Thermographic Measurement of the Temperature of Reactive Power Compensation Capacitors

: An excessive increase in reactive power consumption is unfavorable from the point of view of a power system. For this reason, devices compensating reactive power consumption are used. The capacitor is one such device. Capacitors must be tested regularly during their exploitation. One of the activities that should be performed is testing the degree of heating of the cells of a capacitor bank. Thermography can be used to perform such tests. This non-contact method has its limitations. Due to the angular emissivity and the change in the distance between the lens and the object under observation, the temperature measured with a thermographic camera may differ from the actual temperature. This phenomenon is visible on cylindrical capacitor cases. Consequently, depending on the location of the observation point on the capacitor case, the result of the thermographic temperature measurement may be different. To investigate this phenomenon, experimental work has been undertaken.


Introduction
The operation of electrical devices is related to the power consumption by the power system [1]. In the case of devices powered by alternating current, two components of the consumed power can be distinguished: active power and reactive power. Active power is considered to be useful power [2]. It is a component of power that is used to generate work or heat [3]. On the other hand, reactive power is most often associated with the magnetization of the coils of electrical motors and transformers [4]. The consumption of this power is unfavorable from the point of view of the system, as it reduces the power factor of the system and limits the ability of the generators to deliver useful power [5]. It causes an increase in the value of the consumed electrical current, increasing the load on the supply line [6]. In the case of industrial factories, the consumed reactive power is usually inductive [7].
The consumption of inductive reactive power can be compensated by the consumption of capacitive reactive power [8]. It is possible after adding a device whose operation increases the consumption of capacitive reactive power. The capacitor is such a device [9]. The use of capacitors to compensate the consumption of inductive reactive power is a frequently used solution [10]. Reactive power compensation capacitors must be checked regularly. The regular checking of the capacitors makes it possible to detect their capacity decline below the permissible value, which may be caused by the passage of time [11]. Additionally, it is possible to detect a potential failure before it occurs [12]. One of the activities that should be performed while testing the capacitor bank is checking the degree of heating of individual capacitor cells [13].
A method that allows one to check the degree of heating of the capacitor bank cells is thermography. More and more often, the offers of various manufacturers include inexpensive thermographic cameras with parameters that allow checking the heating of individual capacitor cells. This non-destructive method has been approved by the Polish Office of Technical Inspection [14]. It is worth remembering that capacitors used for compensating reactive power work with voltages, which can be hazardous to human health. For this reason, touching the case of a capacitor with a temperature sensor may cause an electric shock. This problem can be minimized by the use of thermography, which is a non-contact method. Consequently, the use of this method to check the degree of heating of individual capacitor cells increases the safety of the person performing the measurement. The most important factors that influence the result of the thermal imaging measurement of temperature ϑ t are the values of emission factor ε [15], reflected temperature ϑ refl [16], distance between the camera lens and the object under observation d [17], ambient temperature ϑ a [18], external optical system temperature ϑ l [19], transmission of the external optical system τ l [20], and relative humidity ω [21]. Their effect on the temperature ϑ t read from a thermogram has been widely described [22]. Furthermore, these values, if correctly determined, make it possible to minimize their effect on ϑ t . It can be achieved by selecting appropriate coefficients in the thermographic camera software [23].
An additional factor that influences the value of ϑ t is the recorded thermogram's lack of sharpness [24]. Modern thermographic cameras are often equipped with automatic sharpness adjustment systems working similarly to systems used in digital cameras [25]. For this reason, making a thermogram with a satisfactory sharpness is not difficult. It becomes problematic when one uses cameras that do not have such a system.
Another problem that can be encountered during thermographic temperature measurement is the low ε value of the material frequently used to make capacitor cases, i.e., aluminum [26], which has a low value of ε (0.238-0.247) at a spectral range of 8-14 µm [27]. For non-transparent materials, the relationship between ε and the reflection factor ρ can be described by Equation (1) [28].
ε + ρ = 1 (1) Based on Equation (1), it can be concluded that in the case of aluminum, the value of ρ is high. For this reason, while checking the degree of heating of individual capacitor cells, special attention should be paid to ϑ refl compensation.
Another issue that can be encountered is the cylindrical shape of the capacitor case. The value of ϑ t measured on the part of the case in front of the thermographic camera differs from the value of ϑt measured on another part of the case. In order to show this phenomenon, a thermogram of the Pt 1000 temperature sensor with a known temperature of ϑ s = 49.6 • C was made. The temperature of the sensor was selected arbitrarily. The sensor was placed in a metal case with a diameter of φ = 3 mm and a length of l = 6 mm. The recorded thermograms showing the sensor case are presented in Figure 1.
