In this paper I present a complete investigation into a Hall-element elliptical-array current sensor developed for high-voltage battery systems in new-energy vehicles. The increasing demand for smaller, lighter and more cost-effective power batteries requires a current sensing solution that can accurately measure large busbar currents in a narrow installation space. Conventional closed-loop Hall transducers, fluxgate sensors and shunts become bulky, expensive or intrusive. I therefore studied an array-based current sensing approach in which several Hall sensors are placed around a rectangular busbar without any magnetic core. My work covers the mathematical model, array geometry design, hardware realization, full-temperature-range calibration, eccentricity and tilt error optimization, finite-element simulation and experimental verification.
High-voltage battery safety is the central motivation for this study. In electric vehicles the battery management system relies on accurate current information to estimate the state of charge and state of health and to detect dangerous internal faults. A faulty or low-accuracy current sensor can lead to an incorrect state estimation, unbalanced cell equalization, overheating and even thermal runaway. Therefore improving the measurement accuracy and robustness of the current sensor directly improves the safety of the high-voltage battery. The array-type Hall current sensor is particularly suitable because it has no saturable magnetic core, no magnetic hysteresis, a broad linear range, small mass and a small footprint. However, practical issues such as printed-circuit-board assembly tolerances, conductor displacement and conductor tilt still cause unacceptable errors. I addressed all three error sources in this paper.
Modern Hall sensors are often monolithic integrated circuits with internal amplification and temperature compensation. They can measure the magnetic flux density generated by a current-carrying conductor without galvanic contact. I first summarize the fundamental relations used in the work. The Lorentz force acting on a moving charge in a magnetic field leads to the Hall voltage, which is proportional to the product of the driving current and the magnetic flux density. For a rectangular Hall plate placed in a magnetic field perpendicular to its current direction, the Hall voltage can be written as
$$
V_H = \frac{I_C B}{n q t} = K_H B,
$$
where \(I_C\) is the control current, \(B\) is the magnetic flux density, \(n\) is the carrier concentration, \(q\) is the elementary charge, \(t\) is the thickness of the Hall plate and \(K_H\) is the magnetic sensitivity of the Hall element. The Hall voltage changes sign when the magnetic field orientation is reversed, so a Hall element can detect both the direction and the magnitude of the magnetic field.
For a long straight conductor, Ampere’s circuital law is exactly
$$
\oint \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I.
$$
If I place \(N\) Hall sensors along a closed integration path around a rectangular busbar of a high-voltage battery, the measured magnetic field component tangent to the integration path at each sensor can be weighted by the corresponding segment length. The unknown busbar current is reconstructed as
$$
I_{\mathrm{calc}} = \sum_{i=0}^{N-1} H_{\mathrm{MFS},i} \Delta s_i,
$$
where \(H_{\mathrm{MFS},i}\) is the magnetic field component along the sensitive axis of Hall element \(i\) and \(\Delta s_i\) is the arc length assigned to that element. This is the core principle of the array current sensor: each sensor performs a spatial sampling of the line integral of the magnetic field, and the weighted sum approximates the enclosed current.
The choice of the closed integration path is decisive. A circular path is natural for a circular conductor, but the rectangular busbar used in high-voltage battery power distribution has a flat cross-section. If I use a circular array for a rectangular busbar, the vertical clearance is wasted and the lower sensors are too far from the conductor. A rectangular array follows the conductor shape but creates strong field-gradient jumps at the corners of the magnetic integration path. An elliptical array offers a compromise: it encloses the rectangular busbar more tightly than a circle while providing a smoother magnetic-field variation than a rectangle. Therefore I selected an ellipse-based array as the main configuration.
