# Switched capacitor

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A switched capacitor 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]

An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator. The original name is still widely used in many languages, but not commonly in English.

In electrical engineering, a switch is an electrical component that can "make" or "break" an electrical circuit, interrupting the current or diverting it from one conductor to another. The mechanism of a switch removes or restores the conducting path in a circuit when it is operated. It may be operated manually, for example, a light switch or a keyboard button, may be operated by a moving object such as a door, or may be operated by some sensing element for pressure, temperature or flow. A switch will have one or more sets of contacts, which may operate simultaneously, sequentially, or alternately. Switches in high-powered circuits must operate rapidly to prevent destructive arcing, and may include special features to assist in rapidly interrupting a heavy current. Multiple forms of actuators are used for operation by hand or to sense position, level, temperature or flow. Special types are used, for example, for control of machinery, to reverse electric motors, or to sense liquid level. Many specialized forms exist. A common use is control of lighting, where multiple switches may be wired into one circuit to allow convenient control of light fixtures.

## The switched-capacitor resistor

The simplest switched-capacitor (SC) circuit is the switched-capacitor resistor, made of one capacitor C and two switches S1 and S2 which connect the capacitor with a given frequency alternately to the input and output of the SC. Each switching cycle transfers a charge ${\displaystyle q}$ from the input to the output at the switching frequency ${\displaystyle f}$. The charge q on a capacitor C with a voltage V between the plates is given by:

${\displaystyle q=CV\ }$

where V is the voltage across the capacitor. Therefore, when S1 is closed while S2 is open, the charge stored in the capacitor CS is:

${\displaystyle q_{\text{IN}}=C_{S}V_{\text{IN}}.\ }$

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, after which the charge that remains in capacitor CS is:

${\displaystyle q_{\text{OUT}}=C_{S}V_{\text{OUT}}.\ }$

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

${\displaystyle q=q_{\text{IN}}-q_{\text{OUT}}=C_{S}(V_{\text{IN}}-V_{\text{OUT}})\ }$

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

${\displaystyle I=qf.\ }$

(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:

${\displaystyle I=C_{S}(V_{\text{IN}}-V_{\text{OUT}})f\ }$

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

${\displaystyle V=V_{\text{IN}}-V_{\text{OUT}}.\ }$

So the equivalent resistance R (i.e., the voltagecurrent relationship) is:

${\displaystyle R={V \over I}={1 \over {C_{S}f}}.\ }$

Thus, the SC behaves like a resistor whose value depends on capacitance CS and switching frequency f.

A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active elements, and terminate transmission lines, among other uses. High-power resistors that can dissipate many watts of electrical power as heat, may be used as part of motor controls, in power distribution systems, or as test loads for generators. Fixed resistors have resistances that only change slightly with temperature, time or operating voltage. Variable resistors can be used to adjust circuit elements, or as sensing devices for heat, light, humidity, force, or chemical activity.

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.

An integrated circuit or monolithic integrated circuit is a set of electronic circuits on one small flat piece of semiconductor material that is normally silicon. The integration of large numbers of tiny transistors into a small chip results in circuits that are orders of magnitude smaller, cheaper, and faster than those constructed of discrete electronic components. The IC's mass production capability, reliability and building-block approach to circuit design has ensured the rapid adoption of standardized ICs in place of designs using discrete transistors. ICs are now used in virtually all electronic equipment and have revolutionized the world of electronics. Computers, mobile phones, and other digital home appliances are now inextricable parts of the structure of modern societies, made possible by the small size and low cost of ICs.

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.

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.

## The parasitic-sensitive integrator

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 ${\displaystyle C_{fb}}$ in feedback. One of the earliest of these circuits is the parasitic-sensitive integrator developed by the Czech engineer Bedrich Hosticka. [2] Here is an analysis. Denote by ${\displaystyle T=1/f}$ the switching period. In capacitors,

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

Then, when S1 opens and S2 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 ${\displaystyle dV_{fb}\leftarrow V_{fb}(t)-V_{fb}(t-dt)}$ and rewrite the last equation divided by dt:

${\displaystyle {\frac {dV_{fb}(t)}{dt}}=f{\frac {C_{s}}{C_{fb}}}\cdot V_{s}(t)\,}$

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

${\displaystyle V_{OUT}(t)=-V_{fb}(t)=-{\frac {1}{{\frac {1}{fC_{s}}}C_{fb}}}\int V_{s}(t)dt\,}$

This is an inverting integrator with an "equivalent resistance" ${\displaystyle R_{eq}={\frac {1}{fC_{s}}}}$. 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 parasitic insensitive integrator

### Use in discrete-time systems

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:

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

## The multiplying digital to analog converter

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 ${\displaystyle d}$ 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. [3]

## Analysis of switched-capacitor circuits

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 [4] .

## Related Research Articles

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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.

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Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single path, so the same current flows through all of the components. Components connected in parallel are connected along multiple paths, so the same voltage is applied to each component.

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.

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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.

In electronics, a transimpedance amplifier, (TIA) is a current to voltage converter, almost exclusively implemented with one or more operational amplifiers. It is also possible to construct a transimpedance amplifier with discrete components using a Field effect transistor for the gain element. This has been done where a very low noise figure was required. 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, Rf. The gain of the amplifer is set by this resistor and because the amplifier is in an inverting configuration, has a value of -Rf. 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.

## References

1. Switched Capacitor Circuits, Swarthmore College course notes, accessed 2009-05-02
2. 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.
3. Stephen H. Lewis et al., "A 10-bit, 20Msample/s Analog to Digital Converter", IEEE Journal of Solid State Circuits, March 1992
4. 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