vidual capacitor cells. This non-destructive method has been approved by the Polis fice of Technical Inspection [14]. It is worth remembering that capacitors used for pensating reactive power work with voltages, which can be hazardous to human he For this reason, touching the case of a capacitor with a temperature sensor may cau electric shock. This problem can be minimized by the use of thermography, which non-contact method. Consequently, the use of this method to check the degree of he of individual capacitor cells increases the safety of the person performing the mea ment. The most important factors that influence the result of the thermal imaging m urement of temperature ϑt are the values of emission factor ε [15], reflected temper ϑrefl [16], distance between the camera lens and the object under observation d [17], a ent temperature ϑa [18], external optical system temperature ϑl [19], transmission o external optical system τl [20], and relative humidity ω [21]. Their effect on the tem ture ϑt read from a thermogram has been widely described [22]. Furthermore, these ues, if correctly determined, make it possible to minimize their effect on ϑt. It ca achieved by selecting appropriate coefficients in the thermographic camera software An additional factor that influences the value of ϑt is the recorded thermogram's of sharpness [24]. Modern thermographic cameras are often equipped with autom sharpness adjustment systems working similarly to systems used in digital cameras For this reason, making a thermogram with a satisfactory sharpness is not difficult. comes problematic when one uses cameras that do not have such a system.
Another problem that can be encountered during thermographic temperature m urement is the low ε value of the material frequently used to make capacitor cases aluminum [26], which has a low value of ε (0.238-0.247) at a spectral range of 8-14 [27]. For non-transparent materials, the relationship between ε and the reflection fac can be described by Equation (1) [28].
ε + ρ = 1 Based on Equation (1), it can be concluded that in the case of aluminum, the val ρ is high. For this reason, while checking the degree of heating of individual capa cells, special attention should be paid to ϑrefl compensation.
Another issue that can be encountered is the cylindrical shape of the capacitor The value of ϑt measured on the part of the case in front of the thermographic ca differs from the value of ϑt measured on another part of the case. In order to show phenomenon, a thermogram of the Pt 1000 temperature sensor with a known temper of ϑs = 49.6 °C was made. The temperature of the sensor was selected arbitrarily. The sor was placed in a metal case with a diameter of ϕ = 3 mm and a length of l = 6 mm recorded thermograms showing the sensor case are presented in Figure 1. In order to confirm the occurrence of this phenomenon, three additional the grams were made on the cylindrical case of the capacitor, which showed an electro Figure 1. Thermograms presenting the Pt 1000 temperature sensor in a cylindrical case with a diameter of 3 mm. Thermograms were taken at the same distance between the sensor and the lens of the thermal imaging camera. The sensor temperature was 49.6 • C. Temperature measured in S p : (a) 49.6 • C; (b) 48.5 • C; (c) 47.7 • C. In order to confirm the occurrence of this phenomenon, three additional thermograms were made on the cylindrical case of the capacitor, which showed an electrolytic capacitor with a diameter of φ = 4 mm. In this case, the capacitor was painted with Velvet Coating 811-21 with a known value of the emissivity factor ε ranging from 0.970 to 0.975 for temperatures between −36 • C and 82 • C. The uncertainty with which the value of emissivity factor was determined was 0.004 [29]. A layer of the paint was applied to the outer surface of the capacitor. During the making of the thermograms, the capacitor was placed on a heating mat at a temperature of 60 • C. The temperature of the mat was selected arbitrarily so that it differed from the value of ϑ a . The measurement was taken only after the temperature of the capacitor stabilized. The recorded thermograms are shown in Figure 2. capacitor with a diameter of ϕ = 4 mm. In this case, the capacitor was painted with Velvet Coating 811-21 with a known value of the emissivity factor ε ranging from 0.970 to 0.975 for temperatures between −36 °C and 82 °C. The uncertainty with which the value of emissivity factor was determined was 0.004 [29]. A layer of the paint was applied to the outer surface of the capacitor. During the making of the thermograms, the capacitor was placed on a heating mat at a temperature of 60 °C. The temperature of the mat was selected arbitrarily so that it differed from the value of ϑa. The measurement was taken only after the temperature of the capacitor stabilized. The recorded thermograms are shown in Figure  2.
(a) (b) (c) After recording the thermograms, the following question was asked: how big will the error be in measuring the temperature of the capacitor bank cell if the measuring point is not on the surface exactly in front of the thermographic camera lens? It was noticed that the phenomenon shown in Figures 1 and 2 was due to a different angle of view α and a different distance d between the point on the capacitor case and the lens of the thermographic camera. Due to the low ε value of the capacitor case, the following question was asked: how much influence does the reflected temperature have on the measurement result? Following an analysis of the available literature, information on the influence of d on the value of ϑt was found. Information on the influence of the angle of view α on the value of ϑt was also found. The conducted analysis of the literature did not provide an answer to the question about the value of the error of the thermographic temperature measurement on the case of the capacitor intended for reactive power compensation due to the influence of d, α and ϑrefl. For this reason, research has been undertaken, the results of which will make it possible to answer that question.

