2.2. Temperature Model of Working Digital Camera System
According to the composition of the digital camera system, the entire heat transfer process can be classified into three components: heat conduction between the integrated circuit board, case, mount, and lens; heat convection between the camera’s mechanical components and the environment; heat absorption of the camera components. The heat transfer path of the digital camera system under the coupling effect of self-heating and environmental temperature is shown in
Figure 2.
The heat generated by the integrated circuit board is assumed to be
Q, a part of which (
Qicb) changes the temperature of the integrated circuit board, and the rest (
Qicb−c) transfers to the camera case.
Qicb−c can be further divided into three components:
Qc changes the temperature of the camera case, and
Qc−e is exchanged between the case and the environment, and
Qc−m that transfers to the mount.
Qc−m can also be divided into three components:
Qm that changes the temperature of the mount, and
Qm−e is exchanged between the mount and the environment, and
Qm−l that transfers to the lens. Finally,
Qm−l can be divided into two components:
Ql that changes the temperature of the lens, and
Ql−e is exchanged between the lens and the environment. Thus, the heat transfer process can be described by the expressions:
Combined with the fundamental formulae of thermology, including Fourier’s law of heat conduction, Newton’s cooling formula, and the specific heat capacity formula [
19], Equation (1) can be rewritten as:
where
P denotes the thermal power (units: W) of the integrated circuit board.
R1,
R3, and
R5 represent the heat conduction parameters (units: W/°C) of the case, mount, and lens, respectively, and are directly proportional to the material’s thermal conductivity, cross-sectional area, and inversely proportional to the length of the heat conduction direction of the corresponding components.
R2,
R4, and
R6 represent the heat convection parameters (units: W/°C) of the case, mount, and lens, respectively, and are directly proportional to the material’s surface heat transfer coefficient and superficial area of the corresponding components.
K1–
K4 denote specific heat parameters (units: J/°C) of the integrated circuit board, case, mount, and lens, respectively, and are directly proportional to the material’s specific heat capacity and quality of the corresponding components.
Ticb,
Tc, and
Te are the temperature of the integrated circuit board, case, and environment, respectively, all of which are the function of time
t.
Tm and
Tl are the temperature of the mount and lens, respectively, all of which are the function of time
t as well the position coordinate
x whose direction along the optical axis and origin is located on the contact surface between the mount and the case.
Lm and
Ll represent the length of the mount and lens along the optical axis, respectively.
Equation (2) expresses the relationship between the camera component temperatures (
Tc,
Tm, and
Tl), the environmental temperature (
Te), and the thermal power (
P) of the working digital camera. The temperature of the camera components cannot be solved first-hand according to Equation (2); to calculate the camera components temperatures, Equation (2) is firstly simplified to obtain the temperature of the case (
Tc), the boundary temperature of the mount (
), and the boundary temperature of the lens (
). Then, based on analysis of the internal heat transfer of the mount (lens), the temperature expression of the mount (lens) is established. The simplification of Equation (2) is as follows:
where
r1,
r3, and
r5 are the characteristics heat conduction parameters of the case, mount, and lens, respectively;
r2,
r4, and
r6 are the characteristics heat convection parameters of the case, mount, and lens, respectively; and
k1–
k4 are the characteristics specific heat parameters of the integrated circuit board, case, mount, and lens, respectively. The above-defined parameters are related to the thermal power of working digital camera. If the abovementioned parameters are obtained, the temperature of the case (
Tc), the boundary temperature of the mount (
), and the boundary temperature of the lens (
) can be calculated via the environmental temperature.
Next, the temperature expressions of the mount and lens are established. The shape of the mount can be simplified to a cylinder with inner diameter
R0, outer diameter
r0, and length
Lm.
Figure 3a shows cross sections of the simplified mount along the optical axis and perpendicular to the optical axis. The heat transfer path of object
dx is shown in
Figure 3b; according to the conservation of energy, the heat transfer can be expressed as:
where
Qinto is the heat flowing into
dx from the last infinitesimal;
is the heat flowing into the environment from
dx;
is the heat flowing into the next infinitesimal from
dx; and
Qabsorb is the heat increment of
dx.
Combined with the fundamental formulae of thermology, including Fourier’s law of heat conduction, Newton’s cooling formula, and the specific heat capacity formula [
19], Equation (4) can be rewritten as:
where
km,
mm, and
nm represent the heat conduction parameter, the heat convection parameter, and the specific heat parameter of the mount, respectively. If those parameters are obtained, the temperature of the mount (
Tm) can be calculated via the environmental temperature, the initial temperature, and the boundary temperature of the mount.
Similarly, the temperature expression of the lens can be expressed as:
In summary, Equation (2) describes the temperature variation and distribution of a working digital camera system, which can be calculated by Equations (3), (5), and (6). Thus, the temperature model is established.
