Design of a High-Precision Self-Balancing Potential Transformer Calibrator

Abstract

Potential transformers are vital for measuring and protecting the power grid. Their accuracy and reliability directly impact the stability and security of the power system. To address the issues with traditional high-precision potential transformer calibrators, such as cumbersome operation and low efficiency, a high-precision potential transformer calibrator has been developed. The calibrator is based on an embedded system architecture of FPGA and ARM. It uses a high-precision current comparator along with feedback control technology. By monitoring and adjusting the error feedback voltage, it can perform the automated calibration of potential transformers with an accuracy class of 0.0001. The measurement ranges from 0.00001% to 200.0%. This study can be adapted to meet the development needs of modern digital measurement systems.

1. Introduction

Potential transformers are critical components of the power grid’s measurement and protection system. Their accuracy and reliability directly impact the stability, efficiency, and safety of the entire power system [1]. It is essential to inspect and calibrate power transformers regularly to ensure their accuracy and performance meet the operational requirements of the power system [2]. However, for high-precision potential transformers (accuracy class 0.001 and above), error measurements still rely on traditional electrical-type transformer calibrators with a zero indicator. These calibrators can calibrate transformers from accuracy class 5 to 0.0001, but the processes are complex, requiring intricate wiring and repeated adjustments for accuracy and high demands for the sensitivity, accuracy, and stability of the balance indicator [3]. This results in inefficiency and high prices. Therefore, it is necessary to update traditional transformer calibrators to align with the advancement of modern digital measurement systems.

Transformer calibrators are special devices for calibrating and testing various types and levels of potential and current transformers in the laboratory and the field. By applying the same voltage/current on the primary side of the standard and test power transformers, and measuring the difference in the output value of the secondary side between the standard power transformer and test power transformer with a transformer calibrator, the amplitude and phase errors of the transformer under test can be calculated. The amplitude error is a relative error, expressed in %, and the phase error is expressed in minutes (′) or radians (rad) [4].

The electrical-type calibrator is the earliest use of the first generation of calibrators; most can only calibrate 0.1 level of the subsequent transformers. At the end of the 1960s, Zhao Xiu Min developed their own comparative rituals electrical-type calibrator, which can be calibrated to a 10~0.01 level of the transformer, and which has laid the foundation for subsequent transformer calibration technology. An electrotechnical-type calibrator uses the compensated balanced bridge principle, and the key components are conductance and capacitance. By adjusting the amplitude and phase of the error current through the adjustment disk, its error resolution is very high, the error data are relatively reliable, and it has a stable performance, which is especially suitable for calibrating high-precision transformers, but in the process of using it, the operation is cumbersome, the wiring is complicated, and it needs to be adjusted repeatedly to ensure the accuracy; thus, the work efficiency is very low, and the requirements for the balance indicator’s sensitivity, precision, stability, and so on are also very high [5].

The second generation of electronic calibrators, high-efficiency electronic calibrators, were introduced in the 1980s. Electronic and digital analog circuits are used to measure the measured transformer’s ratio error and phase displacement [6].

With the development of microprocessors and computers, the third generation of digital transformer calibrators uses digital acquisition and analysis. It is gradually maturing, for example, by using DSP [7], FPGA [8], microcontrollers [9], and Labview [10]. This type of calibrator has a higher degree of automation and is easy to operate. Electrotechnical and electronic calibrators participate in the operation of electricity, which is an analog signal. A digital calibrator on the reference signal and the difference signal through the ADC sampling participate in the operation of the digital signal [11]. In the digital transformer calibrator ADC signal acquisition and data processing process, a large number of chips and electronic circuits improve the automation at the same time, but also increase a variety of noise interference, restricting the calibration accuracy of the digital calibrator, which generally can only be 0.01 level and the following accuracy level of the transformer calibration work.

The measurement technology of calibrators developed from analog instruments, then digital and microprocessors and has reached the level of the so-called virtual instrument [12]. A digital calibrator for a Labview-based virtual instrument is widely used. A virtual instrument is a product of the combination of modern computer technology and instrumentation technology. A virtual instrument-type calibrator combined with a high-precision data acquisition card can directly collect the measured signals and reference signals, convert them into digital signals by ADCs and then process them with software to obtain the results for the core of the software algorithms of the computer’s function, the realization of the various functions of the calibrator. Virtual instrument-type calibrator hardware is simple, the development of data acquisition cards on the market is relatively mature, it can be utilized directly, and through the Personal Computer (PC), can realize the various functions of the instrument, while the operation is simple and convenient [13].

Based on digital transformer calibrators, in addition to the application of virtual instruments, intelligent transformer calibrators have also been developed. An intelligent transformer calibrator not only uses computer technology but also has a computer communication interface and human–computer interaction unit, with wireless Bluetooth transmission, and directly available host computer management [14]. The biggest advantage of the intelligent calibrator is that it reduces the labor force, does not require too much human operation, and is able to measure and adjust automatically, including data recording, data processing, and certificate printing, which improves the office level [15].

