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 =

$\frac{x (j ω)}{X (j ω)} = \frac{ω^{2}}{R^{2} + ω^{2}} + j \frac{R ω}{R^{2} + ω^{2}}, j = \sqrt{- 1}, ω = 2 π f = a n g u l a r f r e q u e n c y$

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 ${Log}_{10} (ω / R)$ .
The bottom figure plots the loss angle of the transfer function against ${Log}_{10} (ω / R)$ .
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 $(ω / R)$ ≪ 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 $(ω / R)$ ≫ 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 $(ω / R)$ ≫ 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.