# First Order Filter

A first order filter, with cutoff frequency `R`, is used to identify
the dynamic component, `x` , of the input signal,
`X`. The transfer functions of the signal sent to the dynamic
and the static models, assuming `X0`=0, are:

TF of signal to dynamic model =

TF of signal to dynamic model = 1

The following are Bode plots for these transfer functions:

- The top figure plots the magnitude of the transfer function against ${\text{Log}}_{10}\left(\omega /R\right)$ .
- The bottom figure plots the loss angle of the transfer function against ${\text{Log}}_{10}\left(\omega /R\right)$ .
- Plots of the signal sent to the dynamic model are gray-blue.
- Plots of the signal sent to the static model are brick-red.

The log scale used for the x-axis lets you view a wide range of frequencies and filter behavior as follows:

- When
$\left(\omega /R\right)$
≪ 1, that is at low frequencies, then:
- The magnitude of the signal sent to the dynamic model is close to 0.
- The loss angle of the signal sent to the dynamic model is close to 90°.
- The magnitude of the signal sent to the static model is 1.
- The loss angle of the signal sent to the static model is 0°.

The bushing essentially behaves as the static model. The loss angle of the signal sent to the dynamic model is close to 90°, but this is not important since the magnitude of the signal is close to zero.

- When
$\left(\omega /R\right)$
≫ 1, that is at high frequencies, then:
- The magnitude of the signal sent to the dynamic model is close to 1.
- The loss angle of the signal sent to the dynamic model is close to 0° .
- The magnitude of the signal sent to the static model is 1.
- The loss angle of the signal sent to the static model is 0°.

The bushing essentially behaves as a dynamic model superimposed on top of a static model.

- When
$\left(\omega /R\right)$
≫ 1, that is at cut-off frequency, then:
- The magnitude of the signal sent to the dynamic model is $1/\sqrt{2}\approx 0.701$ .
- The loss angle of the signal sent to the dynamic model is 45°.
- The magnitude of the signal sent to the static model is 1.
- The loss angle of the signal sent to the static model is 0°.

The bushing essentially behaves as a dynamic model superimposed on top of a static model.