Switched-capacitor Filter
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A switched capacitor (SC) is an
electronic circuit 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. It is a type of electrical ...
that implements a function by moving
charges Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
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 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 el ...
. This makes them much more suitable for use within
integrated circuit An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
s, where accurately specified resistors and capacitors are not economical to construct. 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 and two switches S and S which alternatively connect the capacitor to either in or out at a switching frequency of f. Recall that
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
can express the relationship between voltage, current, and resistance as: :R = .\ 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_ = 1 / (C_S f).


Equivalent resistance calculation

By definition, the charge q on any capacitor C with a voltage V between its plates is: :q = CV.\ Therefore, when S is closed while S is open, the charge stored in the capacitor C_S will be: :q_ = C_S V_ assuming V_ is an
ideal voltage source A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unli ...
. 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 C_S can be expressed in terms of the unknown variable V_ : :q_ = C_S V_.\ Thus, the charge transferred from in to out during one switching cycle is: :q_ = q_-q_ = C_S(V_-V_) .\ This charge is transferred at a rate of f. So the average
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
(rate of transfer of charge per unit time) from in to out is: :I_ = q_ f = C_S(V_-V_)f .\ The voltage difference from in to out can be written as: :V_ = V_ - V_ .\ 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: :R_ = = = .\ : 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 A loss free resistor (LFR) is a resistor that does not lose energy. The first implementation was due to SingerSinger, S, "Realization of Loss Free Resistive Elements", ''IEEE Transactions on Circuits and Systems'', Vol. CAS-37, No. 1, pp. 54-60, J ...
. 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 A microcontroller (MCU for ''microcontroller unit'', often also MC, UC, or μC) is a small computer on a single VLSI integrated circuit (IC) chip. A microcontroller contains one or more CPUs (processor cores) along with memory and programmable i ...
).


Applications

SC simulated resistors are used as a replacement for real resistors in
integrated circuit An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
s 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 in discrete time systems (such as ADCs) as a sample and hold circuit. During the appropriate clock phase, the capacitor samples the analog voltage through switch S1 and in the second phase presents this held sampled value through switch S2 to an electronic circuit for processing.


Filters

Electronic filters Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
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.


The parasitic-sensitive integrator

Switched-capacitor simulated resistors can replace the input resistor in an op 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.


Analysis

Denote by T = 1 / f the switching period. In capacitors, :\text = \text \times \text Then, when S1 opens and S2 closes (they are never both closed at the same time), we have the following: 1) Because C_s has just charged: : Q_s(t) = C_s \cdot V_s(t)\, 2) Because the feedback cap, C_, is suddenly charged with that much charge (by the op amp, which seeks a virtual short circuit between its inputs): : Q_(t) = Q_s(t-T) + Q_(t-T)\, Now dividing 2) by C_: : V_(t) = \frac + V_(t-T)\, And inserting 1): : V_(t) = \frac \cdot V_s(t-T) + V_(t-T)\, This last equation represents what is going on in C_ - it increases (or decreases) its voltage each cycle according to the charge that is being "pumped" from C_s (due to the op-amp). However, there is a more elegant way to formulate this fact if T is very short. Let us introduce dt\leftarrow T and dV_\leftarrow V_(t)-V_(t-dt) and rewrite the last equation divided by dt: : \frac = f \frac \cdot V_s(t)\, Therefore, the op-amp output voltage takes the form: : V_(t) = -V_(t) = - \frac \int V_s(t)dt \, This is the same formula as the op amp inverting integrator where the resistance is replaced by a SC simulated resistor with an equivalent resistance of: :R_ = .\ This switched-capacitor circuit is called "parasitic-sensitive" because its behavior is significantly affected by parasitic capacitances, which will cause errors when parasitic capacitances can't be controlled. "Parasitic insensitive" circuits try to overcome this.


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: : H(z) = \frac


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


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 A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, ...
analysis, with a method that is very similar for switched-capacitor and continuous-time circuits.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''


See also

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


References

*Mingliang Liu, ''Demystifying Switched-Capacitor Circuits'', {{DEFAULTSORT:Switched Capacitor Capacitors Electronic filter applications Electronic filter topology MOSFETs Voltage regulation