2.00 Introduction to the Mathematical Modelling of a Pyroelectric Detector

This mathematical problem has been solved by Alexander Bondarenko


GET THE 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.

   Time behavior characteristics can be static or dynamic. Static characteristics show relations between an input and an output. Dynamic characteristics explain a system behavior in response to a reference input and a disturbance input. When the transient process is over, a system comes to the steady state.

    For a pyroelectric detector, the reference input is a heat flow to be measured. The disturbance input can, for example, be an ambient temperature. If a pyroelectric detector is not well thermally isolated, the disturbance input will have a day-and-night drift. This drift is a measuring error. A pyroelectric detector is preferred to be thermally isolated from ambient temperature fluctuations. The detector itself is an uncooled device. It means that one has to choose an optimal temperature for high-megohm microelectronics integrated in a housing, Curie point of a pyroelectric material, and thermal noises to reduce. Besides, if possible, it would be desirable to keep the high-megohm electronics thermally isolated. The output is a signal to be researched. The thermal-to-electrical model of a pyroelectric detector includes four links connected in series where the output of a previous link is the input of a next. The primary pyroelectric effect can be described by the following. In the first link, the reference input is a heat flow to be measured, the output being a temperature change. The first link can be approximated by the first order lag. In the second link, the reference input is a heat flow to be measured, the output being the rate of temperature change. Due to the rate of temperature change being the first derivative of the temperature change of the sensitive element, the second link can be approximated by a derivative lag. In the third link, the reference input is the rate of temperature change, the output being a pyroelectric current. The third link can be approximated by a proportional lag. In the fourth link, the reference input is the pyroelectric current, the output being the output voltage. The fourth link can be approximated by the first order lag as well as the first. The tertiary pyroelectric effect has the same structure as the primary, except the second link. It happens due to the tertiary effect is proportional to the temperature change rather than the rate of temperature change [0].

    Both static and dynamic characteristics are based on the law of conservation of mass or energy. Due to a pyroelectric detector being an energy converter, we will use the law of conservation of energy. In the static mode, the quantity of energy coming to the sensitive element, equals the quantity of energy going out of it. In the dynamic mode, the change of quantity of energy coming to the sensitive element, equals the sum of the change of quantity of energy stored in it and the change of quantity of energy going out. Having combined the heat flows coming to the sensitive element as the reference and disturbance inputs, the quantity of energy coming to the sensitive element, equals the sum of quantity of energy stored in it and the quantity of energy going out. We set up an equation for the law of conservation of energy


  (1)

    The incoming infrared radiation changes the temperature of the sensitive element. Due to that, we can set up equations for both the reference input and the disturbance input in the form of the product of power of heat flow and time. We set up corresponding equations for the reference input

,
(2)

and the disturbance input

. (3)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(4)

    We set up an equation for the quantity of energy going out of the sensitive element

.
(5)

    When the sensitive element is heated, the heat flow is considered to be permanent, i.e. approximated by the unit step response function. The lack of heat flow, or when the sensitive element is no longer heated, can, probably, be approximated by some techniques, but the author has chosen the technique of differentiating. In the other words, if a heat flow that changes the temperature of the sensitive element, can be described by a constant value, the lack of heat flow can be approximated by the first time derivative of the constant value that will equal zero. Therefore, we can set up a rule: cooling is proportional to the first time derivative of heating; in order to keep the unit of measurement of energy and provide the mathematical function of the transient response continuous, the first time derivative of heating must be multiplied by the thermal time constant. We set up corresponding equations for the reference input

,
(6)

and the disturbance input

.
(7)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(8)

    We set up an equation for the quantity of energy going out of the sensitive element

.
(9)

    When the sensitive element is cooled, the quantity of energy stored in it, is no longer dependent on its heat capacity, but depends on the heat losses and the temperature at a particular time. The energy going out of the sensitive element, is proportional to the heat losses and the temperature at time squared, and inversely proportional to the thermal time constant which it is cooled with. The second power of said time provides the unit of measurement of energy.

    The rate of temperature change of the sensitive element when heated, is the first time derivative of its temperature change; when cooled, it is the second. We set up corresponding equations for the reference input

,
(10)

and the disturbance input

.
(11)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(12)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(13)

    For cooling, we set up corresponding equations for the reference input

,
(14)

and the disturbance input

.
(15)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(16)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(17)

    When the temperature of the sensitive element changes, the latter generates electrical charges that appear on its flat surfaces. In case of the primary pyroelectric effect, the quantity of charges is proportional to the rate of temperature change of the sensitive element. The tertiary pyroelectric effect is proportional to the temperature change of that. One has to keep in mind that these definitions can’t be mixed.

    We set up corresponding equations for the reference input

,
(18)

and the disturbance input

,
(19)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(20)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(21)

    Equation (20) shows that the quantity of thermal energy stored in the sensitive element, is proportional to its heat capacity, i.e. ability to store the thermal energy, and inversely proportional to the pyroelectric effect. It happens due to the pyroelectric effect “takes” only a part of thermal energy. If one considers the quantity of thermal energy, stored in the sensitive element, then the higher the value of a pyroelectric coefficient, the more thermal energy it takes from heat capacity, the less thermal energy remains there. An ideal pyroelectric detector is that the pyroelectric coefficient of which goes to infinity and, therefore, the thermal energy stored in the sensitive element falls down to zero. In this case, the pyroelectric coefficient converts all thermal energy to electrical with no losses. Later we will see that the part of energy taken by a pyroelectric effect is proportional to the output current or voltage and is useful.

    For cooling, we set up corresponding equations for the reference input

,
(22)

and the disturbance input

.
(23)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(24)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(25)

    The tertiary pyroelectric effect has the same properties as the primary.