Distance Between Thermographic Camera Lens and Observed Object
The phenomenon recorded in Figures 1 and 2 is the result of the occurrence of two phenomena: the change in d and the angular emissivity. The capacitor case has the shape of a cylinder. For this reason, the distances between the lens of the thermographic camera and specific parts of the capacitor case will be different. The damping of the atmosphere layer τa between the observed capacitor and the lens of the thermographic camera changes together with the change in d. The relationship between both quantities is described by Equation (2) [29,30]: where Katm = 1.9 is the atmosphere damping factor, α1 = 0.0066 and α2 = 0.0126 are damping factors for an atmosphere without water vapor, β1 = −0.0023 and β2 = −0.0067 are damping factors for water vapor, and ω is the factor indicating the amount of water vapor in the atmosphere [29,30].
The value of ω in (2) can be determined using Equation (  After recording the thermograms, the following question was asked: how big will the error be in measuring the temperature of the capacitor bank cell if the measuring point is not on the surface exactly in front of the thermographic camera lens? It was noticed that the phenomenon shown in Figures 1 and 2 was due to a different angle of view α and a different distance d between the point on the capacitor case and the lens of the thermographic camera. Due to the low ε value of the capacitor case, the following question was asked: how much influence does the reflected temperature have on the measurement result? Following an analysis of the available literature, information on the influence of d on the value of ϑ t was found. Information on the influence of the angle of view α on the value of ϑ t was also found. The conducted analysis of the literature did not provide an answer to the question about the value of the error of the thermographic temperature measurement on the case of the capacitor intended for reactive power compensation due to the influence of d, α and ϑ refl . For this reason, research has been undertaken, the results of which will make it possible to answer that question.

Distance between Thermographic Camera Lens and Observed Object
The phenomenon recorded in Figures 1 and 2 is the result of the occurrence of two phenomena: the change in d and the angular emissivity. The capacitor case has the shape of a cylinder. For this reason, the distances between the lens of the thermographic camera and specific parts of the capacitor case will be different. The damping of the atmosphere layer τ a between the observed capacitor and the lens of the thermographic camera changes together with the change in d. The relationship between both quantities is described by Equation (2) [29,30]: where K atm = 1.9 is the atmosphere damping factor, α 1 = 0.0066 and α 2 = 0.0126 are damping factors for an atmosphere without water vapor, β 1 = −0.0023 and β 2 = −0.0067 are damping factors for water vapor, and ω is the factor indicating the amount of water vapor in the atmosphere [29,30]. The value of ω in (2) can be determined using Equation (3) [29,30]: where ω% is the relative humidity: h 1 = 1.5587, h 2 = 6.939 · 10 −2 , h 3 = −2.7816 · 10 −4 , and h 4 = 6.845 · 10 −7 [30]. It is worth noting that the value of τa is one of the input variables of the equation that allows us to determine the value of ϑ obj from the total flux of IR radiation reaching the lens of the thermographic camera. The relationship between ϑ obj and τ a is described by Equation (4) [31]: where σ is the Boltzmann constant equal to 5.67 cm × 10 −8 W/(m 2 ·K 4 ), W tot is the total IR radiation reaching the camera lens, and τ a is the atmospheric transmittance [31]. The differences in the d value between the capacitor case and the camera lens are shown in Figure 3.
It is worth noting that the value of τa is one of the input variables of the equation that allows us to determine the value of ϑobj from the total flux of IR radiation reaching the lens of the thermographic camera. The relationship between ϑobj and τa is described by Equation (4) [31]: where σ is the Boltzmann constant equal to 5.67 cm × 10 −8 W/(m 2 ·K 4 ), Wtot is the total IR radiation reaching the camera lens, and τa is the atmospheric transmittance [31]. The differences in the d value between the capacitor case and the camera lens are shown in Figure 3.

Angular Emissivity
In order to explain the relationship between ε and α, refer to Equation (1). The ρ factor is the average of the parallel polarized reflectance factor ‖ and the perpendicularly polarized reflectance factor ⟂ (5) [31]: The values of ‖ and ⟂ depend on the value of α and the refractive factor n. Aluminum, often used to make capacitor cases, is a glossy material, similar to other metals. For glossy materials with damping of the electromagnetic radiation wave, the complex value of the refractive factor n' = n − jk is assumed, where k is the damping factor. The values of factors ‖ and ⟂ can be described by Equations (6) and (7) [31]:

