A **switched capacitor** (**SC**) is an electronic circuit element implementing a filter. It works by moving charges into and out of capacitors when switches are opened and closed. Usually, non-overlapping 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", and depend only on the ratios between capacitances. This makes them much more suitable for use within integrated circuits, where accurately specified resistors and capacitors are not economical to construct.^{ [1] }

- The switched-capacitor resistor
- The parasitic-sensitive integrator
- The parasitic insensitive integrator
- Use in discrete-time systems
- The multiplying digital to analog converter
- Analysis of switched-capacitor circuits
- See also
- References

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.^{ [2] }

The simplest switched-capacitor (SC) circuit is the switched-capacitor resistor, made of one capacitor C and two switches S_{1} and S_{2} which connect the capacitor with a given frequency alternately to the input and output of the SC. Each switching cycle transfers a charge from the input to the output at the switching frequency . The charge *q* on a capacitor *C* with a voltage *V* between the plates is given by:

where *V* is the voltage across the capacitor. Therefore, when S_{1} is closed while S_{2} is open, the charge stored in the capacitor C_{S} is:

When S_{2} is closed (S_{1} is open - they are never both closed at the same time), some of that charge is transferred out of the capacitor, after which the charge that remains in capacitor C_{S} is:

Thus, the charge moved out of the capacitor to the output is:

Because this charge *q* is transferred at a rate *f*, the rate of transfer of charge per unit time is:

(A continuous transfer of charge from one node to another is equivalent to a current, so *I* (the symbol for electric current) is used.)

Substituting for *q* in the above, we have:

Let *V* be the voltage across the SC from input to output. So:

So the equivalent resistance *R* (i.e., the voltage–current relationship) is:

Thus, the SC behaves like a resistor whose value depends on capacitance *C _{S}* and switching frequency

The SC resistor is used as a replacement for simple resistors in integrated circuits because it is easier to fabricate reliably with a wide range of values. It also has the benefit that its value can be adjusted by changing the switching frequency (i.e., it is a programmable resistance). See also: operational amplifier applications.

This same circuit can be used in discrete time systems (such as analog to digital converters) as a track and hold circuit. During the appropriate clock phase, the capacitor samples the analog voltage through switch one and in the second phase presents this held sampled value to an electronic circuit for processing.

Often switched-capacitor circuits are used to provide accurate voltage gain and integration by switching a sampled capacitor onto an op-amp with a capacitor in feedback. One of the earliest of these circuits is the parasitic-sensitive integrator developed by the Czech engineer Bedrich Hosticka.^{ [3] } Here is an analysis. Denote by the switching period. In capacitors,

Then, when S1 opens and S2 closes (they are never both closed at the same time), we have the following:

1) Because has just charged:

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

Now dividing 2) by :

And inserting 1):

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

However, there is a more elegant way to formulate this fact if is very short. Let us introduce and and rewrite the last equation divided by dt:

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

This is an inverting integrator with an "equivalent resistance" . This allows its *on-line* or *runtime* adjustment (if we manage to make the switches oscillate according to some signal given by e.g. a microcontroller).

The delaying parasitic insensitive integrator has a wide use in discrete time electronic circuits such as biquad filters, anti-alias structures, and delta-sigma data converters. This circuit implements the following z-domain function:

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. The multiplying digital to analog converter (MDAC) is an example as it can take an analog input, add a digital value to it, and multiply this by some factor based on the capacitor ratios. The output of the MDAC is given by the following:

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 a Signal-flow graph analysis, with a method that is very similar for switched-capacitor and continuous-time circuits.^{ [5] }

A **multivibrator** is an electronic circuit used to implement a variety of simple two-state devices such as relaxation oscillators, timers and flip-flops. It consists of two amplifying devices cross-coupled by resistors or capacitors. The first multivibrator circuit, the astable multivibrator oscillator, was invented by Henri Abraham and Eugene Bloch during World War I. They called their circuit a "multivibrator" because its output waveform was rich in harmonics.

**Electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term *complex impedance* may be used interchangeably.

A **low-pass filter** (**LPF**) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a **high-cut filter**, or **treble-cut filter** in audio applications. A low-pass filter is the complement of a high-pass filter.

A **rectifier** is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction.

**Johnson–Nyquist noise** is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment such as radio receivers can drown out weak signals, and can be the limiting factor on sensitivity of an electrical measuring instrument. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

In electronics a **relaxation oscillator** is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. The circuit consists of a feedback loop containing a switching device such as a transistor, comparator, relay, op amp, or a negative resistance device like a tunnel diode, that repetitively charges a capacitor or inductor through a resistance until it reaches a threshold level, then discharges it again. The period of the oscillator depends on the time constant of the capacitor or inductor circuit. The active device switches abruptly between charging and discharging modes, and thus produces a discontinuously changing repetitive waveform. This contrasts with the other type of electronic oscillator, the harmonic or linear oscillator, which uses an amplifier with feedback to excite resonant oscillations in a resonator, producing a sine wave. Relaxation oscillators are used to produce low frequency signals for applications such as blinking lights and electronic beepers and in voltage controlled oscillators (VCOs), inverters and switching power supplies, dual-slope analog to digital converters, and function generators.

