Review of magnetic sensors
Recently, magnetic sensors have come to play a key role in many applications. They permeate more and more of our life and work in nearly all engineering and industrial sectors. Thanks to several decades of research, today, the diversity of magnetic sensors give us the ability to measure a wide range of magnetic elds at levels even below several tens of femtotesla. There is a high dependency on magnetic sensors in new technologies, such as military, security, magnetic recording, space research, navigation, bio-medical, geometric measurement, car industry, cell phones, etc.
Magnetic sensors can be sorted into three basic categories: high sensitivity; medium sensitivity and low sensitivity. High sensitivity are those that are able to measure magnetic elds lower than nanotesla level. Low sensitivity are those that are appropriate for detecting magnetic elds greater than tens of millitesla. Finally, the boundaries of these two categories are allocated to those that are appropriate for measuring the medium range of magnetic eld. Apart from this, magnetic sensors can also be split into two other main groups: scalar and vector magnetic sensors. Scalar magnetic sensors are those that measure the total magnitude of the eld and vector magnetic sensors measure the magnitude of the eld only in their sensitive direction.
In a chapter of this length, it is not possible to cover all the magnetic sensors and relative applications. Our aim is to focus on the vector magnetic sensors that can be used to measure the Earth’s magnetic eld for such modern navigation systems. However, some recently developed sensors can perform out of their ordinary range. For instance, although the Hall sensor is dedicated to the low sensitive magnetic sensor, it can also reach the resolution of the medium sensitivity group.
Several articles and books have been published for di erent types of magnetic sensors. In general, books such as  and  cover almost all types of magnetic sensors (except for some modern ones). Magnetoresistance sensors such as AMR and GMR have been well described in  and GMI in ,  and . Hall e ect sensors have also been explained in . Note that our review of the magnetic sensor is based mainly on these mentioned references.
Hall sensors are the magnetic sensor most widely used nowadays in several applications. The car industry is a heavy user of Hall magnetic sensors for such applications as anti-lock braking systems (ABS), wheel speed, crankshaft, etc.
The Hall E ect was discovered by Edwin Hall in 1879 while he was a doctoral candidate. Hall sensors work on the basis of Lorentz force. In this case, when a current-carrying conductor is exposed to the magnetic eld density, a Lorentz force deviates electrons from their initial directions and creates a voltage as output (see Figure 2.1). However, this voltage is a low-level signal, and it needs to be equipped with a low noise electronic design to amplify the signal. Equation (2.1) presents the proportionality of this voltage to the other parameters: VH = RH :I:B cos (2.1)
Where RH is the Hall coe cient, I is the current passing through the strip, t is the thickness of the strip and B cos is the perpendicular component of the magnetic eld intensity.
The sensitivity of Hall sensors is dependent on the mobility of electrons which can be given by Equation (2.2). EH = Hn[Ee B] = RH [J B] (2.2)
Where Ee is an external electrical eld, B is the magnetic ux density, EH is the Hall electric eld, J is the current density and Hn is the Hall mobility of electrons. The frequency limitation of Hall E ect sensor is about 1Mhz  and their resolution can be extended to several hundred micro Gauss. However, using a technique known as magnetic eld ampli cation, can improve the Hall sensor sensitivity to close to 200 pT .
Induction coils or search coils are based on Faradays law of induction. Any variation of the magnetic ux environment of a coiled conductor will cause a voltage. Generally, in order to improve the search coil performances, a rod of ferromagnetic material that so called core is used inside the coil. Sometimes this e ect is called magnetic ampli cation. Since the coil has a high permeability material, the surrounding magnetic elds are con-centrated through the core and the senor provides more signals. Moreover, as depicted in Figure 2.2, some methods proposed using additional ux concentrators to increase the magnetic ampli cation inside the coil .
The frequency response of the search coil magnetometer may be limited by the ratio of the inductance of the winding coil to its resistance. In addition, the sensitivity depends on the turn number of winding coils, the dimensions of the coil and the permeability of the core material. The induced voltage of a search coil can be expressed by Equa-tion (2.3).
where S is the cross-sectional area of the core, N is number of turns, e is the perme-ability of the core, and dBdt represents the time variation of the magnetic eld along the sensitive direction of the sensor.