Angular Emissivity
In order to explain the relationship between ε and α, refer to Equation (1). The ρ factor is the average of the parallel polarized reflectance factor ρ and the perpendicularly polarized reflectance factor ρ ⊥ (5) [31]: The values of ρ and ρ ⊥ depend on the value of α and the refractive factor n. Aluminum, often used to make capacitor cases, is a glossy material, similar to other metals. For glossy materials with damping of the electromagnetic radiation wave, the complex value of the refractive factor n' = n − jk is assumed, where k is the damping factor. The values of factors ρ and ρ ⊥ can be described by Equations (6) and (7) [31]: By inserting Equations (6) and (7) into Equation (5) and then into Equation (1), the relationship between α and ε can be seen.
The outer parts of some capacitors are covered with a thin layer of plastic. In the case of dielectrics, the values of ρ and ρ ⊥ can be described by Equations (8) and (9) [31]: where n 1 is the refractive factor of the first medium and n 2 is the refractive factor of the second medium. Based on Equations (6)-(9), it can be concluded that the characteristics ε = f(α) for metals and dielectrics will be different.

Reflected Temperature
Reflected temperature ϑ refl is also called reflected radiation [32] in the literature. It is the radiation emitted by an object that is in the vicinity of the object under observation. The reflected radiation reaches the lens of the thermographic camera together with the radiation emitted by the object under observation. There is no mathematical model that would accurately describe the value of ϑ refl in every case. For this reason, to minimize the influence of ϑ refl on the value of ϑ obj , the value of ϑ refl should always be measured. The measured value of ϑ refl should be entered in the software of the thermographic camera before checking the degree of heating of individual capacitor cells. The relationship between ϑ obj and ϑ refl can be described by Equation (3).
For a better explanation, the discussed phenomenon is presented in Figure 4. An example of the recorded reflected radiation is shown in Figure 5, where capacitors used for compensating reactive power can be seen. Reflected radiation was registered on both capacitor cases, which were made of aluminum.

The Measurement System
The first part of the experiments described in this article was carried out using a specially constructed stand. The stand consisted of three parts: a chamber with the measuring instruments, a control part, and a laptop with Flir Tools + software. The most important device in the stand was the Flir E50 thermographic camera (Flir, Wilsonville, OR, USA) [33]. In order to obtain appropriate spatial resolution, the camera was equipped with an additional Close-up 2× lens (Flir, Wilsonville, OR, USA). This made it possible to obtain an IFOV (Instantaneous Field of View) of 67 µm [34]. The camera with an additional lens was mounted on a linear module. In turn, the linear module was attached to a metal angle. The linear module was controlled by a stepper motor [35]. Consequently, it was possible to change d with a precision of 0.1 mm. The distance d was controlled by means of an appropriately mounted potentiometric offset sensor MMR30, with a nominal resistance value of 10 kΩ [36]. This sensor made it possible to measure the distance d in the range of 0-30 mm, with a precision of 0.1 mm. At a distance of about 30 mm from the lens, an aluminum block sized 16 mm × 16 mm × 45 mm was placed. The surface of the block, monitored with a thermographic camera, was painted with Velvet Coating 811-21. The block temperature was monitored with a Pt 1000 sensor [37]. The monitored block was connected to a stepper motor. Consequently, it was possible to change α with a precision of 0.9 • . In order to run the indivudual parts of the experiment, the block was replaced with a heating plate of a known temperature. The entire set was placed in a chamber made of plexiglass with the external dimensions of 45 cm × 45 cm × 33 cm. In order to minimize the influence of reflections and ensure proper optical isolation, the interior of the chamber was lined with black polyurethane foam. Due to the porous structure of the foam and its color the value of factor ε equaled 0.95 [38] inside the chamber. Additionally, the metal components inside the chamber were painted black. Using the Siemens S7-1200 1214C DC/DC/DC PLC controller (Siemens AG, Munich, Germany) [39], all actuators and readings were made from the sensors. This controller supported the HMI touch panel (Siemens AG, Munich, Germany) [40], by means of which it was possible to set the required settings and read them. where n1 is the refractive factor of the first medium and n2 is the refractive facto second medium. Based on Equations (6)- (9), it can be concluded that the characteristics ε = metals and dielectrics will be different.

Reflected Temperature
Reflected temperature ϑrefl is also called reflected radiation [32] in the literatu the radiation emitted by an object that is in the vicinity of the object under obse The reflected radiation reaches the lens of the thermographic camera together w radiation emitted by the object under observation. There is no mathematical mo would accurately describe the value of ϑrefl in every case. For this reason, to minim influence of ϑrefl on the value of ϑobj, the value of ϑrefl should always be measur measured value of ϑrefl should be entered in the software of the thermographic before checking the degree of heating of individual capacitor cells. The relations tween ϑobj and ϑrefl can be described by Equation (3).
For a better explanation, the discussed phenomenon is presented in Figure 4   The camera was connected to the laptop via a USB interface. Using the Flir Tools + software during the experiment, the settings of the thermographic camera were changed. It was also possible to record thermograms. The diagram of the described stand is presented in Figure 6.
A photo of the presented stand is shown in Figure 7. An example of the recorded reflected radiation is shown in Figure 5, where cap used for compensating reactive power can be seen. Reflected radiation was registe both capacitor cases, which were made of aluminum.