The use of transformer calibrators can be divided into two types: differential measurement and direct comparison. The direct comparison-type transformer calibrator appeared earlier than the differential measurement-type, but because the difficulty and cost of the technical realization are very large, it was gradually replaced by the differential measurement-type. Differential measurement methods and products are now quite mature. Most of the traditional electrical calibrators and digital calibrators for analog outputs are based on this principle [16].

A differential measurement calibrator measures the potential transformers and standard transformer secondary side voltage or current input calibrator differential circuit, and the secondary operating voltage or operating current to compare the results of vector decomposition, according to the in-phase components can be obtained the ratio error, quadrature components can be obtained the phase displacement [17]. With a differential measurement-type calibrator, because the measurement is a differential pressure or current, the calibrator itself has little effect on the measurement results, so as long as the standard transformer is two accuracy levels higher than the measured transformer, and the resolution of the transformer calibrator meets the requirements, the transformer calibrator can be used to test all levels of the transformer; this method is not only highly accurate, but the value can also be traceable. In particular, the electrotechnical calibrator can calibrate transformers from accuracy classes 5 to 0.0001 [18]. Based on the principle of measuring the difference between the transformer calibrator according to the circuit characteristics can be divided into the Sillen-Albert line, comparator measurements, and differential measurements. The Sillen-Albert line calculation relationship is complex, and the large capacitor shunt effect makes the error increase, and is not suitable for precision measurement, so it has been gradually eliminated [19].

Current comparators were first introduced by Zwiring in 1937. The traditional current comparator needs to adjust the conductance and capacitance so that the comparator detects the winding voltage as 0. The current comparator is essentially an ampere-turn balance indicator, which flows the secondary currents of the standard transformer and the transformer under test in the opposite direction into the comparator winding and judges the balance state of the current comparator by manually adjusting it and detecting whether the coil’s finger-zero meter is 0 [20]. However, in practice, due to the presence of winding leakage impedance and distributed capacitance, the circuit will produce a capacitive error, and there is the need for manual adjustment, as the efficiency is not high. In the 1960s, the proposed compensated current comparator [21] well overcame the difficulties in calibration, is widely used, and has a variety of circuit types. The traceability method of the calibration system of the National Center for High Voltage Measurement (NCHVM) is based on the compensated current comparator [22]. The literature [23] developed a current comparator that utilizes an electronic circuit to compensate the excitation current of the iron core, which detects the proportional winding ampere-turn balance and automatically compensates the excitation current of the current transformer through the electronic circuit, which greatly improves the accuracy. The current comparator does not require the manual adjustment of the ampere-turn balance, the wiring is very simple in use, and the technique has been used for the online calibration of power transformers [24].

The differential branch measurement, first proposed by Silsbee, reverses the secondary side currents of the standard transformer and the transformer under test at the same time in the differential branch in the middle of the two transformers, and then detects the voltage drop across the branch load and measures the error. The differential branch measurement is one of the widely used transformer calibration circuits today because of its simple circuit structure and small introduced error. Many differential branch measurement methods have been developed over the years. Two systems based on this principle were previously developed by the InstituteNikolaTesla based on a standard CT [25] and an electronic data acquisition system [26].

In summary, the calibrators utilizing the differential measurement method have the following advantages:

(1)

Small error and high accuracy.

(2)

Strong anti-interference ability and high stability.

(3)

Wide measurement range, applicable to different calibration needs.

(4)

Simple structure, few components, no need for complicated calculation and derivation.

(5)

Low circuit complexity, low requirements for precision, lower cost.

2. Principle of Measurement of Transformer Error

Transformer calibrations are mainly conducted using the comparison method. The method is typically determined by comparing the voltage (or current) between the measured potential transformers (or current transformers) and the standard transformers using transformer calibrators. The transformer calibrators perform a vectorial decomposition of the two voltages (or currents) to obtain the transformers’ ratio errors and phase displacements. Figure 1 illustrates the calibration method for potential transformers [27].

The standard potential transformer (${VT}_N$) is powered by the regulator ($T_1$) and booster ($T_2$). The primary winding of the standard potential transformer and the primary winding of the measured transformer (${VT}_X$) are connected in parallel. The secondary winding of the standard transformer and the secondary winding of the measured potential transformer are connected in reverse series to obtain the differential voltage, which is then sent to the measurement circuit of the transformer calibrator. The secondary voltage of the standard potential transformer is also fed into the transformer calibrator. The two voltages are measured separately in the calibrator by two sampling resistors. After amplification, switching, filtering, and other processing, the in-phase and quadrature voltages are extracted. Then, the transformer calibrator displays the ratio error and phase displacement of the measured potential transformer. Figure 2 shows the vector diagram of the voltage error.