A **resistor–capacitor circuit**, or **RC filter** or **RC network**, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

In electronics, a **voltage divider ** is a passive linear circuit that produces an output voltage (*V*_{out}) that is a fraction of its input voltage (*V*_{in}). **Voltage division** is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

The **Ćuk converter** is a type of DC/DC converter that has an output voltage magnitude that is either greater than or less than the input voltage magnitude. It is essentially a boost converter followed by a buck converter with a capacitor to couple the energy.

The **RC time constant**, also called tau, the time constant of an RC circuit, is equal to the product of the circuit resistance and the circuit capacitance, i.e.

**Delta-sigma** modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC).

This article illustrates some typical **operational amplifier applications**. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.

A **buck converter** is a DC-to-DC power converter which steps down voltage from its input (supply) to its output (load). It is a class of switched-mode power supply (SMPS) typically containing at least two semiconductors and at least one energy storage element, a capacitor, inductor, or the two in combination. To reduce voltage ripple, filters made of capacitors are normally added to such a converter's output and input.

**Ripple** in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

The **commutation cell** is the basic structure in power electronics. It is composed of an electronic switch and a diode. It was traditionally referred to as a chopper, but since switching power supplies became a major form of power conversion, this new term has become more popular.

A **capacitor** is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

An **integrating ADC** is a type of analog-to-digital converter that converts an unknown input voltage into a digital representation through the use of an integrator. In its basic implementation, the dual-slope converter, the unknown input voltage is applied to the input of the integrator and allowed to ramp for a fixed time period. Then a known reference voltage of opposite polarity is applied to the integrator and is allowed to ramp until the integrator output returns to zero. The input voltage is computed as a function of the reference voltage, the constant run-up time period, and the measured run-down time period. The run-down time measurement is usually made in units of the converter's clock, so longer integration times allow for higher resolutions. Likewise, the speed of the converter can be improved by sacrificing resolution.

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

In electronics, a **transimpedance amplifier**, (TIA) is a current to voltage converter, almost exclusively implemented with one or more operational amplifiers. The TIA can be used to amplify the current output of Geiger–Müller tubes, photo multiplier tubes, accelerometers, photo detectors and other types of sensors to a usable voltage. Current to voltage converters are used with sensors that have a current response that is more linear than the voltage response. This is the case with photodiodes where it is not uncommon for the current response to have better than 1% nonlinearity over a wide range of light input. The transimpedance amplifier presents a low impedance to the photodiode and isolates it from the output voltage of the operational amplifier. In its simplest form a transimpedance amplifier has just a large valued feedback resistor, R_{f}. The gain of the amplifier is set by this resistor and because the amplifier is in an inverting configuration, has a value of -R_{f}. There are several different configurations of transimpedance amplifiers, each suited to a particular application. The one factor they all have in common is the requirement to convert the low-level current of a sensor to a voltage. The gain, bandwidth, as well as current and voltage offsets change with different types of sensors, requiring different configurations of transimpedance amplifiers.

The **operational amplifier integrator** is an electronic integration circuit. Based on the operational amplifier (op-amp), it performs the mathematical operation of integration with respect to time; that is, its output voltage is proportional to the input voltage integrated over time.

- ↑
*Switched Capacitor Circuits*, Swarthmore College course notes, accessed 2009-05-02 - ↑ Allstot, David J. (2016). "Switched Capacitor Filters". In Maloberti, Franco; Davies, Anthony C. (eds.).
*A Short History of Circuits and Systems: From Green, Mobile, Pervasive Networking to Big Data Computing*(PDF). IEEE Circuits and Systems Society. pp. 105–110. ISBN 9788793609860. - ↑ B. Hosticka, R. Brodersen, P. Gray, "MOS Sampled Data Recursive Filters Using Switched Capacitor Integrators", IEEE Journal of Solid-State Circuits, Vol SC-12, No.6, December 1977.
- ↑ Stephen H. Lewis et al., "A 10-bit, 20Msample/s Analog to Digital Converter", IEEE Journal of Solid-State Circuits, March 1992
- ↑ H. Schmid and A. Huber, "Analysis of switched-capacitor circuits using driving-point signal-flow graphs", Analog Integr Circ Sig Process (2018).
*https://doi.org/10.1007/s10470-018-1131-7*.

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

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