To generate the sensor positions on the ellipse, I compared two methods. The first is the parallel-projection method, which maps equally spaced angular points on a circle onto an ellipse. The second is the uniform-curve-segment-length method, called UCSL, which divides the total elliptic circumference into \(N\) equal arc segments. I found by numerical evaluation that UCSL provides a more stable reconstruction for the same number of sensors. For an ellipse with major semi-axis \(a\), minor semi-axis \(b\), and aspect ratio \(AR=b/a\), I approximate the ellipse by a fine polygon containing \(K\) points. The arc length from the X-axis to the \(k\)-th polygon point is
$$
s(k) = \sum_{n=1}^{k}
\sqrt{\left(E_{n,x}-E_{n-1,x}\right)^2 + \left(E_{n,y}-E_{n-1,y}\right)^2},
$$
$$
E_{n,x}=a\cos\frac{360^\circ n}{K},\qquad
E_{n,y}=b\sin\frac{360^\circ n}{K}.
$$
The total perimeter is \(s(K)\), and the length of each uniform curve segment is
$$
\Delta s = \frac{s(K)}{N}.
$$
A small relative offset parameter \(s_{0,r}\) is used to shift the first sensor from the X-axis. For even \(N\) and for odd \(N\) the recommended values are implemented as
$$
s_{0,r}=
\begin{cases}
\dfrac{N}{2}+\dfrac{1}{4}, & N \text{ odd},\\[4pt]
\dfrac{1}{4}, & N \text{ even},
\end{cases}
\ \text{(scaled appropriately by the segment relation).}
$$
The distance from the X-axis to sensor \(i-1\) is
$$
s_{i-1} = s_{0,r}s(K) + (i-1)\Delta s.
$$
If there exists an integer \(m\) such that \(s(m)=s_{i-1}\), then the sensor coordinate is
$$
P_{i-1} = \left(a\cos\frac{360^\circ m}{K},\ b\sin\frac{360^\circ m}{K}\right),
$$
and its tangent direction is
$$
\mathbf{t}_{i-1} =
\frac{\left(-a\sin\dfrac{360^\circ m}{K},\ b\cos\dfrac{360^\circ m}{K}\right)}
{\sqrt{a^2\sin^2\dfrac{360^\circ m}{K}+b^2\cos^2\dfrac{360^\circ m}{K}}}.
$$
The tangent direction is the sensitive-axis direction because the array integral requires the magnetic field component tangential to the integration loop. Using a typical rectangular busbar width of 20 mm and thickness of 4 mm, and considering the limited space in a high-voltage battery pack, I set the major semi-axis of the elliptical array to \(a=30\) mm. I then evaluated the theoretical current error as a function of the number of Hall elements using MATLAB. The simulation showed that the current error decreases as \(N\) increases, but the marginal improvement becomes small after \(N=6\). I also evaluated the current error as a function of the array aspect ratio \(AR\). The results show that, for \(N=6\), the pair \(N=6, AR=0.33\) gives a low theoretical error while keeping the required vertical dimension much smaller than a circular array. For \(N=7\), the pair \(N=7, AR=0.31\) is also acceptable, but it does not justify the additional sensor cost in most high-voltage battery applications. I therefore selected \(N=6\), \(AR=0.33\), \(a=30\) mm, and \(b=9.9\) mm for the final elliptical array.
The designed six-element elliptical array is summarized in the following table.
| Hall element | Position on ellipse (mm) | Unit tangent vector | Tangent angle (degrees) |
|---|---|---|---|
| \(P_0\) | \((26.94,\ 4.40)\) | \((-0.827,\ 0.563)\) | 145.79 |
| \(P_1\) | \((5.57,\ 9.83)\) | \((-0.998,\ 0.063)\) | 176.38 |
| \(P_2\) | \((-16.59,\ 8.33)\) | \((-0.976,\ -0.216)\) | 192.58 |
| \(P_3\) | \((-26.94,\ -4.40)\) | \((0.827,\ -0.563)\) | 325.79 |
| \(P_4\) | \((-5.56,\ -9.83)\) | \((0.998,\ -0.063)\) | 356.38 |
| \(P_5\) | \((16.59,\ -8.33)\) | \((0.976,\ 0.216)\) | 12.58 |
After fixing the array geometry, I developed the magnetic field model of a rectangular busbar. For an untitled busbar whose cross-section lies in the XOY plane and whose current flows along the Z-axis, I treat the busbar cross-section as a set of infinitesimal filaments. If the rectangular busbar has width \(w\) and thickness \(t\), the cross-section area is \(A_c=wt\), and a filament at \((x,y)\) carries the current \(dI = (I/A_c)\,\mathrm{d}x\,\mathrm{d}y\). The magnetic field at an arbitrary point \((a,b,0)\) is obtained by integrating over the cross-section. For a conductor centered at the origin, the X-axis component is
$$
H_x(a,b) = -\int_{-t/2}^{t/2}\int_{-w/2}^{w/2}
\frac{I}{A_c}\frac{b-y}
{2\pi\left[(a-x)^2+(b-y)^2\right]}\,\mathrm{d}x\,\mathrm{d}y,
$$
and the Y-axis component is
$$
H_y(a,b) = \int_{-t/2}^{t/2}\int_{-w/2}^{w/2}
\frac{I}{A_c}\frac{a-x}
{2\pi\left[(a-x)^2+(b-y)^2\right]}\,\mathrm{d}x\,\mathrm{d}y.
