Calibration-Free Current Measurement with Integrated Quantum Sensor ^†

Abstract

This paper presents the application of a compact and fully integrated LED quantum sensor based on the NV centers in diamond for current measurement in a busbar. The magnetic field measurements from the sensor are directly compared with measurements from a numerical simulation, eliminating the need for calibration. The sensor setup achieves an accuracy of 0.28% in the measurement range of 0–30 $\mathrm{A}$ DC. The integration of advanced quantum sensing technology with practical current measurement demonstrates the potential of this sensor for applications in electrical and distribution networks.

1. Introduction

Current sensing techniques are essential for monitoring and controlling electrical systems. Common methods include the use of shunt resistors, which measure the voltage drop across a known resistance, and Hall effect sensors, which detect the magnetic fields generated by current flow. Advanced techniques such as Rogowski coils and fluxgate sensors provide accurate, non-invasive measurements of high current applications. In addition, NV centers in diamonds offer an exciting solution to the highly sensitive and precise measurement of magnetic fields, and therefore currents, through quantum metrology.

In recent years, NV centers in diamond have become prominent in quantum-based sensing. They enable highly sensitive magnetic field detection, even in the $\mathrm{fT}/\sqrt{\mathrm{Hz}}$ range [1,2,3], with spatial resolutions down to the atomic scale [4,5,6,7]. This technology provides accurate magnetic field measurements with minimal energy and space requirements [8]. In addition, NV centers measure temperatures [9,10,11,12] as well as electric fields [13] and have applications in quantum computing [14,15]. NV centers can also operate at room temperature, which simplifies their structure.

An NV center is a point defect in a diamond in which two carbon atoms are replaced by a nitrogen atom (red) and an adjacent vacancy (Figure 1a). NV centers in a diamond can have four orientations within its tetrahedral structure (purple atoms). The negatively charged NV center is a spin $S=1$ system with spin triplets in ground (³${\mathrm{A}}_2$) and excited states (³$\mathrm{E}$) (see Figure 1b). The optical excitation of the ground state is spin-preserving. An $m_s=0$ state fluoresces at $637\mathrm{nm}$ as it decays. However, the $m_s=\pm 1$ state tends towards non-radiative transitions to the ¹$A_1$ singlet state, making the spin state optically detectable. Microwave magnetic fields resonant to electron spin transitions in the ³$A_2$ ground state therefore reduce the NV center’s fluorescence and could be used to detect spin states.

The magnetic sensing capability of the NV center is due to the interaction of an external magnetic field $\overrightarrow{B}$ (the green arrow in Figure 1a) with the electron spin, causing a Zeeman shift in the $m_s=\pm 1$ states that is visible in ODMR measurements. In the absence of an applied magnetic field, zero-field splitting (ZFS) is observed due to internal crystal strain. The ZFS center frequency ($D=2.87\mathrm{GHz}$ at room temperature) also shifts with temperature [9,17], which allows for simultaneous magnetic field and temperature sensing.

For NV ensembles aligned along four crystal axes, sweeping the microwave frequency while observing the fluorescence yields eight dips, corresponding to the $m_s=+1$ and $m_s=-1$ levels, for each of the four NV axes (see Figure 1c).

As high-sensitivity magnetic field sensors, NV-center-based applications have already been used as current sensors [18,19,20,21], reaching dynamic ranges up to $1000\mathrm{A}$ with a step size of $10\mathrm{mA}$ [18]. However, these and other setups are often quite complex and not really portable. Here, we propose a quantum current sensor with all electrical inputs and outputs in a compact form.

2. Materials and Methods

2.1. Measurement Setup

The sensor is designed as described in detail in a previous publication [16]. Its setup is shown in Figure 2a. Basically, the sensor consists of three stacked printed circuit boards (PCBs) that emulate the basic principles of an optical quantum sensor based on NV centers. The LED-PCB contains an LED as the pump light source. The 150 $\mathsf{\mu }\mathrm{m}$ diamond is attached to the LED with an optical adhesive. Microwave excitation is provided by a microwave antenna located on the microwave PCB (MW-PCB). The resulting fluorescence in the red region is formed by a filter foil between the MW-PCB and the photodiode PCB (PD-PCB). The PD-PCB contains the photodiode that converts the fluorescence into a current signal. A clip to hold the sensor and integrate the offset magnet is printed using stereolithography (SLA). Microwave signals are swept using a vector signal generator (SMBV100B, Rhode & Schwarz, Munich, Germany). The LED current is supplied by a lab-built $30\mathrm{mA}$ current source powered by a $9\mathrm{V}$ block battery. The photodiode current is amplified by a lab-built [22] transimpedance amplifier (TIA) and measured with a multimeter (GDM9061, GW-Instek, Taipeh, Taiwan).

