Objectives
To observe experimentally the time response of the output function of the system to a sine function input. To observe the system's amplitude ratio and the system's phase lag. To make the observations for a number of different values for the amplitude and the average value of the input sine function. To make the observations for a variety of system configurations, as appropriate.
Reference: Smith & Corripio, pp 95, 292-295
The input function baseline is Mb. The peak-to-peak amplitude of the sine wave is delta-M. The sine wave has a frequency, f, measured in Hertz (Hz). Hertz is the same as cycles per second.
The output function will have the same frequency but may have a different peak-to-peak amplitude, delta-C. The output function may also be delayed so that it lags in phase compared to the input function.
PROCEDURE FOR GETTING SINE RESPONSE DATA
Choose a value of the "input" variable that you want to be the base line value. Choose a value that you want to be the sine amplitude height of the "input" variable. Choose the frequency of the sine wave that you want. Set these in the appropriate windows in the panel.
You'll probably want to play around a bit before taking any serious data. You'll want to try different values of the sine parameters: frequency, sine wave amplitude and base line value.
When you are ready to get some good data, make the time-response graph AND the input-output graph. This latter graph is the famous and useful Lissajous figure.
The very low frequency Lissajous figure is actually a portion of the same curve you got as the steady state operating curve.
Incidentally, the slope of this line, delta-C/delta-M , is the s.s. gain. The amplitude ratio (AR) is the ratio of the vertical height to the horizontal height of the Lissajous figure. When the input and output are exactly in phase, the Lissajous figure is a single line. This means the phase shift is 0.
At a higher frequency, the AR and phase shift are different. The AR is determined by the ratio of the vertical height to the horizontal height of the oval. At an even higher frequency, the phase shift becomes even larger and the Lissajous figure leans over to the left.
You will benefit by running experiments at about 10 different frequencies. A good place to start is at a frequency of about w = 1/tau (notice that f=w/2*pi, f is the frequency parameter in the LabVIEW panel). Then you might run experiments at successively lower frequencies by approximately halving the frequency each time. Stop going to lower frequencies when the output is nearly in phase with the input.
Then you might run experiments at higher frequencies by approximately doubling your starting frequency and continue doubling the frequency on successive experiments. Stop going to higher frequencies when the output has no perceptible steady oscillation. "No perceptible" oscillation means that the amplitude in the output is smaller than twice the standard deviation of the output measurements.
After you have AR and phase angle data for a number of frequencies, make two plots. These are Bode plots. The AR plot is a log-log plot and the phase angle plot is a semi-log plot. Prepare the results for making the Bode plots by filling in a table of results such as below.
Frequency Amplitude Ratio Phase Angle
..(lowest). .................. ..................
.............. .................. ..................
.............. .................. ..................
..(highest). .................. ..................
There are three important values to get from a Bode plot. 1. The order of the system. This is the negative of the slope of the AR vs. Frequency plot at the high frequencies. 2. The ultimate frequency. This is the frequency for which the phase angle is -180 degrees. 3. The Kcu. This is the 1/(AR) at the ultimate frequency.
"The study of how [the amplitude ratio and the phase angle] vary as w varies is an important part of automatic process control." ÐÐSmithÊ&ÊCorripio,Êp.Ê95
contact Jim Henry - jhenry@utcvm.utc.edu ... Send E-Mail to Jim Henry