[email protected] engr-25_plot_model-3.ppt 1 bruce mayer, pe engineering/math/physics 25:...

[email protected] • ENGR-25_Plot_Model-3.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Chp5 MATLABPlots &

Models 3



Learning Goals List the Elements of a COMPLETE

Plots• e.g.; axis labels, legend, units, etc.

Construct Complete Cartesian (XY) plots using MATLAB• Modify or Specify MATLAB Plot Elements:

Line Types, Data Markers,Tic Marks

Distinguish between INTERPolation and EXTRAPolation



Learning Goals cont

Construct using MATLAB SemiLog and LogLog Cartesian Plots

Use MATLAB’s InterActive Plotting Utility to Fine-Tune Plot Appearance

Use MATLAB to Produce 3-Dimensional Plots, including• Surface Plots• Contour Plots



Logarithmic Plots

Rectilinear Plots do Not Reveal Important Features when one or both of the variables range over several orders of magnitude

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

x

y

10-1

100

101

102

10-2

10-1

100

101

102

x

y

>> x = [0:0.1:100];>> y = sqrt((100*(1-0.01*x.^2).^2 + 0.02*x.^2)./((1-x.^2).^2 + 0.1*x.^2));>> plot(x,y), xlabel('x'), ylabel('y');

222

222

1.01

02.001.01100

xx

xxy

>> loglog(x,y), xlabel('x'), ylabel('y')

Rectilinear Plot Log-Log Plot• LogLog Plot is MUCH More Revealing



Making Logarithmic Plots

Important Points to Remember1. You cannot plot negative numbers on a

log scale – Recall the logarithm of a negative number is

not defined as a real number

2. You cannot plot the number 0 (zero) on a log scale– Recall log10(0) = ln(0) = −

Therefore choose an appropriately small number (e.g., 10−18) as the lower limit on the plot.

3. Tick-mark labels on a log scale are the actual values being plotted; they are not logs of the No.s– The x values in the previous log-log plot

range over 10−1 = 0.1 to 102 = 100.



Making Logarithmic Plots cont

4. Gridlines and tick marks within a decade are unevenly spaced. – If 8 gridlines or tick marks occur within the

decade, they correspond to values equal to 2, 3, 4, . . . , 8, 9 times the value represented by the first gridline or tick mark of the decade.

5. Equal distances on a log scale correspond to multiplication by the same constant – as opposed to addition of the same constant

on a rectilinear scale– e.g.; all numbers that differ by a factor of 10 are

separated by the same distance on a log scale. That is, the distance between 0.3 and 3 is the same as the distance between 300 and 3000. This separation is referred to as a decade or cycle



MATLAB Log & semiLog Plots

MATLAB has three commands for generating plots with log scales:1. Use the loglog(x,y) command to have

both scales logarithmic.

2. Use the semilogx(x,y) command to have the x scale logarithmic and the y scale RECTILINEAR.

3. Use the semilogy(x,y) command to have the y scale logarithmic and the x scale RECTILINEAR



SemiLog Plot Comparisons

Again Plot 222

222

1.01

02.001.01100

xx

xxy

x → log; y → linear x → linear; y → log10

-110

010

110

20

5

10

15

20

25

30

35

x

y

semilogx(x,y), xlabel('x'), ylabel('y')

0 10 20 30 40 50 60 70 80 90 10010

-2

10-1

100

101

102

x

y

semilogy(x,y), xlabel('x'), ylabel('y')



Example Low Pass Filter

Consider a Simple RC “Voltage Divider”

4.7 kΩ

22 nF

By the Methods of Junior-Level EE Find the Voltage “Gain”, Gv

RCjCj

R

CjV

VGv

1

11

1

1

0

21

1||)(

RCGM v

In this Case the Time Constant, RC

µs

RC

4.103104.103

1022107.46

93

Finding the Magnitude of Gv



Example Low Pass Filter Plot

Recall the Mag of G

Lets “Center” out the M(ω) plot at ωτ = 1

Thus ω = 1/τ = 9671 rad/s 104 rad/s

2

2

1

1

1

1||)(

RCGM v

Thus Make a log-log Plot for M(ω) (called a “Bode” Plot) with the Domain• 102 ≤ ω ≤ 106

%7.70

2

1

11

1

4.1039671

1

19671

2

2

SS

M



Low Pass Filter Plot

102

103

104

105

106

10-3

10-2

10-1

100

Angular Frequency, w (rad/sec)

Vol

tage

Gai

n (u

nitle

ss

Bode Plot for RC LowPass Filter

4.7 kΩ

22 nF

1% left at 106

70.7% left at ω = 1/τ

• This Ckt Leaves UNCHANGED, or PASSES, Low Frequency signals, but attenuates High Frequency Versions



