Optimization of Amplitude-Frequency Characteristic of Broadband Voltage Divider Intended for Measurement of Power Quality Parameters

The object of research is the circuit diagram of a broadband capacitive-resistive voltage divider with a series-parallel connection of its resistive and capacitive components. For many years, the use of voltage dividers was limited to measuring various voltages in high-voltage laboratories. However, voltage dividers, compared to voltage transformers, are characterized by a wider bandwidth, therefore they began to be considered as one of the main means of measuring voltages in high-voltage electric networks. One of the catalysts for the implementation of this solution may be the intensive development of the Smart Grid concept, which requires new, more advanced means of monitoring the quality of electric power. Therefore, experimental and theoretical studies aimed at reducing the error of broadband voltage dividers are important.<br><br>The task of optimally adjusting the low voltage arm of the voltage divider is solved by using linear programming elements to study the systematic error function.<br><br>This article presents the results of the study of adjusting the amplitude-frequency characteristics of the voltage divider, which are aimed at reducing its error. For this purpose, a parameter for optimizing the capacitance value of low-voltage arm at which the absolute value of the positive and negative maximum of the systematic error of the capacitive-resistive voltage divider will be the same was found. The calculations are performed for different values of the division ratio of the voltage divider. The resulting data sets are generalized in the form of three-dimensional graphs.<br><br>The work contributes to the further development of the theory of high-voltage voltage dividers. As a result of the studies, the possibility of optimizing the amplitude-frequency characteristics of a broadband capacitive-resistive voltage divider by varying the capacitance value of its low-voltage arm is shown. The studies are relevant due to the fact that this category of high-voltage scale transducers has the potential to become mandatory for determining the quality of electric energy directly in high-voltage networks.


Introduction
Measurement of electric power quality parameters is ne cessary during energy generation, distribution and consump tion. In addition, measurements are necessary to ensure the possibility of controlling the quality of electricity as a type of product. For this purpose, technical and organizational measures are being implemented to achieve compliance of ISSN 2664-9969 electric power quality parameters with the requirements of international standards, for example, IEC 61000430:2015. Under modern conditions, attention to the electric power quality is constantly growing, since electric power qua lity determines in many cases the ability to function of many complicated devices, critical equipment and entire systems. This task requires the development of measuring instruments with an extremely low error and the ability to measure voltages over a wide frequency range. Among highvoltage scale transducers, one should mention voltage transformers and voltage dividers. For many years, the use of voltage dividers was limited to measuring various volt ages [1,2] in highvoltage laboratories. However, voltage dividers, compared to voltage transformers, are characteri zed by a wider bandwidth. Therefore, many researchers began consider them as one of the main means for volt age measurement in highvoltage electric networks [3][4][5]. The authors also believe that voltage dividers have much greater potential for improvement than instrument trans formers. In this connection, studies on the possibility of using voltage dividers to measure electric power quality parameters [6][7][8] were started at the Department of Theo retical Electrical Engineering of the National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» (Ukraine). The work being performed contains both experimental and theoretical studies on the possibility of using voltage dividers instead of voltage transformers for measuring the electric power quality parameters.
Thus, the object of research is the circuit diagram of a broadband capacitiveresistive voltage divider with a seriesparallel connection of its resistive and capaci tive components. In its turn, the main aim of the article is to study the adjustment of the amplitudefrequency characteristics of the voltage divider, which is aimed at reducing its measurement error.

