It is detailed in that, for the same gate size (W/L), the 1/f noise of a PMOS transistor is lower than a NMOS one. To achieve a very low noise performance and a high gain, the amplifier is composed of two stages (Fig. ): the first stage represents the main contribution to the total output noise, while the second one aims to increase the gain of the open loop amplifier. The first stage is a simple PMOS differential pair with resistive charges (R1 and R2). In this configuration the input transistor (M1 and M2) design is related to the low input voltage noise performance, while the combination of the drain resistance (R1) and the transconductance gm1 of the input transistors M1 and M2 is used to set the gain (A) of the differential pair: A=gm1R1.

By considering the thermal noise of the input pair transistor, the low-frequency noise from the input pair transistor and the thermal noise arising from the drain resistance, the power spectrum density of the equivalent input noise (ein12) of the preamplifier's first stage can be obtained: ein12=28kT3gm1+KFId1AFCoxL1W1fgm12+4kTgm12R1, where k is the Boltzmann constant, T is the temperature in K, Id1=I0/2 is the drain resistance, W1 is the channel width, and L1 is the gate length of the input transistors. AF (=1.4) and KF (=1.8e-26) are PMOS noise parameters. Knowing that gm1=2Id1KP′W1L1 and I0=2Id1, we get ein12=16kT3I0KP′W1L1+KF/2AFKP′CoxW12fI0AF-1+8kTI0KP′W1L1R1, where I0 is the bias current and KP′=Coxμp.

As shown in the equation, ein1 depends on the transistor gate size (W1/L1) and the bias current I0. A trade-off between the size and the power consumption of the final circuit must be specified considering the noise objective. Increasing the transistor size allows one to decrease its 1/f noise contribution. The current is also set to help in reducing the noise regarding the power consumption and the gain. In order to achieve 3 nV Hz-1 at 10 kHz of equivalent input noise and a minimum gain A1= 30 dB, I0 was set to 2 mA, W1 = W2= 5000 µm, L1 = L2= 1.2 µm and R1 = R2= 3 kΩ.

The second stage, which is a PMOS differential pair (M4–M5) with a NMOS load (M6–M7), will allow one to enhance the open loop gain to achieve the closed loop gain specification (GdB= 50 dB) over the desired frequency bandwidth (> 50 kHz). This stage contains a minimum number of transistors to save power consumption, silicon area and specifically the noise performance of the first stage. The power spectrum density of the voltage noise of this stage referred to as M4–M5 input is written as ein22=2BPI0AF-1W52f1+KN′BNKP′BPL5W5L72+16kT3I0KP′W5L51+KN′W7L5KP′W5L7, with BN=KF/2KN′Cox and BP=KF/2KP′Cox.

The second stage provides a gain A2= 55 dB if W4 = W5= 5000 µm and W6 = W7= 50 µm.

The total equivalent input noise voltage ePA is finally calculated using the PSDs of each stage ein12 and ein22 and their open loop gains A1 and A2: ePA=eout2A12A22=2A12A22ein12+A22ein22A12A22.

This last equation can be used to get the noise objective for the second stage, ensuring that ePA is equal to 3 nV Hz-1 at 10 kHz.

To make the gain of the amplifier weakly sensitive to temperature variation, the amplifier is used in a closed-loop configuration. The closed-loop gain is set by the R3 to R4 ratio (G=1+R3/R4), while the C1 capacitance is needed to ensure the phase margin. Since this capacitance does not impact the noise analysis, it will not be considered in the rest of the article. As decribed in , in the context of operational amplifier noise analysis, our noise analysis can be summarized as three contributions: the equivalent voltage input noise at the non-inverting input (ePA), the Johnson noise in R4 at the inverting input and the Johnson noise in R3. According to the analysis, two gains should be considered: the inverting (Av-inv) and non-inverting ones (Av-non-inv). However, in the case of an ideal op amp (which is a correct hypothesis in our design since the open loop gain is A1+A2= 85 dB), these two gains are written Av-inv=R3/R4 and Av-non-inv=1+R3/R4. Moreover, in the case of high closed loop gain (i.e. R3/R4≫1), these two gains can be considered to be identical. That allows us to consider the equivalent op amp input noise to be eOpAmp2=ePA2+4kTR4.