$$
These formulas are the basis of the two-dimensional current-reconstruction model. The actual magnetic field of a rectangular busbar in a high-voltage battery has an approximately elliptic contour, which further supports the use of an elliptic Hall array.
My hardware design consists of six Hall elements placed on the elliptic integration path, analog front-end circuits, a microcontroller, isolated CAN communication, and a power supply chain. I selected a linear Hall sensor with a sensitivity of approximately 130 mV/mT and a magnetic measurement range of ±18 mT. The sensor is supplied by a low-noise 5 V source. Because the high-voltage battery environment may involve a 12 V vehicle electrical network, I used a DC-DC isolated module to generate the 5 V rail, and a low-dropout linear regulator to generate the 3.3 V rail for the microcontroller. Each Hall output passes through a voltage follower so that the ADC input impedance does not load the Hall element. The follower outputs are then combined in an adder circuit, and both the individual Hall voltages and the summed voltage are sampled by the internal 12 bit ADC of the microcontroller.
The current range to be measured in the high-voltage battery application is \(-1500\) A to \(+1500\) A. Since the Hall voltage is proportional to the magnetic field, a wide current range would cause the small-current signals to be too small for accurate sampling if the same amplification gain were used for every range. I therefore separated the signal chain into three overlapping ranges based on the expected current magnitude: small currents from 0 A to 90 A use an amplifier gain of 100; medium currents from 91 A to 870 A use a gain of 10; and large currents from 871 A to 1500 A use a gain of 5. The sampled voltage is derived from the ADC code difference as
$$
V_{\mathrm{ADC}} = \frac{\mathrm{ADC_{diff}}\ V_{\mathrm{LSB}}}{K_s\ A_{\mathrm{gain}}},
$$
where \(\mathrm{ADC_{diff}}\) is the digitally converted difference between the two half-cycle samples, \(V_{\mathrm{LSB}}=1\) mV is the least-significant-bit voltage of the sampling system, \(K_s=128\) is the number of samples and \(A_{\mathrm{gain}}\) is the gain of the selected signal path. The current is then obtained as
$$
I_{\mathrm{out}} = \frac{V_{\mathrm{ADC}}}{R_s},
$$
where \(R_s=0.47\ \Omega\) is the sampling resistance used in the measurement circuit. This straightforward calculation gave an initial measurement error that was too large for high-voltage battery management because of component offsets, finite amplifier linearity, and production tolerances. Therefore I designed a calibration algorithm to reduce the initial error.
The calibration procedure is based on least-squares fitting. In the first step, I calibrate the sensor at room temperature. The measured output current \(I_{\mathrm{out}}\) is compared with the reference current \(I_{\mathrm{input}}\) to obtain the current error
$$
\Delta I = I_{\mathrm{out}}-I_{\mathrm{input}}.
$$
For every current segment, I perform a first-order linear fit using \(I_{\mathrm{out}}\) as the independent variable and \(\Delta I\) as the dependent variable:
$$
\Delta I_{\mathrm{fit},i}=k_i I_{\mathrm{out}}+b_i,\qquad i=0,1,\dots,5.
$$
The fitted error is then subtracted from the output current:
$$
I_{\mathrm{mea}} = I_{\mathrm{out}}-\Delta I_{\mathrm{fit},i}.
$$
In my experiments, a first-order fit performed better than a second-order polynomial fit over the full range after considering both numerical stability and simplicity. The residual error after room-temperature calibration was below the target threshold specified for the high-voltage battery application.
