Theoretical and Experimental Study of the Critical Dimension of Low-Cost Low-Frequency Micro Search Coil Magnetometer for IoT Smart Energy Monitoring

Abstract

In this letter, we theoretically calculated the critical dimension for achieving minimum noise equivalent magnetic induction <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(NEMI)</i> in a micro search coil magnetometer (μSCM) with an amplifier circuit. The μSCM consists of a fabricated planar spiral coil on a quartz wafer (to avoid parasitic capacitance effect) and an optimized low-noise readout circuit. An analytical model was developed to study the effect of key parameters [line width ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> ), line spacing ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> ), line thickness ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> ), outer diameter ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_o</i> ), inner diameter ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_i</i> ), and noise sources of an amplifier] on the sensitivity, noise, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NEMI</i> of a μSCM. Simplified formulas are derived for sensitivity and noise of the presented μSCMs by approximating certain key factors, and it is shown that the critical <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> is constant for different die-sizes. It is found that there are critical dimensions to achieving minimum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NEMI</i> . These critical dimensions are a function of layout design and amplifier noise parameters. The critical line width is found from the theoretical analysis as a root of a fourth-degree polynomial, and a generalized contour plot is presented, which can be utilized as a general design guideline for a μSCM. To validate the theoretical modeling, 20 different μSCMs with different dimensions were fabricated using a low-cost single-mask micro-electro-mechanical systems (MEMS) technology. The fabricated sensors were successfully characterized, and the experimental results were in good agreement with our theoretical analysis. Furthermore, the experimental results indicated the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NEMI</i> was 7.9 nT/Hz <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.5</sup> for a uniform sinusoidal magnetic field with a frequency of 100 Hz, which is better than previously reported works and most of the available commercial Hall effect sensors.