Lastly, the input referred noise contribution coming from gain resistance of the preamplifier (R4) is neglected, since its value is small (in our design we got R4=28Ω, which leads to an equivalent voltage noise contribution of about one-tenth of ePA).

The amplifier was fabricated in a standard 0.35 µm four-metal bulk CMOS process. The manufactured circuit has a 1.21 mm2 area. The silicon chip contains one amplifier. Its microphotograph is shown in Fig. .

The gain transfer function and the equivalent input noise have been characterized. A high pass filter, with cut-off frequency at 1 Hz, is inserted to remove DC offset, while the low pass filtering cut-off frequency is due to the combination R3 and C1. Figure  shows that the gain is about 50.7 dB from a few Hz (a high pass filtering is inserted to remove the offset) up to 50 kHz. The measured gain value is consistent with the 1+R3/R4 ratio (R3=10kΩ and R4=28Ω). Figure  demonstrates a measured input voltage noise about 3 nV Hz-1 at 10 kHz when connecting a 50 Ω resistor at the input of the amplifier.

The induction sensor has a very high input impedance. It implies that it is essential to minimize the input noise current of the amplifier since it will lead to a high contribution to the output noise voltage. In our design, the input current noise contribution is less than 20 fA Hz-1, which is achieved thanks to the CMOS technology. It can be concluded, for this part, that the combination of low power consumption, low input voltage noise and low current noise, which are essential to induction sensors, can be achieved using CMOS technology at the price of a significant design work.

The mentioned noise source can be modelled through the concept of complex permeability where the imaginary part of the permeability is related to the ferromagnetic noise source. For high permeability Mn–Zn sintered ferrite, we use the complex susceptibility of resonance type given in . The first fraction in the susceptibility relation (Eq. ) corresponds to the frequency dispersion of domain wall motion contribution, while the second fraction represents the magnetic moment rotation contribution.

μ=1+ωd2χd0ωd2-ω2+iωβ+ωs+jωαωsχs0ωs+iωα2-ω2, where χd0 and χs0 are the static susceptibilities for domain wall motion and magnetic moment rotation, ωd=2πfd and ωs=2πfs are resonance frequencies of domain wall motion and magnetic moment rotation, β and α are the damping factors, and f=ω/(2π) is the operating frequency.

The apparent permeability can be written as μ=μ′-jμ′′.

So, the real and imaginary parts, deduced from Eq. (), are expressed as follows: μ′=1+ωd2χd0ωd2-ω2ωd2-ω22+(ωβ)2+χs0ωs2ωs2-ω2+α2ω2ωs2-ω2(1+α2)2+2ωωsα2,

μ′′=χd0ωβωd2ωd2-ω22+(ωβ)2+χs0ωsωαωs2+ω2(1+α2)ωs2-ω2(1+α2)2+2ωωsα2.

The expressions of real and imaginary parts of susceptibilities are quite similar to the one given by at a sign near in the numerator of the real component of the susceptibility. We will now consider Mn–Zn ferrite from Ferroxcube of 3C95 type, which appears to be a good candidate for designing an induction sensor thanks to its high relative permeability (μr>2000), its availability in different shapes and its stability over a wide temperature range (from -100 up to +200 ∘C). For this material, we have determined on a toroidal core sample the values of the complex susceptibility model parameters (ωd, ωs, χd0, χd0, and ωr). These parameters are summarized in Table , while the measured and fitted susceptibility dispersions (real and imaginary parts) are plotted in Fig. . The obtained values are in the same magnitude range as those reported in .

The magnetic gain produced by the ferromagnetic core, known as apparent permeability , allows one to increase the induced voltage. This one results in the combination of the relative permeability of the material (μr) and its shape, through the demagnetizing coefficient (Nx,y,z) in a given direction (x, y or z). For a long cylinder core of length to diameter ratio m, the approximation of the ellipsoid demagnetizing coefficient, given in , is repeated here: Nz(m)=1m2(ln⁡(2m)-1).

In the current study, a diabolo core shape (shown in Fig. ) is used, whose apparent permeability (μapp given in ) is expressed as μapp=μr1+Nz(m)d2DO2(μr-1), where Nz(m=LC/DO) is the demagnetizing coefficient in the z direction for a cylinder of length LC and diameter DO.