Use the length of OA as the unit length 1 in Figure 2. In this case, f = BC and δ = CA. Here, δ represents the phase displacement between the measured voltage $({\dot{U}}_X$) and the standard voltage $({\dot{U}}_N$), and θ is the phase displacement between the differential voltage ($∆\dot{U}$) and the measured voltage $({\dot{U}}_X$). The calculation of the error is

$${\displaystyle \frac{∆\dot{U}}{{\dot{U}}_N}}={\displaystyle \frac{∆U(-cos\theta +jsin\theta )}{U_N}}=f+j\delta$$

(1)

In fact, the definition of the transformer error, as given in Equations (2)–(4), indicates the error of the calibrator according to the regulation.

$$\epsilon ={\displaystyle \frac{∆\dot{U}}{{\dot{U}}_N}}=\overrightarrow{BA}$$

(2)

$$f=\left|\overrightarrow{OA}\right|-\left|\overrightarrow{OB}\right|=\sqrt{1+{\epsilon }^2-2\epsilon cos\theta }-1$$

(3)

$$\delta =\mathrm{a}\mathrm{r}\mathrm{c}sin{\displaystyle \frac{\epsilon sin\theta }{\sqrt{1+{\epsilon }^2-2\epsilon cos\theta }}}$$

(4)

Thus, Equations (5) and (6) show the effect of the calibrator error on the actual ratio error and phase displacement of the potential transformer, respectively.

$$1-cos\delta \approx {\displaystyle \frac{\delta }{2}}$$

(5)

$$\delta -sin\delta \approx {\displaystyle \frac{{\delta }^3}{6}}$$

(6)

When using the comparison method, if the accuracy class of the reference transformer is two classes higher than that of the measured transformer and the resolution of the transformer calibrator meets the requirements, then the calibrator will have minimal influence on the measurement results. In this case, the calibrator can be used to test transformers of all accuracy classes [28].

3. Principle of the High-Precision Transformer Calibrator

Figure 3 shows the designed transformer calibrator operating on the current comparator principle. The calibrator takes the differential voltage ($∆U$) from the measured potential transformer ($U_{TX}$) and the standard potential transformer ($U_{TN}$) and sends it to the current comparator coil. The asterisk (*) on the coil is the dotted terminal. The dotted terminal represents the existence of a coupling relationship between the coils. The reference voltage ($U_{REF}$) from the reference transformer’s secondary side is input into the in-phase and quadrature channels. The in-phase voltage goes through the digital-to-analog converter (DAC), whereas the quadrature voltage passes through the phase shifter, obtains a 90° shift, and goes through the DAC. Then, both voltages are fed into the adder. The sum voltage from the adder ($∆U^{\prime }$) is then sent to the current comparator for comparison with the actual differential voltage ($∆U)$. The current comparator output is the differential voltage ($U_Z$) between the sum voltage and the actual difference voltage. After the voltage $U_Z$ is amplified, filtered, and conditioned, the voltage is sampled by the microcontroller unit (MCU) through the analog-to-digital converter (ADC) to determine the balance state of the current comparator’s ampere-turns.

When the voltage $U_Z$ is not 0, it means that the current comparator is unbalanced. The MCU in the calibrator drives the network coding of the DACs using the unbalanced voltage $U_Z$, so it controls the analog signal to balance with the digital signal. It can internally output an in-phase voltage and a quadrature voltage proportional to the reference voltage signal [29]; that is, according to the voltage $U_Z$ sampling results, the two DAC channels control the amplitude of the in-phase voltage and quadrature voltage, which can be continuously adjusted until the voltage $∆U^{\prime }$ is equal to the voltage $∆U$ and the voltage $U_Z$ is equal to 0. In this way, the calibrator forms a self-balancing and closed-loop feedback control system. When the voltage $U_Z$ is 0, the MCU processes the data code from the DACs to produce the final ratio error and phase displacement.

From the above, it can be seen that in the design of the high-precision calibrator, the core lies in the closed-loop feedback control to achieve stable and accurate output. The feedback system is mainly divided into two parts: the sampling results of the voltage $U_Z$ and the output results of the DACs, which are controlled by the MCU. Figure 4 shows the closed-loop feedback control system.

In the feedback system, the voltage $U_Z$ is used as the feedback signal. The voltage $U_Z$ is first amplified K times by a programmable amplifier and then fed into the ADC through a filter. To ensure the accuracy of the sampled signal and enhance the filtering effect, a two-stage second-order low-pass filter is used to sample the voltage. The transfer function of the filter is

$$G_F\left(s\right)={\left({\displaystyle \frac{{{\omega }_n}^2}{s^2+2\xi {\omega }_n·s+{{\omega }_n}^2}}\right)}^2$$

(7)

After the system sampling is completed, the MCU selects the one-half look-up table method based on the sampling results to update the outputs for the DACs of the in-phase and quadrature channels, respectively. During the processes of the ADC and DACs, the MCU drives the network coding using the unbalanced voltage and controls the analog signal to balance with the digital signal. It is internally capable of outputting an in-phase and a quadrature voltage proportional to the reference voltage.

