LORENTZ FORCE MAGNETOMETER WITH QUADRATURE FREQUENCY MODULATION

In this paper, a Lorentz force magnetometer demonstrates quadrature frequency modulation (QFM) operation. The Lorentz force magnetometer consists of a conventional 3-port resonator, which is put into oscillation by electrostatic driving and sensing. The bias current flowing through the resonator is proportional to the displacement, and generates Lorentz force in quadrature with the electrostatic force. As a result, the Lorentz force acts as an equivalent spring and the magnetic field can be measured by reading the change in oscillation frequency. The sensor has a sensitivity of 500 Hz/T with a short-term noise floor of 500 nT/√Hz. The bandwidth of the sensor is increased to 50 Hz, a factor of 12 greater than that of the same resonator operating in amplitude-modulated (AM) mode. INTRODUCTION Many Lorentz force magnetometers have been proposed in recent years [1-3]. The MEMS magnetometer is entirely silicon, sharing the same fabrication process as commercially available accelerometers and gyroscopes. A single-structure 3-axis magnetometer has also been reported [3], giving it great potential to be used as an electronic compass in smart phones. Compared to the commercially available Hall-Effect sensors currently used in smart phones [4], MEMS magnetometers do not require flux-concentrators for 3-axis measurement and are free from magnetic hysteresis. This work demonstrates a resonant Lorentz force magnetometer with quadrature frequency modulation (QFM) readout. Readout control electronics based on amplitude modulation (AM) and fabrication of the device were reported in [1]. A conventional AM magnetometer modulates low frequency magnetic field to a frequency near the device’s resonance frequency. The motion resulting from the Lorentz force is therefore amplified by the mechanical quality factor (Q) of the sensor, and the motion’s amplitude is used as a measure of the input magnetic field. Although Brownian-limited noise floor can be achieved, AM magnetometers suffer from small bandwidth and large temperature sensitivity. In an FM magnetometer, input magnetic field results in a change in the oscillation frequency by varying the stiffness (k) of the sensor. Two different methods for FM modulation have been reported in the literature. A straightforward method is to use Lorentz force to generate axial stress [5] which results in a change in k of the oscillator and therefore the resonance frequency of the oscillator changes. The other method is to use quadrature frequency modulation (QFM), where an external force having the same frequency as, but in quadrature with, the self-sustaining force creates a phase shift in the oscillation loop. This phase shift results in a change in the oscillation frequency, since oscillation always occurs at the frequency that satisfies 0° phase shift around the loop. MEMS QFM gyroscopes (having Coriolis force as the external force) [6] and magnetometers (having Lorentz force as the external force) [7] were demonstrated. Using the same MEMS magnetometer structure, magnetic field can be either amplitude modulated (AM) or frequency modulated (FM) by controlling the phase difference between the Lorentz force and electrostatic force. Relative to earlier AM magnetometers, QFM operation extends the sensor’s bandwidth from a few Hz to 50 Hz, which is independent from the sensor’s mechanical bandwidth. QFM operation also significantly increases the dynamic range. THEORY System Dynamics The MEMS Lorentz force magnetometer is based on a traditional three-port MEMS resonator operating as an oscillator. The resonance frequency serves as the system reference clock for the bias current generation. The system dynamics of the Lorentz force magnetometer can be modeled as a second order mass-spring-damper system: + + = + (1) where m is the effective mass, b is the damping coefficient, k is the spring constant, FE is the electrostatic driving force and FL is the Lorentz force. In the operation of a QFM magnetometer, FL is always in phase with the displacement : = = (2) where the bias current gain AI is used here to represent the bias current to displacement ratio, AI=I/x, which is controlled by the electronics. By replacing the Lorentz force FL according (2), the equation of motion (1) can be written as + + ( − ) = (3) It is clearly seen from (3) that the magnetic field modifies the effective spring constant of the device, . Therefore, when the resonator operates in closed-loop, the magnetic field signal can be observed through a change in oscillation frequency. To demonstrate the principle of operation, the open-loop force-to-displacement frequency response of the QFM magnetometer is measured. Figure 2 shows the amplitude (top) and phase (bottom) characteristics. In this measurement, the bias current gain is set to AI = -3200 A/m. The black trace in Figure 2 shows the response of the magnetometer in the absence of an external magnetic field, whereas the blue and red traces show the Figure 1: Amplitude (top) and phase (bottom) characteristics of the QFM magnetometer with no external magnetic field applied (black trace) and ±50 mT magnetic field (blue and red traces) applied. r r o th c a d b q L th e