Aswitched capacitor (SC) is anelectronic circuit that implements afunction by movingcharges into and out ofcapacitors whenelectronic switches are opened and closed. Usually, non-overlappingclock signals are used to control the switches, so that not all switches are closed simultaneously.Filters implemented with these elements are termedswitched-capacitor filters, which depend only on the ratios between capacitances and the switching frequency, and not on preciseresistors. This makes them much more suitable for use withinintegrated circuits, where accurately specified resistors and capacitors are not economical to construct, but accurate clocks and accuraterelative ratios of capacitances are economical.^[1]

SC circuits are typically implemented usingmetal–oxide–semiconductor (MOS) technology, withMOS capacitors andMOS field-effect transistor (MOSFET) switches, and they are commonlyfabricated using thecomplementary MOS (CMOS) process. Common applications of MOS SC circuits includemixed-signal integrated circuits,digital-to-analog converter (DAC) chips,analog-to-digital converter (ADC) chips,pulse-code modulation (PCM) codec-filters, and PCMdigital telephony.^[2]

The simplest switched-capacitor (SC) circuit is made of one capacitor${\displaystyle C_S}$ and two switchesS₁ andS₂ which alternatively connect the capacitor to eitherin orout at a switching frequency of${\displaystyle f}$.

Recall thatOhm's law can express the relationship between voltage, current, and resistance as:

${\displaystyle R=\frac{V}{I}.}$

The following equivalent resistance calculation will show how during each switching cycle, this switched-capacitor circuit transfers an amount of charge fromin toout such that it behaves according to a similarlinearcurrent–voltage relationship with${\displaystyle R_{\text{equivalent}}=1/(C_{S}f).}$

By definition, the charge${\displaystyle q}$ on any capacitor${\displaystyle C}$ with a voltage${\displaystyle V}$ between its plates is:

${\displaystyle q=CV.}$

Therefore, whenS₁ is closed whileS₂ is open, the charge stored in the capacitor${\displaystyle C_S}$ will be:

${\displaystyle q_{\text{in}}=C_S{V}_{\text{in}}}$

assuming${\displaystyle V_{\text{in}}}$ is anideal voltage source.

WhenS₂ 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${\displaystyle C_S}$ can be expressed in terms of the unknown variable${\displaystyle V_{\text{out}}}$:

${\displaystyle q_{\text{out}}=C_S{V}_{\text{out}}.}$

Thus, the charge transferred fromin toout during one switching cycle is:

${\displaystyle q_{\text{in-out}}=q_{\text{in}}-q_{\text{out}}=C_S(V_{\text{in}}-V_{\text{out}}).}$

This charge is transferred at a rate of${\displaystyle f}$. So the averageelectric current (rate of transfer of charge per unit time) fromin toout is:

${\displaystyle I_{\text{in-out}}=q_{\text{in-out}}f=C_S(V_{\text{in}}-V_{\text{out}})f.}$

The voltage difference fromin toout can be written as:

${\displaystyle V_{\text{in-out}}=V_{\text{in}}-V_{\text{out}}.}$

Finally, the current–voltage relationship fromin toout 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:

${\displaystyle R_{\text{equivalent}}=\frac{V_{\text{in-out}}}{I_{\text{in-out}}}=\frac{(V_{\text{in}}-V_{\text{out}})}{C_S(V_{\text{in}}-V_{\text{out}})f}=\frac{1}{C_{S}f}.}$

This circuit is called aparallel resistor simulation becausein andout are connected in parallel and not directly coupled. Other types of SC simulated resistor circuits arebilinear resistor simulation,series resistor simulation,series-parallel resistor simulation, andparasitic-insensitive resistor simulation.

Charge is transferred fromin toout 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 thebandlimit of the inputsignal.

The SC circuit modeled here using ideal switches with zero resistance does not suffer from theohmic heating energy loss of a regular resistor, and so ideally could be called aloss free resistor. However real switches have some small resistance in their channel orp–n junctions, so power is still dissipated. The capacitors are not ideal either and dissipate power as well.

Because the resistance inside electric switches is typically much smaller than the resistances in circuits relying on regular resistors, SC circuits can have substantially lowerJohnson–Nyquist noise. However,harmonics of the switching frequency may be manifested as high frequencynoise that may need to be attenuated with alow-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. Thusonline orruntime adjustment can be done by controlling the oscillation of the switches (e.g. using an configurable clock output signal from amicrocontroller).

SC simulated resistors are used as a replacement for real resistors inintegrated circuits because it is easier to fabricate reliably with a wide range of values and can take up much less silicon area.

This same circuit can be used indiscrete time systems (such as ADCs) as asample and hold circuit. During the appropriate clock phase, the capacitor samples the analog voltage through switchS₁ and in the second phase presents this held sampled value through switchS₂ to an electronic circuit for processing.