During sampling, the ADC has a range of sampling voltages ($U_{RANGE}$). The voltage $U_Z$ is used as the unbalanced voltage, and the ratio of the sampling voltage $U_{\mathrm{Z}}$ to the voltage $U_{RANGE}$ is used to represent the effect of the unbalanced voltage $U_Z$ on the output results of the two DACs. The in-phase voltage $∆{U_f}^{\prime }$ and quadrature voltage $∆{U_{\delta }}^{\prime }$ are proportional to the reference voltage, so we set the proportionality adjustment coefficients of the in-phase and quadrature channels to be $k_1$ and $k_2$, respectively. Equations (8) and (9) show the in-phase and quadrature voltages, respectively. And $A_1$ and $A_2$ range from 0 to 1. In Equation (9), $1/(1+\tau s)$ represents the transfer function of the 90° phase shifter in the quadrature channel. Equation (10) shows the result of adding the two voltages. And Equation (11) shows the relationship between the voltage $U_Z$ and the voltage $∆U$.

$$∆{U_f}^{\prime }=A_1·U_{REF}$$

(8)

In Equation (8), $A_1=k_1·\left({\displaystyle \frac{K·G_F\left(s\right)·U_Z}{U_{RANGE}}}\right)$.

$$∆{U_{\delta }}^{\prime }=A_2{·U}_{REF}$$

(9)

In Equation (9), $A_2=k_2·\left({\displaystyle \frac{K·G_F\left(s\right)·U_Z}{U_{RANGE}}}\right)·{\displaystyle \frac{1}{1+\tau s}}$.

$$∆U^{\prime }={\sqrt{{A_1}^2+{A_2}^2}·U}_{REF}=A{·U}_{REF}$$

(10)

In Equation (10), $A=\sqrt{{k_1}^2+{\left({\displaystyle \frac{k_2}{1+\tau s}}\right)}^2}·{\displaystyle \frac{K·G_F\left(s\right)·U_Z}{U_{RANGE}}}$.

$$G\left(s\right)={\displaystyle \frac{U_Z}{∆U}}={\displaystyle \frac{1}{1+G}}$$

(11)

In Equation (11), $G=\sqrt{{k_1}^2+{\left({\displaystyle \frac{k_2}{1+\tau s}}\right)}^2}·{\displaystyle \frac{K·G_F\left(s\right){·U}_{REF}}{U_{RANGE}}}$.

In Equation (7), ${\omega }_n$ is the natural frequency, which indicates the undamped oscillation frequency of the system. $\xi$ is the damping ratio, which indicates the degree of damping of the system and affects the stability and response speed of the system. ${\omega }_n$ = 862.58 and ξ = 5.80 × 10⁵. The phase shifter is a quadrature phase shifter, τ = 0.0032, with the programmable amplifier amplification K of 1–64³. The device uses a standard reference voltage of 100 V or 57.7 V, which is converted through a potential transformer and operational amplifiers to a range of 0.7 V to 7.2 V. The voltage $U_{RANGE}$ can be selected as either ±10 V or ±5 V, depending on the application. And the values of $k_1$ and $k_2$, determined by $A_1$ and $A_2$, fall within the range of 0 to 1.

So, from Equation (11), the system has five zeros, including two pairs of conjugate complex roots and one real root, and all of them lie in the left half-plane. The denominator of the transfer function after the system is simplified has one more nonlinear component than the numerator, and the nonlinear component is introduced by the proportionality adjustment coefficient of the in-phase and quadrature voltages. The Equation also contains high powers, which make the system show highly sensitive behavior in some specific regions, and there are several similar pole values, but the real part of these similar pole points is all negative so that the number of system poles is larger than the number of zeros, and all the poles are located in the left half-plane. The dynamic response of the system under these pole conditions does not show divergence or exhibit unstable behavior, and the closed-loop feedback control system is a stable system. The calibrator can be based on the voltage $∆U$, provide feedback control from the voltage $U_Z$ to the voltage $∆U^{\prime }$, and finally make the voltage $U_Z$ equal to 0 or close to 0. So, the calibrator can output stable and accurate results.

4. Hardware Design of the Calibrator

The system designs hardware for each link based on design principles and transfer functions, to achieve self-balancing and closed-loop feedback control. The calibrator utilizes a robust embedded system, which is under the control of the STM32 and FPGA, to output the results. The calibrator not only increases the level of automation but also enhances the accuracy and stability of the device [30]. Figure 2 shows the diagram of the potential transformer calibrator.

The MCU selects the 32-bit high-performance processor STM32F407 with Cortex-M4, which clocks up to 168 MHz. The self-balancing transformer calibrator system requires high processing accuracy for weak signals and precise control timing. To meet these requirements and improve the ARM processing speed, we use the STM32 + FPGA embedded system. The FPGA handles the control timing and internal and external clock synchronization, while the STM32 manages the main program algorithm, data precision sampling, and feedback control parameter adjustment. The STM32 also handles the external communication for automatic measurements and human–computer interaction. The FPGA (LCMXO2-1200HC-4TG100C) has a quick startup time, built-in hardware acceleration logic, and up to 6864 LUT4, which helps reduce the application costs and meets the design requirements [31].

4.1. Regulation Circuit of the Current Comparator Output Voltage

The key to the high-precision potential transformer calibrator is to provide feedback based on the sampling results of the voltage $U_Z$. For potential transformers of accuracy class 0.001 and above, the voltage $U_Z$ may be at the microvolt level [32]. To accurately sample these weak voltages, the system needs to be designed with amplification and filtering circuits so that the voltage $U_Z$ is within the voltage range of the ADC.