The Measurement System
The first part of the experiments described in this article was carried out using cially constructed stand. The stand consisted of three parts: a chamber with the mea instruments, a control part, and a laptop with Flir Tools + software. The most imp device in the stand was the Flir E50 thermographic camera (Flir, Wilsonville, OR [33]. In order to obtain appropriate spatial resolution, the camera was equipped w additional Close-up 2× lens (Flir, Wilsonville, OR, USA). This made it possible to an IFOV (Instantaneous Field of View) of 67 µ m. [34]. The camera with an addition was mounted on a linear module. In turn, the linear module was attached to a metal The linear module was controlled by a stepper motor [35]. Consequently, it was p to change d with a precision of 0.1 mm. The distance d was controlled by mean appropriately mounted potentiometric offset sensor MMR30, with a nominal res value of 10 k [36]. This sensor made it possible to measure the distance d in the ra 0-30 mm, with a precision of 0.1 mm. At a distance of about 30 mm from the le aluminum block sized 16 mm × 16 mm × 45 mm was placed. The surface of the monitored with a thermographic camera, was painted with Velvet Coating 811-2 block temperature was monitored with a Pt 1000 sensor [37]. The monitored blo connected to a stepper motor. Consequently, it was possible to change α with a pr of 0.9°. In order to run the indivudual parts of the experiment, the block was replace a heating plate of a known temperature. The entire set was placed in a chamber m plexiglass with the external dimensions of 45 cm × 45 cm × 33 cm. In order to mi the influence of reflections and ensure proper optical isolation, the interior of the ch was lined with black polyurethane foam. Due to the porous structure of the foam color the value of factor ε equaled 0.95 [38] inside the chamber. Additionally, the components inside the chamber were painted black. Using the Siemens S7-1200 DC/DC/DC PLC controller (Siemens AG, Munich, Germany) [39], all actuators and ings were made from the sensors. This controller supported the HMI touch panel (Si AG, Munich, Germany) [40], by means of which it was possible to set the required s and read them.
The camera was connected to the laptop via a USB interface. Using the Flir T software during the experiment, the settings of the thermographic camera were ch It was also possible to record thermograms. The diagram of the described stand sented in Figure 6.  The second part of the experiment was carried out on a stand consisting of a Flir E50  The second part of the experiment was carried out on a stand consisting of a Flir E50 thermographic camera (without an additional lens) and the tested capacitors for reactive  The second part of the experiment was carried out on a stand consisting of a Flir E50 thermographic camera (without an additional lens) and the tested capacitors for reactive power compensation. The capacitors were star-connected to increase their temperature to the operating temperature, higher than the ambient temperature. The distance between the capacitors and the lens was measured with a precision of 1 cm. In the last part of the work, the capacitors were changed to a cylinder made of aluminum with an adjustable diameter. A piece of metal (aluminum), which was industrial waste, measuring 90 cm × 4 cm was used. Using that method, attempts were made to recreate the conditions prevailing during the thermographic temperature measurement of the operated capacitor bank. A diagram of the measuring system is shown in Figure 8. A photo of the system, which has been described and showed sented in Figure 9.

Results of Simulation Tests
As a result of the simulation work carried out, it was checked ho A photo of the system, which has been described and showed in Figure 8, is presented in Figure 9. A photo of the system, which has been described and showed in Figure 8, is pre sented in Figure 9.

Results of Simulation Tests
As a result of the simulation work carried out, it was checked how the result of the ϑt value would change due to the change in d caused by placing the measurement poin on various parts of the case. Based on the analysis of the available technical documenta tion, it was assumed that the diameter of the capacitor case intended for reactive power compensation would not exceed 3000 mm. This means that the difference between the largest and smallest d values will not be bigger than 1500 mm [41]. The simulation was performed for standard conditions (ϑa = 25 °C). The half of the range of values recorded in the laboratory (45%) was adopted as the value of ω%. The given criteria were followed

