Non-Contact Measurement Method of Phase Current Based on Magnetic Field Decoupling Calculation for Three-Phase Four-Core Cable

Abstract

As one of the important parameters characterizing the cable’s operation status, an accurate measurement of the current is particularly important for the cable’s reliable operation and status monitoring. Aiming at the problem that the sum of the current phasors of multiphase cables is 0 when running in a steady state, and that the traditional mutual inductance current measurement method cannot be used for the phase current measurement of such cables, or that the insulation layer needs to be damaged, this paper proposes a non-contact measurement method of three-phase cable current, based on the magnetic field decoupling calculation. When combined with the actual parameters of the three-phase cable, the analytical calculation model of the magnetic field distribution of the three-phase cable is established. The relationship between the output ratio of the sensor and the deflection angle is obtained through theoretical derivation, the magnetic field coupling coefficient matrix is determined, and the transfer relationship between the output of the magnetic sensor and the current of each phase is clarified; an array magnetic field sensor is designed, which can sense the information of the adjacent magnetic field independently and is used for the reconstruction of three-phase currents. The effectiveness of the proposed method was tested on the built three-phase four-core cable current measurement test platform. The experimental results show that under the three-phase current balance, the measurement error of phase A, phase B and phase C is less than 2.8%, and the waveform and phase angle of the three-phase current can be well restored, which verifies the three-phase current measurement method proposed in this paper.

1. Introduction

With the transformation of the traditional power grid to the smart grid, in addition to building a flexible, stable and safe energy network, the realization of the smart grid is more important in the real-time monitoring feedback and dynamic adjustment of the grid status [1,2,3]. At the same time, the power cable is one of the more important carriers of power transmission. Its operation reliability directly affects the power supply safety of the power system, and the three-phase current and other parameters directly reflect the operation status and health level of the power cable, which is the main basis for evaluating the power quality, calculating the transmission power and diagnosing faults [4,5]. Therefore, it is of great significance that we detect cable transmission lines safely, accurately and conveniently and evaluate the operation status of power cables [6,7,8,9,10].

The traditional current transformer needs to strip the insulation layer of the cable and cut the power off to ensure measurement safety. Compared with the direct contact measurement method, the non-contact measurement realizes the inverse calculation of the electrical parameters of the cable by acquiring the electromagnetic distribution characteristics near the cable, which has the advantages of low invasion, simple and easy installation, and no power cut during the measurement process [11].

At present, the non-contact measurement of a three-phase cable phase current is difficult because: (1) a single mutual inductance element cannot directly measure a multiphase current; (2) there lacks a fast and accurate forward analytical calculation model of the three-phase cable magnetic field distribution for a reverse calculation of three-phase currents [12,13]; (3) three-phase currents are coupled with each other. For the non-contact measurement method of three-phase currents, the literature [14] designed a Rogowski coil current sensor for a three-phase three-wire cable. The core sensing unit of the sensor comprises three independent coils wound on the same circular non-magnetic core. The current of each core in the cable is reconstructed by three induced electromotive forces’ output by the three coils. The sensor can simultaneously measure the current of each core of the three-phase three-wire cable without stripping the measured cable. However, the Rogowski coil has no iron core for magnetic focusing, which makes the magnetic flux density in the Rogowski coil winding very small, resulting in a maximum error of 9.07% when measuring the current below 20 A. Therefore, this method cannot achieve good measurement results in a low-frequency and small-current measurement environment. Document [15] proposes a current measurement method based on Hall current sensor for non-contact three-phase three-wire transmission lines. This method obtains the output signals of multiple Hall current sensors at the same time, uses closed-loop integration to sum the output response of the transient magnetic field at the location of each Hall sensor, and finally uses decoupling algorithm to reconstruct the three-phase three-wire current. However, this measurement method is only applicable to overhead lines, not to cables sharing a layer of insulation. When measuring three-phase current of 5A with this method, the measurement error has reached 5%. The literature in [16] proposes to use the piezoelectric cantilever current sensor to measure the current of the three-phase four-wire current-carrying straight wire and verifies the feasibility of its measurement through simulations and experiments. However, the sensitivity of the sensor is not high, the decoupling process is complex, and the external interference load cannot be completely overcome, thus affecting the accuracy of the sensor output response. Document [17] reconstructs the spatial distribution theory of the surrounding magnetic field of the current-carrying straight wire, modifies the voltage output theory of the current sensor, and proposes a decoupling calculation method for the current detection when the three-phase five-wire cable is connected to any load. However, the array structure, composed of multiple identical cantilever beam sensors, is used to calculate the current on each cable, resulting in many unknown parameters and also required cantilever beams, a complicated calculation process, and the sensitivity of the sensor is seriously affected by the distance between the sensor and the cable. The literature [18,19] also uses the cantilever beam composite array to measure the current of multi-core cables, but this paper can only measure the three-phase four-wire current of about 8 A at most. The literature by [20] uses power frequency magnetic field measurement data in the space below the line to inverse the line current value through the inverse optimization algorithm. They then conduct curve fitting and extract the characteristic parameters. This method can realize the non-contact measurement of a line current, but due to the strong coupling between three-phase and four-wire cable currents, this method is not applicable to cables sharing a common insulation layer.