In the amplifier circuit, the ADG409 serves as the key component, integrated with the operational amplifier OP07 and controlled by the MCU to achieve automatic gain adjustment. Figure 5 shows the amplifier circuit.

When the gain control signals are different, the internal switch of the ADG409 states varies accordingly, and combined with external integrated operational amplifiers, the gain can be controlled [33].

The circuit has a fast response speed and high linearity and is not affected by the analog switch-on resistance. In the measurement process, because the voltage measurement ranges from 10⁻⁶ to 10 V, to ensure the accuracy of the sampling signal and to ensure that the ADC can sample different signals, the programmed gain circuit is set, which includes the three stages of the above amplification circuit.

To improve the circuit’s ability to eliminate the third and fifth harmonic disturbances in the voltage $U_Z$, the system is designed with a second-order voltage-controlled low-pass filter, which is illustrated in Figure 6. The circuit is also designed with a bidirectional limiter circuit, which can filter and isolate the output signal for protection, and the amplification of the filter is designed to be 1 [34]. The filter circuit will produce a slight phase shift on the sampled signal, so the phase is adjusted in the hardware circuit by a sliding varistor, which can improve the accuracy of the sampled signal.

Equations (12) and (13) are the cutoff frequency and transfer function of the circuit, respectively. According to the designed parameters, $R_1=24 \mathrm{K}\mathsf{\Omega }$, $R_2=56 \mathrm{K}\mathsf{\Omega }$, $R_3=24 \mathrm{K}\mathsf{\Omega }$, $R_4=47 \mathrm{K}\mathsf{\Omega }$, $C_1=0.1 \mathsf{\mu }\mathrm{F}$, and $C_2=0.01 \mathsf{\mu }\mathrm{F}$, the cutoff frequency can be obtained as 43 Hz and the amplification is 1.

$$f_Z={\displaystyle \frac{1}{2\pi \sqrt{R_1{R}_2{C}_1{C}_2}}}$$

(12)

$$G_1\left(s\right)=-{\displaystyle \frac{R_3}{R_1}}·{\displaystyle \frac{1}{1+s{R}_1{R_{3}C}_2\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right)+s^2{R}_1{R_3{C}_{1}C}_2}}$$

(13)

4.2. Design of the Data Sample System

The input of the ADC is analog and the output is digital, so its selection needs to be determined according to the size and nature of the input analog signal and the required accuracy. The system’s sampling circuit uses a 16-bit, 8-channel ADC, the AD7606, powered by a 5 V supply. The logic level on the ADC’s RANGE pin controls the analog input range (±5 V or ±10 V) [35], and the ADC’s REFIN/REFOUT pins have a built-in 2.5 V reference voltage source that allows the converter to choose to use either an internal 2.5 V REF reference or an external 2.5 V REF reference [36]. The system uses the external reference voltage, so an external reference voltage source, the AD780, was chosen to drive the REFIN/REFOUT connector on the AD7606.

The output of the ADC is encoded in a binary complement, and Equation (14) shows the transfer function characteristics.

$$CODE={\displaystyle \frac{VIN}{RANGE}}\times \mathrm{32,768}\times {\displaystyle \frac{REF}{2.5V}}$$

(14)

The sampling circuit utilizes an SPI serial bus for control, with the bus itself being electrically isolated by a four-channel digital isolator (π141M61). The isolator enhances the circuit’s reliability and stability. To achieve rapid readout and high data transfer rates, the system selects 128 signal measurement points. The digital isolator supports a maximum data transfer rate of 10 Mbps, ensuring compatibility with the SPI data flow requirements of the system.

4.3. Construction and Injection of Weak Signals

In the system, instead of sampling the differential voltage and operating voltage, a weak voltage is constructed to balance with the actual differential voltage to indirectly obtain the voltage errors. As shown in Figure 7, the circuit is based on electronics and software programming techniques, including weak voltage construction and injection. The voltage construction includes voltage generation and conditioning, as well as summation, and is then injected into a high-precision current comparator. The circuit is based on the Cartesian coordinate system. The secondary voltage of the reference potential transformer is passed through a 100/3 potential transformer as the reference voltage for the DACs. The reference voltages of the two DACs are one in phase with the voltage of the reference potential transformer, and one phase-shifted 90° out of phase with the reference potential transformer. The MCU controls the RMS values of the in-phase and quadrature voltages $V_{out1}$ and $V_{out2}$ through the output voltage of the current comparator to achieve voltage ratio adjustment and adds them together to obtain a voltage with an adjustable phase range of 0 to 360° and the amplitude of the voltage $V_{out3}$.

