Switched capacitor
Encyclopedia
A switched capacitor is an electronic circuit
element used for discrete time
signal processing
. It works by moving charges into and out of capacitors when switch
es are opened and closed. Usually, non-overlapping signals are used to control the switches, so that not all switches are closed simultaneously. Filter
s implemented with these elements are termed 'switched-capacitor filters'. switched capacitor filters depend only on the ratios between capacitances. This makes them much more suitable for use within integrated circuit
s, where accurately specified resistors and capacitors are not economical to construct.
from the input to the output at the switching frequency 
. Recall that 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 is closed while S is open, the charge stored in the capacitor C is:
When S is closed, some of that charge is transferred out of the capacitor, after which the charge that remains in capacitor C 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:
Note that we use I, the symbol for electric current, for this quantity. This is to demonstrate that a continuous transfer of charge from one node to another is equivalent to a current. 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 lossless resistor
whose value depends on capacitance C and switching frequency f.
The SC resistor is used as a replacement for simple resistors in integrated circuit
s 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.
the switching period. Recall that in capacitors charge = capacitance x voltage. Then, at the instant when S1 opens and S2 closes, we have the following:
1) Because
has just charged:
2) Because the feedback cap,
, is suddenly charged with that much charge (by the opamp, 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:
Note that this is an 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 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.
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...
element used for discrete time
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...
signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
. It works by moving charges into and out of capacitors when switch
Switch
In electronics, a switch is an electrical component that can break an electrical circuit, interrupting the current or diverting it from one conductor to another....
es are opened and closed. Usually, non-overlapping signals are used to control the switches, so that not all switches are closed simultaneously. Filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
s implemented with these elements are termed 'switched-capacitor filters'. switched capacitor filters depend only on the ratios between capacitances. This makes them much more suitable for use within integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s, where accurately specified resistors and capacitors are not economical to construct.
The switched capacitor resistor
The simplest switched capacitor (SC) circuit is the switched capacitor resistor, made of one capacitor C and two switches S and S 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 . Recall that 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 is closed while S is open, the charge stored in the capacitor C is:
When S is closed, some of that charge is transferred out of the capacitor, after which the charge that remains in capacitor C 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:
Note that we use I, the symbol for electric current, for this quantity. This is to demonstrate that a continuous transfer of charge from one node to another is equivalent to a current. 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 lossless resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
whose value depends on capacitance C and switching frequency f.
The SC resistor is used as a replacement for simple resistors in integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s 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
Operational amplifier applications
This article illustrates some typical applications of operational amplifiers. A simplified schematic notation is used, and the reader is reminded that many details such as device selection and power supply connections are not shown....
.
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 Cfb in feedback. One of the earliest of these circuits is the parasitic-Sensitive integrator developed by the Czech engineer Bedrich Hosticka. Let us analyze what happens in this case. Denote by the switching period. Recall that in capacitors charge = capacitance x voltage. Then, at the instant when S1 opens and S2 closes, we have the following:1) Because
has just charged:2) Because the feedback cap,
, is suddenly charged with that much charge (by the opamp, 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:
Note that this is an 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).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: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 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.