Log Interactive 3D - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

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Engr/Math/Physics 25

Chp5 MATLAB

Plots & 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

Logarithmic Plots

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

(^00 10 20 30 40 50 60 70 80 90 )

5

10

15

20

25

30

35

x

y

1010 -2-1 100 101 102

10 -

100

101

102

x

y

**>> x = [0:0.1:100];

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

( ) ( 2 )^2

2 2 2 1 0. 1

1001 0. 01 0. 02 x x

y x x − +

= − +

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

 Rectilinear Plot  Log-Log Plot

• LogLog Plot is MUCH More Revealing

Making Logarithmic Plots

• Important Points to Remember
  1. 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 log 10 (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 10 2 = 100.

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 (^ ) ( 2 )^2

2 2 2 1 0. 1

1001 0. 01 0. 02 x x

y x x − +

= − +

 x → log; y → linear  x → linear; y → log

100 -1 100 101 10

5

10

15

20

25

30

35

x

y

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

10 -2 0 10 20 30 40 50 60 70 80 90 10

10 -

100

101

102

x

y

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

Example  Low Pass Filter Plot

• Recall the Mag of G

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

 Thus ω = 1/ τ = 9671 rad/s ≈ 10 4 rad/s

( )

( )^2

2

1

1

1

( ) | |^1

ωτ

ω

ω

=

= = RC

M Gv

 Thus Make a log-log Plot for M (called a “Bode” Plot) with the Domain

• 102 ≤ ω ≤ 106

70. 7 % 2

1 1 1

1

1 9671103. 4

9671 1

2

2

= =

=

 

  

+

= S S

M μ

Low Pass Filter Plot

102 103 104 105 106 10 -

10 -

10 -

100

Angular Frequency, w (rad/sec)

Voltage Gain (unitless

Bode Plot for RC LowPass Filter

4.7 kΩ 22 nF

1% left at 10 6

70.7% left at ω = 1/τ

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

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

• Recall the Sagging Cantilever Beam

 Go From This  To This

• Plot Sag vs Time Using Interactive to

(^00 5 10 15 20 )

2

4

6

8

10

12

(^00 5 10 15 20 )

2

4

6

8

10

12

Load Application Time (minutes)

Vertical Deflection (mm)

Polystrene Cantilever Beam Creep-Test

Significant "Kink" 931 mN Load

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 N 2 , QN2 , and the Chem Vapor, Qv

Bubbler-OutPut Physics

• The Details of Bubbler Operation Found in

 Chemical Vapor Output

 

  

 −

= hs v

v v N P P

P

• P (^) hs ≡ Absolute Pressure in Q^ Q 2 Bubbler HeadSpace
• P (^) v = Thermodynamic Vapor Pressure

Bubbler Physics cont

• In many cases C ≈ 0
• In This Case The Antione Eqn Reduces to the Clapeyron Eqn^2

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

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

ln ( Pv ) = A − B T

 Thus Pv (T)

B T

A B T

A B T v

De

e e

P e

−

−

−

=

= ⋅

=

 

  

 −

= (^) −

− B T hs

B T v N P De

De Q Q /

/ 2  or ( ) (^) B T hs

B T o hs N

v P De

De Q P T 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.
• B = 5562.30 Kelvins