2.2. Simulation of Magnetic Field Distribution

The calculation of magnetic fields around conductors that are not round and therefore do not follow the simplified formulas of Biot–Savart is not trivial. Hence, a numerical approach is chosen and the field distribution is simulated numerically. COMSOL Multiphysics 6.1 is a simulation software for modeling and analyzing complex physical systems. In this study, COMSOL is used to create a three-dimensional model of the measurement setup. The busbar is modeled as a $6.5\mathrm{mm}\times 6.5\mathrm{mm}\times 60\mathrm{mm}$ piece of copper. A current of 0–30 $\mathrm{A}$ is applied via the coil feature of the magnetic field physics module in COMSOL. The diamond is approximated as a sphere of 150 $\mathsf{\mu }\mathrm{m}$ and is located $1.8\mathrm{mm}+0.5\mathrm{mm}=2.3\mathrm{mm}$ above the center of the busbar. The measurement $1.8\mathrm{mm}$ comes from the fact that the diamond is located $1.8\mathrm{mm}$ from the nearest edge. A $0.5\mathrm{mm}$ thick insulation layer was taken into account during printing. Additionally, a cylindrical permanent magnet with a diameter of $2\mathrm{mm}$ and a height of $3\mathrm{mm}$ was simulated. The material is labeled by the manufacturer as NdFeB, with a magnetization of N45. This magnet is positioned at the height of the diamond and shifted to one side by $5.5\mathrm{mm}$. Simulations are performed without an activated magnet and with currents between 0–30 $\mathrm{A}$. The maximum average volume in the diamond sphere is $1.051\mathrm{mT}$ at $30\mathrm{A}$ (see Figure 3a). The magnet produces a theoretical magnetic field of $5.46\mathrm{mT}$ (see Figure 3b). The current and the offset magnetic field are then applied together (see Figure 3c). The uniformity of the resulting magnetic field in the x-direction, i.e., the axis on which the diamond and the offset magnet are aligned, is given by the angle $\theta$ and is calculated as

$$\theta =acos\left({\displaystyle \frac{|B_x|}{\left|\right|B\left|\right|}}\right).$$

(1)

Averaged over the volume, a homogeneous field with an average deviation of $\theta =2.{15}^{\circ }$ from the x-axis is found. This setup would therefore be suitable for measurements using one axis of the NV center, which allows high sampling rates that are easy to obtain. For this DC application, however, a complete sweep of all NV axes is used.

2.3. Extraction of Magnetic Field Information from ODMR Measuremnts

To extract the information of the external magnetic field, the procedure described in [23] is used. The first eight superimposed Lorenz functions, such as

$$L_i={\displaystyle \frac{C}{1+{(2·(f-f_i)/\mathsf{\Delta }{v}_i)}^2}}$$

(2)

are fitted to a spectrum where C is the ODMR contrast, $f_i$ and $i\in [1,...,8]$ are the eight resonance frequencies, and $\mathsf{\Delta }{v}_i$ is the full width at half maximum (FWHM) of the resonance dips.

Now an attempt is made to fit a B-field vector that would produce exactly these resonance frequencies. To do this, a matrix is defined that represents the structure of the NV center as unit vectors in the crystal structure and follows from the trahedral arrangement of the crystal lattice:

$$NVs={\displaystyle \frac{1}{3}}\left(\begin{array}{cccc}0& 2\sqrt{2}& -\sqrt{2}& -\sqrt{2}\\ 0& 0& \sqrt{6}& \sqrt{6}\\ 3& -1& -1& -1\end{array}\right)$$

(3)

As described in the introduction, an external magnetic field projects on to the axis of the NV center, resulting in a parallel and a perpendicular component, $B_{\Vert }$ and $B_{\perp }$, for each axis. As described in [24], this results in specific resonance frequencies

$$f_{\pm }=D+{\displaystyle \frac{{\left(\gamma B_{\perp }\right)}^2}{D}}\pm \xi \sqrt{{\left(\gamma B_{\Vert }\right)}^2+E^2}$$

(4)

with $\xi$ as

$$\xi =\sqrt{1+{\displaystyle \frac{{\gamma }^4{B}_{\perp }^4}{4D^2({\gamma }^2{B}_{\Vert }^2+E^2)}}}$$

(5)

Here, D is the zero-field splitting parameter which allows us to obtain information about the crystal temperature [9], $\gamma$ is the gyromagnetic ratio, and E is the crystal strain parameter.

With a spherical magnetic field vector $\overrightarrow{B}=(B_0,\varphi ,\theta )$, a fit function is defined that optimizes $B_0,\varphi ,\theta$, and D so that the measured resonances fit the ones calculated using Equation (4). Thus, it is possible to extract the absolute magnetic field $B_0$ if the orientation of the diamond is known the direction of the vector can also be determined. In the present application, this is not necessary, and the measurement can be performed with randomly oriented diamonds.

