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

• 

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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, $f_c$, to pass.

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

• Band pass — Allows signals, $f$, only in the range of frequencies between two cutoff frequencies, $f_{c1}$ and $f_{c2}$, to pass.

• Band stop — Prevents signals, $f$, only in the range of frequencies between two cutoff frequencies, $f_{c1}$ and $f_{c2}$, from passing.

Filter Type	Frequency Range, $f$
Low-Pass		$f<f_c$
High-Pass		$f>f_c$
Band-Pass		$f_{c1}<f<f_{c2}$
Band-Stop		$f_{c1}<f<f_{c2}$

Equations

The second order derivative state equation for the filter is:

$\frac{d^{2}x}{d{t}^2}=u-2\zeta {\omega }_n\frac{dx}{dt}-{\omega }_n^{2}x$

Where:

• x is the filter internal state.

• u is the filter input.

• ω_n 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 Type	Output, $y(x)$	Transfer Function, $G(s)$
Low-Pass	${\omega }_n^{2}x$	$\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$
High-Pass	$\frac{d^{2}x}{d{t}^2}$	$\frac{s^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$
Band-Pass	$2\zeta {\omega }_n\frac{dx}{dt}$	$\frac{2\zeta {\omega }_{n}s}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$
Band-Stop	$\frac{d^{2}x}{d{t}^2}+x$	$\frac{s^2+{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$

For Initialization:

$\dot{x}(0)={\frac{dx}{dt}|}_{t=0}$

$u\left(0\right)=u_1\left(0\right)+u_2(0)$

$u_1\left(0\right)=A_0{e}^{j{\phi }_0}$

$u_2\left(0\right)=b_0{e}^{j\frac{\pi }{2}}$

Where:

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

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

• $u_1(0)$ is the AC component of the steady-state initial input.

• $A_0$ is the initial amplitude.

• ${\phi }_0$ is the initial phase.

• $u_2(0)$ is the DC component of the steady-state initial input.

• $b_0$ is the initial bias.

In the s-domain $s=j{\omega }_0$. Therefore, for the initial frequency, ${\omega }_0$:

$\dot{x}(0)=Im\left(\frac{j{\omega }_0{u}_1\left(0\right)}{-{\omega }_0^2+j{\omega }_{0}2\zeta {\omega }_n+{\omega }_n^2}\right).$

$$x(0)=Im\left(\frac{\dot{x}\left(0\right){\omega }_n^2}{j{\omega }_0}+u_2\left(0\right)\right)$$

Ports

Input

u — Filter input

scalar

Filter input.

Data Types: single | double

Output

y — Filtered output

scalar

Filtered output.

Data Types: single | double

Parameters

Main

Filter type — Filter type

Low-pass (default) | High-pass | Band-pass | Band-stop

Type of second-order filter.

Natural frequency (Hz) — Natural frequency

60 (default) | positive scalar

Natural frequency, in Hz.

Initial Conditions

Damping factor — Damping factor

0.707 (default) | nonnegative scalar

Damping factor of the filter.

Sample time (-1 for inherited) — Block sample time

-1 (default) | 0 | positive scalar

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).

Initial amplitude — Initial amplitude

0 (default) | nonnegative scalar

Amplitude at the start of simulation.

Initial phase (rad) — Initial phase

0 (default) | nonnegative scalar

Phase, in rad, at the start of simulation.

Initial frequency (Hz) — Initial frequency

0 (default) | scalar

Frequency, in Hz, at the start of simulation.

Initial bias — Initial bias

0 (default) | nonnegative scalar

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™.