In view of the above analysis, this paper proposes a non-contact current sensing technology using magnetic sensors to measure three-phase current [21,22,23,24]. The current measurement method can realize the measurement of sensor and cable at any relative position, and eliminate the interphase interference through decoupling [25,26,27].

It is applicable to power cables with different section sizes, and can also measure the unbalanced current. First of all, the measurement model is established: four magnetic sensors are installed on the cable surface, 90° apart from each other in the circumferential direction. Then, according to the established mathematical and physical models, the relationship between the sensor deflection angle and the sensor ratio is obtained, the decoupling calculation of the three-phase current and magnetic field is completed, and the quantitative relationship between the sensor output and phase current is defined. Then, the three-phase current reconstruction is realized by conducting a numerical simulation, which verifies the correctness of the model. Finally, the designed fluxgate array sensor is used for experimental verification. The results show that the measurement error of the three-phase current balance of phases A, B and C is less than 2.2%, and the measurement error of the unbalanced current is less than 2.8%. The three-phase current waveform and phase angle can be well recovered, which verifies the three-phase current measurement method proposed in this paper.

2. Measurement Structure and Principle

2.1. Determination of Measurement Model

The structure of the three-core symmetrical power cable used in the modeling is shown in Figure 1. The center of the cable is O, and A, B, C and N are the positions of the three-phase core wires. The distance from the center of the cable to the three-phase core wires of A, B and C is d, and the distance from the cable to the insulation layer is D. As shown in Figure 1, the tangential component of the magnetic field on the surface of the four-core cable along the circumferential direction is measured through four fluxgate sensors installed on the surface of the power cable, so as to realize the effective measurement of the core current of each phase in the power cable. Since the current of three live wires in the three-phase current will not be positive or negative at the same time, but the current of any two live wires must be positive or negative, while the current of the other one is opposite, so it is assumed that the current of phase A and phase B is positive and the current of phase C is negative at a certain time. The amplitude of the sinusoidal current passing through A, B, C and N is I, respectively, $i_1$, $i_2$, $i_3$ and $i_4$.

Without losing generality, the distance between any point P on the sensor and the origin coordinate is D, and then the coordinate of point P is ($D·cos\theta$, $D·sin\theta$), and $\theta$ is the deflection angle between sensor P1 and the origin coordinate. The coordinate of point A is (0, d), and the included angle from the cable center to the three-phase core wires of B, C and N is 0°, 180° and 270°, respectively. Therefore, the coordinates of point B are (d, 0), point C is (−d, 0), and point N is (0, d).

2.2. Principle of Magnetic Field Coupling Measurement and Determination of Decoupling Coefficient

As shown in Figure 1, four magnetic sensors are installed at the positions of P1, P2, P3 and P4, respectively, so that the magnetic induction intensity measured by the magnetic sensors is the tangential component of the magnetic induction intensity at the four points of P1, P2, P3 and P4 along the circumference of the outer surface of the cable. P is any point on the P1 sensor. According to Ampere’s loop theorem, when the phase A current is $i_1,$ the magnetic induction intensity generated by the phase A current at point P can be decomposed into magnetic induction intensity along the x-axis and y-axis:

$$\begin{array}{l}{B}_{P1-A-x}=\frac{{\mu }_0}{2\pi }\frac{i_1\cdot (D\cdot \sin \theta -d)}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta -d)}^2}\\ B_{P1-A-y}=\frac{{\mu }_0}{2\pi }\frac{i_1\cdot (D\cdot \cos \theta )}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta -d)}^2}\end{array}$$

(1)

Among ${\mu }_0$ is the vacuum permeability, $i_1$ is the current generated by phase A, D is the distance from the cable center to the P1 sensor, d is the distance from the cable center to the phase A core, and $\theta$ is the deflection angle between the sensor and the cable center.