Temperature variation is one of the main challenges in a high-voltage battery current sensor. The sensitivity and offset of a Hall element drift with temperature, and the analog front end also changes its gain. To reduce this effect, I developed a high/low-temperature calibration module. The sensing system was placed in a temperature chamber and tested at \(-40^\circ\)C, \(25^\circ\)C and \(125^\circ\)C. At each temperature, the measured current and current error are again fitted with first-order lines. Let the low-temperature fit be
$$
y_{L,i}=l_i x+c_i,
$$
the room-temperature fit be
$$
y_{C,i}=k_i x+b_i,
$$
and the high-temperature fit be
$$
y_{H,i}=h_i x+a_i.
$$
The temperature drift coefficient is calculated separately for the low-temperature side and high-temperature side. Since \(-40^\circ\)C is \(65^\circ\)C below \(25^\circ\)C and \(125^\circ\)C is \(100^\circ\)C above \(25^\circ\)C, I compute
$$
k_{\mathrm{low},i} = \frac{l_i-k_i}{65},\qquad
k_{\mathrm{high},i} = \frac{h_i-k_i}{100},\qquad
\bar{k}_i = \frac{k_{\mathrm{low},i}+k_{\mathrm{high},i}}{2}.
$$
After combining the fitted error and the temperature drift, the compensated current error is written as
$$
\Delta I_{\mathrm{fit,temp},i} = \left(k_i+\bar{k}_i T\right) I_{\mathrm{out}}+b_i,
$$
where \(T\) is the environmental temperature in degrees Celsius. The final compensated current is
$$
I_{\mathrm{mea,temp}} = I_{\mathrm{out}}-\Delta I_{\mathrm{fit,temp},i}.
$$
I applied this full-temperature-range calibration to the elliptic-array Hall sensor and verified that the remaining current error is less than 3‰ over the full range at \(-40^\circ\)C, \(25^\circ\)C and \(125^\circ\)C. The following table summarizes representative fit slopes and intercepts before and after temperature compensation for the positive current direction in my prototype.
| Temperature point | Small-current slope | Small-current intercept | Medium-current slope | Medium-current intercept | Large-current slope | Large-current intercept |
|---|---|---|---|---|---|---|
| \(-40^\circ\)C | \(-3.49\times10^{-3}\) | \(3.59\times10^{-2}\) | \(-3.80\times10^{-3}\) | \(-2.32\times10^{-1}\) | \(-2.72\times10^{-3}\) | \(-9.91\times10^{-1}\) |
| \(25^\circ\)C | \(-5.05\times10^{-3}\) | \(4.66\times10^{-2}\) | \(-3.67\times10^{-3}\) | \(6.02\times10^{-2}\) | \(-2.84\times10^{-3}\) | \(-1.13\) |
| \(125^\circ\)C | \(3.69\times10^{-3}\) | \(-7.63\times10^{-2}\) | \(4.21\times10^{-3}\) | \(-6.54\times10^{-1}\) | \(6.18\times10^{-3}\) | \(-2.03\) |
Although the calibration algorithm guarantees low initial error in ideal centered and vertical conditions, the busbar in a high-voltage battery can move relative to the sensor array because of vibration, mechanical tolerances and thermal expansion. When the busbar is no longer at the center of the elliptic path, each Hall element samples a different portion of the magnetic field distribution, so the discrete sum deviates from the ideal line integral. This deviation is called eccentricity error. Similarly, if the busbar is tilted rather than perpendicular to the sensor plane, the cross-section projection becomes distorted and the magnetic field distribution changes in all three axes. This leads to tilt error.