The thermal equilibrium time and thermal equilibrium temperature during the working process of a camera system are important values in mechanics measurement. In this paper, the thermal equilibrium time refers to the time period during which the temperature of working camera remains constant and the thermal equilibrium temperature refers to the camera’s temperature during the thermal equilibrium time. The precise value of the thermal equilibrium time and temperature can effectively guide the selection of the ‘measurement time window’ so as to eliminate thermal-induced errors of mechanics measurement. After the working camera system reaches thermal equilibrium, the temperature variation of any point (
i) of the camera satisfies:
In other words, the slope and curvature of the temperature curve over time are all zero for any measurement point on the camera system, all of which can be calculated by differential algorithms.
2.3. Experimental Verification
In this subsection, taking the digital camera system consisting of an IPX-16M3-L CCD camera and Sigma DG 28-300 mm lens, the accuracy of the camera component’s temperature variation and distribution calculated using the temperature model was verified experimentally. Firstly, in order to obtain the specific temperature model, the thermal parameters in Equations (3), (5), and (6) were calibrated by the optimization algorithm of nonlinear least square (the MATLAB’s built-in nonlinear least square function, i.e., lsqnonlin). Then, under the coaction of camera self-heating and time-varying environmental temperature, the temperatures of the camera components were calculated via the above obtained temperature model, and compared to the measured camera component temperatures measured using thermal sensors (i.e., the reference values for experimental verification). Finally, based on the verified temperature model, the thermal equilibrium time and thermal equilibrium temperature of the working digital camera were calculated.
First, the obtaining of specific temperature model is introduced in detail. The thermal parameters of the camera system were calibrated using the camera component temperature variation value induced by camera self-heating. A schematic and layout of the calibration-experiment setup are shown in
Figure 4a. The camera system was placed in a temperature box (an electric heating equipment named CINITE MAC3) that achieved the constant environmental temperature. The temperature data of the camera case, the different measurement points of the mount, the different measurement points of the lens and environment were measured using thermal sensors (K-type thermocouple, precision of 0.01 °C) during the process of camera self-heating. The experimental results are shown in
Figure 4b. In the whole process, the average value and variance of environmental temperature are −0.01 and 0.01 °C, respectively. Therefore, the influence of environmental temperature in those thermal parameters’ calibration was ignored.
According to the experimental temperature data of the case (
Tc), the boundary of the mount (
), and the boundary of the lens (
), the thermal parameters of Equation (3) were obtained via the optimization algorithm; the results are given in
Table 1. According to the measured temperature data of the mount (
,
, and
) and assuming that
nm = 1.00, the thermal parameters (
km,
mm) of Equation (5) were obtained via the optimization algorithm; the results are given in
Table 2. According to the measured temperature data of the lens (
,
, and
) and assuming that
nl = 1.00, the thermal parameters (
kl,
ml) of Equation (6) were obtained via the optimization algorithm; the results are given in
Table 3. The specific relationship between the environmental temperature and the camera components temperature was determined by substituting the calibration results into Equations (3), (5), and (6).
Next, the validity of the temperature model was verified experimentally with the coaction of camera self-heating and time-varying environmental temperature. The layout of the verification experiment is the same as shown in
Figure 4. During the process of camera self-heating, the environmental temperature controlled by the temperature box changed over time. The environmental temperature and the camera component temperatures were measured by thermal sensors. According to the measured environmental temperature and the obtained temperature model, the theoretical camera component temperatures were calculated, and compared to the measured temperature of the corresponding components.
Figure 5 shows the variation of the measured camera component temperatures and the calculated temperatures over the whole verification experiment.
Figure 6 shows the errors between the calculated temperatures and the measured temperatures over the whole experimental process. The average errors range from just −0.6 to 0.5 °C and the variances range from 0.2 to 0.3 °C, confirming the accuracy of the proposed temperature model to describe camera component temperature variations and distribution.
Finally, the thermal equilibrium time and thermal equilibrium temperature were investigated via the temperature model. By controlling the temperature box, the environmental temperature firstly rose to nearly 20 °C, then maintained for a period of time, and finally gradually decreased to a stable state. During the experiment, the camera component temperatures and environmental temperature were measured by means of thermal sensors. Through the measured environmental temperature, the temperature model, Equation (7), in which the threshold of slope and curvature were set to 0.001, the thermal equilibrium time, and the equilibrium temperature were calculated, as shown in
Figure 7. The experimental results confirm that our temperature model can accurately calculate the thermal equilibrium time and equilibrium temperature. Once the camera’s thermal equilibrium state is obtained, the time period of performing the mechanics measurement without thermal-induced error can be effectively determined.