Methods of research
The voltage divider, constructed according to principle of voltage distribution over the complex impedances, is usu ally called a mixed capacitiveresistive voltage divider with a seriesparallel connection of its resistive and capacitive components. A generalized equivalent circuit for a voltage divider of this type, which also gives a general idea of the voltage divider component layout in space, is shown in Fig. 1.
There is no doubt that a careful selection of resistors and capacitors the highvoltage arm is assembled of, can improve the transfer properties of the voltage divider. In addition to the fact that such an approach is very labour consuming, even after the most careful selection of com ponents, there will always be some kind of nonidentity of the components. For this reason, it can't be considered that the voltage divider will consist of the same compo nents. Hence, all components of the voltage divider will be characterized by different thermal stresses and different dependencies on atmospheric conditions (humidity, pressure). As a result of the nonidentity of the highvoltage arm components, the divider's voltage ratio, in addition to the dependence on the frequency and amplitude of the applied voltage, also becomes dependent on the temperature, hu midity, and am bient pressure. This is especially true, since modern voltage dividers are quite dimensional devices with a vertical arrangement of components. However, it is pos sible to reduce the negative effect of the nonidentity of the highvoltage arm components by making some adjust ments to the lowvoltage arm components of the voltage divider [6]. According to the theory of voltage dividers [9], the amplitudefrequency characteristic of the voltage divider is determined by the expression: where A 1 g ( ) -the reduced amplitudefrequency charac teristic, which in its turn is determined by the following expressions: In expressions (1)-(4): K -divider's nominal voltage ratio; g -dimensionless parameter of angular frequency w. Parameter g is determined by the following expressions: where R 0 and C 0 -average values of resistive and capacitive components of the highvoltage arm of the voltage divider, respectively; n -the total number of these components.
It is shown in [6] that the nonidentity of the resis tive components of the highvoltage arm of the voltage divider is negligible compared to the nonidentity of the capacitive components.
Then in the expression (3) In its turn, in expression (4), the function δ is deter mined by the expression: In the two above expressions parameter α i depends on the capacitance values of the highvoltage arm of the voltage divider (refer to Fig. 1) as follows: The parameter ′ δ in (2)-(4) corresponds to the maxi mum value of δ (6) when g → ∞, that is: The selection of the lowvoltage arm components is being performed according to the common expressions: what corresponds to the value of Θ = 0 in formulas (2)-(4).
Herewith, this case corresponds to the absence of the low voltage arm adjustment of the voltage divider. The value of Θ = 1 in formulas (2)-(4) corresponds to the maximum (or, in other words, ultimate) adjustment of the lowvoltage arm capacitance of the voltage divider, which takes the value: As preliminary calculations show that the use of the maximum adjustment of the lowvoltage arm capacitance allows reducing the maximum value of the systematic error of the voltage divider by more than 2 times: However, this maximum value Δ A , in its turn, can again be reduced by almost 2 times by choosing the optimal parameter value 0 1 < < Θ opt . The task of the work is to search for this optimal value Θ opt , as well as to study the dependence of the «minimized» error of the voltage divider on a number of factors.
It should be noted that the optimized value of the low voltage arm capacitance of the voltage divider can be deter mined by the ratio: and its amplitudefrequency characteristic is determined by the expressions (1)-(4) when substituting Θ Θ = opt . As a model of capacitive components' nonidentity of the highvoltage arm of the voltage divider, a symmetric «triangular» distribution is used [6]. For this distribution the maximum deviations of the capacitances C i from C 0 are characterized by the ratio: where Δ C -the parameter specified in this research, which can take values in the range from 0 to 0.2. All the above formulas form a mathematical background for the research. In general, the graph of the systematic error of the capacitiveresistive voltage divider is a multi modal function containing both a positive and a negative maximum. The task of the research is to find a value of the parameter Θ, at which the absolute value of the positive and negative maximum will be the same. Such an adjustment of the lowvoltage arm is called optimal.
This problem is solved using linear programming. The key fragment of the program for finding the optimal pa rameter value Θ is shown in Fig. 2. Programming is performed with a help of the Mathcad software [10]. The program works as follows. First, the program searches for the maximum of the systematic error curve. Then, the program searches for the minimum of the systematic error curve. The program then compares the absolute values of the maximum and minimum. If these values are different, the variable Θ increments 0.0001 and the search loop repeats. The loop will end when equality is reached between the absolute value of the maximum and minimum of the studied function. The program allows finding the value of Θ opt accurate to the fourth digit after the decimal point. The results of calculations obtained with a help of this program are given in the next section.

Research results and discussion
The calculations were performed for various values of the divider's voltage ratio (K = 10 1 ; 10 2 ; 10 3 ; 10 4 ; 10 5 ) and for various values of the maximum deviation of the highvoltage arm capacitances from the average value (Δ C = = 0.01…0.20). Under such conditions, authors obtained sur face graphs of the variable Θ opt (refer to Fig. 3) and the systematic error Δ A (refer to Fig. 4). In Fig. 3 and Fig. 4, for the divider's voltage ratios K the logarithmic scale is used (the orders of magnitude of the divider's voltage ratio are plotted along the axis).
The surface graph in Fig. 3 depicts what value a vari able Θ opt should have so that for given K and Δ C the absolute values of the maximum and minimum of the systematic error are the same.
The surface graph in Fig. 4 depicts that with an increase in the Δ C parameter value, almost independently of the value of the divider's voltage ratio K, the value of the amplitude error Δ A increases in a parabolic dependence on Δ C . The graphs in Fig. 3 and Fig. 4 summarize a huge array of computational data.
Let's show in more details one of the results of optimiz ing the amplitudefrequency characteristics of the voltage divider in Fig. 5. This graph shows the dependence of the systematic error Δ A (%) on the generalized parameter g. The shape of the curve is practically the same in the entire studied range of parameters Δ C , K, Θ opt . Only the absolute values of the maximum and minimum differ. . and K = 10 2 . Using the above search algorithm, a value Θ opt = 0 656 . is obtained. With this optimal value the ab solute values of the maximum and minimum amplitude errors are the same and equal to 0.01495 %.
The further development of the theory of voltage di viders is promising due to the fact that this category of highvoltage scale transducers has the potential to become mandatory for determining the quality parameters of elec tric energy directly at high voltage. One of the catalysts for this may be the intensive development of the Smart Grid concept, which requires new, more advanced means of monitoring the quality of electric power [11,12]. Therefore, experimental and theoretical studies aimed at reducing the error of broadband voltage dividers are important.

Conclusions
As a result of the performed research, the possibility of optimizing the amplitudefrequency characteristics of a broadband capacitiveresistive voltage divider by varying the value of its lowvoltage arm capacitance is shown.
The optimization parameter of the lowvoltage arm capacitance value of the voltage divider in the entire range of the studied parameters K=10 1 -10 5 , Δ C = − 0 0 2 . can be characterized by a constant value Θ opt = 0 656 . .

ISSN 2664-9969
The dependence of the systematic error of the opti mized voltage divider on the dimensionless frequency parameter g is universal in form with the difference in the maximum ultimate values of Δ A .
For the values of divider's voltage ratio in the range 1 10 < < K , an additional research is required to optimize the amplitudefrequency characteristic.