2.03 Rate of Temperature Change (Heating)

This mathematical problem has been solved by Alexander Bondarenko

https://www.slideshare.net/AlexanderBondarenko17/03-rate-of-temperature-change-heating

THE COMPLETE, VERIFIED VERSION OF THIS ARTICLE HAS BEEN PUBLISHED AS A BOOK "MATHEMATICAL MODELING OF A PYROELECTRIC DETECTOR" FREE AVAILABLE ON 
I HAVE DEVELOPED THE SIMULATOR OF A PYROELECTRIC DETECTOR
VIDEOS OF WHICH ARE FREE AVAILABLE ON
lIKE THE VIDEOS. SUBSCRIBE THE CHANNEL. SHARE IT WITH YOUR FRIENDS.
FOR ANY QUESTIONS, PLEASE, CONTACT ME ON LINKEDIN
MY PERSONAL SITE IS




THIS SITE IS NO LONGER SUPPORTED.

In accordance with the logical data model, we set up
an equation for the law of conservation of energy 
(1)

with

(2)

being the amount of energy coming to the detector from a heat source;

(3)

being the amount of energy coming to the detector from the environment;

(4)

being the amount of energy stored in the volume of the sensitive element;

(5)

being the amount of energy going out of the sensitive element to the environment.

Taking into account the parameters determined in equations 2-5, we set up an equation for power for which one has to divide each parameter by dt squared

(6)

and

(7)

and

(8)

and

(9)

and

(10)

We extract static terms from the family of equations 7-10

(11)

Equation 11 is a static model. We extract equation 11 from the family of Equations 7-10

(12)

Equation 12 is a dynamic model. We find the proper unit of measurement for which one has to divide both left and right terms of equation 12 by G2T/CT


(13)

We multiply and divide both left and right terms by corresponding variables


(14)

We replace relative increments with variables of input, disturbance, and output


We insert the variables obtained above into equation 14

(15)

We divide both left and right terms by the variable that determines the unit of measurement

(16)

We apply the Laplace Transform to the variables determined above


We replace the static terms with static coefficients


(17)

We write an equation for the law of conservation of energy in terms of the Laplace variables


(18)

We write the transfer functions for the input output” channel

(19)

and “disturbance output” channel

(20)

We replace the Laplace variable s with jω


We apply the replaced variable jω to equations 19 and 20

(21)

with

(22)

being the numerator for equations 19 and 20;

(23)

being the denominator for equations 19 and 20. We define the corresponding terms R, I, M and N

   

We define the amplitude-frequency response

(24)

or

(25)


 
 Fig. 1 Amplitude-frequency response (τT=0.159 s)

This image has been created by Alexander Bondarenko 

with LibreOffice Calc ver. 5.2


We multiply the numerator and denominator of equation 23 by a conjugate M(ω)-jN(ω)

(26)

with

(27)

and

(28)

We define the phase response

(29)


 Fig. 2 Phase response (τT=0.159 s)

This image has been created by Alexander Bondarenko 

with LibreOffice Calc ver. 5.2

We apply the Inverse Laplace Transform to equations 21 and 22 for which the former and the latter must be divided by s

(30)

In equation 32, we put the τT outside the brackets

(31)

As can be seen, the pole is s1=-1/τT

(32)

We write the function f(t) as

(33)

Once equation 35 has been multiplied by Ψ, the rate of temperature change of the sensitive element when heated, as a function of time equals

  (34)


  Fig. 3 Transient response (τT=0.159 s)

This image has been created by Alexander Bondarenko 

with LibreOffice Calc ver. 5.2

THIS ARTICLE HAS BEEN VERIFIED BY

the world's first 
team of students
at the Department 
of Design Automation
at Kharkiv National University of Radioelectronics, 
May, 2017

EGOROV DENIS ALEKSANDROVICH
email: 
denthejoker()gmail.com

ORLIK VALENTIN VIKTOROVICH
email:
vin4eester()gmail.com

TENDITNYK KATERYNA
email:
katetenditnyk()
gmail.com

BUBLYK OLEKSANDRA
email:
sashkabublik95()
gmail.com

OLIINYK OLEKSANDR
email:
sasha.oleinik96()
gmail.com

MAKSYM ZALOZNYI
email:
maksim.zaloznuy()
gmail.com

SEMENIKHIN
DMYTRYI
email:
grows2line()gmail.com
TEAM LEAD