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Switched Capacitor Filters


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About Switched Capacitor Filters Switched Capacitors Biquad Architectures
High/Low Q Biquads IIR Z Transforms FIR Z Transforms
Band Pass Architectures Net Lists User Modification Analysis
Monte Carlos Real and Quadruplet Zeros Delay Equalization  

About Switched Capacitor Filters

Switched capacitor filters are normally used in an integrated circuit product environment, resistors are expensive and not easily controlled.  Switched capacitors may be used to simulate resistors to an acceptable degree of accuracy using capacitors and mosfets, and are generally less costly than resistors.  The switching functions of the mosfets produces a discrete response rather than a continuous response from the filter.  Therefore, Z Transforms are employed rather than S Transforms, and, just as in digital filters, aliasing effects occur.  Any Z Transform approximation to a continuous function may be used to design a switched capacitor filter.  However, it is generally desirable to use the Z Transform that results in the smallest capacitor spread..  Filter Solutions offers six IIR Z Transforms to choose from, and has the option of selecting the design with the smallest capacitor spread.  Care must be taken to insure that the Z Transform is acceptably accurate given the design requirements.  The design with the best capacitor spread is not necessarily the best design to use due to unacceptable accuracy.

Single Op Amp Switched Capacitor StagesSingle Op Amp Switched Capacitor Stages

Double Op Amp Switched Capacitor StagesDouble Op Amp Switched Capacitor Stages

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Switched Capacitors

Switched capacitors may simulate positive resistors or negative resistors.  Positive resistors may float, and are treated as constants in the Z Transform.  Negative resistors may not float and are treated as a constant with a single frame delay in the Z Transform.

Positive Switched Capacitors and Modal Negative Switched Capacitors and Modal
Positive Switched Capacitor Symbol Negative Switched Capacitor Symbol With Op Amp
Positive Switched Capacitor Modal Negative Switched Capacitor Modal

Note that the negative switched capacitors do not float.  It is also critical that the clock DOES NOT OVERLAP with its invert.  Switch polarities are user optional, but the polarity definition must be followed everywhere in the filter.

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Biquad Architectures

Filter Solutions supports both cascade and parallel biquads.  The cascade architecture produces higher quality stop band zeros, and the parallel architecture may minimize errors to do op amp parasitics.  The following are examples of a 1MHz  third order Elliptic filter.

Cascade Switched Capacitor Arcitecture
Cascade Architecture

 

Parallel Switched Capacitor Arcitecture
Parallel Architecture

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High/Low Q Biquads

Second order biquad stages may be arranged in a high Q or low Q configuration.  In general, the low Q configuration produces a more desirable capacitor spread for Q below 1.0, and the high Q configuration does better for Q above 1.0.  This rule does not always apply though, so Filter Solutions calculates both stages, and then uses the stage with the lowest Q.  The user has the capability of selecting the other stage configuration by left clicking on an op amp.  The following example is a fourth order Butterworth filter composed of a low Q stage and a high Q stage.

Loq Q and High Q Switched Capacitor Stages
Low Q Stage                                                         High Q Stage

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IIR Z Transforms

Filter Solutions offers six IIR Z Transforms to choose from to approximate a continuous filter.  In Summary, they are:

Bilinear
The Bilinear uses trapezoidal integration to implement the digital Z transform.  It is usually the most accurate digital implementation for filtering a continuous variable.  However, significant warping of the frequency spectrum occurs.  Prewarping is also an available option.

 Matched Z
The Matched Z transform employs a basic simple method of translating analog poles and zeros to digital poles and zeros on the cascaded continuous transfer function..

 Impulse Invariant
The Impulse Invariant transformation retains the exact impulse response of the analog system.  It is desirable in cases where the impulse time response of the filter is important to be retained.  The transformation is properly performed on the parallel transfer function.

Step Invariant
The Step Invariant transformation retains the exact impulse response of the analog system.  It is desirable in cases where the step time response of the filter is important to be retained.  The transformation is properly performed on the parallel transfer function.