The input to the DAC is the reference voltage and the output is a voltage signal proportional to the reference voltage. Both the input and output are analog, and the selection of the DAC is determined by the size and nature of the analog signal, but the internal operation is controlled by the digital signal. The MCU automatically sequences the serial address data according to the clock cycle, multiplies them by the gain, and sends them to the DACs to complete the conversion. The DACs use two 16-bit serial input DACs (AD7849), which have excellent differential linearity and monotonicity guaranteed to be 16 bits. Equations (15) and (16) represent the output analog voltage and reference voltage of the DAC, respectively.

$$V_{out}=p{·V}_{REF+}$$

(15)

$$V_{REF+}=U_{REF}\times \left({\displaystyle \frac{U}{U_{REF}}}\times 100\%\right)\times p_u$$

(16)

In Equation (15), p is the proportionality factor for the DAC.

In Equation (16), $U_{REF}$ represents the secondary standard reference voltage, typically 100 V or 57.7 V. $U/U_{REF}\times 100\mathrm{\%}$ denotes the operational voltage percentage signal, with the potential transformer calibration range from 20% to 120%. $p_u$ is the total ratio after passing through the potential transformer and operational amplifier, and the voltage $V_{REF+}$ ranges from 0.7 V to 7.2 V.

The output voltage of the DAC depends on its reference voltage $V_{REF+}$ and the digital reference value returned based on the sampling results (D), where D determines the magnitude of the adjustment factor $p$. The reference voltage $V_{REF+}$ is a sinusoidal signal with an industrial frequency of 50 Hz. The STM32 controls the digital value D to output a series of smoothly adjustable sinusoidal voltages. One of the DACs is in phase with the voltage $V_{REF+}$, while the other DAC has a 90° phase difference relative to the voltage $V_{REF+}$. Equations (17) and (18) represent the two voltages, respectively. In Equation (18), 1/(1 + τs) is the transfer function of the integrator. Equation (19) represents the sum of the two voltages. When the output voltage of the current comparator is 0, Equations (17) and (18) can be used to obtain the ratio error and phase displacement of the measured transformer.

$$V_{out1}=p_1{·V}_{REF+}$$

(17)

$$V_{out2}=p_2{·V}_{REF+}·{\displaystyle \frac{1}{1+\tau s}}$$

(18)

$$∆U^{\prime }=V_{REF+}·\sqrt{{p_1}^2+{\left({\displaystyle \frac{p_2}{1+\tau s}}\right)}^2}$$

(19)

5. Software Design of the Calibrator

The system software comprises two main units: the STM32 software design unit and the FPGA software design unit. The version number of the STM32 software is STM32F407VET6. The STM32 is primarily responsible for data acquisition, data analysis, control of DAC data addresses, and communication modules. These communication modules include LCD, RS232, RS485, Bluetooth, and Ethernet modules, enabling reception of remote-control commands, transmission of measurements, and execution of results, thus facilitating enhanced human–machine interaction. The FPGA module is primarily used for controlling the timing, internal and external clock synchronization, and latch control of the amplified data. The version number of the FPGA software is LCMXO2–1200HC–4TG100C.

The program adopts a modular and structured design, with all components uniformly scheduled or executed via interrupts within the main program. Figure 8 shows the main program flowchart.

Firstly, the system undergoes initialization procedures, which include configuring the main clock frequency, setting up timers, initializing the DAC, and initializing the serial ports. According to the voltage output from the current comparator, the STM32 controls the address of the DAC so that the sum of the in-phase and quadrature voltages is balanced with the actual difference voltage, at which point the current comparator outputs a voltage of 0. Then, the calibrator processes and analyzes the data in the DACs. The core functionalities of the system are mainly divided into sampling and measurement modules.

• Sampling module: The software uses interrupt mode to measure the input voltage and complete the data calculation and processing. The two voltages, the current comparator output voltage, and the actual operating voltage are converted into digital quantities by the ADC. The MCU controls the ADC according to the clock signal from the FPGA to sample the voltages with 128 points in a full cycle, thus obtaining the discrete digital signals [37].
• Measurement module: The STM32 uses the one-half query method to quickly find out the voltage that can make the current comparator at zero flux. The measurement module includes the control of the output voltage of the DACs and the measurement of the output signal of the current comparator, the selection of amplification, and so on.

6. Results

To verify the validity of the self-balancing calibrator, the experiment was conducted using the transformer calibrator rectifier setup with a given error applied to the current comparator [38].

The experiment used a transformer calibrator seized the entire device as the standard device and used it to measure the accuracy class 2 transformer calibrator. The technical parameters of the transformer calibrator seized the entire device are shown in

Table 1. The technical parameters of the transformer calibrator seized the entire device.

Equipment Name	Manufacturer	Model Number	Factory Number	Accuracy Class *
Transformer calibrator seized the entire device	Ningbo Sanwei Electric Measurement Equipment Co., Ltd. (Ningbo, China)	HESW-2	1,706,006	0.1

* Errors given by the transformer calibrator seized the entire device.

.

6.1. The Transformer Calibrator Test Results

Figure 9 and Figure 10 show the schematic and physical diagrams of the experimental setup, respectively [39]. Ten sets of tests were conducted.

Table 2. The results of the self-balancing calibrator.