Similarly, the magnitudes of the magnetic induction generated at P1 by phase B current $i_2$ and phase C current $i_3$ are:

$$\begin{array}{l}{B}_{P1-B-x}=\frac{{\mu }_0}{2\pi }\frac{i_2\cdot (D\cdot \sin \theta )}{{(D\cdot \cos \theta -d)}^2+{(D\cdot \sin \theta )}^2}\\ B_{P1-B-y}=\frac{{\mu }_0}{2\pi }\frac{i_2\cdot (D\cdot \cos \theta -d)}{{(D\cdot \cos \theta -d)}^2+{(D\cdot \sin \theta )}^2}\end{array}$$

(2)

$$\begin{array}{l}{B}_{P1-C-x}=\frac{{\mu }_0}{2\pi }\frac{i_3\cdot (D\cdot \sin \theta )}{{(D\cdot \cos \theta +d)}^2+{(D\cdot \sin \theta )}^2}\\ B_{P1-C-y}=\frac{{\mu }_0}{2\pi }\frac{i_3\cdot (D\cdot \cos \theta +d)}{{(D\cdot \cos \theta +d)}^2+{(D\cdot \sin \theta )}^2}\end{array}$$

(3)

where, $i_2$ is the current generated by phase B and $i_3$ is the current generated by phase C.

The magnetic induction intensity B at point P is formed by the current in the four live lines:

$$B=B_A+B_B+B_C+B_N$$

(4)

Among them, $B_A$, $B_B$, $B_C$ and $B_N$ are the magnetic induction intensities of phases A, B, C and N at point P, which can be obtained by Ampere’s loop theorem. Therefore, the vector sum of magnetic induction intensity generated by three-phase current at point P is decomposed into horizontal component and vertical component as follows:

$$\begin{array}{l}{B}_{P1-x}=\frac{{\mu }_0}{2\pi }(\frac{i_1\cdot (D\cdot \sin \theta -d)}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta -d)}^2}+\frac{i_2\cdot (D\cdot \sin \theta )}{{(D\cdot \cos \theta -d)}^2+{(D\cdot \sin \theta )}^2}\\ +\frac{i_3\cdot (D\cdot \sin \theta )}{{(D\cdot \cos \theta +d)}^2+{(D\cdot \sin \theta )}^2}+\frac{i_4\cdot (D\cdot \sin \theta +d)}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta +d)}^2})\end{array}$$

(5)

$$\begin{array}{l}{B}_{P1-y}=\frac{{\mu }_0}{2\pi }(\frac{i_1\cdot (D\cdot \cos \theta )}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta -d)}^2}+\frac{i_2\cdot (D\cdot \cos \theta -d)}{{(D\cdot \cos \theta -d)}^2+{(D\cdot \sin \theta )}^2}\\ +\frac{i_3\cdot (D\cdot \cos \theta +d)}{{(D\cdot \cos \theta +d)}^2+{(D\cdot \sin \theta )}^2}+\frac{i_4\cdot (D\cdot \cos \theta )}{{(D\cdot \cos \theta )}^2+{(D\cdot \sin \theta +d)}^2})\end{array}$$

(6)

Therefore, the magnetic induction intensity of fluxgate sensor P1~P4, along the sensitive magnetic direction, can be obtained as follows:

$$\begin{array}{l}{B}_1=B_{P1-y}\cdot \cos \theta -B_{P1-x}\cdot \sin \theta \\ B_2=B_{P2-y}\cdot \cos (\theta +\frac{\pi }{2})-B_{P2-x}\cdot \sin (\theta +\frac{\pi }{2})\\ B_3=B_{P3-y}\cdot \cos (\theta +\pi )-B_{P3-x}\cdot \sin (\theta +\pi )\\ B_4=B_{P4-y}\cdot \cos (\theta +\frac{3\pi }{2})-B_{P4-x}\cdot \sin (\theta +\frac{3\pi }{2})\end{array}$$

(7)

Similarly, the magnetic induction intensity generated by phase B, phase C and the zero line can be obtained, and the expression written as the correlation coefficient is:

$$\begin{array}{l}{B}_2=\frac{{\mu }_0}{2\pi }\left[K_{21}\cdot i_1+K_{22}\cdot i_2+K_{23}\cdot i_3+K_{24}\cdot i_4\right]\\ B_3=\frac{{\mu }_0}{2\pi }\left[K_{31}\cdot i_1+K_{32}\cdot i_2+K_{33}\cdot i_3+K_{34}\cdot i_4\right]\\ B_4=\frac{{\mu }_0}{2\pi }\left[K_{41}\cdot i_1+K_{42}\cdot i_2+K_{43}\cdot i_3+K_{44}\cdot i_4\right]\end{array}$$

(8)

where K is related to the cable radius D, the distance d from the center of the cable to the core wire, and the deflection angle $\theta$.