Assuming that apparent permeability owns real and imaginary parts, it can be written under the following form: μapp=μapp′-jμapp′′.

By substituting, in apparent permeability (Eq. ), the equation of complex permeability derived for ferrites (Eq. ), and by identifying it with Eq. (), we deduce the real and imaginary parts of the apparent complex permeability, respectively Eqs. () and (). μapp′=μ′1+Nz(m)d2DO2(μ′-1)+Nz(m)d2DO2μ′′21+Nz(m)d2DO2(μ′-1)2+Nz(m)d2DO2μ′′2 μapp′′=μ′′1-Nz(m)d2DO21+Nz(m)d2DO2(μ′-1)2+Nz(m)d2DO2μ′′2

In the case of a ferromagnetic core induction sensor, the inductance equation is Ł=λN2μ0μappSLC, where (S) is the ferromagnetic core section, μ0 is the vacuum permeability and λ=(LC/Lw)2/5 is a correction factor. Thus, the inductance will also have a real part (L′) and an imaginary part (L′′): Ł=L′-jL′′, which are written as follows: Ł′=λN2μ0μapp′SLC, Ł′′=λN2μ0(μapp′′)SLC.

Finally, the noise source contribution arising from the ferromagnetic core will look like a Johnson noise whose power spectrum density can be written as PSDL=4kTℜ(jLω), which becomes PSDL=4kTL′′ω.

In the same way, the mutual inductance will exhibit real and imaginary parts; however, since the mutual inductance is much smaller than the self-inductance, its imaginary part will be neglected and the mutual inductance will be assumed to be a real number.

The block diagram of Fig.  is used to facilitate the computation of the output noise contribution for each noise source. The transmittance of the feedback flux amplifier, given by Eq. (), is modified to take into account the contribution of the complex inductance: T(jω)=VOUTB=-jNSGμappω(1-LCω2)+jω((R+L′′ω)C+GMRfb).

In this block diagram, the noise source coming from the ferromagnetic core is directly added to the thermal noise of the coil resistance. Since this block diagram is dedicated to noise analysis, it is assumed that measured flux (φ) is null. For the reasons given in Sect. , the noise contribution coming from the input resistance of the preamplifier (R4) is neglected.

The block diagram permits one to determine the transfer function between the output noise contribution (referred to as the VOUT node) and each of the noise sources. The method is the following: the block diagram is drawn for a given source, while the other noise sources are cancelled thanks to the superposition theorem (for instance, see the block diagram for the feedback resistance noise source shown in Fig. ).

Then, the closed loop transfer function seen by the Rfb noise is obtained: T(jω)Rfb=jωMGRfb1-L′Cω2+j(R+L′′ω)Cω+jωMGRfb. Using the general relation between input and output PSD (namely, PSDOUT=∣T(jω)∣2PSDIN), we deduce the output noise contribution of the feedback resistance: PSDRfb=4kTRfbωMGRfb2(1-L′Cω2)2+(R+L′′ω)Cω+ωMGRfb2.

This latter expression can be simplified in the frequency range where the feedback flux operates: PSDRfb≃4kTRfb.

In a similar manner, the noise source contribution from the coil's resistance is derived: PSDR=4kTG2(R+L′′ω)(1-L′Cω2)2+(R+L′′ω)Cω+GMωRfb2.

The 1/f noise contribution of the preamplifier input voltage noise being neglected, the noise source contribution of the preamplifier input voltage noise is PSDePA=ePA2G2(1-L′Cω2)2+(Cω(R+L′′ω))2(1-L′Cω2)2+(R+L′′ω)Cω+GMωRfb2.

Similarly, the noise source contribution of the preamplifier input current noise is obtained: PSDiPA=∣Z∣iPA2G2(1-L′Cω2)2+(Cω(R+L′′ω))2(1-L′Cω2)2+(R+L′′ω)Cω+GMωRfb2, where |Z| is the equivalent impedance modulus of the induction sensor seen at the positive input of the amplifier, which is expressed (after some computations) as |Z|=(R+L′′ω+(Mω)2Rfb)2+(L′ω)2(1-L′Cω2)2+(R+L′′ω+M2ω2Rfb)Cω+GMωRfb2.

