WEEK 7
MODELING FREQUENCY RESPONSE
Objectives
To compare the experimental Bode plots with the approximate FOPDT
model's Bode plots for the system. To observe the impact of parameter values
on the model's Bode plots. To learn how to generate sine responses with
Excel. To observe the sine responses of the approximate mathematical model
for a system.
Modeling Assignment
If the input function to a FOPDT is a sine function, having an amplitude = A
and frequency = f, then
m(t) = A•sin(2ft)
then the steady oscillation part of the time response of the FOPDT system is
c(t ) 
A K
1   2 2
sin(   t0  arctan(  ))
where K is the gain, to is the dead-time and  is the first-order time constant,
or characteristic time, of the FOPDT system. The variables m(t) and c(t) in
these equations are deviation variables. This means that you have to add the
input baseline and output baseline to the values to get them to agree with the
experimental data.
The model equations for the Bode plots are
AR  amplitude ratio 
K
1  2  2
and
  phase angle  to  tan 1 ( )
We want to have Excel draw a graph of these equations so we can see the
effect of the various parameters and see what parameters best match our
experimental results.
Comparing Bode Plot for Model and Experiment
From last week, you got some values of the Amplitude Ratio (AR) and Phase
Angle (PA) for the sine inputs to your system at some different frequencies.
Enter these into a new Excel spreadsheet, like this: (These are ONLY
example numbers)
A
1
2
3
4
5
6
7
8
9
B
Freq
C
D
AR
0.011
0.02
0.05
0.1
0.2
0.5
1
E
PA
50
49
40
21
10
4
2
-8
-25
-55
-88
-150
-190
-255
You are leaving columns C and D blank for now; later you'll put the model
equations in there. It will be useful if you enter column D by using a formula
and setting it to column A. The formula for cell D1 is =A1.
For now, add to your spreadsheet, in columns G and H, the names and values
of the approximate model parameters for your system: K, to and tau. (These
are ONLY example numbers)
G
1
2
3
H
K=
to=
tau=
7.2
0.65
0.33
Now enter the values of frequency that you want to have for you model Bode
plot. Suggestion: have these values' range a bit wider than your experimental
frequency range. Put these in column A, starting BELOW where your last
experimental value is, say cell A10. Copy these into column D, also.
In the cell in column C BELOW the last experimental point, let's say, C10,
enter this formula for the amplitude ratio
=K/SQRT(1+(2*PI()*A10)^2*tau^2)
In the cell in column F BELOW the last experimental point, let's say, F10,
enter this formula for the phase angle
=(ATAN(-2*PI()*A10*tau)-2*PI()*A10*to)*180/PI(0)
Now plot the amplitude ratio part of the Bode plot by highlighting columns A,
B and C; choosing the ChartWizard, selecting Scatter and making the axes
log-log.
Now plot the phase angle part of the Bode plot by highlighting columns D, E
and F; choosing the ChartWizard, selecting Scatter and making the axes semilog.
VARYING PARAMETERS
Now go back to the parameter cells and change one of them and watch the
graph change. Find the values of the parameters that make the model agree
well with the experimental results. Here is the suggested order:
Which Bode
Graph
AR vs f
Parameter(s)
to vary
K
Objective
AR vs f
tau
get model's AR-vs-f curve to agree near the "corner" of the curve
PA vs f
to
get model's PA-vs-f curve to agree at PA=-180
get model's AR-vs-f curve to agree at low frequencies
Save the file.
Label the graphs.
Print the graphs.
Save the file. These values are now the best estimates for your approximate
model's parameters to fit your system.
Comparing Time Responses for Model and Experiment
We're going to build a time-response model in a file that has some
experimental sine-response data in it. Begin by choosing the file that has data
in it for the lowest frequency sine-input experiment that you did that has good
data.
Open an Excel by double clicking on the Excel icon. Open the data file from
Week 6 by using the File/Open... menu item.
Part of the spreadsheet will look like this:
A
B
C
D
E
F
1
Time (sec) Input Value (%)
Pressure (cm-H2O)
2
0.011
50
0.152
3
0.18
50
0.173
4
0.196
50
0.152
5
0.213
50
0.132
6
0.213
50
0.132
7
0.213
50
0.132
Save this now in a new file named "W7-something-or-other." Do this with
8
0.213
50
0.132
the File/Save As... menu.
We're going to enter in two columns of numbers to have Excel plot for us.
The columns are "Input" and "Output." Type each of these names in the top
row of the spreadsheet in columns D and E, respectively. Move to the right in
the spreadsheet by touching the "tab" key or using the  key or by clicking
where you want to move to.
We're going to put formulas in cells D2 and E2 for the functions m(t) and
c(t). In columns F and G, we're going to put names and values for A, f, K,
to,, input baseline and output baseline. Let's use 2, 3, 0.2, 0.3, 0.4, 40 and
11, respectively. So those columns will look like this:
F
A
f
K
to
tau
in-baseline
out-baseline
G
2
3
0.2
0.3
0.4
40
11
Use the Insert/Names >/Define to define the names for all these parameters.
Click on D2 and put in this formula:
=A*sin(2*PI()*f*A2)+inbl
Click on E2 and put in this formula:
=A*K/(SQRT(1+2*PI()*f*2*PI()*f*tau*tau))*SIN(2*PI()*f*A2
+ATAN(-2*PI()*f*tau)-2*PI()*f*to)+outbl
Now copy these formulas down the spreadsheet. Click in the last time-value
in column D and drag across to column E and release the mouse. Both should
be selected. Now click on the square at the right bottom of that selection and
drag it down (off the screen and it will scroll down) to the end of the
experimental data.
Now you should have 5 columns with the values in them that we want.
Adjust amplitude, frequency, "inbl" and "outbl" so the curves agree.
Label the curves and print the curves. Save your file.
G
Repeat these time response models for 2 additional frequencies. Use one at a
frequency near the "corner" frequency and one at a frequency of one of the
highest frequencies you did.
File Suggestion: For all your data files that you save this week, start their
names with "W7" (meaning week #7) and save them on "Student files on the
Web" in your team's folder.
By the way, there are files like the one you've developed that are available on
the Internet at http://chem.engr.utc.edu/329.