Write the above four Formulas (5)–(8) in matrix form. The matrix relationship between the current and magnetic induction strength is as follows:

$$\left[\begin{array}{c}{B}_1\\ B_2\\ B_3\\ B_4\end{array}\right]=\frac{{\mu }_0}{2\pi }\left[\begin{array}{cccc}{K}_{11}& K_{12}& K_{13}& K_{14}\\ K_{21}& K_{22}& K_{21}& K_{24}\\ K_{31}& K_{32}& K_{33}& K_{34}\\ K_{41}& K_{42}& K_{43}& K_{44}\end{array}\right]\cdot \left[\begin{array}{c}{i}_1\\ i_2\\ i_3\\ i_4\end{array}\right]$$

(9)

Therefore, the matrix for solving the three-phase current can be obtained by an inverse calculation of the above matrix, as follows:

$$\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\\ i_4\end{array}\right]=\frac{2\pi }{{\mu }_0}\cdot {\left[\begin{array}{cccc}{K}_{11}& K_{12}& K_{13}& K_{14}\\ K_{21}& K_{22}& K_{21}& K_{24}\\ K_{31}& K_{32}& K_{33}& K_{34}\\ K_{41}& K_{42}& K_{43}& K_{44}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{B}_1\\ B_2\\ B_3\\ B_4\end{array}\right]$$

(10)

In the three-phase four-wire power cable, the current of the zero line is the sum of the current flowing through other wires, according to Kirchhoff’s current law:

$$i_1+i_2+i_3+\cdot \cdot \cdot +i_N=0$$

(11)

Therefore, for the three-phase four-wire power cables, the matrix has 16 elements, but KCL reduces the number of unknown currents by 1. Only the nine-element matrix of three sensors can be used to determine all currents. The decoupling matrix of three-phase four-wire cable phase currents is:

$$\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\end{array}\right]=\frac{2\pi }{{\mu }_0}\cdot m\cdot \left[\begin{array}{ccc}{K}_{11}{}^{\prime }& K_{12}{}^{\prime }& K_{13}{}^{\prime }\\ K_{21}{}^{\prime }& K_{22}{}^{\prime }& K_{23}{}^{\prime }\\ K_{31}{}^{\prime }& K_{32}{}^{\prime }& K_{33}{}^{\prime }\end{array}\right]\cdot \left[\begin{array}{c}{B}_1\\ B_2\\ B_3\end{array}\right]$$

(12)

where m and the coefficients in the matrix are obtained, as shown in the Appendix A, Formula (A1).

For the zero line current:

$$i_4=i_1+i_2+i_3$$

(13)

According to the above analysis, when measuring the phase current of multiphase cable, the magnetic induction intensity generated by other phases will be superimposed on the magnetic induction intensity of that phase, and a single sensor can detect a part of each phase current flowing through the cable. Thus, when the deflection angle is determined $\theta$, after the value of cable radius D and d, the phase current of the three-phase four-wire power cable can be calculated according to the above theoretical solution from the linear relationship between current I and magnetic induction intensity B.

3. Current Reconstruction Technology

3.1. Determination of Deflection Angle

According to the above analysis, we can reduce the three-phase current through the decoupling matrix. However, in the decoupling matrix, the decoupling coefficient and deflection angle $\theta$ are relevant, but we cannot predict this. Therefore, in decoupling reconstruction, the deflection angle must be solved first. We used COMSOL to simulate the 360-degree magnetic induction intensity around the cable and to obtain the magnetic induction intensity values of four sensors at any position. The default upper right corner is the P1 sensor. Through the ratio of four sensors, we obtained data and the curve of sensor ratio. From Figure 2, we can see that each ratio corresponds to an angle, and the unique deflection angle can be determined by consulting three groups of data. At 45 degrees, 135 degrees, 225 degrees and 315 degrees, the ratio is 1. It needs to be discussed separately to determine the relative position of the sensor and each phase. Other angles can be determined by consulting the table. The specific array of table lookup method is shown in

Table 1. Sensor ratio table.