Results of Simulation Tests
As a result of the simulation work carried out, it was checked how the result of the ϑ t value would change due to the change in d caused by placing the measurement point on various parts of the case. Based on the analysis of the available technical documentation, it was assumed that the diameter of the capacitor case intended for reactive power compensation would not exceed 3000 mm. This means that the difference between the largest and smallest d values will not be bigger than 1500 mm [41]. The simulation was performed for Energies 2021, 14, 5736 9 of 16 standard conditions (ϑ a = 25 • C). The half of the range of values recorded in the laboratory (45%) was adopted as the value of ω % . The given criteria were followed by selecting the ϑ refl value of 30 • C. The ε was chosen for aluminum (ε = 0.238). It was assumed that the range d would be changed from 0.85 m to 0.15 m. It was an arbitrarily adopted distance of 1 m changed by the range resulting from the largest assumed capacitor diameter. During the simulation, the value of d was changed with the step of i = 0.01.
In the first part of the simulation, the W tot value was determined for the ϑ obj value in the range from −20 • C to 70 • C, changed in steps h = 5 • C. The lower value of the assumed range, ϑ obj , is the assumed ambient temperature. The upper limit of the range is the highest possible temperature of the capacitor case that can be found in the catalogs. During the simulation, the W tot values were determined according to the algorithm shown in Figure 10 for all variable values. The values of the coefficients used were taken from [29].
After determining the value of W tot , the values of ϑ cam were determined for each value of d and ϑ obj . The graph obtained as a result of the simulation, showing the changes of ϑ cam as a function of d, for different values of ϑ obj , is shown in Figure 11.

Influence of Angle of View on the Value of Thermographic Temperature Measurement of Small Diameter Cases
In order to check the influence of α on the value of ϑ t , it was decided to repeat the experiment carried out by one of the authors of this article and described in [42]. This was performed to obtain the data that would allow us to estimate the influence of the α value on the ϑ cam value in the case of thermographic temperature measurements on the cases capacitors intended for reactive power compensation. An aluminum block with a known temperature of 57.35 • C was observed. In order to minimize the impact of the changing conditions during the measurement, measurements were made by automatic changing of α and simultaneous film recording with a thermographic camera. The results were obtained by analyzing individual frames of the recorded film. The results were made for three series of measurements. The achieved results are shown in Figure 12. In the figure, the black lines indicate the value of the known temperature of the block under observation increased or decreased by the measurement error of the thermographic camera [34].
The aluminum block was then replaced with a temperature-controlled heating plate. During the measurements, the plate was placed parallel to the lens of the thermographic camera. The capacitors covered with Velvet Coating 811-21 paint were put on the plate. The paint was applied to a thin plastic layer that was on the capacitor. After the temperature of the capacitor was established, a thermogram was made. Examples of thermograms are presented in Figure 13.
The ϑ t values recorded in the thermogram were exported to Excel. The recorded values of ϑ t as a function of the position on the case x are shown in Figures 14 and 15. The initial value (x = 0 mm) was the point of the capacitor case that was closest to the thermographic camera lens (in the middle of the capacitor case).

Influence of Angle of View on the Value of Thermographic Temperature Measurement of a Typical Case
At the beginning, it was decided to experimentally check the simulation results presented in Figure 6. For that purpose, the KJF-0.83/400 capacitors were star-connected and loaded with the rated current ( Figure 9). The distance d was changed in the range from 0.85 m to 1.15 m. With a step equal to 0.01 m and for each distance, a thermogram was made. The results of thermographic temperature measurements are shown in Figure 16.
Then, the edge length of the field of view of a single detector of the thermographic camera I was checked, while the distance d was 1 m. The formula (10) was used. The determined value of I was 1.82 mm: where IFOV is the value of the angle between the adjacent walls of the pyramid, read from the technical documentation of the thermographic camera, the apex of which is in the IR detector of the camera and the base of the object under observation. For Flir E50, IFOV equals 1.82 mrad. Knowing the value of I, the temperature distribution (function ϑ t = f(x)) was checked on the metal case of the KJF-0.83/400 capacitors. An example thermogram is presented in Figure 17.  Figure 11.

Influence of Angle of View on the Value of Thermographic Temperature Measurement of Small Diameter Cases
In order to check the influence of α on the value of ϑt, it was decided to repeat t experiment carried out by one of the authors of this article and described in [42]. This w performed to obtain the data that would allow us to estimate the influence of the α val on the ϑcam value in the case of thermographic temperature measurements on the cas capacitors intended for reactive power compensation. An aluminum block with a know temperature of 57.35 °C was observed. In order to minimize the impact of the changi conditions during the measurement, measurements were made by automatic changing α and simultaneous film recording with a thermographic camera. The results were o tained by analyzing individual frames of the recorded film. The results were made f three series of measurements. The achieved results are shown in Figure 12. In the figu the black lines indicate the value of the known temperature of the block under observati increased or decreased by the measurement error of the thermographic camera [34].