$\mathit{\epsilon }$ 1 (10−6)	$\mathit{f}$ 2 (10−6)	$\mathit{\delta }$ 3 (10−6)
1	1.01	0.99
2	2.01	2.03
3	3.04	3.05
4	4.02	4.07
5	5.08	5.10
6	6.08	6.12
7	7.10	7.13
8	8.13	8.15
9	9.14	9.15
10	10.17	10.19

¹ Errors given by the transformer calibrator seized the entire device. ² Ratio errors for the transformer calibrator. ³ Phase displacements for the transformer calibrator.

and Figure 11 show the results and graph of the self-balancing calibrator, respectively.

6.2. Uncertainty Analysis of the Transformer Calibrator

In this paper, in addition to the measurement of the transformer calibrator, the uncertainty of its measurement results is also evaluated.

6.2.1. Measurement Model

$${∆f}_X=f_X-f_N$$

(20)

$${∆\delta }_X={\delta }_X-{\delta }_N$$

(21)

In Equations (20) and (21), $f_X$ is the value of the in-phase component of the calibrator under test; $f_N$ is the value of the quadrature component of the standard device; ${\delta }_X$ is the value of the in-phase component of the calibrator under test; and ${\delta }_N$ is the value of the quadrature component of the standard device.

6.2.2. Measurement Uncertainty Source Analysis

The main points for analyzing the uncertainty elicitors are as follows:

• The standard uncertainty component introduced by the measurement repeatability of the calibrator of the transformer under test.
• Uncertainty components of the power supply influence elicitor.
• Uncertainty components induced by external electromagnetic interference.
• Uncertainty components introduced by the transformer calibrator seized the entire device.
• Uncertainty components introduced by the load of the calibrator.
• Uncertainty components introduced by changes in the error of the transformer calibrator seized the entire device during the calibration cycle.
• Standard uncertainty components introduced by the resolving power of the transformer calibrator under test.

In response to these points, the different components are specified and rated below.

6.2.3. Evaluation of Standard Uncertainty

• Determination of basic error limits for transformer calibrator

Equations (22) and (23) represent the allowable values of the basic errors of the in-phase and quadrature components of the transformer calibrator, respectively.

$${\epsilon }_f=K\left(f_i·a\mathrm{\%}+\delta ·a\mathrm{\%}+D_f\right)=K·a\mathrm{\%}\left(f_i+0.1R_k\right)$$

(22)

$${\epsilon }_{\delta }=K\left({\delta }_i·a\mathrm{\%}+f·a\mathrm{\%}+D_{\delta }\right)=K·a\mathrm{\%}\left({\delta }_i+0.1R_k\right)$$

(23)

${\epsilon }_f$—Limit of the ratio error of the basic error of the tested transformer calibrator under test.

${\epsilon }_{\delta }$—Limit value of the phase displacement of the basic error of the tested transformer calibrator.

$K$—Instrument constant, taken as a multiple of a range.

$f_i$—The absolute value of the ratio error measurement.

${\delta }_i$—The absolute value of the phase difference measurement.

$D_f$,$D_{\delta }$—In measuring the disk’s minimum index value or quantitative value, the regulations specify D = 0.1$R_{k}a\mathrm{\%}$, $R_k$ for the full-scale value; $a\mathrm{\%}$ for the calibrator accuracy-level index.

2.

Uncertainty components introduced by measurement repeatability

Standard uncertainty can be evaluated by the values obtained from continuous measurements. At the ambient temperature of 22 °C and relative humidity of less than 80% of the measurement conditions, the choice of using the transformer calibrator seized the entire device at the accuracy class of 0.1 as the standard device was evaluated and used to measure the accuracy class 2 transformer calibrator. The transformer calibrator seized the entire device output secondary voltage is 100 V, the percentage meter is 100%, the reference in-phase component is 1 × 10⁻⁷, and the reference quadrature component is 1 × 10⁻⁷. The experiment was performed on the transformer calibrator with 10 repetitive measurements and was rated using the Class A methodology. At the time of calibration, the uncertainty introduced by electromagnetic interference, power fluctuation, and other factors has been included in $u_A$, and these components will not be evaluated separately.

3.

Uncertainty introduced by the standardizer

The experiment used the transformer calibrator seized the entire device as the standard device at the accuracy class 0.1. The actual error of the standard is not greater than 1/3 of the basic error limit of the inspected transformer calibrator, and the error limit of the inspected calibrator is ε. Equation (24) shows the uncertainty component introduced by the standard.

$$u_{B1}={\displaystyle \frac{0.3\epsilon }{\sqrt{3}}}$$

(24)

4.

Uncertainty introduced by the load of the transformer calibrator

The transformer calibrator seized the entire device in access to the load of the transformer calibrator, and the error generated by the load shall not exceed the transformer calibrator seized the entire device’s basic error 1/3. Calibrator load on the measurement results of the impact of 1/10 of the examined calibrator error limit. This interval obeys the uniform distribution, distribution coefficient k = 3. Equation (25) shows the uncertainty component introduced by the load of the transformer calibrator.

$$u_{B2}={\displaystyle \frac{0.1\epsilon }{\sqrt{3}}}$$

(25)

5.

