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 and two switches S and S which alternatively connect the capacitor to either in or out at a switching frequency of .
Recall that Ohm's law can express the relationship between voltage, current, and resistance as:
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
Equivalent resistance calculation
By definition, the charge on any capacitor with a voltage between its plates is:
Therefore, when S is closed while S is open, the charge stored in the capacitor will be:
assuming is an ideal voltage source.
When S is closed (S 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 can be expressed in terms of the unknown variable :
Thus, the charge transferred from in to out during one switching cycle is:
This charge is transferred at a rate of . So the average electric current (rate of transfer of charge per unit time) from in to out is:
The voltage difference from in to out can be written as:
Finally, the current–voltage relationship from in to out can be expressed with the same form as Ohm's law, to show that this switched-capacitor circuit simulates a resistor with an equivalent resistance of:
This circuit is called a parallel resistor simulation because in and out are connected in parallel and not directly coupled. Other types of SC simulated resistor circuits are bilinear resistor simulation, series resistor simulation, series-parallel resistor simulation, and parasitic-insensitive resistor simulation.
Difference with real resistor
Charge is transferred from in to out as discrete pulses, not continuously. This transfer approximates the equivalent continuous transfer of charge of a resistor when the switching frequency is sufficiently higher (≥100x) than the bandlimit of the input signal.
The SC circuit modeled here using ideal switches with zero resistance does not suffer from the ohmic heating energy loss of a regular resistor, and so ideally could be called a loss free resistor. However real switches have some small resistance in their channel or p–n junctions, so power is still dissipated.
Because the resistance inside electric switches is typically much smaller than the resistances in circuits relying on regular resistors, SC circuits can have substantially lower Johnson–Nyquist noise. However harmonics of the switching frequency may be manifested as high frequency noise that may need to be attenuated with a low-pass filter.
SC simulated resistors also have the benefit that their equivalent resistance can be adjusted by changing the switching frequency (i.e., it is a programmable resistance) with a resolution limited by the resolution of the switching period. Thus online or runtime adjustment can be done by controlling the oscillation of the switches (e.g. using an configurable clock output signal from a microcontroller).