Finally, the total output noise contribution (PSDout) is computed by adding the individual power spectral density contribution of each noise source (under the hypothesis of uncorrelated noise): PSDout=PSDZ+PSDePA+PSDiPA+PSDRfb.

Finally, the noise equivalent magnetic induction (NEMI), which is the square root of the power spectrum density of the total output noise (PSDOUT) related to the transfer function modulus of the induction magnetometer (T(jω) given by Eq. () for feedback flux magnetometer) can be determined.

A single axis induction magnetometer has been built with an induction sensor using a diabolo core shape made of 3C95 Mn–Zn ferrite from Ferroxcube. The sensor has been combined with the MAGIC2 ASIC amplifier. The parameters of the induction sensor design and the preamplifier are summarized in Table .

The parameters of the sensor lead to the following value of the electrokinetic modelling: R=48Ω (copper wire operating at 300 K is assumed, i.e. ρ=1.7×10-8Ωm), L=0.306 H (assuming λ equal to 1), M=3 mH, C=150 pF and μapp=420. The sensor weight is lower than 30 g, while the ASIC amplifier power consumption supplied with a 12 V battery is lower than 30 mW. The noise measurement (PSDout) of the induction magnetometer (i.e. sensor connected to its preamplifier) has been performed inside a shielded box consisting of three layers of mu-metal materials and one of conductive material (connected to the preamplifier ground in a way that minimizes the current loop via ground connection. The thickness of each layer is 1 mm and the inner box side length is 40 cm. Each layer of the shielding box is separated by 1 cm air gaps (the size of the magnetic shielded box should be much wider than the one of the sensor). The transfer function (T(jω)) of the induction magnetometer has been measured in gain and phase in a large diameter Helmholtz coil (1 m) mounted on a wood structure to ensure a homogeneous magnetic field at the scale of the sensor. The accuracy of the facility was verified using a small air-core coil whose theoretical transfer function is fully known. For both measurements (transfer function and noise), an Agilent 35670 spectrum analyser was used. The sensor was equipped with a very thin electrostatic shielding to be insensitive to the electric field component of the electromagnetic waves. The electrostatic shielding was designed to minimize the additional noise from induced current . The measurements have been done in two configurations: one with the electrostatic shielding and the other one without (in this case, the shielded box plays the role). The simulated NEMI curve, using the modelling of the complex permeability, is compared to the measured one (in the configuration without electrostatic shielding) in Fig. .

The result is that the theoretical NEMI (computed for both real and complex permeability) leads to an extremely low minimum NEMI value (< 2 fT √(Hz)-1), while practical measurement leads to a higher NEMI value (∼ 4 fT √(Hz)-1) in the same frequency range (namely, 20 and 30 kHz). The contribution of the complex permeability increases weakly the NEMI (at least in the frequency range between 10 and 100 kHz). A significant difference between the measured and computed NEMI remains, which suggests that a noise source other than ferromagnetic core contribution dominates and limits the NEMI value in the frequency range where the feedback operates (namely, from 10 to 100 kHz in this design). Since the coil was wound directly on the ferromagnetic core, magnetostriction has been suspected of modifying the complex permeability dispersion and thus the higher NEMI measurement. In this aim, a sensor wound on an epoxi tube was realized and a ferrite core with comparable apparent permeability was compared to the sensor reference, but no significant differences have been noticed.

Next, the occurence of an extra noise coming from the coil AC resistance is suspected of increasing the Johnson noise contribution coming from the coil resistance (namely, PSDR). The AC resistance increase of the coil comes from the skin effect enhanced by the proximity effect. This effect is taken into account by designers of transformers . In these devices, the AC resistance increase causes extra losses and thus temperature elevation of the transformer. The model proposed by is mono-dimensional and assumes a skin depth depending on the distance between wires. However, a lateral skin effect occurring at the end of the winding is also expected , making Dowell's model unusable. The contribution of the skin effect enhanced by proximity and the lateral skin effect is also well known to increase strongly the AC resistance and thus to reduce the signal-to-noise ratio of induction sensors for nuclear magnetic resonance . Consequently the skin effect enhanced by the proximity and lateral effect in the case of a multi-layer winding is one of the possible causes which could explain a part of the difference between measurement and modelling.