$\mathbf{Angle} \left(\mathit{\theta }\right)$	Array1 (P2/P1)	Array2 (P3/P1)	Array3 (P4/P1)
0	0.734912	0.993191	0.363180
1	0.739019	0.997070	0.364397
2	0.737446	1.001517	0.366266
…	…	…	…
357	0.734047	0.992519	0.367433
358	0.731474	0.991843	0.364359
359	0.732534	0.993064	0.363779

, the specific table lookup method and program block diagram are shown in Figure 3.

3.2. Three-Phase Current Simulation and Reconstruction

In order to further verify the feasibility of the method in this paper, COMSOL finite element simulation is used to simulate a three-phase four-core power cable under three-phase unbalanced current conditions, and the magnetic field strengths of P1, P2, P3 and P4 are calculated. In the finite element simulation process, according to the cables used in the laboratory, the cable parameters are set as follows: D = 8.636 mm, d = 2.54 mm, the deflection angle is set as 30 degrees, and the sine current expression through A, B, C and the zero line is as follows:

$$\begin{array}{l}{i}_1=5\sqrt{2}\sin (2\pi ft)\\ i_2=4\sqrt{2}\sin (2\pi ft-{120}^{°})\\ i_3=3\sqrt{2}\sin (2\pi ft+{120}^{°})\\ i_4=i_1+i_2+i_3=2.45\sin (2\pi ft-{30}^{°})\end{array}$$

(14)

where the power frequency f = 50 Hz is taken here.

The three-wire current waveform given in COMSOL simulation software is shown in Figure 4.

According to the above simulation, it can be seen from Figure 5 that when t = 0.058 s, the magnetic induction intensity distribution at sensors P3 and P4 is the largest, and the magnetic induction intensity distribution at P1 is the smallest, which is consistent with the results of the magnetic induction density curve in Figure 6 at the same time, indicating that the magnetic induction intensity is sinusoidal. In addition, when the deflection angle is 30 degrees, the simulation of asymmetric current shows that the magnetic induction intensity curve has obvious characteristics, which can be used to determine whether the sensor output value is correct during the experiment.

3.3. Evaluation of Reconstruction Current Accuracy

Take the magnetic induction intensity curve shown in Figure 6 as the known condition and use the mathematical and physical model and decoupling matrix established above to restore the current in the three-phase four-wire power cable. If the restored current amplitude is consistent with the input current amplitude in the simulation calculation, the method proposed in this paper is proven to be feasible.

In this paper, the Mathematica tool is used to solve the above Formula (12), and the curve of the phase current in a three-phase four-wire power cable with time is obtained, as shown in Figure 7. In order to evaluate the accuracy of the established model, the following formula is used to carry out binary linear fitting for the solved current curve:

$$I_p=a\cdot \cos (2\pi ft)+b\cdot \sin (2\pi ft)$$

(15)

where I_p is the phase current obtained from the above analytical model, a and b are the cosine and sine coefficients, and f is the current frequency.

Therefore, the amplitude and phase angle of a three-phase four-wire phase current can be reconstructed from the following Formula (16):

$$\{\begin{array}{c}I=\sqrt{a^2+b^2}\\ \phi =\arctan (\frac{a}{b})\end{array}$$

(16)

The data obtained from the analytical model is shown in

Table 2. Three-phase current reconstructed under simulation conditions.

Phase Sequence	a	b	$\sqrt{{\mathit{a}}^2+{\mathit{b}}^2}$	$\mathit{\phi }/°$
A	7.09	0.02	7.10	0.5
B	−2.81	−4.90	5.64	−121.8
C	−2.11	3.67	4.23	122.5
N	2.13	−1.22	2.45	−28.8

. The table shows that the current results obtained from the decoupling matrix are consistent with the finite element simulation results. The relative errors of the amplitude calculation of phase A, B, C and N currents are 0.41%, 0.30%,0.30% and 0.41%, respectively, and the phase angle errors are 0.5°, −1.8°, 2.5° and 1.2°, respectively.

From the above indicators, it can be seen that the mathematical and physical model and analysis model proposed in this paper are feasible and can be used to measure the unbalanced current of three-phase four-wire power cables.

3.4. Reduction of Cable Current with Different Section Sizes

In order to verify the practicability of the method proposed in this paper, the same simulation and reduction current method as the above is used to simulate and verify the two types of national standard three-phase four-wire cables, YJV4 × 16 and YJV4 × 25. The input current of 16 mm² cable is 20A and 100A, and its reduction current curve is shown in Figure 8. The input current of 25 mm² cable is 50A and 150A, and its reduction current curve is shown in Figure 9.

