# Second-Order Filter

Implement second-order filter

## Library

Simscape / Electrical / Specialized Power Systems / Control & Measurements / Filters

## Description

Based on the Filter type selected in the block menu, the Second-Order Filter block implements the following transfer function:

Low-pass filter:

`$H\left(s\right)=\frac{{\omega }_{n}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`

High-pass filter:

`$H\left(s\right)=\frac{{s}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`

Band-pass filter:

`$H\left(s\right)=\frac{2\zeta {\omega }_{n}s}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`

Band-stop (notch) filter:

`$H\left(s\right)=\frac{{s}^{2}+{\omega }_{n}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`

The key characteristics of the Second-Order Filter block are:

• Input accepts a vectorized input of N signals, implementing N filters. This feature is particularly useful for designing controllers in three-phase systems (N = 3).

• Filter states can be initialized for specified DC and AC inputs.

• It enables you to compute and plot filter response.

## Parameters

Filter type

Specify the type of filter: `Lowpass`, `Highpass`, `Bandpass` (default), or `Bandstop (notch)`.

Natural frequency fn (Hz)

Specify the natural frequency of the filter, in hertz. This value must be a scalar or a vector. Default is `120`.

Damping ratio Zeta (Q = 1/(2*Zeta))

Specify the damping ratio of the filter. The damping ratio is typically a value between 0 and 1. Default is `0.707`.

The damping ratio is related to the filter quality factor Q:

`$Q=\frac{1}{2\zeta }$`

For a bandpass or a bandstop filter, the 3 dB bandwidth is given by

`$BW=\frac{{f}_{n}}{Q}=2\zeta {f}_{n}$`

Sample time

Specify the sample time of the block, in seconds. Set to 0 to implement a continuous block. Default is `0`.

Initialize filter states

When this check box is selected, filter states are initialized according to the AC initial input and DC initial input parameters. Default is selected.

AC initial input: [ Mag, Phase (degrees), Freq (Hz) ]

Specify the magnitude of the initial AC component of the input signal, its phase, in degrees, and its frequency, in hertz. Default is `[0, 0, 60]`.

When the input is vectorized (N signals), specify an N-by-3 matrix, where each row of the matrix corresponds to a particular input.

The AC initial input parameter is visible only when the Initialize filter states parameter is selected.

DC initial input

Specify the value of the initial DC component of the input signal. When the input signal is vectorized, specify a 1-by-N vector, where each value corresponds to a particular input. Default is `0`.

The DC initial input parameter is visible only when the Initialize filter states parameter is selected.

Plot filter response

When this check box is selected, the filter step response and its Bode diagram (magnitude and phase of transfer function as a function of frequency) are plotted in a figure. Default is cleared.

Frequency range (Hz): [Start, End, Inc.]

Specify the frequency range for plotting the filter Bode diagram. Specify a vector containing the starting frequency, the end frequency, and the incremental frequency, in hertz. Default is ```[0, 500, 1]```.

The Frequency range parameter is visible only when the Plot filter response parameter is selected.

## Characteristics

 Direct Feedthrough Yes Sample Time Specified in the Sample Time parameter Continuous if Sample Time = 0 Scalar Expansion Yes, of the parameters States Two states per filter Dimensionalized Yes

## Examples

The `power_SecondOrderFilter` example shows the Second-Order Filter block using two Filter type parameter settings (`Lowpass` and `Bandstop`).

The model sample time is parameterized with variable Ts (default value Ts = 50e-6). To simulate continuous filters, specify Ts = 0 in the MATLAB® Command Window before starting the simulation.