PT100 newton_algorithm.pdf

Solving Nonlinear Equations

with Newton’s Algorithm

- 1 -

Application Note

Solving Nonlinear Equations with Newton’s Algorithm

Example: Temperature measurement with resistive platinum sensor Pt100

1. Introduction

Solving nonlinear equations is a difficult task in general. Unless the solution can be expressed in analytical form one

must solve the equations numerically. Even if the solution is given as an expression (e.g. the roots of a quadratic

polynomial) this expression contains nonlinear functions that are not readily available on standard microprocessors and

microcontrollers. Consider the roots of a quadratic polynomial ax 2 +bx+c=0. Although the solution is given by the famous

formula x 1,2 = (-b ± (b 2 – 4ac) 1/2 )/2a, it contains a square root operation that must be implemented in software.

We will show in this application note how solutions of nonlinear equations can be obtained using Newton’s algorithm that

uses only elementary operations like addition, multiplication and division. It is always a good idea to try the Newton

algorithm to solve nonlinear equations instead of trying hard to get an analytical expression that contains non-elementary

operations that must be implemented in software.

2. Newton Algorithm

The Newton algorithm finds a zero x*, f(x*)=0 of a function f(x) by iteratively improving an initial guess. The algorithm is

applicable to functions of one or several variables. For the sake of simplicity we will concentrate here on functions with

one variable only. The general case can be found in text books on numerical mathematics.

Figure 1: How the Newton algorithm works. Starting from an initial guess x 0 , the algorithm makes a step towards the solution by finding

the intersection of the tangent in point x 0 with the x-axis. This intersection, labeled x 1 , is an improved guess of the real solution x*. By

repeated application of this procedure the algorithm eventually converges to the solution x*.

How the Newton algorithm works is shown in Figure 1 . Starting from an initial guess x 0 , the algorithm makes a step

towards the solution by finding the intersection of the tangent in point x 0 with the x-axis. This intersection, labeled x 1 , is

an improved guess of the real solution x*. We can now repeat the procedure using the tangent at x 1 to find x 2 . By

repeated application of this procedure the algorithm eventually converges to the solution x*. The choice of a suitable

initial guess is very important for fast convergence or convergence to the right solution at all. The convergence order of

ErSt Electronic GmbH

Aeschstrasse 171

CH-8123 Ebmatingen

Switzerland

Phone +41 44 980 61 44

Fax

+41 44 980 61 30

Internet info@erst.ch

http://www.erst.biz

http://www.erst.ch

Solving Nonlinear Equations

with Newton’s Algorithm

- 2 -

the Newton algorithm is 2, meaning a doubling of significant binary digits with each iteration. Usually a few iteration

cycles will be sufficient for practical purposes if the initial guess is within 10% of the final solution.

The general formulation of the Newton algorithm for finding a zero of a function of one variable is given in Figure 2

below.

Newton Algorithm

1) Make an initial guess x 0

2) Given an n-th guess x n , calculate an improved guess

x n+1 by setting

x n+1 = x n – f(x n ) / f’(x n )

3) Repeat step 2 until x n+1 meets a predefined error

tolerance, e.g. |x n+1 – x n | < 10 -8

Figure 2: Newton algorithm for finding a zero of a function of one variable.

3. Temperature Measurement with Pt100

The usual Pt100 temperature sensor has a nominal resistance of R(0) = 100Ȫ at zero degrees Celsius. The resistance

depends nonlinearly on the temperature. Over the full measurement range of -200 … +850 o C the sensor characteristic

can be expressed with two polynomials of 2 nd and 4 th order:

For the range 0 … +850 o C we have the equation

R(T) = R(0) * (1 + A*T - B*T 2 )

and for the range -200 … 0 o C we have

R(T) = R(0) * (1 + A*T - B*T 2 - C*(T - 100)*T 3 )

with the constants

A = 3.90802e-3 o C -1

B = 0.580195e-6 o C -1

C = 4.27350e-12 o C -1

T is the temperature in degree Celsius and R(T) the resistance of the sensor at temperature T. R(0) is the resistance of

the sensor at zero degrees Celsius, i.e. 100Ȫ for Pt100.

3.1. Application of the Newton Algorithm

Our task is to calculate the temperature for a given resistance, i.e. given R(T) we want to know T. As we have seen in the

discussion of the Newton algorithm we have to reformulate the problem such that we want to find the zeros of a function.

We can easily do that by setting

f(T) = R(T) – R measured

and finding T such that f(T)=0.