Because of the winding coil, a capacitance exists between the conductors that causes a resonance on the sensor output. This resonance can be considered a drawback, as it saturates the output of electronics and limits the frequency response. Due to this e ect, depending on application, feedback ux can be used to suppress the resonance of the search coil magnetometer. Finally, apart from the search coil’s size, these sensors can be used in numerous applications due to their wide sensitivity range and frequency response, from 1Hz to 1Mhz. However, the sensitivity is reduced by decreasing the frequency, and even by using the integrator the DC magnetic eld can not be measured.
Flux-gate sensors measure weak magnetic elds in the range of 0.01nT to 1mT and they are the most widely used sensor for navigation systems. The most common type of ux-gate magnetometer is called the second harmonic device. This ux-gate magnetometer is based on using two coils around a common high permeability ferromagnetic core. A premagnetization winding or excitation coil produces an alternative eld in order peri-odically to saturate the core of the ux-gate and then a pick-up winding or sensing coil is used to measure an external magnetic eld. (see Figure 2.3)
In the absence of the external eld, the sensor reading only relates to the magnetic eld induced by the excitation coil at a frequency of f. To be more precise, the sensor output is a voltage that corresponds to the sum of di erent odd harmonics of the excitation frequency. Once the external eld is applied, the even harmonics are added to the sensor reading as well (pick-up coil). Thus, the amplitude of these even harmonics can be used to identify the intensity of the external magnetic eld.
The frequency responses of the ux-gate magnetometers are limited by the excitation eld that is provided by the excitation coil and the response time of the ferromagnetic material. Their operating frequency is limited to a range from DC eld measurement to about a few tens of kHz.
However, the ux-gate magnetometers are basically big because of the large ferromag-netic materials, and they su er from a limited operating range and high power require-ment. Due to this, recently several e orts have been made to miniaturize the ux-gate sensor for the sake of reducing the size, weight, cost and power consumption . This type of ux-gate sensor can be considered comparable (in term of resolution, size and power consumption) to anisotropic magnetoresistance (AMR) sensors.
Magnetoresistance and Magnetoimpedance magnetometer
The principle and functionality of AMR (anisotropic magnetoresistance) sensor will be detailed in the next chapter. Here, we limit ourselves to presenting some advantages of this type of magnetometer. AMR magnetometers have a simple fabrication process regarding the number of layers and materials used. They are widely available nowadays and several companies participate in this segment, such as Honeywell, Philips, Sensitec, Memsic, etc. Compared to other magnetometers, AMR sensors are also the most stable magnetoresistance sensor in term of bias and sensitivity. Because of MEMS technologies, triple axes of this type of sensor are available in a tiny package. These are the reasons that AMR sensors are still mostly used for low-cost inertial navigation systems.
GMR and TMR
Gain Magneto Resistance (GMR) was discovered in 1980s . This phenomenon is based on the principle of spin dependent scattering. The most frequently used type of GMR is a spin-valve sensor that exhibits magnetoresistance of 8-20%. The simple spin-valve GMR structure contains four-layers with di erent speci cations (see Figure 2.4). Two layers of ferromagnetic thin layer material are separated by conducting non-ferromagnetic interlayer. The forth layer is an antiferromagnetic eld that is used to pin the magnetization of the ferromagnetic layers. The main key point of GMR perfor-mances is the thickness of the layers. This thickness causes di erent electrical resistance as a functionality of external magnetic eld.
Typically, the layers should be thinner than the mean free path of the electrons (less than 10nm). The antiferromagnetic layer of GMR structure is used to x the mag-netization direction of one of the ferromagnetic layers that is called the pinned layer. In contrast, the highly conducting nonmagnetic layer separates the other ferromagnetic layer that is called the free layer. Then, by assuming that the magnetic moment of the pinned layer is xed with the help of the antiferromagnetic layer, the magnetic moments of the free layer only change with the external magnetic eld. Figure 2.4 (c) depicts a schematic illustration of the density of electron states in two ferromagnetic layers. Here the current contains spin down and spin up elements and as a consequence of the fer-romagnetic material the density of states at the Fermi level is asymmetric. Meanwhile, assuming that the spin down electrons are scattered more strongly than the spin up electrons. In this case, when the magnetization of the two ferromagnetic layers is in an aligned state the spin down electrons as mentioned are scattered in both ferromagnetic layers and the spin up electrons can pass from one ferromagnetic layer to another almost without scattering. Therefore, the total resistance of the multilayer appears to be low (Figure 2.4 (c) top).
