Implementation of a Feedback I²-Controlled Constant Temperature Environment Temperature Meter

Introduction

Figure 1. Feedback Control System

Formulation

Static analysis of the feedback system

Using the same methodology described for feedback I²-controlled constant solar radiation meter [1], and applying to environment temperature meter, measure variable could be expressed as:

x = I_s² = R_s⁻¹ hS(T_s − T_a)

(1)

With Rs being the sensor resistance, h the sensor heat transfer coefficient, S the sensor superficial area, T_s the sensor temperature and T_a the environment temperature.

Exciting sensor with PWM current pulse, equation (1) could be written as [3],

(2)

With T₁ the modulated pulse width, T the system control period and I_m the PWM current pulse amplitude.

Linear relationship between pulse width variations, the measure variable, and environment temperature, the variable to be estimated, was obtained [1,2,3].

Model of small signal of thermo-resistive sensor

The dynamic thermal equilibrium equation of sensor, (NTC thermo-resistive sensor), was linearized around a quiescent point (R_so, T_so). Laplace transformer was used to obtain sensor small signal transfer function [1]:

(3)

In which,

With k_x being the function gain, τ_a the sensor apparent time constant and k_t the temperature coefficient.

Controller transfer function

PI controller was used to obtain a feedback system with a response time less then sensor constant time and s-domain function transfer is given by equation (4).

(4)

Model of small signal of a thermo-resistive sensor-based feedback environment temperature meter

Feedback system scheme can be observed in Figure (2).

Figure 2. Analog feedback system

Using equations (3) and (4) we could get system transfer function using pole-zero cancellation:

System transfer function had a pole at − kx.Ki of a time constant of

Feedback system constant time, τ_sr, was chosen to be much less than sensor small model apparent constant time, τ_a , and much less than sensor intrinsic constant time, τ.

Conclusion

Experimental

General

Controller system was implemented in FPGA, Figure (3), to permit higher working frequencies, and because digital dedicated systems improves performance to real time systems when compared to micro-processed systems. Numeric PI controller was based on analog PI controller and uses the same principle.

Figure 3. System implementation with FPGA

Thermo-resistive sensor transient response to a change on reference level is shown in Figure (4). Sensor temperature operation point was defined at T_so = 74 °C and it was suddenly changed to 80 °C. Sometime later to 68 °C and sometime later returned to original level of 74 °C. In this figure it can be observed that measured values converging to simulating references levels.

The linear relationship between environment temperature and measure variable can be observed in Figure (5). A linear approximation was realized and maximum difference between measured pulse width duration and straight line was equal to -2.2358.

Project Data

For a NTC thermo-resistive sensor with A = 5.9936e-2 Ω, B = 3423.3 K, ( ), hS = 7.011e- 4 W/K, mc = 1.4092e-3 J/K, τ = 2.01 s, sensor analog resistance reference R_so = 1149 Ω, sensor analog temperature reference T_so = 74 °C, I_so = 5.5733 mA, I_mo= 7.3552 mA, k_t= -3.0637e-2 K/Ω, k_x = - 2.2908e7 Ω/A², τ_a= 0.861s, τ_sr = 0.1 s, analog PI controller constants K_p = -3.7575e-7 A²/Ω and K_i = - 4.3652e-7 A²/(Ω.s), numeric PWM resolution n_PWM = 10 bits, Analog/Digital resolution n_CAD= 12 bits, numeric PI controller constants K_pD= -4 and K_iD = -5, numeric reference y_rD = 1762, Clock = 1 MHz, T = 1.024 ms, T_1o = 50% T.

Figure 4. Thermo-resistive Sensor Response Time to a Sudden Change on Reference

Figure 5. Linear Relationship Between Measure Variable (Pulse Duration) and Environment Temperature

References and Notes

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