Modified Impulse Invariant
The Modified Impulse Invariant transformation performs an Impulse Invariant computation on the cascaded transfer function biquads instead of the parallel.  This introduces slight errors in the impulse response, but has the benefit of sometimes resulting in a smaller capacitor spread.

Modified Step Invariant
The Modified Step Invariant transformation performs an Step Invariant computation on the cascaded transfer function biquads instead of the parallel.  This introduces slight errors in the impulse response, but has the benefit of sometimes resulting in a smaller capacitor spread.

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FIR Z Transforms

Switched capacitor filters may generally be designed to meet any desired FIR Z transform.  All FIR Z Transforms supported in the digital filters are also available for switched capacitor filter design.  Finite impulse response filters have an advantage over the above IIR filters in that the group delay may be held constant.  FIR filters have a disadvantage is that the pass band is less controlled and they may be excessively large to be of practical use in switched capacitor filters.  Filter Solutions supports all FIR filters that are supported with digital filters

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Band Pass Architectures

Just as in Active Filters, Filter Solutions allows the user to architect band pass filters from band pass stages or low pass and high pass stages.  Generally, band pass stages are more suited for narrow band filters, and low pass and high pass stages are more suited for wide band filters.  Selection of the incorrect stage type may result in excessively high stage gains.

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Net Lists

Filter Solutions allow the user to create spice executable net lists for all switched capacitor filters that are designed, including those with user modifications.  This allows for the easy validation of the software and its output, as well as to perform analysis on modifications to the filter beyond what Filter Solutions support.  Only transient analysis is supported for switched capacitor filters.

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User Modification Analysis

Users have the capability to alter capacitor values and reanalyze the filter.  Left click on any capacitor, as shown, and enter the new desired value in the pop up window.  Click any analysis button on the schematic to regenerate frequency, time, input impedance, or Z Transform analysis.

Modify Switched Capacitor Value
Modify a Capacitor Value

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Monte Carlos

A Monte Carlo statistical analysis may easily be performed visually with Filter Solutions. After creating and displaying a switched capacitor filter and filter frequency,  impedance, or time response, left click a capacitor you would like to study.  You may study one capacitor, or all capacitors at once. In the Change Control Panel, select "Random", and enter the maximum tolerance or standard deviation in percent of the desired random change. Monte Carlos may be done manually by repetitious clicks of "Apply", or automatically by entering the desired number of simulations.  Graphical traces may be overwritten or retained as desired.  Both Uniform and Gaussian distributions are provided for inserting element value error.

Below is an example of the effect of random error from 2% capacitors has on the magnitude of a third order Elliptic Filter

Switched Capacitor Monte Carlo Display
Random Error Due to 5% Capacitors

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Real and Quadruplet Zeros Delay Equalization

Phase angle and group delay may be altered by the presence of dual and quadruplet off axis zeros. Unlike all pass stages, the mere addition of dual and quadruplet off-axis zeros also effects the pass band magnitude response, so additional calculations are needed to adjust the pole locations as needed to restore the pass band.  Delay equalization with real and quadruplet zeros result in a flatter Chebyshev pass band and steeper attenuation near the cut off frequency than a comparable size filter equalized with traditional all pass stages.  This may provide a more efficient filter, depending on the specific filter design requirements.

Filter Solutions offers a fast and easy approach to real and quadruplet delay equalization for low pass, high pass, and band pass switched capacitor filters.  Poles and group delay are updated in real time in response user zeros manipulation to flatten the pass band back into an equiripple (Chebyshev I) or maximally flat (Butterworth) shape, and switched capacitor filters are calculated instantly with the positioned zeros.

    
Quadruplet Zero Equalized Low Pass Chebyshev Passive Filter, Frequency Response and Pole/Zero Plane
 

Filter Solutions offers efficient switched capacitor designs for this filter.


Quadruplet Zero Equalized Low Pass Chebyshev Passive Filter Schematic
 

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