Interactive Plotting in MATLAB The “SemiAutomatic” interface

can be very convenient when You• Need to create a large number of different

types of plots,• Construct plots involving many data sets,• Want to add annotations such as

rectangles and ellipses• Desire to change plot characteristics such

as tick spacing, fonts, bolding, italics, and colors



MATLAB Interactive Plots cont

The interactive plotting environment in MATLAB Includes tools for• Creating different types of graphs,• Selecting variables to plot directly

from the Workspace Browser• Creating and editing subplots,• Adding annotations such as lines, arrows,

text, rectangles, and ellipses, and• Editing properties of graphics objects, such

as their color, line weight, and font



Interactive Plotting

Go From This To This

Recall the Sagging Cantilever Beam• Plot Sag vs Time

Using Interactive to

0 5 10 15 20 250

2

4

6

8

10

12

0 5 10 15 20 250

2

4

6

8

10

12

Load Application Time (minutes)

Ver

tical

Def

lect

ion

(mm

)

Polystrene Cantilever Beam Creep-Test

931 mN LoadSignificant "Kink"



0 5 10 15 20 250

2

4

6

8

10

12

Load Application Time (min)

Ver

tica

l Def

lect

ion

(m

m)

Styrofoam Beam Creep Test

931 mN Load

Significant Kink



Format Plots by Coding

A Tedious Process for ONE-Time Use• HELP must be consulted a LOT to

implement Complex Formatting

Useful forConstructinga Personal “Standard Format” for Plots

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

x

y =

f(x)

MTH15 • Bruce Mayer, PE

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m



Code for Previous Plot% Bruce Mayer, PE% MTH-15 • 23Jun13% XY_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m%% The FUNCTIONx = linspace(-6,6,500); y = -x.^2/3 +5.5;% % The ZERO Lineszxh = [-6 6]; zyh = [0 0]; zxv = [0 0]; zyv = [-6 6];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y, zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 3),axis([-6 6 -6 6]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Bruce Mayer, PE',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)



3D Surface Plots

Example Consider a Humidification Vapor-Generator used to Fabricate Integrated Circuits A Carrier Gas, Nitrogen in this case, “bubbles” thru the Liquid Chemical, Becoming Humidified in the Process

The “Bubbler OutPut”, Qmix, is the sum of Carrier N2, QN2, and the Chem Vapor, Qv



Patent 5,078,922 Bubbler in Operation

Carrier N2 Flow Rate

in slpm

Bubble

6.35 mm

Sparger Tube

Water Surface



Bubbler-OutPut Physics The Details of Bubbler Operation Found

in

Chemical Vapor Output

vhs

vNv PP

PQQ 2• Phs Absolute Pressure in

Bubbler HeadSpace• Pv = Thermodynamic Vapor Pressure



Bubbler Physics

Over a substantial Range of Temperatures Between Freezing & Boiling The ThermoDyamic Vapor Pressure of the Liquid Chemical Can be described by the Antoine Eqn1

CT

BAPvln

Where• T Absolute

Temperature• A, B, C are

CONSTANTS in Units consistent with T & Pv

1. R. C. Reid, J. M. Prausnitz, B. E. Poling, Properties of Gases & Liquids, 4th Ed., New York, McGraw-Hill, 1987, pg 208



Bubbler Physics cont

In many cases C 0

In This Case The Antione Eqn Reduces to the Clapeyron Eqn2

Then the Bubbler Eqn in terms of the Independent Vars QN2, Phs & T

2. R. C. Reid, J. M. Prausnitz, B. E. Poling, Properties of Gases & Liquids, 4th Ed., New York, McGraw-Hill, 1987, pg 206

TBAPv ln Thus Pv(T)

TB

TBA

TBAv

De

ee

eP

TBhs

TB

Nv DeP

DeQQ

/

/

2

or

TB

hs

TB

hsoN

v

DeP

DeTPQ

Q

Q/

/

2

,



Bubbler Physics cont

Thus the Normalized Output, Qo, Can be Modulated by Pressure and Temperature Control

We would Now Like to Plot Qo(Phs,T) for The Chemical TEOS

From the Manufacturer’s Data A summarized in [Mayer96], Find the Antoine/Clapeyron Constants for Pv in Torr • A = 19.3197• B = 5562.30 Kelvins



TEOS Chemical/Physical Data

General • Synonyms: ethyl

silicate, tetraethoxysilane, silicic acid tetraethyl ester, TEOS, tetraethyl silicate

• Molecular formula: (C2H5O)4Si

Physical data• Appearance:

colorless liquid with an alcohol-like odor

Physical data• Melting point: −86

C• Boiling point: 169

C• Vapor density: 7.2

(air = 1)• Vapor pressure:

