Wednesday, April 26, 2017

HP Prime and TI-84 Plus: Basic Wheatstone Full Bridge Circuit

HP Prime and TI-84 Plus:  Basic Wheatstone Full Bridge Circuit

The program WHEATSTONE (HP Prime) and WHTSTONE (TI-84 Plus CE) deals with the full Wheatstone circuit:



Formulas

EA = E * (1 – R4/(R4 + R3))
EB = E * (1 – R1/(R1 + R2))
eO = EA – EB

Variables used:  A = R1, B = R2, C = R3, D = R4, L = EA, R = EB, O = eO

Resistance is measured in ohms (Ω), current in volts (V).

HP Prime Program WHEATSTONE

EXPORT WHEATSTONE()
BEGIN
// 2017-04-25 EWS

LOCAL K,A,B,C,D,L,R,O;

INPUT({A,B,C,D,E},
"Basic Wheatstone",
{"R1: ","R2: ","R3: ","R4: ",
"E: "});
L:=E*(1-D/(D+C));
R:=E*(1-A/(A+B));
O:=L-R;
PRINT();
PRINT("EA: "+L);
PRINT("EB: "+R);
PRINT("e0: "+O);
END;

TI 84 Plus CE Program WHTSTONE

"2017-04-25 EWS"
Input "R1: ",A
Input "R2: ",B
Input "R3: ",C
Input "R4: ",D
Input "E: ",E
E*(1-D/(D+C))→L
E*(1-A/(A+B))→R
L-R→O
Disp "EA: ",L
Disp "EB: ",R
Disp "EO: ",O

Example

Inputs:  R1:  200 Ω, R2:  150 Ω, R3:  180 Ω, R4:  200 Ω, E:  16 V

Results:   EA ≈ 7.57895 V, EB ≈ 6.85714 V, eo ≈ 0.742180 V

Source:

Strain Gages and Instruments.  Tech Note TN-514  Vishay Precision Group.  February 1, 2013.  Link:  http://www.vishaypg.com/docs/11064/tn514.pdfhttp://www.vishaypg.com/docs/11064/tn514.pdf

Note:

I will be taking some time off from blogging in early May. I am going to work on a presentation that I hope to present in HHC 2017 this September (link: http://hhuc.us/2017/ ).  I encourage anyone to come attend the HHC conferences, they are a lot of fun.  This year the conference will be in Nashville.

Eddie

This blog is property of Edward Shore, 2017



Sunday, April 23, 2017

HP Prime and TI-84 Plus CE: Shallow Wave Analysis

HP Prime and TI-84 Plus CE:  Shallow Wave Analysis

Introduction



The program H2OWAVES calculates wave speed, impedance, and wave flux for shallow ocean waves.  The following are assumed:

1. The criteria of λ < D/20 is assumed where λ is the length of the wave.
2. The water assumed to be 0°C, where the density (ρ) is 1,000 kg/m. 
3. SI units are used.  For gravity, g = 9.80665 m/s^2 is used.

Formulas Used

Wave Speed (m/s):  v = √(g * D)
Wave Impedance (Mks):  Z = ρ * v
Wave Energy Flux (W/m):  I = (ρ * g * H)^2/(2 * Z)
Maximum Possible Length (m):  λ = 20 * D

HP Prime Program H2OWAVES

EXPORT H2OWAVES()
BEGIN
// EWS 2017-04-21
// Shallow Wave Analysis
// for D/L<1/20
// SI Units

LOCAL D,H;
LOCAL v,Z,I;

INPUT({D,H},
"Wave Analysis",
{"D: ","H: "},
{"Depth (m)","Wave Height (m)"});

v:=√(9.80665*D);
Z:=1000*v;
I:=(9806.65*H)^2/(2*Z);

PRINT();
PRINT("Wave speed (m/s): "+v);
PRINT("Impedance (Mks): "+Z);
PRINT("Wave Flux (W/m): "+I);

RETURN {v,Z,I};

END;

TI-84 Plus CE Program H2OWAVES

"EWS 2017-04-23"
"SHALLOW WAVES"
"D<F/20"
Disp "SHALLOW WAVES"
Input "DEPTH (M): ",D
Input "HEIGHT (M): ",H
√(9.80665*D)→V
1000*V→Z
(9806.65*H)^2/(2*Z)→I
Disp "WAVE SPEED (M/S):",V
Disp "IMPEDANCE (WKS):",Z
Disp "WAVE FLUX (W/M):",I

Example

Input: 
Depth:  3.2 m
Height:  0.49 m

Output:
Wave Speed: 5.601899678 m/s
Impedance:  5601.899678 Wks
Wave Flux:  2060.953478 W/m

Source:

Ingard, K.U. Fundamental of Waves and Oscillations Cambridge University Press:  New York 1988.  IBSN 0 521 32734

Surf’s up!  Eddie


This blog is property of Edward Shore, 2017.

Wednesday, April 19, 2017

HP Prime and TI-84 Plus CE: Simple Logistic Regression

HP Prime and TI-84 Plus CE:  Simple Logistic Regression

The program SIMPLOGI attempts to fit two lists of data (X, Y) to the equation:

y = 1 / (A + B*e^(-x))

by using the translation:  X’ = e^-X and Y’ = 1/Y and performing linear regression analysis on X’ and Y’.   This is good for all data except when y = 0.

