Frequency diagrams

Frequency response

The frequency diagrams allows you to analyze the frequency response of the equivalent linear system. The frequency response function describes the relationship between the inputs (control variables) and the outputs (observer variables) of a system in the frequency domain. A representation of the response function has the form:

where, Y(s) and U(s) represent the Laplace transforms of the output and the input vectors respectively, and G(s) represents the transfer function that defines the relationship between the input and the output.

For more information, see the appendix on linear algebra.

The frequency response is characterized by the magnitude of the system response, typically in decibels (dB) or as a decimal, and the phase in degree, versus the frequency in radians/sec (rad/s) or Hertz (Hz).

These response functions can be plotted in 3 ways: by plotting the magnitude and phase measurements on two rectangular plots as functions of frequency to obtain a Bode plot; by plotting the magnitude and phase angle on a single rectangular plot with frequency as a parameter to obtain a Nichols plot; or by plotting magnitude and phase on a single polar plot with frequency as a parameter to obtain a Nyquist plot.

To generate these plots which are obtained at a linearization time, select the Frequency diagram item in the Linear Analysis menu.

Figure 45: Frequency diagram item in the toolbar

It opens a dedicated window that lists the control and observer variables of your system and the settings to generate the frequency response diagram.

Figure 46: Frequency diagram tool

The plot is generated according to the linearization time and the result set selected from the drop-down lists.

The Update button allows you to refresh the drop-down lists with the latest available simulation results.

Control and observer variable selection

If you select one control variable and one observer variable, you can generate a frequency response plot that describes the relationship between the selected control variable and the selected observer variable.

If you select multiple control variables and one observer variable, you can generate as many frequency response plots as there are control variables selected. The plots are on the same graph.

If you select one control variable and multiple observer variables, you can generate as many frequency response plots as there are observer variables selected. The plots are on the same graph.

You can't select several control variables and several observer variables.

Parameters of the diagram

To calculate the magnitude of the frequency response function, an evaluation of the function H(s) is performed at several frequency values.

The calculation and the plot differ depending on the selected scale type:

• If you select decibel, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to decimal.
  

• If you select linear, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to linear
  

• If you select logarithmic, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to logarithmic
  

Start frequency, Final frequency and Number of points allow you to define the frequency range and its discretization for the calculation.