# 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 =

$\frac{x\left(j\omega \right)}{X\left(j\omega \right)}=\frac{{\omega }^{2}}{{R}^{2}+{\omega }^{2}}+j\frac{R\omega }{{R}^{2}+{\omega }^{2}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}j=\sqrt{-1},\text{\hspace{0.17em}}\omega =2\pi f=angular\text{\hspace{0.17em}}frequency$

TF of signal to dynamic model = 1

The following are Bode plots for these transfer functions:

The Bode plots show the magnitude and loss angle of the transfer functions over a range of operating frequencies:
• 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:

1. 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.

2. 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.

3. 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.