Electronic filters consisting of resistors and capacitors can have their resistors replaced with equivalent switched-capacitor simulated resistors, allowing the filter to be manufactured using only switches and capacitors without relying on real resistors.

Switched-capacitor simulated resistors can replace the input resistor in anop amp integrator to provide accurate voltage gain and integration.

One of the earliest of these circuits is the parasitic-sensitive integrator developed by the Czech engineer Bedrich Hosticka.^[3]

Denote by${\displaystyle T=1/f}$ the switching period. In capacitors,

${\displaystyle \text{charge}=\text{capacitance}\times \text{voltage}}$

Then, whenS₁ opens andS₂ closes (they are never both closed at the same time), we have the following:

1) Because${\displaystyle C_s}$ has just charged:

${\displaystyle Q_s(t)=C_s\cdot V_s(t)}$

2) Because the feedback cap,${\displaystyle C_{fb}}$, is suddenly charged with that much charge (by the op amp, which seeks a virtual short circuit between its inputs):

${\displaystyle Q_{fb}(t)=Q_s(t-T)+Q_{fb}(t-T)}$

Now dividing 2) by${\displaystyle C_{fb}}$:

${\displaystyle V_{fb}(t)=\frac{Q_s(t-T)}{C_{fb}}+V_{fb}(t-T)}$

And inserting 1):

${\displaystyle V_{fb}(t)=\frac{C_s}{C_{fb}}\cdot V_s(t-T)+V_{fb}(t-T)}$

This last equation represents what is going on in${\displaystyle C_{fb}}$ - it increases (or decreases) its voltage each cycle according to the charge that is being "pumped" from${\displaystyle C_s}$ (due to the op-amp).

However, there is a more elegant way to formulate this fact if${\displaystyle T}$ is very short. Let us introduce${\displaystyle dt\leftarrow T}$ and${\displaystyle d{V}_{fb}\leftarrow V_{fb}(t)-V_{fb}(t-dt)}$ and rewrite the last equation divided by dt:

${\displaystyle \frac{d{V}_{fb}(t)}{dt}=f\frac{C_s}{C_{fb}}\cdot V_s(t)}$

Therefore, the op-amp output voltage takes the form:

${\displaystyle V_{\text{out}}(t)=-V_{fb}(t)=-\frac{1}{\frac{1}{f{C}_s}{C}_{fb}}\int V_s(t)dt}$

This is the same formula as the op ampinverting integrator where the resistance is replaced by a SC simulated resistor with an equivalent resistance of:

${\displaystyle R_{\text{equivalent}}=\frac{1}{C_{s}f}.}$

This switched-capacitor circuit is called "parasitic-sensitive" because its behavior is significantly affected byparasitic capacitances, which will cause errors when parasitic capacitances can't be controlled. "Parasitic insensitive" circuits try to overcome this.

The delaying parasitic insensitive integrator^{[clarification needed]} has a wide use in discrete time electronic circuits such asbiquad filters, anti-alias structures, anddelta-sigma data converters. This circuit implements the following z-domain function:

${\displaystyle H(z)=\frac{1}{z-1}}$

One useful characteristic of switched-capacitor circuits is that they can be used to perform many circuit tasks at the same time, which is difficult with non-discrete time components (i.e. analog electronics).^{[clarification needed]} The multiplying digital to analog converter (MDAC) is an example as it can take an analog input, add a digital value${\displaystyle d}$ to it, and multiply this by some factor based on the capacitor ratios. The output of the MDAC is given by the following:

${\displaystyle V_{Out}=\frac{V_i\cdot (C_1+C_2)-(d-1)\cdot V_r\cdot C_2+V_{os}\cdot (C_1+C_2+C_p)}{C_1+\frac{(C_1+C_2+C_p)}{A}}}$

The MDAC is a common component in modern pipeline analog to digital converters as well as other precision analog electronics and was first created in the form above by Stephen Lewis and others at Bell Laboratories.^[4]

Switched-capacitor circuits are analysed by writing down charge conservation equations, as in this article, and solving them with a computer algebra tool. For hand analysis and for getting more insight into the circuits, it is also possible to do aSignal-flow graph analysis, with a method that is very similar for switched-capacitor and continuous-time circuits.^[5]

• Aliasing
• Charge pump
• Nyquist–Shannon sampling theorem
• Switched-mode power supply
• Thyristor-switched capacitor (TSC)

• Mingliang Liu,Demystifying Switched-Capacitor Circuits,ISBN 0-7506-7907-7

• Capacitors
• Electronic filter applications
• Electronic filter topology
• MOSFETs
• Voltage regulation

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from September 2022