Introduction
Induction magnetometers are used in a wide range of
applications to measure extremely weak magnetic
fields over a wide frequency range (from mHz up to GHz). At 1 Hz, for
magnetotelluric waves observation purposes, noise equivalent magnetic
induction about 0.2 pT Hz-1 is reported in . The
context of this work concerns the study and the design of an induction
magnetometer in the very low–low frequency (VLF–VF) range to investigate plasma waves in space in Jupiter's
environment for an ESA mission. For this purpose, the goal of electromagnetic
wave measurement, given in terms of noise equivalent magnetic induction (NEMI
in T Hz-1), is challenging. An ability to reach NEMI lower than
10 fT Hz-1 at 10 kHz is mandatory. Due to the severe radiation
environment, it has been considered to locate the preamplifier either inside
the hollow ferromagnetic core of the induction sensor or
inside the mechanical tri-axis structure to take advantage of an efficient
radiation shielding provided by the sensor itself. An ASIC preamplifier
designed in 0.35 µm CMOS technology offers
a possibility of achieving very efficient induction magnetometers. In the
context of this work we designed an ASIC low noise preamplifier (called
MAGIC2) which offers especially low noise parameters which make possible the
investigation of the noise source of the induction magnetometer. This work
aims to extend the induction magnetometer modelling presented in
by introducing the noise source arising from the
ferromagnetic core based on physical modelling of the complex permeability.
Usually this noise source, which is frequency dependent, can be hidden by
other dominating noise sources (especially the equivalent input current noise
from the preamplifier). However, the use of a low input current noise
preamplifier permits one to enhance the noise source from the sensor itself
near the resonance. For this purpose, a comparison between the modelling and
the measurement of the NEMI is performed on a 12 cm length sensor using a
commercial Mn–Zn ferrite core (3C95 from Ferroxcube) of diabolo shape.
Induction magnetometer using feedback flux: generalities
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 N turns coil of section S. When the coil is wound around a
ferromagnetic core, the induced voltage is multiplied by a factor
μapp known as apparent permeability (described in
Sect. ). In harmonic regime at angular frequency ω, the
induction voltage is written as
e=-jωNSμappB,
where j2=-1 is the imaginary unit and B is the magnetic flux density
to be measured. The electrokinetic modelling assumes that the induced voltage
e is in series with the resistance R and the inductance L, while the
accessible voltage (Vout) is got at the capacitance C
terminals. The transmittance of the induction sensor exhibits a resonance at
angular frequency ω0=1/√(LC). In order to remove the resonance,
two kinds of electronic conditioning are classically implemented: a feedback
flux amplifier or a transimpedance amplifier . In this work,
we will focus only on the feedback flux amplifier schematically presented in
Fig. .
Feedback flux principle.
The transmittance of the feedback flux amplifier is expressed as
T(jω)=VOUTB=-jNSGμappω(1-LCω2)+jω(RC+GMRfb),
where j is the unity imaginary number, G is the voltage gain of the
amplifier, M is the mutual inductance between the measurement winding and
the feedback one and Rfb is the feedback resistance. In the following
section we will focus on the ASIC amplifier design and its noise parameters.
Low voltage and current noises ASIC amplifier design for feedback flux induction magnetometers
To preserve the sensor noise performances in terms of NEMI, the equivalent
input voltage noise (ePA) and the input current noise
(iPA) of the amplifier must be as low as possible with a special
awareness of 1/f noise. The requirement of the ASIC amplifier design is to
satisfy 3 nV Hz-1 of equivalent input voltage noise and a few tens of
fA Hz-1 of equivalent input current noise on a frequency range from
10 kHz up to 1 MHz. The gain needs to be about 50 dB to be suited to the
16 bit ADC and the power consumption should be lower than 30 mW. In this
context, CMOS technology, which is mainly composed of MOSFET transistors, is
an adequate solution. In the following section, design steps, voltage noise
modelling and some measurement results of the low noise ASIC preamplifier are
given.
Open loop noise considerations
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.
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 (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.
Closed loop noise considerations
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).
Measurement results and performances
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. .
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 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.
Amplifier transfer function (in dB) of MAGIC2.
Equivalent input referred noise (in nV (Hz)-1)
measured using a 50 Ω input resistor.
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.
Modelling of the ferromagnetic core noise source contribution
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 or a set of quality factors to
take into account the NEMI increase near to the resonance. In the first
quoted paper, the coefficient of the correlation is determined experimentally
for a given core size, which does not allow one to take it into account in a
preliminary design stage, while, in the second paper, the quality
factor values
are given from a tentative estimation. At low ambient field (< mT), the
noise in the ferromagnetic core comes either from an eddy current or from
magnetization mechanisms like domain wall relaxation and magnetization
rotation. At a high magnetic field (typ. mT), Barkhausen noise, related to
domain wall jumps, will occur. The usual domain of application of an
induction magnetometer is related to a quiet electromagnetic environment;
thus, Barkhausen noise will not be considered in this study.
Complex permeability of the Mn–Zn ferrite
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
.
Susceptibility dispersion parameters for spin and domain wall
resonance of Ferroxcube 3C95 Mn–Zn ferrite.
χd0
fd (MHz)
β
χs0
fs (MHz)
α
1400
1.4×106
7.5×106
900
8×106
5
Measured (μr′_Meas and μr′′_Meas) and
fitted (μr′_Fit and μr′′_Fit) susceptibility
dispersions of 3C95 Mn–Zn ferrite.
Complex apparent permeability
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.
Diabolo core induction sensor.
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.
Modelling and experimental results comparison
The noise equivalent magnetic induction
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.
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 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. ).
Block diagram representation of the feedback resistance noise
source.
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.
Experimental results and discussion
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.
Sensor length
LC=120 mm
Winding length
Lw=100 mm
Sensor diameter
d=4 mm
Diabolo ends diameter
DO=14 mm
Turns number
N=2350
Feedback coil turns number
N2=24
Copper wire diameter
dw=0.12 mm
Insulator thickness
t=25 µm
Layer number
nl=4
Feedback resistance
Rfb=10kΩ
Amplifier gain
GdB=50.7 dB
Voltage noise
ePA=3.3 nV √(Hz)-1
Current noise
iPA=20 fA √(Hz)-1
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. .
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
(< 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.