One method of modeling or analyzing a linear, time-invariant tension control system is with gain and phase vs. frequency. A mathematical model describing the phase and magnitude of gain is called a Transfer Function. The simplest method is the open loop graphical technique named after Hendrik Bode in the 1930s. Bode plots graph the absolute value of gain and phase vs. frequency. The absolute value of gain is expressed on a logarithmic scale in decibels (dB). Frequency is also expressed on a log scale. The phase shift is graphed normally.
The advantage of using logarithmic scales on the graphs is that multiplications become additions resulting in very simple techniques for combining parts of a control system. The parts could be the “Plant” or the “Control” portions of the system. The plant consists of the fixed portions of the equipment. For a web handling example, the plant includes the web, the rollers and the motors, brakes, and clutches and traction providing tension. The control may consist of a PID regulator or other controller.
There are two methods of determining the transfer functions for parts of the tension control system. One method is to measure the gain and phase shift for a part. For instance, it is possible to excite a motor by itself with torque varying in frequency. Specifically, the torque reference must vary from zero to positive to negative and back to zero. The frequency of the torque signal forms the X-axis on the Bode plot. Then the torque and speed of the motor can be charted using a two channel scope. At various frequencies, the technologist will calculate the magnitude of the speed variations and the phase shift compared with the torque reference. It will be necessary to plot the results for frequencies of 0.1 radian/second 100 radians/second. These results can be plotted on a log graph to produce the Bode plot of the motor’s transfer function. I didn’t say it would be easy!
The second method is to recognize that transfer functions have been determined for most parts of the control system. The transfer function can be researched and parameters filled in from data provided with the motor. We learn that
RPM (s) = PowerIn(s)_______
(1 + s * InertiaTimeConstant)
There are several recognizable features of Bode plots. Constant gain is shown as a horizontal line. Integrators are shown as lines sloping downward to the right at -20dB/decade. Derivatives are shown as lines sloping upward to the right at 20db/decade.
Of course, the purpose of the Bode Plot is to assist in setting the PID gains to tame the tension regulator.
Below is a Bode plot for the motor.