    For heating, we set up corresponding equations for the reference input

,
(26)

and the disturbance input

.
(27)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(28)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(29)

    For cooling, we set up corresponding equations for the reference input

,
(30)

and the disturbance input

.
(31)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(32)

    We set up an equation for the quantity of thermal energy going out of the sensitive element

.
(33)

    When the sensitive element is connected to a high-megohm electronics, new parameters such as electrical capacitance and electrical resistance appear. In this case one has to take into account that there are two modes in which a pyroelectric detector can work. In the current mode, both electrical capacitance and electrical resistance are the feedback elements. In the voltage mode, the electrical capacitance is formed by the electrodes deposited on the sensitive element, the electrical resistance being a high-megohm resistor that is approximately two orders of reduction in the value than the active resistance of the sensitive element.

    The fourth link is the high-megohm electronics. It doesn’t take part in signal formation. The sign (positive or negative) is formed by the pyroelectric current depending on heating or cooling. By this reason, the mathematical model of a pyroelectric detector applied to the output voltage, is not considered separately for heating and cooling. 

    In case of the primary pyroelectric effect, we set up an equation for the law of conservation of energy

(34)

    We set up corresponding equations for the reference input

,
(35)

and the disturbance input

.
(36)

    We set up an equation for the quantity of energy that converts from thermal to electrical, being stored in the sensitive element

.
(37)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(38)

    We set up an equation for the quantity of electrical energy stored in the sensitive element (or feedback capacitance)

.
(39)

    We set up an equation for the quantity of electrical energy going out of the sensitive element

.
(40)

   In case of the tertiary pyroelectric effect, we set up an equation for the law of conservation of energy

(41)
    We set up corresponding equations for the reference input
,
(42)

and the disturbance input

.
(43)

    We set up an equation for the quantity of energy that converts from thermal to electrical, being stored in the sensitive element

.
(44)

    We set up an equation for the quantity of thermal energy stored in the sensitive element

.
(45)

    We set up an equation for the quantity of electrical energy stored in the sensitive element (or feedback capacitance)

.
(46)
    We set up an equation for the quantity of electrical energy going out of the sensitive element
.
(47)

    When all equations of the law of conservation of energy have been set up, we divide each part of a corresponding equation by a part that includes the highest power of time (dt). As a result, the energy balance equations become general nonlinear differential equations which can be further solved.

    In practice, it’s pretty hard to solve nonlinear differential equations. To remove this problem, one has to linearize them. Linearization can be performed by taking a Taylor series expansion around a constant value. The second order derivative and higher are not taken into account. The first order derivative terms are used only. If a constant term is a derivative, then it equals zero. The zero terms are used by the author in square brackets to make reading easier. Further, we take static terms out of the family of linear differential equations. For a pyroelectric detector, all static equations equal zero, except one for temperature change and one for tertiary pyroelectric effect. Both are true when the detector is heated. This can be treated by the presence of thermal current flowing through the sensitive element and that the tertiary pyroelectric current is proportional to the temperature change of the sensitive element. Dynamic equations can be obtained by extracting static terms from the family of linear differential equations. The dynamic equations exist for all processes in a pyroelectric detector. In order to provide a proper unit of measurement, each term of a linear differential equation must be divided by a term that includes the part of energy going out of the sensitive element. As a result, said term will have the unity gain.

    Now it’s time to set up the variables of input "x", disturbance "z" and output "y" (see Fig. 1). In order to achieve it, one has to multiply and divide increment of variable by a corresponding constant variable of the same unit of measurement. The relative increments obtained are replaced with said variables of input, disturbance and output. Each term of the linear differential equations must be divided by the term of energy that goes out of the sensitive element. The differential equations now become dimensionless.


 Fig. 1 Logical data model

This image has been created by Alexander Bondarenko 

with LibreOffice Draw ver. 5.2, Inkscape ver. 9.1

     Next, we apply the Laplace transformation to the variables of input, disturbance and output. The three now become the variables of s. Also, we replace the operator of differentiating d/dt with the Laplace operator s. One can replace the constant parameters with constant coefficients. Finally, we determine the time constants.

    After all variables and constants having been determined, we can set up an equation of the law of conservation of energy transformed by Laplace. This equation will give the transfer functions of interest.

    In order to determine frequency response of a pyroelectric detector, we replace the Laplace variable s with the frequency operator jω. The amplitude-frequency response can be found as square root out of R square omega plus I square omega divided by square root out of M square omega plus N square omega, with R being the real part of the numerator, I the imaginary part of the numerator, M the real part of the denominator, N the imaginary part of the denominator.

    In order to find the phase-frequency response, one has to multiply both the numerator and the denominator of a transient function by a conjugate of the denominator. The phase-frequency response can be found as inverse tangent of the imaginary part over the real part. Both amplitude-frequency response and phase-frequency response are plotted in the logarithmic scale.

    Transient responses can be found by applying the Inverse Laplace transformation to the transient functions. We just simply divide a transfer function by the Laplace operator s. If the denominator includes the common monomial factor s, then, through author’s personal experience, one can resolve such a transient function into partial fractions. This will let us find the residues one of which always equals zero. If the denominator does not include the common monomial factor s, then, still through author’s personal experience, one can put the thermal time constants outside the brackets of the denominator. We insert the residues obtained into the powers of exponents. The transient response f(t) equals the sum of the exponents. The resulted function f(t) is dimensionless. In order to release its unit of measurement, we have to multiply both left and the right terms of the function f(t) by a variable that has the unit of measurement of interest.

THIS ARTICLE NEEDS EXPERTS FOR VERIFICATION