I analyzed the eccentricity error first. The eccentricity can be decomposed into an X-axis offset \(X_p\) and a Y-axis offset \(Y_p\). For an untitled busbar shifted by \((X_p,Y_p)\), the limits in the two-dimensional magnetic field integral are no longer symmetric. The X-axis component becomes
$$
H_{x,p}(a,b)= -\int_{-t/2+Y_p}^{t/2+Y_p}\int_{-w/2+X_p}^{w/2+X_p}
\frac{I}{A_c}\frac{b-y}{2\pi\left[(a-x)^2+(b-y)^2\right]}\,\mathrm{d}x\,\mathrm{d}y,
$$
and the Y-axis component becomes
$$
H_{y,p}(a,b)= \int_{-t/2+Y_p}^{t/2+Y_p}\int_{-w/2+X_p}^{w/2+X_p}
\frac{I}{A_c}\frac{a-x}{2\pi\left[(a-x)^2+(b-y)^2\right]}\,\mathrm{d}x\,\mathrm{d}y.
$$
Using this model, I computed the theoretical eccentricity error curves for the six-element elliptic array with \(AR=0.33\), the seven-element elliptic array with \(AR=0.31\), and the corresponding circular arrays. The elliptic arrays and circular arrays have different allowable geometric eccentricity ranges because of their different dimensions. The following table compares the geometric limits and the worst-case theoretical errors.
| Array configuration | Weight (g) | Footprint area (mm\(^2\)) | X-eccentric error (%) | Y-eccentric error (%) |
|---|---|---|---|---|
| Fluxgate sensor | 69.5 | 3290 | — | — |
| Circular Hall array | 15.5 | 2827.4 | ≤17.08 | ≤2.21 |
| My elliptic Hall array | 11.9 | 908.5 | ≤4.73 | ≤0.35 |
The elliptic array reduces the installation footprint by 72.4% and the mass by 82.9% compared with a conventional fluxgate sensor used for high-voltage battery current monitoring. It also reduces the eccentricity error significantly compared with a circular array because the practical eccentricity range is constrained by the smaller available space.
I then analyzed the tilt error. A complete description of an inclined busbar requires the tilt elevation angle \(\alpha\) with respect to the positive Z-axis and the azimuth direction angle \(\beta\) in the XOY plane. The unit vector along the busbar is
$$
\mathbf{u}_l = (m,n,p),
$$
$$
m=\sin\alpha\cos\beta,\qquad n=\sin\alpha\sin\beta,\qquad p=\cos\alpha.
$$
The cross-section of an inclined busbar projected onto the sensor plane changes its shape and orientation, and the field point on the ellipse must be described in three dimensions. For a source point \((x,y)\) in the projected cross-section, the vector from the source to a Hall element at \((a,b,0)\) is
$$
\mathbf{r}_s = (a-x,\ b-y,\ 0).
$$
The magnetic field produced by the current flowing along \(\mathbf{u}_l\) at the Hall element can be written as
$$
\mathbf{H}(a,b) = \int_S
\frac{I}{2\pi A_c}
\frac{\mathbf{u}_l \times \mathbf{r}_s}
{|\mathbf{u}_l \times \mathbf{r}_s|^2}
\,\mathrm{d}x\,\mathrm{d}y.
$$
Expanding the components yields
$$
H_x = \int_S
\frac{I}{2\pi A_c}
\frac{-p(b-y)}
{p^2(a-x)^2+p^2(b-y)^2+\left[n(a-x)-m(b-y)\right]^2}
\,\mathrm{d}x\,\mathrm{d}y,
$$
$$
H_y = \int_S
\frac{I}{2\pi A_c}
\frac{p(a-x)}
{p^2(a-x)^2+p^2(b-y)^2+\left[n(a-x)-m(b-y)\right]^2}
\,\mathrm{d}x\,\mathrm{d}y,
$$
$$
H_z = \int_S
\frac{I}{2\pi A_c}
\frac{m(b-y)-n(a-x)}
{p^2(a-x)^2+p^2(b-y)^2+\left[n(a-x)-m(b-y)\right]^2}
\,\mathrm{d}x\,\mathrm{d}y.
$$
When \(\mathbf{u}_l=(0,0,1)\), these expressions reduce to the two-dimensional formulas for an untitled rectangular busbar. I therefore used the three-dimensional model to reconstruct the busbar current and evaluate both eccentricity and tilt errors. The reconstruction is based on the elliptic integration path, and it can be expressed as
$$
I_{\mathrm{calc}} = \sum_{i=0}^{5} H_{\mathrm{MFS},i}\,\Delta s,
$$
$$
H_{\mathrm{MFS},i} = \mathbf{H}(P_i)\cdot \mathbf{t}_i,
$$
where \(\mathbf{t}_i\) is the tangent unit vector at sensor \(i\) and \(\Delta s\) is the uniform curve segment length of the elliptic array.
