-out}}=q_{\text{in-out}}f=C_{S}(V_{\text{in}}-V_{\text{out}})f.\ }
The voltage difference from in to out can be written as:
V in-out = V in − V out . {\displaystyle V_{\text{in-out}}=V_{\text{in}}-V_{\text{out}}.\ }
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