The Newton algorithm needs the derivative, f’(T) = df/dT, of f(T) with respect to T. For the range 0 … +850 o C we get the

equation

f’(T) = df/dT = dR(T)/dT = R(0)*(A – 2*B*T)

and for the range -200 … 0 o C we get

f’(T) = df/dT = dR(T)/dT = R(0)*(A – 2*B*T +300*C*T 2 - 4*C*T 3 )

ErSt Electronic GmbH

Aeschstrasse 171

CH-8123 Ebmatingen

Switzerland

Phone +41 44 980 61 44

Fax

+41 44 980 61 30

Internet info@erst.ch

http://www.erst.biz

http://www.erst.ch

Solving Nonlinear Equations

with Newton’s Algorithm

- 3 -

with the constants A, B and C as above.

For the initial guess T 0 we use the linear approximation

R(T) = R(0)*(1 + A*T)

and get

T 0 = (R measured / R(0) – 1) / A

Starting with n=0 we can now apply the iterative process from Figure 2 to get better approximations for the measurement

temperature:

T n+1 = T n – f(T n ) / f’(T n )

The data in the following two tables was created with the C code program in the Appendix (see Figure 3). They show the

power of the Newton algorithm. Starting with a given temperature the Pt100 resistance R(T) is calculated. Then an initial

guess of the temperature using the linear approximation is made. Successive iterations converge rapidly to the solution.

3.2. Demonstration of the Newton algorithm for T in -200...0 degrees

Given temperature Calculated R(T)

Initial guess T 0

(linear approximation)

1st iteration

T 1

2nd iteration

T 2

3rd iteration

T 3

-200.000000

18.493180

-208.562955

-200.032435

-200.000000

-150.000000

39.713685

-154.263068

-150.005979

-150.000000

-100.000000

60.254135

-101.703331

-100.000695

-100.000000

-50.000000

80.306838

-50.391660

-50.000027

-50.000000

-10.000000

96.086131

-10.014967

-10.000000

0.000000

100.000000

0.000000

Table 1: Demonstration of the Newton algorithm for temperatures below zero. Starting with a given temperature we calculate the Pt100

resistance R(T). We then make an initial guess of the temperature using the linear approximation. Successive iterations converge

rapidly to the solution.

3.3. Demonstration of the Newton algorithm for T in 0...+850 degrees

Given temperature Calculated R(T)

Initial guess T 0

(linear approximation)

1st iteration

T 1

2nd iteration

T 2

3rd iteration

T 3

20.000000

107.792832

19.940615

19.999999

20.000000

100.000000

138.500005

98.515374

99.999663

100.000000

200.000000

175.839620

194.061494

199.994444

200.000000

300.000000

212.018845

286.638362

299.971029

300.000000

400.000000

247.037680

376.245976

399.905694

399.999999

400.000000

500.000000

280.896125

462.884338

499.762894

499.999990

500.000000

600.000000

313.594180

546.553447

599.493755

599.999954

600.000000

700.000000

345.131845

627.253302

699.034502

699.999825

700.000000

800.000000

375.509120

704.983905

798.304823

799.999441

800.000000

Table 2: Demonstration of the Newton algorithm for temperatures above zero. Starting with a given temperature we calculate the

Pt100 resistance R(T). We then make an initial guess of the temperature using the linear approximation. Successive iterations

converge rapidly to the solution.

4. Conclusion

The iterative Newton algorithm to find a zero of a function of one variable is both fast and easy to implement. It makes

use of only elementary operations (addition, multiplication and division) and evaluations of the function and its first

derivative. The algorithm is well suited for implementations on standard microprocessors and microcontrollers. As an

example we applied the algorithm to the calculation of the measurement temperature from the resistance of a Pt100

sensor. The simplicity and accuracy of the Newton algorithm is clearly evident from the examples (see Table 1 and Table

2 above).

ErSt Electronic GmbH

Aeschstrasse 171

CH-8123 Ebmatingen

Switzerland

Phone +41 44 980 61 44

Fax

+41 44 980 61 30

Internet info@erst.ch

http://www.erst.biz

http://www.erst.ch

Solving Nonlinear Equations

with Newton’s Algorithm

- 4 -

5. Appendix: C Code Program that demonstrates the Newton algorithm

/* Demonstration of the application of the Newton algorithm to calculate

* the temperature from the measured resistance of a Pt100 sensor.