It can be seen from the above figure that the method proposed in this paper can well recover three-phase four-wire cables with different section sizes and different current sizes, and the overall error is less than 1%. Therefore, the method proposed in this paper is universal for cables with different cross-sectional areas, balanced current and unbalanced current.

4. Design of Magnetic Array Current Sensor

4.1. Measuring Principle of Fluxgate Sensor

According to the mathematical and physical model analyzed above, four DRV425 fluxgate sensors are used to form a magnetic array current sensor. Each sensor has a single-terminal output voltage, and a single sensor can measure the maximum magnetic induction intensity of ±2 mT. The working principle of the fluxgate magnetic field sensor is that the external magnetic induction intensity B generates residual flux ${\Phi }_1$ in the magnetic core inside the sensor, and the output of the fluxgate sensor module is connected to the integrator and then the output to the built-in differential driver. The differential driver will generate a voltage and compensation current on the shunt resistor $R_{SHUNT}$, which will generate the opposite magnetic flux ${\Phi }_2$ and make the residual magnetic flux inside the magnetic core return to zero. The compensation current is set to 12.2 mA/mT. Use an integrated differential amplifier with a fixed gain of 4 V/V to measure the voltage at both ends of the shunt resistor $R_{SHUNT}$, and output a voltage that is proportional to the magnetic field. The maximum magnetic induction intensity that can be measured is ±2 mT. Therefore, the fluxgate magnetic field sensor can achieve extremely high magnetic field measurement accuracy, and its output voltage is:

$$V_{VOUT}[\mathrm{V}]=B\times G\times R_{SHUNT}\times G_{AMP}=B[\mathrm{mT}]\times 12.2\mathrm{mA}/\mathrm{mT}\times R_{SHUNT}[\Omega ]\times 4[\mathrm{V}/\mathrm{V}]$$

(17)

where G is the compensation current of 12.2 $\mathrm{m}\mathrm{A}/\mathrm{m}\mathrm{T}$, and $G_{AMP}$ is the fixed gain of the fluxgate chip of 4 $\mathrm{V}/\mathrm{V}$.

In addition, as shown in Figure 10, the fluxgate magnetic field sensor is a single-axis sensitive magnetic field measurement sensor, which can only detect the magnetic induction intensity along its sensitive axis. The magnetic flux generated by the current in the magnetic core is a closed loop along the magnetic core. The direction of the sensitive axis of the fluxgate sensor must be along the radial tangent direction of the magnetic core so that the direction of the magnetic flux is consistent with the direction of the sensitive axis. Therefore, the DRV425 fluxgate sensor meets the requirements of magnetic array sensor design in this paper.

4.2. Design of Peripheral Parameters of Magnetic Array Sensor

For the DRV425 fluxgate magnetic field sensor, the output voltage VOUT is limited, and the output of the sensor has a constant DC bias, which is half of the supply voltage. Therefore, when the power supply voltage is 3.3 V, the maximum output of $V_{OUT}$ is 1.65 V. The value of shunt resistance $R_{SHUNT}$ will affect the maximum magnetic induction intensity $B_{MAX}$ that can be measured by the sensor. In order to ensure that the sensor meets the measurement requirements, the maximum magnetic induction intensity $V_{OUT}$ must be greater than the magnetic induction intensity generated by the current to be measured. The parameters of shunt resistance $R_{SHUNT}$ meet the following formula:

$$R_{SHUNT}=\frac{V_{OUTMAX}}{4\times 12.2\times B_{MAX}}$$

(18)

According to the above simulation analysis, when the current is a 5 A effective value, the maximum magnetic induction intensity of point P at any point in the three-phase four-core cable is about 0.16 mT, and thus the resistance value of $R_{SHUNT}$ can be determined as 211 $\mathsf{\Omega }$. In order to leave a certain margin for the output of the sensor, the resistance value of the shunt resistor can be selected as 100 $\mathsf{\Omega }$.

Then, the voltage output of each fluxgate sensor designed in this paper is:

$$V_{OUT}=4\times 12.2\times B_{Synthetic}\times R_{SHUNT}$$

(19)

$B_{Synthentic}$ is the synthetic magnetic field of a three-phase current detected by the sensor.

