WEEK 6 FREQUENCY RESPONSE

Week 6 Report

Submission of final report and oral presentation (10 min limit).

Background

Brief description of system, "input" and "output"

Brief description of performance curves (SSOC)

Objective of controller design

Theory

Brief review of system transfer function (FOPDT) (include parameter values)

Modelling

Model & parameters

Results

Sample time-response graphs for experiment & approximate model

Conclusions

"The experimental results showed

The approximate model results showed ."

Some suggested slides for Week 6 Report

Background Theory Modelling Results Conclusions	Theory Transfer function Parameters	Results Time response Experimental Approximate model
Background System Input Output SSOC Operating Range	Modelling Model equations Parameters	Conclusions

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 purpose of this lab is to get frequency response data as shown in Figure 12.

Figure 12. Frequency response input and output functions

Figure 12(a) shows an input to the system that is a sine wave. The input function baseline is Mb. The peak-to-peak amplitude of the sine wave is ÆM. The sine wave has a frequency, f, measured in Hertz (Hz). Hertz is the same as cycles per second.

A typical output function is shown in Figure 12(b). The output function will have the same frequency but may have a different peak-to-peak amplitude, ÆC. The output function may also be delayed so that it lags in phase compared to the input function.

A sine-input operating curve can be plotted, also, as shown in Figure 12(c). More about this later.

PROCEDURE FOR GETTING SINE RESPONSE DATA

Prepare system for operation

Open LabVIEW program labeled "(Sine)". This program emulates the operation of a programmable controller. You should get a panel somewhat like the one shown in Figure 13.

Figure 13. Sine wave input controller panel

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. Click on the RUN arrow. The chart on the right of the screen is emulating a strip chart recorder.

When you want to stop, click on the STOP button. As before, it will ask if you want to save the data on a disk file, if you want to draw a time-response graph of the data and if you want to draw an input-output graph of the data. If you've just been playing around, click "NO" to all of these requests.

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, set the parameter values and start the instrument (click on the RUN arrow). After the sine response has ended, click the STOP button and save the data, make the time-response graph AND the input-output graph. This latter graph is the famous and useful Lissajous figure.

When you complete a run you will get a graph similar to that in Figure 14. The level-control system will get a different graph due to the "reverse" nature of the system.

Figure 14. Very low frequency sine response graphs

The very low frequency Lissajous figure is actually a portion of the same curve you got as the steady state operating curve in Weeks 1 & 2. Incidentally, the slope of this line, , 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 as in Figure 14(b). This means the phase shift is 0.

At a higher frequency, the AR and phase shift are different. A Lissajous figure like Figure 15(b) results. The AR is determined by the ratio of the vertical height to the horizontal height of the oval.

Figure 15. Medium frequency sine response graphs

At an even higher frequency, the phase shift becomes even larger and the Lissajous figure leans over to the left. (To the right for level-control.) An example is shown in Figure 16.

Figure 16. Medium-high frequency sine response Lissajous figure

You will benefit by running experiments at about 10 different frequencies. A good place to start is at a frequency of about w = . 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 found in weeks 1 & 2.

With Speed, Pressure and Flow systems, the "TURBO" switch allows you to get more data at the higher frequencies. For frequencies greater than about 0.5 Hz, start the experiment with TURBO off and then turn TURBO on for several cycles of oscillation.

After you have AR and phase angle data for a number of frequencies, make two plots like those shown in Figure 17. These are Bode plots. Notice Figure 17(a) is a log-log plot and Figure 17(b) is a semi-log plot. Prepare the results for making the Bode plots by filling in a table of results such as below.

Frequency ..(lowest).. .............. .............. .............. .............. ..(highest).

Amplitude Ratio .............. .............. .............. .............. .............. ..............

Phase Angle .............. .............. .............. .............. .............. ..............

Figure 17. Example Bode plot obtained from experimental data

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°.

3. The Kcu. This is the 1/(AR) at the ultimate frequency.

Hints:

XRC/248, SRC/249, PRC/308 & FRC/309 are probably higher than 1st order.

LRC/307 & TRC/303 are probably about 1st order.

Disk File Suggestion: For all your data files that you save this week, start their names with "W6" (meaning week #6)

"The study of how [the amplitude ratio and the phase angle] vary as w varies is an important part of automatic process control."

ññSmith & Corripio, p. 95