The performance
of induction magnetometers, in terms of resolution, depends both on the
induction sensor and the electronic circuit. To investigate accurately the
sensor noise sources, an induction sensor, made of a ferrite ferromagnetic
core, is combined with a dedicated low voltage and current noise
preamplifier, designed in CMOS 0.35

Induction magnetometers are used in a wide range of
applications

In this section we briefly remind the reader of the basis of an induction
magnetometer using feedback flux. Induction sensors are basically built with
an

Feedback flux principle.

The transmittance of the feedback flux amplifier is expressed as

To preserve the sensor noise performances in terms of NEMI, the equivalent
input voltage noise (

It is detailed in

Schematic of the ASIC amplifier design.

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 (

As shown in the equation,

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 (

The second stage provides a gain

The total equivalent input noise voltage

This last equation can be used to get the noise objective for the second
stage, ensuring that

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

Lastly, the input referred noise contribution coming from gain resistance of
the preamplifier (

The amplifier was fabricated in a standard 0.35

Photographs of the low noise ASIC amplifier named MAGIC2.

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

Amplifier transfer function (in dB) of MAGIC2.

Equivalent input referred noise (in nV

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

The NEMI reaches its minimum value in the decade around the resonance
frequency. The usual modelling of the NEMI will underestimate its value in
this frequency range. In rare works, to our best knowledge, noise sources
related to the ferromagnetic material are evoked either through an empirical
correlation

The mentioned noise source can be modelled through the concept of complex
permeability

The apparent permeability can be written as

So, the real and imaginary parts, deduced from Eq. (

The expressions of real and imaginary parts of susceptibilities are quite
similar to the one given by

Susceptibility dispersion parameters for spin and domain wall resonance of Ferroxcube 3C95 Mn–Zn ferrite.

Measured (

The magnetic gain produced by the ferromagnetic core, known as apparent
permeability

In the current study, a diabolo core shape (shown in Fig.

Diabolo core induction sensor.

Assuming that apparent permeability owns real and imaginary parts, it can be
written under the following form:

By substituting, in apparent permeability (Eq.

In the case of a ferromagnetic core induction sensor, the inductance equation

Finally, the noise source contribution arising from the ferromagnetic core
will look like a Johnson noise whose power spectrum density can be written as

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.

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
(

Noise sources in the feedback flux induction configuration and block diagram representation.

The block diagram permits one to determine the transfer function between the
output noise contribution (referred to as the

Block diagram representation of the feedback resistance noise source.

Then, the closed loop transfer function seen by the

This latter expression can be simplified in the frequency range where the
feedback flux operates:

In a similar manner, the noise source contribution from the coil's resistance
is derived:

The

Similarly, the noise source contribution of the preamplifier input current
noise is obtained:

Finally, the total output noise contribution (

Finally, the noise equivalent magnetic induction (NEMI), which is the square
root of the power spectrum density of the total output noise
(

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

Design parameters.

The parameters of the sensor lead to the following value of the
electrokinetic modelling:

NEMI curves comparison: NEMI with real permeability (pink), NEMI with complex permeability (green) and NEMI measured on a prototype (blue).

The result is that the theoretical NEMI (computed for both real and complex
permeability) leads to an extremely low minimum NEMI value
(

Next, the occurence of an extra noise coming from the coil AC resistance

While fitting methods usually assume that the extra noise from an induction
magnetometer comes from the ferromagnetic core, we have undertaken a
modelling attempt of the noise source contribution from a high-permeability
Mn–Zn ferrite core. The way to take it into account has been achieved by
modelling the apparent complex permeability through the susceptibility
frequency dispersion of the domain wall motion and magnetization rotation. We
have assumed that the machining of the core did not modify the complex
permeability. The comparison of the NEMI measurement on a prototype with the
NEMI modelling has shown a significant difference around the frequency
resonance. Thus, the ferromagnetic core noise seems too weak to explain the
difference between the model and the measured NEMI. Thus, the occurrence of
an extra noise due to the AC resistance increase is suspected of playing a
role. The ferromagnetic core noise contribution (through the apparent complex
permeability modelling) should be studied on other ferromagnetic core
materials (especially Ni–Fe alloy ferromagnetic material). The accurate and
rigorous modelling of the NEMI around the resonance frequency remains an
issue for fT

The authors would like to thank the reviewers for the time they spent to perform the review and for the stimulating discussions. The authors would also like to thank A. Grosz for the fruitful exchange concerning the delicate NEMI measurements. Edited by: B. Jakoby Reviewed by: three anonymous referees