4.3. Matrix Relationship between Sensor Output Voltage and Three-Phase Current

After the sensor is designed, the output voltage of each sensor can be obtained by substituting the magnetic induction intensity at P1, P2, P3 and P4 into Formula (19) and then by substituting Formula (19) into matrix Formula (12) to obtain the matrix relationship between the three-phase current and output voltage as follows:

$$\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\end{array}\right]=m\cdot \mathrm{g}\cdot \left[\begin{array}{ccc}{K}_{11}{}^{\prime }& K_{12}{}^{\prime }& K_{13}{}^{\prime }\\ K_{21}{}^{\prime }& K_{22}{}^{\prime }& K_{23}{}^{\prime }\\ K_{31}{}^{\prime }& K_{32}{}^{\prime }& K_{33}{}^{\prime }\end{array}\right]\cdot \left[\begin{array}{c}{V}_1\\ V_2\\ V_3\end{array}\right]$$

(20)

where m, the matrix coefficient and Formula (9) are the same,

$$g=\frac{1}{4\times 12.2\times R_{SHUNT}},R_{SHUNT}= 100 \Omega$$

Substitute the specific parameters D = 8.636 mm, d = 2.54 mm, G = 12.2 $\mathrm{m}\mathrm{A}/\mathrm{m}\mathrm{T}$, $G_{AMP}$ = 4 $\mathrm{V}/\mathrm{V}$, $R_{SHUNT}$ = 100 $\mathsf{\Omega }$, when the deflection angle is 30 degrees according to the above method, and the specific matrix of the relationship between the three-phase current and the sensor output is as follows:

$$\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\end{array}\right]=\left[\begin{array}{ccc}9.34& -5.30& -6.77\\ 29.63& 19.81& 1.68\\ -24.04& -8.65& 10.19\end{array}\right]\cdot \left[\begin{array}{c}{V}_1\\ V_2\\ V_3\end{array}\right]$$

(21)

In this way, according to the matrix relationship between the sensor output voltage and the three-phase input current, the time domain waveform and effective value of the phase current of the three-phase four-wire cable can be calculated by the method shown in Figure 11.

5. Experimental Verification

5.1. Establishment of Experimental Platform

In this section, the proposed magnetic array sensor is used to measure the phase current of a three-phase four-wire power cable, and the experimental system platform, as shown in Figure 12, is built. The three-phase four-wire AC experimental system is mainly composed of a three-phase AC power supply, three-phase four-wire cable, resistive load and oscilloscope. In the experiment, the resistance value of the resistance load is a constant 10. The output current is adjusted by adjusting the output voltage of the three-phase AC power supply. The resistance load is connected in a star shape. The probe of the oscilloscope is connected to the output end of the sensor. The oscilloscope is used to store data and waveforms. The adjustable current range of 0~6.6 A and 0~3.3 A can be obtained through two voltage ranges (0~150 V and 0~300 V) on the three-phase AC source.

In the actual measurement, because the current detection method proposed in this paper is non-contact, the cable will not be stripped to determine the position of the zero line and the three live lines. Even if the approximate position of the fire line and zero point can be seen visually, the positioning error of the sensor will be caused by human parallax. According to the linear relationship between the output voltage of the fluxgate sensor and the magnetic induction intensity, the deflection angle can be obtained by the ratio of the voltage output values of the four sensors. In order to verify the consistency between the experiment and simulation, we can adjust the position of the sensor by rotating the PCB in the experiment to obtain the relationship between the voltage output value and the reflection angle. We used a stepper motor to rotate the PCB. Through the fixed bearing, the single-chip microcomputer drives the stepper motor to drive the belt to rotate, and then the PCB sensor installed on the drive wheel rotates. Rotate the PCB 360 degrees to obtain the relationship curve consistent with the simulation model. Three-phase four-wire current can recover the value of each phase current from the measured voltage value. The actual experimental platform is shown in Figure 13.

5.2. Analysis and Discussion of Experimental Results

The preliminary experiments show that the current sensor has high measurement accuracy when the current is greater than 3 A. Taking the output of a three-phase unbalanced current with the effective values of 5 A, 4 A and 3 A as an example, the calculated deflection angle is 30 degrees, and the output voltage waveform of the current sensor is shown in Figure 14. It can be seen from Figure 14 that when measuring the current of different current sizes, the output voltage of the current sensor is always a sine signal that changes the sine with time, and its frequency is the same as the measured current. Therefore, it can be inferred that the measured current and output voltage of the magnetic array sensor are sinusoidal signals with the same frequency and different phases, and the voltage output waveform of the sensor is consistent with the waveform of the magnetic induction intensity in the simulation, which verifies the measurement model proposed in this paper. The waveform in the figure has some distortion and burr, which is caused by the noise and cross distortion of the three-phase power supply itself.