Influence of Angle of View on the Value of Thermographic Temperature Measurement of Small Diameter Cases
In order to check the influence of α on the value of ϑt, it was decided to repeat the experiment carried out by one of the authors of this article and described in [42]. This was performed to obtain the data that would allow us to estimate the influence of the α value on the ϑcam value in the case of thermographic temperature measurements on the cases capacitors intended for reactive power compensation. An aluminum block with a known temperature of 57.35 °C was observed. In order to minimize the impact of the changing conditions during the measurement, measurements were made by automatic changing of α and simultaneous film recording with a thermographic camera. The results were obtained by analyzing individual frames of the recorded film. The results were made for three series of measurements. The achieved results are shown in Figure 12. In the figure, the black lines indicate the value of the known temperature of the block under observation increased or decreased by the measurement error of the thermographic camera [34].  The aluminum block was then replaced with a temperature-controlled heating plate. During the measurements, the plate was placed parallel to the lens of the thermographic camera. The capacitors covered with Velvet Coating 811-21 paint were put on the plate. The paint was applied to a thin plastic layer that was on the capacitor. After the temperature of the capacitor was established, a thermogram was made. Examples of thermograms are presented in Figure 13. The ϑt values recorded in the thermogram were exported to Excel. The recorded values of ϑt as a function of the position on the case x are shown in Figures 14 and 15. The initial value (x = 0 mm) was the point of the capacitor case that was closest to the thermo-  The ϑt values recorded in the thermogram were exported to Excel. The recorded values of ϑt as a function of the position on the case x are shown in Figures 14 and 15. The initial value (x = 0 mm) was the point of the capacitor case that was closest to the thermographic camera lens (in the middle of the capacitor case).

Influence of Angle of View on the Value of Thermographic Temperature Measurement of a Typical Case
At the beginning, it was decided to experimentally check the simulation results presented in Figure 6. For that purpose, the KJF-0.83/400 capacitors were star-connected and loaded with the rated current ( Figure 9). The distance d was changed in the range from 0.85 m to 1.15 m. With a step equal to 0.01 m and for each distance, a thermogram was made. The results of thermographic temperature measurements are shown in Figure 16. The capacitors were star-connected and loaded with the rated current. The distance d was 1 m. The remaining parameters corresponded to the conditions prevailing during the measurement (ε = 0.24, ϑ refl = 26 • C, ϑ l = 26 • C, τ l = 1, ω % = 45%). The value of ϑ refl was measured by placing a bent aluminum foil on the case of the observed capacitor and changing the settings of ε = 1, d = 0. The value of ϑ refl was the one indicated by the thermographic camera. The obtained functions ϑ t = f(x) for three capacitors are shown in Figure 18. The point located in the middle of the capacitor case was assumed as x = 0.
In order to make the work more universal, it was decided to check whether a similar temperature distribution (the function ϑ t = f(x)) would also be recorded on cylindrical aluminum surfaces with different cross-sections. For this purpose, an aluminum strip measuring 4 cm by 90 cm was rolled up. It was decided to check the temperature distribution on a cylinder with a diameter of 100 mm and 130 mm. The center of the observed surface was taken as x = 0 again. The thermographic camera settings were as follows: ε = 0.24, ϑ refl = 26 • C, ϑ l = 26 • C, τ l = 1, ω % = 45%, d = 1 m. The obtained results are shown in Figure 19.

Influence of Angle of View on the Value of Thermographic Temperature Measurement of Typical Case
At the beginning, it was decided to experimentally check the simulation results pre sented in Figure 6. For that purpose, the KJF-0.83/400 capacitors were star-connected and loaded with the rated current ( Figure 9). The distance d was changed in the range from 0.85 m to 1.15 m. With a step equal to 0.01 m and for each distance, a thermogram wa made. The results of thermographic temperature measurements are shown in Figure 16. Then, the edge length of the field of view of a single detector of the thermographi camera I was checked, while the distance d was 1 m. The formula (10) was used. The de termined value of I was 1.82 mm: where IFOV is the value of the angle between the adjacent walls of the pyramid, read from the technical documentation of the thermographic camera, the apex of which is in the IR detector of the camera and the base of the object under observation. For Flir E50, IFOV equals 1.82 mrad. Knowing the value of I, the temperature distribution (function ϑt = f(x)  The capacitors were star-connected and loaded with the rated cu d was 1 m. The remaining parameters corresponded to the condition the measurement (ε = 0.24, ϑrefl = 26 °C, ϑl = 26 °C, τl = 1, ω% = 45%). T measured by placing a bent aluminum foil on the case of the obse changing the settings of ε = 1, d = 0. The value of ϑrefl was the one indic graphic camera. The obtained functions ϑt = f(x) for three capacitors a 18. The point located in the middle of the capacitor case was assumed was checked on the metal case of the KJF-0.83/400 capacitors. An example thermogram is presented in Figure 17. The capacitors were star-connected and loaded with the rated current. The distance d was 1 m. The remaining parameters corresponded to the conditions prevailing during the measurement (ε = 0.24, ϑrefl = 26 °C , ϑl = 26 °C , τl = 1, ω% = 45%). The value of ϑrefl was measured by placing a bent aluminum foil on the case of the observed capacitor and changing the settings of ε = 1, d = 0. The value of ϑrefl was the one indicated by the thermographic camera. The obtained functions ϑt = f(x) for three capacitors are shown in Figure  18. The point located in the middle of the capacitor case was assumed as x = 0. In order to make the work more universal, it was decided to check whether a similar temperature distribution (the function ϑt = f(x)) would also be recorded on cylindrical aluminum surfaces with different cross-sections. For this purpose, an aluminum strip measuring 4 cm by 90 cm was rolled up. It was decided to check the temperature distribution