My numerical results show that the eccentricity error and tilt error are strongly nonlinear functions of the conductor state parameters \(X_p\), \(Y_p\), \(m\), \(n\), and \(p\). To compensate these errors in real time, I need to estimate the conductor state from the Hall measurements, re-derive the projection of the tilted conductor, and then compute the current integral using the projected integration region. Estimating all five parameters from the six three-axis Hall outputs is a high-dimensional nonlinear regression problem.
I initially tested a back-propagation neural network called BP for this regression task. A BP network can approximate the nonlinear relationship between the magnetic field samples and the conductor state parameters. However, the BP network is sensitive to the initial values of its weights and thresholds. If the initialization is not suitable, the training process converges slowly or falls into a local minimum, resulting in large prediction errors. To solve this problem, I used the Grey Wolf Optimizer, abbreviated GWO, to optimize the initial weights and thresholds of the BP network before training. The resulting network is called the GWO-BP model. The grey wolf optimizer imitates the hunting and social hierarchy of grey wolves. In the algorithm, the best solution is denoted as \(\alpha\), the second-best as \(\beta\), the third-best as \(\delta\), and all other candidate solutions as \(\omega\). During each iteration, the wolves update their positions according to
$$
\mathbf{D} = \left|\mathbf{C}\mathbf{X}_p(t)-\mathbf{X}(t)\right|,
$$
$$
\mathbf{X}(t+1) = \mathbf{X}_p(t)-\mathbf{A}\mathbf{D},
$$
where \(\mathbf{A}\) and \(\mathbf{C}\) are coefficient vectors. The coefficients are given by
$$
\mathbf{A} = 2a\,\mathbf{r}_1-a,
$$
$$
\mathbf{C} = 2\mathbf{r}_2,
$$
where \(a\) decreases linearly from 2 to 0 during the search, and \(\mathbf{r}_1\), \(\mathbf{r}_2\) are random vectors in \([0,1]\). The \(\alpha\), \(\beta\) and \(\delta\) wolves guide the other wolves to promising regions of the search space. This mechanism allows the global optimizer to find better initial values for the BP neural network and, consequently, improves the regression accuracy.
I constructed the GWO-BP conductor-state estimator using an 18-12-5 architecture. The input layer contains 18 neurons because the final six-element array contains six Hall elements and each element provides three magnetic field components. The hidden layer contains 12 neurons, which I selected after trial experiments. The output layer contains five neurons corresponding to \(X_p\), \(Y_p\), \(m\), \(n\), and \(p\). The population size of the grey wolf optimizer is set to 20 and the maximum iteration number is 50. The BP network is then trained with the optimized weights and thresholds. Once the conductor state parameters are obtained, the projection of the busbar section is reconstructed, and the current is calculated numerically using the three-dimensional magnetic field integral and the elliptic line integral. This GWO-BP method effectively combines physical current reconstruction with data-driven state estimation.
I evaluated the performance of the GWO-BP estimator using multiple regression metrics. The mean absolute error is defined as
$$
\mathrm{MAE} = \frac{1}{n}\sum_{i=1}^{n}\left|y_i-\hat{y}_i\right|,
$$
the mean squared error is defined as
$$
\mathrm{MSE} = \frac{1}{n}\sum_{i=1}^{n}\left(y_i-\hat{y}_i\right)^2,
$$
and the mean absolute percentage error is defined as
$$
\mathrm{MAPE} = \frac{1}{n}\sum_{i=1}^{n}\frac{\left|y_i-\hat{y}_i\right|}{\left|y_i\right|}\times 100\%.
$$
The table below gives the performance comparison between the standard BP estimator and the proposed GWO-BP estimator for the five conductor-state parameters.