2 mm-Hg at 20 C– H2O → 17.54 mm-

Hg

• Liquid Density (g/cm3): 0.94

• Flash point: 39 C (closed cup)

http://ptcl.chem.ox.ac.uk/M

SD

S/T

E/tetraethyl_orthosilicate.htm

l



Bubbler OutPut - TEOS

Thus for TEOS the Clapeyron Eqn

We Now Want to Make a MATLAB Plot of Qo,TEOS for these Conditions

Now the TEOS Bubbler Normally Operates under these Conditions• T: 60-85 °C

= 333-358 K• Phs: 250-750 Torr

T

TTEOSv

e

eP3.55628

3.55623197.19,

Torr 10457.2

TPP

TP

DeP

DeQ

TEOSvhs

TEOSv

TBhs

TB

TEOSo

,

,

/

/

,



mesh Plot Example

The Command Session>> Trng = linspace(333,358,25);>> Prng = linspace(250,750,25);>> [T,Phs] = meshgrid(Trng,Prng);>> A = 19.3197; B = 5562.30;>> Pv = exp(A - B./T);>> Qo = Pv./(Phs - Pv);>> mesh((T-273),Phs,Qo), xlabel('T (°C)'), ylabel('Phs (Torr)'),...zlabel('Qo (slpm-TEOS/Slpm-N2)'), grid on,...title('Vapor Output From TEOS Bubbler')

SQUARE XY Grid of 252 (225) points

Bubbler_Qo_of_TPhs_1010.m



mesh Plot Result

6065

7075

8085

300400

500600

700

0

0.05

0.1

0.15

0.2

0.25

T (°C)

Vapor Output From TEOS Bubbler

Phs (Torr)

Qo

(slp

m-T

EO

S/S

lpm

-N2)



mesh Plot Result: Swap X↔Y

250350

450550

650750

6065

7075

80850

0.05

0.1

0.15

0.2

0.25

Phs (Torr)


T (°C)

Qo

(slp

m-T

EO

S/S

lpm

-N2)

mesh(Phs,(T-273),Qo), xlabel('Phs (Torr)'), ylabel('T (°C)'),...zlabel('Qo (slpm-TEOS/Slpm-N2)'), grid on,...title('Vapor Output From TEOS Bubbler')



Mesh Plot Caveats

To Make the Simple Surface Plot Shown the X-Y Grid Must be SQUARE• i.e.; [No. X-pts] = [No. Y-pts]

– 25 in this case

Do NOT make the grid too DENSE• I tried the Qo Plot with a 500x500 Grid →

250 000 Points• Along with the 250 000 Qo calc Points,

MATLAB had to operate on a Half a MILLION pts (took “forever”)



contour Plot Example The Command Session

>> Trng = linspace(333,358,25);>> Prng = linspace(250,750,25);>> [T,Phs] = meshgrid(Trng,Prng);>> A = 19.3197; B = 5562.30;>> Pv = exp(A - B./T);>> Qo = Pv./(Phs - Pv);>> contour(Phs,(T-273),Qo), xlabel('Phs (Torr)'), ylabel('T (°C)'),...zlabel('Qo (slpm-TEOS/Slpm-N2)'), grid on,...title('Vapor Output From TEOS Bubbler')>> contour((T-273),Phs,Qo), xlabel('T (°C)'), ylabel('Phs (Torr)'),...zlabel('Qo (slpm-TEOS/Slpm-N2)'), grid on,...title('Vapor Output From TEOS Bubbler')



contour Plot Result

X → Phs X → T

Phs (Torr)

T (°

C)


250 300 350 400 450 500 550 600 650 700 75060

65

70

75

80

85

T (°C)

Phs

(Tor

r)


60 65 70 75 80 85250

300

350

400

450

500

550

600

650

700

750



Other 3D Plot CommandsCommand Plot Description

[X,Y] = meshgrid(x,y)

Creates the matrices X and Y from the vectors x and y to define a rectangular grid

[X,Y] = meshgrid(x) Same as [X,Y]= meshgrid(x,x).

mesh(x,y,z) Creates a 3D mesh surface plot

meshc(x,y,z) Same as mesh but draws contours under the surface

meshz(x,y,z) Same as mesh but draws vertical reference lines under the surface



Other 3D Plot Commands cont

Command Plot Description

contour(x,y,z) Creates a contour plot.

surf(x,y,z) Creates a shaded 3D mesh surface plot

surfc(x,y,z) Same as surf but draws contours under the surface

waterfall(x,y,z) Same as mesh but draws mesh lines in one direction only

250350

450550

650750

6065

70

7580

850

0.05

0.1

0.15

0.2

Phs (Torr)


