WOLFRAM SYSTEMMODELER

Filter

Continuous low pass, high pass, band pass or band stop IIR-filter of type CriticalDamping, Bessel, Butterworth or ChebyshevI

Wolfram Language

In[1]:=
SystemModel["Modelica.Blocks.Continuous.Filter"]
Out[1]:=

Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

This blocks models various types of filters:

low pass, high pass, band pass, and band stop filters

using various filter characteristics:

CriticalDamping, Bessel, Butterworth, Chebyshev Type I filters

By default, a filter block is initialized in steady-state, in order to avoid unwanted oscillations at the beginning. In special cases, it might be useful to select one of the other initialization options under tab "Advanced".

Typical frequency responses for the 4 supported low pass filter types are shown in the next figure:

LowPassOrder4Filters.png

The step responses of the same low pass filters are shown in the next figure, starting from a steady state initial filter with initial input = 0.2:

LowPassOrder4FiltersStepResponse.png

Obviously, the frequency responses give a somewhat wrong impression of the filter characteristics: Although Butterworth and Chebyshev filters have a significantly steeper magnitude as the CriticalDamping and Bessel filters, the step responses of the latter ones are much better. This means for example, that a CriticalDamping or a Bessel filter should be selected, if a filter is mainly used to make a non-linear inverse model realizable.

Typical frequency responses for the 4 supported high pass filter types are shown in the next figure:

HighPassOrder4Filters.png

The corresponding step responses of these high pass filters are shown in the next figure:

HighPassOrder4FiltersStepResponse.png

All filters are available in normalized (default) and non-normalized form. In the normalized form, the amplitude of the filter transfer function at the cut-off frequency f_cut is -3 dB (= 10^(-3/20) = 0.70794..). Note, when comparing the filters of this function with other software systems, the setting of "normalized" has to be selected appropriately. For example, the signal processing toolbox of MATLAB provides the filters in non-normalized form and therefore a comparison makes only sense, if normalized = false is set. A normalized filter is usually better suited for applications, since filters of different orders are "comparable", whereas non-normalized filters usually require to adapt the cut-off frequency, when the order of the filter is changed. See a comparison of "normalized" and "non-normalized" filters at hand of CriticalDamping filters of order 1,2,3:

CriticalDampingNormalized.png
CriticalDampingNonNormalized.png

Implementation

The filters are implemented in the following, reliable way:

  1. A prototype low pass filter with a cut-off angular frequency of 1 rad/s is constructed from the desired analogFilter and the desired normalization.
  2. This prototype low pass filter is transformed to the desired filterType and the desired cut-off frequency f_cut using a transformation on the Laplace variable "s".
  3. The resulting first and second order transfer functions are implemented in state space form, using the "eigen value" representation of a transfer function:
    
      // second order block with eigen values: a +/- jb
      der(x1) = a*x1 - b*x2 + (a^2 + b^2)/b*u;
      der(x2) = b*x1 + a*x2;
           y  = x2;
         
    The dc-gain from the input to the output of this block is one and the selected states are in the order of the input (if "u" is in the order of "one", then the states are also in the order of "one"). In the "Advanced" tab, a "nominal" value for the input "u" can be given. If appropriately selected, the states are in the order of "one" and then step-size control is always appropriate.

References

Tietze U., and Schenk C. (2002):
Halbleiter-Schaltungstechnik. Springer Verlag, 12. Auflage, pp. 815-852.

Parameters (13)

analogFilter

Value: Modelica.Blocks.Types.AnalogFilter.CriticalDamping

Type: AnalogFilter

Description: Analog filter characteristics (CriticalDamping/Bessel/Butterworth/ChebyshevI)

filterType

Value: Modelica.Blocks.Types.FilterType.LowPass

Type: FilterType

Description: Type of filter (LowPass/HighPass/BandPass/BandStop)

order

Value: 2

Type: Integer

Description: Order of filter

f_cut

Value:

Type: Frequency (Hz)

Description: Cut-off frequency

gain

Value: 1.0

Type: Real

Description: Gain (= amplitude of frequency response at zero frequency)

A_ripple

Value: 0.5

Type: Real (dB)

Description: Pass band ripple for Chebyshev filter (otherwise not used); > 0 required

f_min

Value: 0

Type: Frequency (Hz)

Description: Band of band pass/stop filter is f_min (A=-3db*gain) .. f_cut (A=-3db*gain)

normalized

Value: true

Type: Boolean

Description: = true, if amplitude at f_cut = -3db, otherwise unmodified filter

init

Value: Modelica.Blocks.Types.Init.SteadyState

Type: Init

Description: Type of initialization (no init/steady state/initial state/initial output)

nx

Value: if filterType == Modelica.Blocks.Types.FilterType.LowPass or filterType == Modelica.Blocks.Types.FilterType.HighPass then order else 2 * order

Type: Integer

x_start

Value: zeros(nx)

Type: Real[nx]

Description: Initial or guess values of states

y_start

Value: 0

Type: Real

Description: Initial value of output

u_nominal

Value: 1.0

Type: Real

Description: Nominal value of input (used for scaling the states)

Connectors (3)

u

Type: RealInput

Description: Connector of Real input signal

y

Type: RealOutput

Description: Connector of Real output signal

x

Type: RealOutput[nx]

Description: Filter states

Used in Examples (7)

Filter

Modelica.Blocks.Examples

Demonstrates the Continuous.Filter block with various options

FilterWithDifferentiation

Modelica.Blocks.Examples

Demonstrates the use of low pass filters to determine derivatives of filters

FilterWithRiseTime

Modelica.Blocks.Examples

Demonstrates to use the rise time instead of the cut-off frequency to define a filter

SMEE_Rectifier

Modelica.Electrical.Machines.Examples.SynchronousInductionMachines

Test example: ElectricalExcitedSynchronousInductionMachine with rectifier

ThreePhaseTwoLevel_PWM

Modelica.Electrical.PowerConverters.Examples.DCAC.MultiPhaseTwoLevel

Test of pulse width modulation methods

ThreePhaseTransformerWithRectifier

Modelica.Magnetic.FluxTubes.Examples.Hysteresis

3 Phase transformer (including hysteresis effect) with rectifier

SMEE_Rectifier

Modelica.Magnetic.FundamentalWave.Examples.BasicMachines

Test example: ElectricalExcitedSynchronousInductionMachine with rectifier