After obtaining the time domain waveform of the measured voltage, the actual current time domain waveform can be obtained by substituting the curve into matrix Formula (21), as shown in Figure 15. The current amplitude of the theoretical current curve is set as $5\sqrt{2}$A, $4\sqrt{2}$A and $3\sqrt{2}$A, the zero line current is the sum of the three live line currents, and the three-phase phase angle difference is 120°. FFT (Fast Fourier Transform) is used to obtain the effective value, phase angle and error of phase A, B and C currents, as shown in

Table 3. The error between the reduction current and the theoretical current.

Phase Sequence	Valid Value	Amplitude Error (%)	Phase Angle Error (°)
A	5.0619	1.44	−1.8
B	4.0825	2.06	1.98
C	3.0845	2.81	2.78
N	2.3867	2.56	1.3

.

It can be seen from the above indicators that the overall result of the three-phase current obtained from the matrix model is good. The current calculation model and magnetic array sensor designed in this paper are feasible and can be used for measuring the current of three-phase four-wire power cables.

5.3. Linear Error and Sensitivity of Sensor

In order to obtain the linearity and sensitivity of the magnetic array sensor, the current effective value of the three-phase current balance is 500 mA to 5 A when the deflection angle is 0 degrees. The drawn sensor input/output curve and linear fitting curve are shown in Figure 16.

It can be seen from Figure 16 that the overall linearity of the sensor measuring the three-phase current is good. When measuring phase B and C currents, the output of the sensor is basically the same. When measuring the A-phase current, the output voltage of the sensor is slightly less than the other two phases, and the N-phase voltage value is far less than the other three phases. According to the input–output theoretical model of the sensor and the internal structure of the three-phase four-wire cable, this is because the spatial distribution of the three-phase live wire in the three-phase four-wire cable is different, and there is no current on the zero line.

In order to verify that the method proposed in this paper can effectively recover current under different input current conditions, this paper verifies the balance current of 1 A~5 A when randomly placing PCB sensors, as shown in Figure 17.

It can be seen in Figure 18 above that this method can better restore the current waveform. The error between the reduction current and the ideal current calculated by FFT is shown in

Table 4. The error between the reduction current and the theoretical current.

Phase Sequence/Current	1 A	2 A	3 A	4 A
	Error (%)
A	1.8	1.6	1.53	1.38
B	1.8	1.8	1.74	1.42
C	2.1	2.1	1.69	1.48

, and the error curve of the reduction current is shown in Figure 19.

From the above error results, when the effective value of the input current is increasing, the error of the magnetic array sensor becomes smaller and smaller, which is related to the sensitivity of the sensor. When the input current is large, the interference effect of the surrounding magnetic field on the sensor becomes smaller. Due to the small amplitude of the three-phase current generated in our laboratory, a large current can be used for subsequent research.

6. Conclusions

Aiming at the problem that the traditional current measurement method cannot measure the phase current of four-core cable, a non-contact measurement method of three-phase four-wire phase current, based on a magnetic field decoupling calculation, is proposed. The decoupling matrix can be applied to multiphase cables with different section sizes. The sensor can be placed randomly, and the output signal of the fluxgate sensor can be reconstructed using a decoupling matrix. Firstly, the measurement model and principle are analyzed in detail, and the transfer relationship between the three-phase current and the output signal of the sensor is obtained, which provides theoretical guidance for the design of the magnetic sensor and the reduction experiment of the three-phase current. Then, the correctness of the proposed model is verified by a finite element simulation. Finally, the voltage output waveform of the sensor is restored to the actual current waveform through the experiment. Within the three-phase balance current range of 1~5 A, the overall measurement amplitude error is less than 2.2%, and the phase angle error is less than 3°. In the case of a three-phase unbalanced current, the measurement error is less than 2.8%, which proves the feasibility and effectiveness of this method.

Compared with the existing technology, the three-phase four-wire power cable phase current measurement method proposed in this paper can better solve the problem where the traditional inductance measurement device is unable to measure the phase current in the past and thus realize the random and accurate measurement of the four-core cable phase current. However, due to the limited experimental conditions in this paper, all the experiments were carried out under current conditions of below 5 A, with a frequency of 50 Hz. The follow-up research can conduct a three-phase three-wire system, a three-phase five-wire system and a large current measurement with a higher frequency and wider frequency band.