3. Results and Discussion

The magnetic fields are measured in 3 A increments. A frequency sweep is performed between 2.6 and 3.1 GHz. The LED in the sensor head is driven with a constant current of $30\mathrm{mA}$. The output of the TIA is measured with a multimeter. The spectra of the different busbar currents are shown in Figure 4a.

A comparison between the resulting measurements and the simulated values is shown in Figure 5a. The constant offset is measured as $B_o=4.40\mathrm{mT}$, which is slightly different from the simulated value of $B_{os}=5.46\mathrm{mT}$. By compensating for the offset, the slope of the simulation and the measurement fit, with a standard deviation of ${\sigma }_B=0.0027\mathrm{mT}$ (see Figure 5b).

The values obtained with the magnetic field-based current sensor indicate that the current may be measured with a fully linear slope of $I=28.53{\displaystyle \frac{A}{mT}}$ and a current value standard deviation of ${\sigma }_I=0.085\mathrm{A}$. With respect to the measurement range, this results in an accuracy of $0.25\%$. The measuring range of 0–30 $\mathrm{A}$ was selected due to the availability of a current source with this current capacity. The mean absolute error

$$MA{E}_B={\displaystyle \frac{1}{n}}\sum _{i=0}^n|B_{meas,i}-B_{sim,i}|$$

(6)

is calculated as $MA{E}_B=2.97\mathsf{\mu }\mathrm{T}$. If converted into a current value by the simulated slope, this would result in a mean absolute error for the current values of $MA{E}_I=0.085\mathrm{A}$. The current method is predicated upon a precise interplay between the simulation and the positioning accuracy of the sensor. The measurement data are then compared with the simulation data in which the diamond is moved relative to the current simulated position. The permanent magnet is disregarded, and only the gradient resulting from the current is analyzed. Consequently, the simulation can be conducted in two dimensions, as no alterations are anticipated in the y-direction, which represents the axis of the busbar. An area of (1 × 1) mm² with a resolution of $50\mathsf{\mu }\mathrm{m}$ is selected on each axis. Figure 6a illustrates the correlation between the standard deviation of the simulated and measured values. On both graphs, a black dot indicates the initial simulation location based on the production data of the sensor and clip. It is evident that the sensor is in a satisfactory position for integrating measurement and simulation data; however, it is not optimally positioned to achieve the best possible result. This discrepancy may be attributed to the diamond’s non-central positioning with respect to the LED, resulting in a slight deviation from the assumed position or, alternatively, the sensor’s suboptimal placement within the holder. The accuracy of the sensor in the measuring range 0–30 $\mathrm{A}$ in relation to the position of the diamond is shown in Figure 6b. In order to achieve an accuracy of 0.5%, the diamond must be positioned in an almost horizontal band, at a height of 160 $\mathsf{\mu }\mathrm{m}$, and in the center.

Regarding the general measurement range, the NV center, and therefore the sensing technique, is not the limiting factor. Only from about a $102\mathrm{mT}$ external field do the excited and ground state overlap, rendering the extraction of the resonance frequencies no longer easily possible. If an additional magnetic field of $21\mathrm{mT}$ caused by the current is added to the existing offset, the measuring range is already 0–600 $\mathrm{A}$. The previously described fitting process also includes the ZFS parameter D, which is mainly caused by a change in temperature of the diamond. The sensor should therefore be able to function and measure magnetic fields correctly even when the temperature changes. The non-linear shift of the resonances due to off-axis magnetic fields is also covered by the parameter $\xi$. Therefore, we assume that the accuracy of the current sensor might be $0.014\%$ in a 0–600 $\mathrm{A}$ range and even lower in higher measurement ranges. However, this is a very theoretical consideration, as the conductor rail would have to be changed for this load and the field distribution would have to be recalculated. According to tables given in DIN 43 671, a load of $160\mathrm{A}$ for the used busbar, which has a cross-section of $A=42.5{\mathrm{mm}}^2$, can be assumed even without a correction factor for an increased accepted temperature. In this range, the sensor could be used with an accuracy of $0.05\%$. These results underscore the potential of the sensor for high-precision current measurements in practical applications, contingent upon the adaptation of its physical and magnetic configurations to support the desired current ranges. Moreover, a workflow was presented that demonstrates highly accurate measurements in practical applications using only numerical simulation results. Additionally, it was demonstrated that the accuracy of the measurements can be enhanced through the implementation of a reference measurement and the adjustment of the simulation parameters. This workflow can be further utilized when measuring currents in high-current applications such as DC supply cables using quantum sensors without contact.