Uncertainty components introduced by changes in the error of the transformer calibrator seized the entire device during the calibration cycle

The change in the measurement standards in the calibration cycle error is not greater than the transformer calibrator seized the entire device error limit value of 1/3. Equation (26) indicates the measurement of each point in the calibration cycle due to error changes induced by the uncertainty component.

$$u_{B3}={\displaystyle \frac{0.1\epsilon }{\sqrt{3}}}$$

(26)

6.

Standard uncertainty components introduced by the resolving power

The standard uncertainty component introduced by the resolving power of the transformer calibrator obeys a uniform distribution, and the error limit is equal to half of the resolving power of the calibrator, and its uncertainty is expressed in Equations (27) and (28).

$$u_{B4}\left(f\right)={\displaystyle \frac{D_f}{2\sqrt{3}}}$$

(27)

$$u_{B4}\left(\delta \right)={\displaystyle \frac{D_{\delta }}{2\sqrt{3}}}$$

(28)

Since the uncertainty introduced by the repeatability of each measurement point is duplicated by the uncertainty introduced by the resolution, the larger of the two values is taken in synthesizing the standard uncertainty.

7.

Sensitivity factor

Equations (29) and (30) represent the calculation of the sensitivity factor.

$$c_{if}={\displaystyle \frac{\partial f}{\partial f_X}}$$

(29)

$$c_{i\delta }={\displaystyle \frac{\partial \delta }{\partial {\delta }_X}}$$

(30)

From Equations (21) and (22), we obtain $c_{if}=c_{i\delta }=1$.

8.

Synthetic standard uncertainty

Since the sensitivity coefficients are all 1 and all the inputs are independent or uncorrelated with each other, the synthetic standard uncertainty is shown in Equation (31).

$$u_C=\sqrt{{{u_A}^2+u_{B1}}^2{+u_{B2}}^2{+u_{B3}}^2{+u_{B4}}^2}$$

(31)

Table 3. The uncertainty analysis of the transformer calibrator.

Uncertainty Component	Uncertainty Type	Probability Distribution	$\mathbf{Inclusion} \mathbf{Factor} \mathit{k}$	Sensitivity Factor	Uncertainty (In-Phase Component)	Uncertainty (Quadrature Component)
$u_A$	A	normal distribution	3	1	7.27 × 10−9	6.89 × 10−9
$u_{B1}$	B	uniform distribution	$\sqrt{3}$	1	7.07 × 10−6	7.07 × 10−6
$u_{B2}$	B	uniform distribution	$\sqrt{3}$	1	2.36 × 10−6	2.36 × 10−6
$u_{B3}$	B	uniform distribution	$\sqrt{3}$	1	2.36 × 10−6	2.36 × 10−6
$u_{B4}$	B	uniform distribution	$\sqrt{3}$	1	2.89 × 10−8	2.89 × 10−8

demonstrates the results of the uncertainty analysis of the transformer calibrator.

9.

Evaluation of Extended Uncertainty

The extended uncertainty is calculated by multiplying the synthetic standard uncertainty by the inclusion factor, as shown in Equation (32).

$${U=ku}_C$$

(32)

In Equation (32), take the inclusion factor $k=2$, inclusion probability $p=95.45\mathrm{\%}$, and $u_C$ is the synthetic standard uncertainty. The extended uncertainty of the in-phase and quadrature components of the transformer calibrator was calculated to be 1.56 × 10⁻⁵.

In this paper, the uncertainty of the measurement results of the transformer calibrator is evaluated, and the standard uncertainty and the extended uncertainty are calculated, respectively. Among them, the standard uncertainty can be used as a component of the uncertainty assessment of the potential transformers it calibrates, and the uncertainty introduced by this calibrator can be taken into account when the uncertainty assessment of the transformers is carried out later. The extended uncertainty may be used to determine a reasonable inclusion interval for the calibration certificate or uncertainty report issued.

7. Conclusions

The error measurement of high-precision potential transformers typically relies on traditional electrical calibrators and zero indicators. However, this method is cumbersome due to complex wiring and low operational efficiency. And traditional digital calibration instruments primarily offer a narrow measurement range and lower calibration accuracy.

To address these challenges, this paper presents a high-precision, self-balancing potential transformer calibrator. It incorporates a high-performance ARM processor and high-accuracy current comparator, along with feedback control technology. The calibrator enables the automated calibration of potential transformers at an accuracy class of 0.0001, with a measurement range from 0.00001% to 200.0% and a resolution of 0.00001%. The system enhances the accuracy and efficiency through improved sampling and signal processing methods, making it suitable for modern digital measurement systems.

Furthermore, this paper introduces the main principles and design of the calibrator and analyzes the key technologies of the high-precision potential transformer calibrator based on the current comparator:

• The calibrator is combined with a high-precision current comparator to design a closed-loop feedback control system.
• The calibrator incorporates programmable amplifier circuits, hardware multi-stage filtering technology, and software filtering technology to improve the accuracy of the calibrator.
• The calibrator changes the traditional digital transformer calibrator to directly sample and calculate the differential voltage and operating voltage. It constructs a weak voltage balancing transformer error using the secondary voltage of the reference potential transformer as a reference, which uses an indirect way to obtain the error result.