T (°C)

Qo

(slp

m-T

EO

S/S

lpm

-N2)

Qmin = 0.0186Qmax = 0.2132

meshz

200

400

600

800

60

70

80

900

0.05

0.1

0.15

0.2

0.25

Phs (Torr)


T (°C)

Qo

(slp

m-T

EO

S/S

lpm

-N2)

surf



Caveat on 3D Surface Plots

3D Surfaces are Difficult for Many ENGINEERS/SCIENTISTS to Quickly Interpret• If you have a

NonTechnical Audience for your Plots, I suggest Sticking with 2D, Cartesian Plots

5060

7080

90100

200250

300350

400450

500550

600650

700

4.4%

4.6%

4.8%

5.0%

5.2%

5.4%

5.6%

5.8%

6.0%

6.2%

6.4%

6.6%

6.8%

Bubbler Temperature (°C)

Chamber Pressure (Torr)

TEOS Bubbler Vapor Generator Temp Sensitivity

file =VapGen_T-P_Sens.xls



All Done for Today

Excel Plot:BubblerOutPut

60

65

70

75

80

85

200

250

300

350

400

450

500

550

600

650

700

0

25

50

75

100

125

150

175

200

225

250

275

300

BubblerTemperature (°C)HeadSpace

Pressure (Torr)

TEOS Bubbler Vapor Output (1 slpm Carrier N2)

file =VapGen_T-P_Sens.xls



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Appendix 6972 23 xxxxf



Interactive-1

Starting Commands

>> delY_mm = [0, 2, 4, 4.5, 5.5, 6, 6.5, 8, 9, 11];>> t_min = [0, 2, 4, 6, 9, 12, 15, 18, 21, 24];>> plot(t_min, delY_mm)



Interactive-2



Interactive-3



Interactive-4



Interactive-5



Interactive-6



Interactive-7



Interactive-8

Activate the FIGURE PALETTE

DoubleClick



Interactive-9



Interactive 10

Activate Axis-Title Format Box by Double-Clicking the Title



Interactive-11 Change the

Plot BackGround Color to Match the PowerPoint BackGround



Interactive 12



Interactive 13



COPY FIGURE result

0 5 10 15 20 250

2

4

6

8

10

12

Load Application Time (minutes)

Ver

tical

Def

lect

ion

(mm

)

Polystrene Cantilever Beam Creep-Test

931 mN LoadSignificant "Kink"



WJ’s Patented Bubbler

C. C. Collins, M. A. Richie, F. F. Walker, B. C. Goodrich, L. B. Campbell, “Liquid Source Bubbler”, United States Patent 5,078,922 (Jan 1992)



WJ Bubbler Design

Schematic diagram of a the WJ chemical vapor generating bubbler system used in CVD applications. Note the use of the dilution MFC to maintain constant mass flow in the output line. An automatic temperature controller sets the electric heater power level

Cut-away view of a WJ chemical source vapor bubbler. The bubbler features a total internal volume of 0.95 liters, and a 25 mm thick isothermal mass jacket with an exterior diameter of 180 mm.



Graphics from [Mayer96]



Why Plot?

Engineering, Math, and Science are QUANTITATIVE Endeavors, we want NUMBERS as Well as Words

Many times we Need to• Understand The (functional) relationship

between two or More Variables• Compare the Values of MANY Data sets



Sys3 2X200 MultiBlok, 997671 250-13.8 PreWeld Pi Tube-1

0

25

50

75

100

125

150

175

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Hole Number (1 = closest to Manifold Block)

Ind

ivid

ual

Ho

le

P (

10X

To

rr)

DNS Tube-1 BMayer Tube1

DNS Normalized BMayer Normalized

PARAMETERS• For Single Tube Manifold• Flow = ??/0.24 slpm/hole• Exh to Atm Pressure (~750Torr)• Test Engr = DNStoddard, BMayer• Test Date = 09Mar00/10Mar

file = HbH997671PreW09Mar00.xls

Plot Title

Axis Title

Tic Mark

Tic Mark Label

Lege

nd

Data Symbol

Annot

atio

ns

Axis UNITS Connecting Line



0 5 10 15 20 250

2

4

6

8

10

12


Ver

tica

l Def

lect

ion

(m

m)

Polystyrene Creep Test

931 mN Load

Significant Kink



0 5 10 15 20 250

2

4

6

8

10

12


Ver

tical

Def

lect

ion

(mm

)

Polystyrene Beam Creep

931 mN Load

Significant "Kink"

[email protected] engr-25_plot_model-3.ppt 1 bruce mayer, pe engineering/math/physics 25:...

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