In contrary, in the absence of the external magnetic eld, the two ferromagnetic layers have antiparallel magnetization direction. In this case, both spin up and down electrons are scattered with one of the ferromagnetic layers and then the total resistance appears to be high. (Figure 2.4 (c) bottom).
The GMR sensor can provide high sensitivity and temperature stability. However, the sensor output is basically unipolar and they have a hysteresis. They can also be de-stroyed easily in a strong magnetic eld.
Tunneling Magnetoresistance (TMR) has a similar structure to the spin valve GMR sensors. They also consist of two ferromagnetic layers separated by an ultra-thin inter-layer. An antiferromagnetic layer also helps to hold the magnetization of the adjacent ferromagnetic layer xed in direction. (see Figure 2.5)
However, compared to the GMR sensor, there are two main di erences. First, the ultra-thin insulating metal oxide material (also called tunnel barrier), is replaced between the ferromagnetic layers. Second, in the TMR sensor electrons pass from one layer to the other through the insulator layer. This is also why this sensor is called tunneling magne-toresistance, because of the behavior of electrons when they can apparently pass across some sort of a barrier.
The resistance of TMR sensor changes in a manner similar to what we have discussed for GMR sensors due to the spin-valve e ect. In the absence of an external magnetic eld, the two ferromagnetic layers have anti-parallel magnetization. This con guration causes low tunneling probability and consequently a higher resistance value for TMR sensor. In contrast, parallel magnetization leads to a higher tunneling probability and lower resistance for TMR sensor.
Table of contents :
2 Review of magnetic sensors
2.1 Hall sensors
2.2 Search coil
2.4 Magnetoresistance and Magnetoimpedance magnetometer
2.4.2 GMR and TMR
2.5 Magneto-Electric sensor
3 Anisotropic Magnetoresistance Sensor
3.2 Temperature eect
3.3 Cross-eld eect
3.4.1 Cross-eld error compensation
3.4.2 Temperature drift on the bias measurement
3.4.3 Power consumption of ipping method
3.5 Low-cost electronic design
3.6 Sensor performances and equivalent magnetic noise
4 Calibration algorithm and sensor error modeling
4.1 Vector magnetometer error modeling
4.1.1 Scale factor
4.1.2 Misalignment error
4.1.3 Soft iron error
4.1.4 Hard iron error
4.1.5 Sensor bias
4.2 Calibration process
5 Novel compensation method of the cross-axis eect
5.1.1 Methods using additional electronic design
220.127.116.11 Flipping method
18.104.22.168 Feedback loop
5.1.2 Sensor fabrication process
5.1.3 Methods using numerical computation
5.2 Numerical compensation method of cross axis in Earth’s magnetic eld
5.2.1 Compensation method without ipping
5.2.2 Compensation method using ipping
5.3 Experimental result
5.3.1 Sensor board
5.3.2 Scale factors
5.3.3 Method to nd scale factors
22.214.171.124 Non-ipped sensor
126.96.36.199 Flipped sensor
6 Indoor calibration
6.1 Indoor magnetometer calibration system (IMCS)
6.1.1 Theory of operation
6.1.2 Hardware Overview
188.8.131.52 Driver board
184.108.40.206 Study of the Helmholtz coil design
220.127.116.11 Mu-metal box design
18.104.22.168 Sensor board
6.1.3 Experimental results
22.214.171.124 Evaluate the performance of IMCS
126.96.36.199 Calibration results
188.8.131.52 Arbitrary position
6.2 On-board solution (auto-calibration)
6.2.1 Theory of operation
6.2.2 Experimental results
184.108.40.206 Evaluation of the performance of the oset coil for calibrating the AMR sensors
A Helmholtz coil
A.1 Magnetic eld provided by a current loop
A.2 Magnetic eld provided by a combination of two coils
A.3 Simulating the eect of an angular error of one coil on the uniformity of the magnetic eld of two coil combinations
B Mu-metal box
B.3 Distance between the layers
B.4 Number of layers