HP Prime Program SIMPLOGI

EXPORT SIMPLOGI(L1,L2)
BEGIN
// EWS 2017-04-18

LOCAL S:=SIZE(L1);
LOCAL L0:=MAKELIST(1,X,1,S);
L1:=e^(−L1);
L2:=1/L2;

LOCAL M1,M2,M3;
M1:=list2mat(CONCAT(L0,L1),S);
M1:=TRN(M1);
M2:=list2mat(L2,S);
M2:=TRN(M2);
M3:=CAS.LSQ(M1,M2);


RETURN {"Y=1/(A+Be^(−X))",M3};

END;

TI-84 Plus CE Program SIMPLOGI

Disp "LOGISTIC FIT"
Disp "Y=1/(A+B*e^(­X))"
Input "X: ",L
e^(­L)→L
Input "Y: ",L
1/L→L
LinReg(a+bx) L,L
Disp "A=",a
Disp "B=",b

Example

Data:  (X’ and Y’ are provided for reference)

X
Y
X’ = e^(-X)
Y’ = 1/Y
0.5
0.384
0.6065306597
2.604166667
1.0
0.422
0.3678794412
2.369668246
1.5
0.450
0.2231301601
2.222222222
2.0
0.468
0.1353352832
2.136752137
2.5
0.480
0.0820849986
2.083333333

X = {0.5, 1, 1.5, 2, 2.5}, Y = {0.384, 0.422, 0.45, 0.468, 048}

Results (Matrix [ [ A ]. [B ] ]):

A = 2.00185929
B = 0.9942654005

Hence y = 1 / (2.00185929 + 0.9942654005*e^(-x))

Eddie



This blog is property of Edward Shore, 2017

Wednesday, April 12, 2017

Approximating y = sin x (0 to 90 degrees, 0 to pi/2 radians)



Approximating y = sin x (0 to 90 degrees, 0 to pi/2 radians)


Calculation

I used the HP Prime to fit polynomials to data by using the vandermonde and LSQ commands.

vandermonde(vector):  creates the matrix consisting of rows [ 1, (n_i), (n_i)^2, (n_i)^3, …, (n_i)^(n-1) ] where the vector has n elements for each i.  Available from the Math-Matrix-Create submenu.

LSQ(X, y):  returns the coefficients [ [a_0], [a_1], [a_2], … , [a_n] ], the minimum norm least squares vector from the system X*a=y.  In this case, the vector a can also be estimated by the operation (X^T X)^-1 X^T y.  Available from the Math-Matrix-Factorize submenu.

Case 1:  Four Points of Data

Fit cubic polynomial to the following data.  Remember we are working with the range of 0° ≤ x ≤ 90° or 0 ≤ x ≤ π/2 radians

X
sin(x)
0
30°
1/2 = 0.5
60°
√3/2 ≈ 0.866025403785
90°
1

Example 1:  Full precision approximation

y = 1.78098409219E-2 x – 1.99435471445E-5 x^2 – 6.05408712068E-7 x^3

Note:  E-n stands for 10^(-n).

Absolute maximum error, data measured in 5° tick marks:  0.002371558429
Absolute maximum error, data measured in 1° tick marks:  0.002392465593



Example 2:  If we round all coefficients to six digits, we get:

y = 0.017810 x – 0.000020 x^2 – 0.000001 x^3

Absolute maximum error, data measured in 5° tick marks:  0.2881 

In comparison between Example 1 and Example 2, keeping the decimal places makes a big difference. 



Example 3:  Let’s see if we do better with using radians instead of degrees.

X
sin(x)
0
0
π/6
1/2 = 0.5
π/3
√3/2 ≈ 0.866025403785
π/2
1

y = 1.02042871863 x – 0.065470803224 x^2 – 0.113871899065 x^3

Maximum absolute error, data measured in π/36 radian tick marks: 0.002371558428
Maximum absolute error, data measured in π/180 radian tick marks: 0.002392465661

The maximum error in Example 3 matched Example 1.

Example 4:  Round the coefficients to six digits:

y = 1.020429 x – 0.065471 x^2 – 0.113872 x^3

Maximum absolute error, data measured in π/36 radian tick marks:  0.002371292924
(2 decimal places)

The difference between Example 2 and Example 4 is that less information is “lost” when rounding the coefficients to a set amount of digits, in this case, 6.

Case 2:  Seven Points of Data

x (degrees)
x (radians)
sin(x)
0
0
15°
π/12
(√6 - √2)/4 ≈ 0.258819045102
30°
π/6
1/2 = 0.5
45°
π/4
√2/2 ≈ 0.707106781185
60°
π/3
√3/2 ≈ 0.866025403785
75°
5π/12
(√6 + √2)/4 ≈ 0.965925826288
90°
π/2
1

Example 5:  Full Precision – Degrees

y = 1.74539026131E-2 x – 1.00636910048E-7 x^2 – 8.79821584089E-7 x^3 – 1.93973162632E-10 x^4 + 1.66950110845E-11 x^5 – 2.72879457224E-14 x^6

Absolute maximum error, data measured in 5° tick marks:  1.2072093E-6
(pretty darn good)

Example 6:  Full Precision – Radians

y = -6.98792887569E-12 + 1.0000349642 x – 3.30435720904E-4 x^2 – 0.165486304503 x^3 – 2.09062241321E-3 x^4 + 1.03087220879E-2 x^5  - 9.65423458482E-4 x^6

Maximum absolute error, data measured in π/36 radian tick marks:  0.00049429473
(3 decimal places)

I think the degrees approximation wins.  But what if we round the radians version (Example 6) to 6 decimal places?

Example 7:  6 Decimal Places – Radians

y = 1.000035 x – 0.00033 x^2 – 0.165486 x^3 – 0.002091 x^4 + 0.010309 x^5 – 0.000965 x^6

Maximum absolute error, data measured in π/36 radian tick marks:  9.03153E-6

Interesting result here, and a nice one at that!



Eddie

This blog is property of Edward Shore, 2017

Next Week... and Plans for October 2017

I'm so excited, can't want for next week's HHC 2017 calculator conference in Nashville!  It is my annual calculator conference ...