| Parameter | Algorithm | MAE | MSE | MAPE (%) |
|---|---|---|---|---|
| \(X_p\) | BP | 0.2272 | 0.0771 | 0.8419 |
| GWO-BP | 0.0498 | 0.0041 | 1.0411 | |
| \(Y_p\) | BP | 0.0619 | 0.0043 | 5.2646 |
| GWO-BP | 0.0146 | 0.0004 | 1.0621 | |
| \(m\) | BP | 0.0124 | 0.0003 | 10.3490 |
| GWO-BP | 0.0037 | 0.0001 | 2.8585 | |
| \(n\) | BP | 0.0071 | 0.0002 | 3.9567 |
| GWO-BP | 0.0028 | 0.0001 | 0.8626 | |
| \(p\) | BP | 0.0143 | 0.0005 | 2.2479 |
| GWO-BP | 0.0029 | 0.0001 | 0.4173 |
For most parameters, the GWO-BP estimator significantly reduces the MAE and MSE. The MAPE for the X-direction parameter \(X_p\) is slightly higher under GWO-BP than under BP, but the MAE and MSE are better, which indicates that the small number of near-zero samples has a disproportionate influence on the percentage metric. In practical high-voltage battery current sensing, the absolute deviations are more important than percentage deviations near zero, because very small conductor offsets are not dangerous. Thus the GWO-BP model provides stronger overall robustness for high-voltage battery systems.
I performed finite-element simulations with a three-dimensional model of the rectangular busbar and the Hall sensor array. The finite-element model was used to generate a data set that contains many combinations of eccentricities and tilt angles. The data set is divided into a training set and a testing set. The simulation also verified that the magnetic flux-density distribution around a rectangular busbar resembles an elliptic contour. This result agrees with my geometric choice of an elliptic array for high-voltage battery current measurement.
After the simulation, I built a prototype and tested it in the laboratory. The test platform includes a host computer, a programmable DC current source, a current reversing unit, a data acquisition unit, a protected DC power supply for the sensor, and a temperature chamber. The current source supplies the rectangular busbar with currents from \(-1500\) A to \(1500\) A. The current reversing unit changes the output polarity under computer control. The data acquisition unit records the reference current, the temperature, and the sensor output in real time. In the software platform, I created a program module, device configuration module, test sequence control module, and data recording module. The software allows the sensor to be tested at multiple current points and multiple temperature points automatically. This automation is useful for evaluating high-voltage battery current sensors because the battery management system must operate over a very wide current range and temperature range.
I first tested the calibration algorithm at room temperature. The current error before and after calibration was measured for the small-current, medium-current and large-current segments. The calibration process reduced the initial current error by 85.58%, and the residual error remained below 3‰ over the full measured current range. For the small-current segment, the negative-direction and positive-direction fit results are different because the amplifier offset and Hall offset are not perfectly symmetric. A single global linear compensation function is therefore not sufficient; I used separate fit coefficients for the positive and negative directions in each of the three current segments.
I then placed the sensor prototype inside the temperature chamber and repeated the calibration at \(-40^\circ\)C and \(125^\circ\)C. At the low temperature point, the calibration reduced the current error by 62.59%. At the high temperature point, the calibration reduced the current error by 42.99%. After the full temperature-range calibration, the current error remained below 3‰ at all three test temperatures. This satisfies the accuracy requirement for high-voltage battery current measurement in my design.
The next experiments evaluate the eccentricity and tilt error compensation algorithm. In the first test, the rectangular busbar is intentionally displaced along the X-axis while the Y-axis displacement remains zero. The X-direction offset is changed from 0 mm up to the geometric limit of the elliptic array. Before compensation, the current error grows almost linearly with the offset and reaches about 4.18% at 10 mm. The current error after BP compensation is significantly lower, but the error still increases with increasing offset. After GWO-BP compensation, the current error at 10 mm is reduced to about 1.46%. This result represents a reduction of 65.07% compared with the uncompensated sensor. In the second test, the busbar is displaced along the Y-axis. For Y offsets smaller than 0.4 mm, the compensation has little effect because the uncompensated error is already small. For larger Y offsets, the GWO-BP method reduces the error significantly. At a Y offset of 2.4 mm, the uncompensated error is 0.94% and the compensated error is 0.51%, corresponding to a reduction of 45.74%. When the busbar is displaced in both X and Y directions, the errors accumulate. The maximum error is about 2.54% at the extreme corner of the allowable eccentricity range.
