Second-Order Filter

Discrete-time or continuous-time low-pass, high-pass, band-pass, or band-stop second-order filter

  • Library:
  • Simscape / Electrical / Control / General Control

Description

The Second-Order Filter block implements different types of second-order filters. Filters are useful for attenuating noise in measurement signals.

The block provides these filter types:

  • Low pass — Allows signals, f, only in the range of frequencies below the cutoff frequency, fc, to pass.

  • High pass — Allows signals, f, only in the range of frequencies above the cutoff frequency, fc, to pass.

  • Band pass — Allows signals, f, only in the range of frequencies between two cutoff frequencies, fc1 and fc2, to pass.

  • Band stop — Prevents signals, f, only in the range of frequencies between two cutoff frequencies, fc1 and fc2, from passing.

Filter TypeFrequency Range, f
Low-Pass

f<fc
High-Pass

f>fc
Band-Pass

fc1<f<fc2
Band-Stop

fc1<f<fc2

Equations

The second order derivative state equation for the filter is:

d2xdt2=u2ζωndxdtωn2x

Where:

  • x is the filter internal state.

  • u is the filter input.

  • ωn is the filter natural frequency.

  • ζ is the filter damping factor.

For each filter type, the table maps the block output, y(x), as a function of the internal state of the filter, to the s-domain transfer function, G(s).

Filter TypeOutput, y(x)Transfer Function, G(s)
Low-Passωn2xωn2s2+2ζωns+ωn2
High-Passd2xdt2s2s2+2ζωns+ωn2
Band-Pass2ζωndxdt2ζωnss2+2ζωns+ωn2
Band-Stopd2xdt2+xs2+ωn2s2+2ζωns+ωn2

For Initialization:

x˙(0)=dxdt|t=0

u(0)=u1(0)+u2(0)

u1(0)=A0ejφ0

u2(0)=b0ejπ2

Where:

  • x(0) is the initial state of the filter.

  • u(0) is the initial input to the filter.

  • u1(0) is the AC component of the steady-state initial input.

  • A0 is the initial amplitude.

  • φ0 is the initial phase.

  • u2(0) is the DC component of the steady-state initial input.

  • b0 is the initial bias.

In the s-domain s=jω0. Therefore, for the initial frequency, ω0:

x˙(0)=Im(jω0u1(0)ω02+jω02ζωn+ωn2).

x(0)=Im(x˙(0)ωn2jω0+u2(0))

Ports

Input

expand all

Filter input.

Data Types: single | double

Output

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Filtered output.

Data Types: single | double

Parameters

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Main

Type of second-order filter.

Natural frequency, in Hz.

Initial Conditions

Damping factor of the filter.

Time between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state. For more information, see What Is Sample Time? (Simulink) and Specify Sample Time (Simulink).

For inherited discrete-time operation, specify -1. For discrete-time operation, specify a positive integer. For continuous-time operation, specify 0.

If this block is in a masked subsystem, or other variant subsystem that allows you to switch between continuous operation and discrete operation, promote the sample time parameter. Promoting the sample time parameter ensures correct switching between the continuous and discrete implementations of the block. For more information, see Promote Parameter to Mask (Simulink).

Amplitude at the start of simulation.

Phase, in rad, at the start of simulation.

Frequency, in Hz, at the start of simulation.

Bias at the start of simulation.

References

[1] Agarwal, A. and Lang, J. H. Foundations of Analog and Digital Electronic Circuits. New York: Elsevier, 2005.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Introduced in R2018b