Natural frequency and damping ratio
damp(sys)
[Wn,zeta]
= damp(sys)
[Wn,zeta,P]
= damp(sys)
damp(
displays
a table of the damping ratio (also called damping factor),
natural frequency, and time constant of the poles of the linear model sys
)sys
.
For a discretetime 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
.
[
returns the natural frequencies, Wn
,zeta
]
= damp(sys
)Wn
,
and damping ratios,zeta
, of the poles of sys
.

Any linear dynamic system model. 

Vector containing the natural frequencies of each pole of If 

Vector containing the damping ratios of each pole of If 

Vector containing the poles of 
The natural frequency, time constant, and damping ratio of the system poles are defined in the following table:
Continuous Time  Discrete Time with Sample Time Ts  

Pole Location 
$$s$$ 
$$z$$ 
Equivalent ContinuousTime Pole 
$$\text{Notapplicable}$$ 
$$s=\frac{ln(z)}{{T}_{s}}$$ 
Natural Frequency 
$${\omega}_{n}=\lefts\right$$ 
$${\omega}_{n}=\lefts\right=\left\frac{ln(z)}{{T}_{s}}\right$$ 
Damping Ratio 
$$\zeta =cos(\angle s)$$ 
$$\begin{array}{lll}\zeta \hfill & =cos(\angle s)\hfill & =cos(\angle ln(z))\hfill \end{array}$$ 
Time Constant 
$$\tau =\frac{1}{{\omega}_{n}\zeta}$$ 
$$\tau =\frac{1}{{\omega}_{n}\zeta}$$ 