Conclusions
Thermography is a method that makes it possible to check the degree of heating the cells in a capacitor bank. Regular performance of thermograms enables safe detect of the increase in surface heating of the observed capacitor. It is worth remembering compare thermograms made with the same settings of the thermographic camera.
The temperature values measured on different parts of the capacitor case may di from each other. This phenomenon may be due to the local properties of the material fr ment. Regarding the observed cylindrical cases, the influence of two phenomena was c sidered: the different distances of the thermographic camera lens, the object under ob vation, and the angular emissivity.
When analyzing the graphs in Figures 14-16, it can be noticed that in the case of sm capacitors (φ ≤ 30 mm), the error of the thermographic temperature measurement cau by an incorrect selection of a point on the capacitor case will not exceed 3 °C . In the c of large capacitors (φ ≥ 30 mm), this value will not exceed 1 °C (Figures 18 and 19). It ne to be emphasized that in the case of thermographic observations of small capacitors additional macro lens was used. One should also remember the error of the thermal aging camera, amounting to ±2 °C or ±2% of reading, for ambient temperature between °C and 35 °C.
In the conducted works, the Flir E50 thermographic camera was used. In the cam software, you can compensate the influence of the distance between the lens and the ob under observation with a resolution of one meter. In the case of the observed phenom non, changes in this distance are smaller. At the stage of simulation tests, it was alrea shown that a change in the distance of the thermographic camera lens from the ob under observation, amounting to less than 15 cm, would not cause a significant chang the temperature indicated by the thermographic camera ( Figure 11). Angular emissivity has a bigger influence on the change in the result of therm graphic temperature measurement. As a result of the conducted experimental works, dependence of the result of thermographic measurement of capacitor surface temperat on the angle of view was demonstrated. In addition, it was shown that the shape of both characteristic-the result of the thermographic temperature measurement and angle of view-would be different for a metal surface of the capacitor and different the surface of the capacitor covered with plastic. The change in the thermographic te perature measurement result above the error value of the thermographic camera occ

Conclusions
Thermography is a method that makes it possible to check the degree of heating of the cells in a capacitor bank. Regular performance of thermograms enables safe detection of the increase in surface heating of the observed capacitor. It is worth remembering to compare thermograms made with the same settings of the thermographic camera.
The temperature values measured on different parts of the capacitor case may differ from each other. This phenomenon may be due to the local properties of the material fragment. Regarding the observed cylindrical cases, the influence of two phenomena was considered: the different distances of the thermographic camera lens, the object under observation, and the angular emissivity.
When analyzing the graphs in Figures 14-16, it can be noticed that in the case of small capacitors (ϕ ≤ 30 mm), the error of the thermographic temperature measurement caused by an incorrect selection of a point on the capacitor case will not exceed 3 • C. In the case of large capacitors (ϕ ≥ 30 mm), this value will not exceed 1 • C (Figures 18 and 19). It needs to be emphasized that in the case of thermographic observations of small capacitors, an additional macro lens was used. One should also remember the error of the thermal imaging camera, amounting to ±2 • C or ±2% of reading, for ambient temperature between 10 • C and 35 • C.
In the conducted works, the Flir E50 thermographic camera was used. In the camera software, you can compensate the influence of the distance between the lens and the object under observation with a resolution of one meter. In the case of the observed phenomenon, changes in this distance are smaller. At the stage of simulation tests, it was already shown that a change in the distance of the thermographic camera lens from the object under observation, amounting to less than 15 cm, would not cause a significant change in the temperature indicated by the thermographic camera ( Figure 11). Angular emissivity has a bigger influence on the change in the result of thermographic temperature measurement. As a result of the conducted experimental works, the dependence of the result of thermographic measurement of capacitor surface temperature on the angle of view was demonstrated. In addition, it was shown that the shape of the both characteristic-the result of the thermographic temperature measurement and the angle of view-would be different for a metal surface of the capacitor and different for the surface of the capacitor covered with plastic. The change in the thermographic temperature measurement result above the error value of the thermographic camera occurs for large viewing angles (approx. 67 • ).
It can be concluded that the change in the thermographic temperature measurement result due to changes in the position of the observed point on the case is mainly caused by angular emissivity. Only extreme positions of the observed point on the capacitor case should be avoided. The effect of reflected radiation is random and difficult to determine.