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# damp

Natural frequency and damping ratio

## Syntax

damp(sys)
[Wn,zeta] = damp(sys)
[Wn,zeta,P] = damp(sys)

## Description

damp(sys) displays a table of the damping ratio (also called damping factor), natural frequency, and time constant of the poles of the linear model sys. For a discrete-time model, the table also includes the magnitude of each pole. Frequencies are expressed in units of the reciprocal of the TimeUnit property of sys. Time constants are expressed in the same units as the TimeUnit property of sys.

[Wn,zeta] = damp(sys) returns the natural frequencies, Wn, and damping ratios,zeta, of the poles of sys.

[Wn,zeta,P] = damp(sys) returns the poles of sys.

## Input Arguments

 sys Any linear dynamic system model.

## Output Arguments

 Wn Vector containing the natural frequencies of each pole of sys, in order of increasing frequency. Frequencies are expressed in units of the reciprocal of the TimeUnit property of sys. If sys is a discrete-time model with specified sampling time, Wn contains the natural frequencies of the equivalent continuous-time poles (see Algorithms). If sys has an unspecified sampling time (Ts = -1), then the software uses Ts = 1 and calculates Wn accordingly. zeta Vector containing the damping ratios of each pole of sys, in the same order as Wn. If sys is a discrete-time model with specified sampling time, zeta contains the damping ratios of the equivalent continuous-time poles (see Algorithms). If sys has an unspecified sampling time (Ts = -1), then the software uses Ts = 1 and calculates zeta accordingly. P Vector containing the poles of sys, in order of increasing natural frequency. P is the same as the output of pole(sys), except for the order.

## Examples

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### Natural Frequency, Damping Ratio, and Poles of Continuous-Time System

Calculate the natural frequency, damping ratio, time constant, and poles of the continuous-time transfer function:

$H\left(s\right)=\frac{2{s}^{2}+5s+1}{{s}^{2}+2s+3}.$

```H = tf([2 5 1],[1 2 3]);
```

Display the natural frequencies, damping ratios, time constants, and poles of H.

```damp(H)
```
```         Pole              Damping       Frequency      Time Constant

-1.00e+00 + 1.41e+00i     5.77e-01       1.73e+00         1.00e+00
-1.00e+00 - 1.41e+00i     5.77e-01       1.73e+00         1.00e+00   ```

Obtain vectors containing the natural frequencies and damping ratios of the poles.

```[Wn,zeta] = damp(H);
```

Calculate the associated time constants.

`tau = 1./(zeta.*Wn);`

### Natural Frequency, Damping Ratio, and Poles of Discrete-Time System

Calculate the natural frequency, damping ratio, time constant, and poles of a discrete-time transfer function.

```H = tf([5 3 1],[1 6 4 4],0.01);
```

Display information about the poles of H.

```damp(H)
```
```         Pole             Magnitude     Damping       Frequency      Time Constant

-3.02e-01 + 8.06e-01i     8.61e-01     7.74e-02       1.93e+02         6.68e-02
-3.02e-01 - 8.06e-01i     8.61e-01     7.74e-02       1.93e+02         6.68e-02
-5.40e+00                 5.40e+00    -4.73e-01       3.57e+02        -5.93e-03    ```

The Magnitude column displays the discrete-time pole magnitudes. The Damping, Frequency, and Time Constant columns display values calculated using the equivalent continuous-time poles.

Obtain vectors containing the natural frequencies and damping ratios of the poles.

```[Wn,zeta] = damp(H);
```

Calculate the associated time constants.

`tau = 1./(zeta.*Wn);`

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### Algorithms

The natural frequency, time constant, and damping ratio of the system poles are defined in the following table:

Continuous TimeDiscrete Time with Sample Time Ts
Pole Location

$s$

$z$

Equivalent Continuous-Time Pole

$s=\frac{ln\left(z\right)}{{T}_{s}}$

Natural Frequency

${\omega }_{n}=|s|$

${\omega }_{n}=|s|=|\frac{ln\left(z\right)}{{T}_{s}}|$

Damping Ratio

$\zeta =-cos\left(\angle s\right)$

$\begin{array}{lll}\zeta \hfill & =-cos\left(\angle s\right)\hfill & =-cos\left(\angle ln\left(z\right)\right)\hfill \end{array}$

Time Constant

$\tau =\frac{1}{{\omega }_{n}\zeta }$

$\tau =\frac{1}{{\omega }_{n}\zeta }$