I then tested the tilt compensation. In the first tilt test, I set the azimuth angle \(\beta=20^\circ\) and changed the elevation angle \(\alpha\) from \(0^\circ\) to \(45^\circ\). Without compensation, the current error increases slowly when \(\alpha<15^\circ\), but after this point it grows quickly. At \(\alpha=45^\circ\), the uncompensated error reaches 12.37%. After applying the GWO-BP compensation, the error at the same angle is reduced to 2.95%, which is a reduction of 76.15%. The BP compensation also improves the accuracy, but the GWO-BP method achieves a substantially larger reduction. In the second tilt test, I set \(\alpha=0^\circ\) and varied \(\beta\) between \(0^\circ\) and \(360^\circ\). The uncompensated error has a local minimum near \(\beta=20^\circ\) and increases rapidly on both sides. The GWO-BP compensation reduces the maximum error in this test by 62.92%. The following table summarizes the experimental error reductions achieved by the GWO-BP algorithm for the high-voltage battery busbar current sensor.
| Error source | Test condition | Uncompensated error | GWO-BP error | Error reduction |
|---|---|---|---|---|
| X eccentricity | \(X_p=10\) mm | 4.18% | 1.46% | 65.07% |
| Y eccentricity | \(Y_p=2.4\) mm | 0.94% | 0.51% | 45.74% |
| Tilt angle | \(\alpha=45^\circ,\ \beta=20^\circ\) | 12.37% | 2.95% | 76.15% |
| Tilt angle | varied \(\beta\) | maximum error | — | 62.92% |
I also tested the final accuracy at three temperature points with three identical prototypes to confirm reproducibility. At the room-temperature point \(25^\circ\)C, all three sensor samples have a current error below 3‰. At the low-temperature point \(-40^\circ\)C and at the high-temperature point \(125^\circ\)C, the error again remains below 3‰. The accuracy experiment confirms that the calibration and compensation algorithms are stable over the complete operating range of a high-voltage battery.
In conclusion, I have designed and verified a Hall-element elliptic-array current sensor for high-voltage battery applications. The elliptic array is generated by the uniform-curve-segment-length method, which allows the sensor to match the flat rectangular conductor geometry while occupying a much smaller installation area than a circular array. The sensor hardware includes the necessary analog conditioning, isolated power supply, microcontroller and CAN communication interface. A full-temperature-range calibration algorithm was developed to reduce the initial error caused by component tolerances and temperature drift. After calibration, the measurement accuracy is better than 3‰ over the full current range at \(-40^\circ\)C, \(25^\circ\)C and \(125^\circ\)C. To address eccentricity and tilt errors caused by real operating conditions inside the high-voltage battery pack, I constructed a three-dimensional magnetic field model and a current-integration model. I then applied a grey wolf optimizer to train a BP neural network for conductor-state estimation. The GWO-BP estimator greatly improves the accuracy of estimating the conductor offset and tilt parameters, and the subsequent current reconstruction reduces the eccentricity error and tilt error effectively. The experimental results demonstrate that the proposed GWO-BP compensation method improves the robustness of the Hall array current sensor in demanding high-voltage battery environments.
Future work can extend this approach to handle additional interference sources. For example, nearby busbars in the high-voltage battery pack produce crosstalk magnetic fields that can disturb the current measurement. A bent conductor or a connector close to the sensor also creates an inhomogeneous magnetic field distribution that is not fully described by the infinite straight conductor model. Real-time performance could also be improved by optimizing the neural-network structure or by using a more lightweight estimation algorithm for embedded implementation. Nevertheless, the work presented in this paper provides a solid foundation for a compact, high-accuracy, and mechanically tolerant current sensor solution for high-voltage battery systems in new-energy vehicles.