#include <stdio.h>

/* Constants for temperature dependence of resistance of Pt100 sensor */

#define A 3.90802e-3

#define B 0.580195e-6

#define C 4.27350e-12

#define R0 100

/* Pt100 resistance for the range 0 ... +850 degrees Celsius */

double pt100_above_zero ( double T) {

return R0*(1 + T*(A - B*T));

}

/* Pt100 resistance for the range -200 ... 0 degrees Celsius */

double pt100_below_zero ( double T) {

return R0*(1 + T*(A + T*(-B + T*(100*C - C*T))));

}

/* Derivative of Pt100 resistance for the range 0 ... +850 degrees Celsius */

double derivative_above_zero ( double T) {

return R0*(A - 2*B*T);

}

/* Derivative of Pt100 resistance for the range -200 ... 0 degrees Celsius */

double derivative_below_zero ( double T) {

return R0*(A + T*(-2*B + T*(300*C - 4*C*T)));

}

/* Initial guess of measurement temperature */

double initial_guess ( double R) {

return (R / R0 - 1) / A;

}

/* Newton algorithm step for the range 0 ... +850 degrees Celsius */

double newton_step_above_zero ( double temp, double rmeas) {

return temp - (pt100_above_zero(temp) - rmeas)/derivative_above_zero(temp);

}

/* Newton algorithm step for the range -200 ... 0 degrees Celsius */

double newton_step_below_zero ( double temp, double rmeas) {

return temp - (pt100_below_zero(temp) - rmeas)/derivative_below_zero(temp);

}

/* Demonstrates the application of the newton algorithm for temperatures

* above zero.

void demo_above_zero ( double temp) {

double rmeas, t0, t1, t2, t3;

/* Calculate Pt100 resistance at given temperature */

rmeas = pt100_above_zero(temp);

printf("Given temperature: %f\n", temp);

printf("R(T): %f\n", rmeas);

/* Calculate Pt100 resistance at given temperature */

rmeas = pt100_above_zero(temp);

/* Recalculate temperature from resistance */

t0 = initial_guess(rmeas);

printf ("Initial guess: %f\n", t0);

t1 = newton_step_above_zero (t0, rmeas);

printf ("1st iteration: %f\n", t1);

ErSt Electronic GmbH

Aeschstrasse 171

CH-8123 Ebmatingen

Switzerland

Phone +41 44 980 61 44

Fax

+41 44 980 61 30

Internet info@erst.ch

http://www.erst.biz

http://www.erst.ch

Solving Nonlinear Equations

with Newton’s Algorithm

- 5 -

t2 = newton_step_above_zero (t1, rmeas);

printf ("2nd iteration: %f\n", t2);

t3 = newton_step_above_zero (t2, rmeas);

printf ("3rd iteration: %f\n", t3);

printf("\n");

}

/* Demonstrates the application of the newton algorithm for temperatures

* below zero.

void demo_below_zero ( double temp) {

double rmeas, t0, t1, t2, t3;

/* Calculate Pt100 resistance at given temperature */

rmeas = pt100_below_zero(temp);

printf("Given temperature: %f\n", temp);

printf("R(T): %f\n", rmeas);

/* Recalculate temperature from resistance */

t0 = initial_guess(rmeas);

printf ("Initial guess: %f\n", t0);

t1 = newton_step_below_zero (t0, rmeas);

printf ("1st iteration: %f\n", t1);

t2 = newton_step_below_zero (t1, rmeas);

printf ("2nd iteration: %f\n", t2);

t3 = newton_step_below_zero (t2, rmeas);

printf ("3rd iteration: %f\n", t3);

printf("\n");

}

void main() {

printf("Demonstration of the Newton algorithm for T in -200...0 degrees\n");

demo_below_zero(-200);

demo_below_zero(-150);

demo_below_zero(-100);

demo_below_zero(-50);

demo_below_zero(-10);

demo_below_zero(0);

printf("Demonstration of the Newton algorithm for T in 0...+850 degrees\n");

demo_above_zero(0);

demo_above_zero(20);

demo_above_zero(100);

demo_above_zero(200);

demo_above_zero(300);

demo_above_zero(400);

demo_above_zero(500);

demo_above_zero(600);

demo_above_zero(700);

demo_above_zero(800);

}

Figure 3: Example C code that demonstrates the application of the Newton algorithm to calculate the temperature from the measured

resistance of a Pt100 sensor.

ErSt Electronic GmbH

Aeschstrasse 171

CH-8123 Ebmatingen

Switzerland

Phone +41 44 980 61 44

Fax

+41 44 980 61 30

Internet info@erst.ch

http://www.erst.biz

http://www.erst.ch

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: