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Advanced Lab Course
∆𝑬 Effect Sensors M214
Stand: Juni 2016
Objective: Introduction to the working principle of ∆𝐸 effect sensors and generally important
sensor characteristics.
Contents
1 Introduction 1
1.1 Objective 1
1.2 Prerequisites 1
2 Theory 2
2.1 Repetition of magnetoelasticity 2
2.2 ∆𝑬 Effect 3
2.3 Working principle of ∆𝑬 sensor 5 2.3.1 Sensitivity 6
2.4 Amplitude modulation 7 2.4.1 Linearity 8
2.5 Other Sensor Parameters 9 2.5.1 Total harmonic distortion (THD) 9 2.5.2 Quality factor and losses 9
2.5.3 Signal to Noise Ration (SNR) and Limit of detection (LOD) 9
2.5.4 Bandwidth 9
3 Experiment 10
3.1 Sensor 10
3.2 Setup 10 3.3 Tasks 12 3.3.1 General remarks 12 3.3.2 Working point 12 3.3.3 SNR (change excitation voltage) 12 3.3.4 THD 12 3.3.5 Bandwidth 12 3.3.6 LOD 12
4 References 13
M214: ∆𝐸 Effect Sensors
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1 Introduction
Magnetic measurements are used in various fields of application. Especially in geosciences
and aerospace they are extensively used to detect the earth magnetic field or magnetic
anomalies. Satellites employ flux gate sensors for reference measurements in positioning
devices. But also in consumer devices magnetometers are commonly present, e.g.
magnetoresistive sensors in reading heads of hard discs or as positioning devices in smart
phones. Simple Hall-effect sensors can be used to find underground power lines. In medicine
and diagnostics measurements of biomagnetic fields, e.g. for magnetocardiography (MKG) or
magnetoenzephalography (MEG) is accompanied with great advantages compared to their
electrical counterparts, like a superior spatial resolution [1] or improved sensor array
positioning and less exogenous distortions by the patient due to contactless measurement [2].
In medical diagnostics magnetic measurement is still used rarely, because these signals are
characterized by amplitudes below 100 pT and low frequency components [3] in the range of
0.1 Hz - 100 Hz [4]. This requires high sensitivity sensors with sufficient band width down to
the dc range. Additionally, these fields are orders of magnitudes lower than disturbance
sources like the earth magnetic field (𝐵earth ≈ 50 µT) or fields originating from power cables
(𝐵cable ≈ 0.1 µT) [5] and other electronic devices. Performing the measurements in a shielded
environment is therefore necessary.
State-of-the art magnetic sensor systems with satisfying characteristics are mainly based on
super-conducting quantum interference devices (SQUIDs) which require liquid
Helium/Nitrogen cooling to maintain superconductivity. Consequently, SQUID
measurements are expensive and extensive in handling, preventing bedside diagnostics and
wide spread utilization [3]. As a promising new approach ∆𝐸 effect sensors utilize a magnetic
field induced resonance frequency shift, which arises mainly by the ∆𝐸 effect in
ferromagnetic materials. The ∆𝐸 effect describes the non linearity of Young’s modulus,
which results in a field and stress dependency of the mechanical properties [6].
1.1 Objective
This Lab course serves as an introduction to the working principle of magnetic field sensors
based on the ∆𝐸 effect. During this course, measurements of low frequency magnetic fields
will be performed to investigate sensor characteristics and to become familiar with its
properties.
1.2 Prerequisites
As this is an advanced lab course, the students must be familiar with the following topics:
Piezo electricity, magnetostriction, magnetic anisotropy, magnetic anisotropy energy, Fourier
series/transformation, harmonic oscillator, including resonance frequency, damping, phase
shift and amplitude modulation.
M214: ∆𝐸 Effect Sensors
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2 Theory
2.1 Repetition of magnetoelasticity
The ∆𝐸 effect is strongly related to magnetoelasticity, which involves all effects with origin in
magnetostriction. In practice, often Joule magnetostriction and inverse Joule magnetostriction
is relevant, referring to the anisotropic induction of strain in a direction, relative to the axis of
magnetization or vice versa (inverse). The anisotropic magnetostrictive strain 𝜆 relative to the
direction of magnetization may be described for an isotropic material by the relation [7]:
𝜆 =Δ𝐿
𝐿=
3
2𝜆s (cos² 𝜃 −
1
3) (1)
This equation holds under certain condition, with 𝜆 measured at an angle 𝜃 relative to the
saturation magnetization direction in which the saturation magnetostrictive strain 𝜆𝑠 is
measured. So along direction of magnetization the magnetostriction is
𝜆∥ =3
2𝜆s (1 −
1
3) = 𝜆s (2)
And perpendicular to the magnetization:
𝜆⊥ =3
2𝜆s (0 −
1
3) = −
𝜆s
2 (3)
If 𝜆∥ is negative, we refer to it as negative Joule magnetostriction, denoting the contraction of
the sample in direction of magnetizsation. Origin of this effect is the coupling of atomic
magnetic moments to the orientation of atomic orbitals resulting in a different interatomic
distance in the magnetized state. Very schematically this is depicted in Fig. 1.
Fig. 1 Schematic of magnetic field induced magnetostriction. A change of
length ∆𝐿 occurs by applying a magnetic field H that rotates the magnetic
moment (indicated by arrows) out of the easy axis (EA)
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2.2 ∆𝑬 Effect
The ∆𝐸 effect is fundamental for the working principle of our sensor. It can be defined as the
deviation of Young’s modulus from Hooks law. Instead of a linear stress strain response a
nonlinear curve is observed. This phenomenon occurs due to stress induced anisotropy, which
rotates the magnetization while the material is strained. The rotated magnetization results in
an additional magnetostrictive strain, which adds up to the linear elastic Hookean strain. If we
define the Young’s modulus as the slope of the stress strain curve, it is therefore not
equivalent to the interatomic spring constant [8]. Not only stress, but also a magnetic field can
rotate the magnetization of a ferromagnetic material. Thereby the non linearity of the stress
strain curve is a function of applied field, too. A simple example is provided in Fig. 2. In
magnetic saturation (𝐻 = 𝐻sat) an applied stress cannot induce an additional magnetostrictive
strain, therefore the stress strain curve is linear and equal to the Hookean response. Below
magnetic saturation (𝐻 = 0), stress can induce magnetostrictive strain, which results in a
nonlinear stress strain curve. With increasing stress the material is successively magnetized
until it is saturated by the applied stress. Then the stress strain curve is linear again and the
Young’s modulus is constant.
Fig. 2: Stress-Strain curves for a tensile test with a sample magnetized up to
saturation in stress direction (𝐻 = 𝐻𝑠𝑎𝑡) and with no magnetic field applied
(𝐻 = 0). Since no additional magnetization can occur in magnetic saturation
the behavior is linear for the first case. If no magnetic field is applied the
magnetization continuously rotates with increasing strain by inverse
magnetostriction, resulting in an increasing Young’s modulus.
Depending on the material, magnetostrictive strains are in order of 10−6 up to 10−3, where
the largest strains are observed for rare earth elements [9]. Consequently, the effect is only
relevant for small deformations, because large strains saturate the material magnetically. The
principle course of Young’s modulus with magnetic field can be qualitatively derived from
the hysteresis curve, if the influence of stress is known. Consider the hard axis magnetization
process that is completely dominated by the rotation of magnetization. In this case the
absolute |�̅�| of the magnetization vector �̅� equals the saturation magnetization 𝑀s. The
M214: ∆𝐸 Effect Sensors
4
component 𝑀 of �̅�, which is measured in direction of the applied magnetic field is given by
the direction cosine of �̅� to that direction, so it is:
cos 𝜃 =𝑀
𝑀s (4)
From (1) the magnetostrictive strain is proportional to the square of magnetization:
𝜆 =3
2𝜆s (cos² 𝜃 −
1
3) ∝ 𝑀² (5)
The total strain 𝜀 that occurs for a certain applied stress 𝜎 can be composed of a superposition
of stress induced magnetostrictive strain 𝜆 and linear Hookean strain 𝑒:
𝜀 = 𝑒 + 𝜆 (6)
Therewith an effective Young’s modulus 𝐸 can be defined as the derivative of applied stress
with respect to the total strain 𝜀. Using the inverse, it is:
1
𝐸=
𝜕𝜀
𝜕𝜎=
𝜕(𝑒 + 𝜆)
𝜕𝜎 (7)
Note that 𝜕𝑒/𝜕𝜎 is simply the constant Hookean Young’s modulus 𝐸m, which can be
measured in saturation. Substituting 𝐸m and 𝜆 yields:
1
𝐸=
𝜕𝜀
𝜕𝜎=
1
𝐸m+
𝜕𝜆
𝜕𝜎∝
1
𝐸m+
𝜕𝑀2
𝜕𝜎 (8)
Consequently 𝐸 is reduced relative to 𝐸m, if 𝜕𝑀²/𝜕𝜎 > 0. A schematic course of 𝑀(𝐻) is
sketched in Fig. 3 for a hard axis magnetization at 𝜎 = 0 and 𝜎 > 0.
Fig. 3: Schematic course of magnetization for different applied stresses. A
tensile stress tilts the magnetization curve due to a reduced effective
anisotropy energy in stress direction. This results in a positive ∆𝑀².
Because ∆𝑀2 > 0 reduces 𝐸, the expected course of Young’s modulus as
function of magnetic field is similar to −∆𝑀2 as indicated by the dashed
line.
M214: ∆𝐸 Effect Sensors
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For 𝜎 > 0 the effective anisotropy energy density in stress direction is reduced, which results
in a ∆𝑀2 > 0. From (8) the principle course of Young’s modulus is expected to follow
qualitatively −∆𝑀2. This results in a 𝑤 shaped course as indicated by the dashed line in
Fig. 3. Note that ∆𝑀2 = 0 at 𝐻 = 0 because the magnetization curve tilts around this point.
This is only true for an exact hard axis magnetization process and if no domain wall motion is
possible. In all other cases the magnetization can also be changed by a stress at 𝐻 = 0. Then
the center maximum at 𝐻 = 0 is not at 𝑀 = 0 but at 𝑀 < 0. Consequently, this simple
qualitative consideration must not be overinterpreted. For a quantitative description, magnetic
models must be used to obtain 𝜕𝜆/𝜕𝜎, which is beyond the scope of this manual.
2.3 Working principle of ∆𝑬 sensor
The sensor consists of a hetero structure of magnetoelastic and piezoelectric material on a
silicon cantilevers (see section 3.1 and 3.2 for the detailed setup). By field annealing a
uniaxial anisotropy is induced with magnetic easy axis oriented perpendicular to the
cantilever. For operating the sensor, a sinusoidal voltage is applied to the piezoelectric layer,
exciting the cantilever to oscillate at 𝑓0. Utilizing the same electrodes for readout a current
with characteristic amplitude is measured. This resonance curve differs from the course of a
typical mechanical resonance. It is called electromechanical resonance. The typical course of
amplitude as function of frequency is shown in Fig. 4 (left).
Fig. 4: Amplitude of measured current vs. excitation frequency applied to the
piezoelectric layer (left); Shift of resonance curve by application of a (DC or AC)
magnetic field at constant excitation frequency results in lower measured current
amplitude (right)
Apparent from Fig. 4 the electromechanical resonance curve obeys a resonance -
antiresonance behavior, which indicates the presence of coupled oscillators. The resonator can
be modeled by an equivalent circuit of two coupled parallel oscillators (BVD-Model). One is
a series LCR circuit to describe the mechanical resonance, the other an RC parallel circuit to
describe the piezoelectric contribution [6]. This oscillator circuit exhibits capacitive behavior
below the resonance frequency and inductive behavior above. Upon application of a DC
magnetic field the effective Young’s modulus changes according to Fig. 3 due to the
∆𝐸 effect. Thereby the resonance frequency of the device changes and a different amplitude is
measured as schematically indicated in Fig. 4 (right). The change of resonance frequency that
occurs if E changes can be understood considering a simple harmonic oscillator:
M214: ∆𝐸 Effect Sensors
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𝑓0 = 1
2𝜋√
𝑘
𝑚∝ √𝐸 (9)
with the effective Young’s modulus 𝐸 the spring constant 𝑘 and equivalent mass 𝑚.
Consistently the course of resonance frequency with magnetic field (Fig. 5) is similar to the
change of Young’s modulus, 𝑤 shaped. This does not change principally when applying an
alternating magnetic field. The AC field 𝐻ac(𝑡), necessarily results in a time dependent
current amplitude, depending on the momentary value of 𝐻ac(𝑡) and leads to an oscillation of
current amplitude around the amplitude measured without 𝐻ac(𝑡).
Fig. 5: Resonance frequency of the device as function of applied magnetic
field, originating directly from the induced change of Young’s modulus. At
the working point (WP) the sensitivity of the sensor is at its maximum. It is
fixed by the point of highest slope and defines the optimum excitation
frequency 𝑓r,w and optimum bias field 𝐻w. The reduced maximum indicates
domain wall motion.
2.3.1 Sensitivity
The two functions in Fig. 4 and Fig. 5 are of fundamental importance, since they determine
together the output signal of the sensor for a given input signal. In this context, it is
appropriate to define a sensor parameter referred to as the sensitivity S. The sensitivity can be
defined as the change (𝑑�̂�) of output signal amplitude with physical input parameter (𝑑𝐻)
𝑆 =𝑑�̂�
𝑑𝐻|
WP
=𝑑𝑓R
𝑑𝐻|
𝑓R(𝐻w)∙
𝑑�̂�
𝑑𝑓|𝐴(𝑓r,w)
(10)
Where 𝑑𝑓R
𝑑𝐻 is called the magnetic sensitivity and
𝑑�̂�
𝑑𝑓R the frequency sensitivity, respectively.
Apparently, the highest change of output amplitude 𝑑�̂� with change of magnetic field strength
M214: ∆𝐸 Effect Sensors
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𝑑𝐻 is obtained, when both factors become maximum. The frequency sensitivity, as slope of
current amplitude over excitation frequency is maximum at the electromechanical resonance
frequency 𝑓0. Therefore, the optimum excitation frequency is chosen to be 𝑓r,w = 𝑓0 always.
However, since it is a function of 𝐻 it convenient to choose the magnetic bias field 𝐻w at
which the magnetic sensitivity is maximum, first. As indicated in Fig. 5 the overall working
point (WP) is then defined by 𝐻w and 𝑓r,w. Thus, in the experimental part we will first
determine the resonance frequency and the working point, before other sensor characteristics
will be determined.
2.4 Amplitude modulation
Operating the sensor, a sinusoidal excitation voltage of (circular) frequency 𝜔 is applied to
the piezoelectric layer. If no additional time dependent magnetic field is applied, it results in a
measured sensor output current 𝐴C(𝑡) = �̂�C ∙ cos (𝜔𝑡), presuming linear piezoelectric
response. This signal is shown schematically in Fig. 6 (left, blue curve). It is called carrier in
terms of signal processing nomenclature. Its amplitude �̂�C is proportional to the amplitude of
the excitation signal and a function of the (carrier) frequency as described by the
electromechanical resonance curve (Fig. 4). Upon application of a lower frequency magnetic
AC field 𝐻(𝑡), the amplitude of the carrier signal �̂�C is modulated by a signal 𝐴inf(𝑡) that
originates from the field. The resulting new, modulated signal 𝐴(𝑡) is measured at the sensor
output. It is schematically depicted in Fig. 6 (right).
Fig. 6: Carrier signal (blue) and information signal (red) (left) superpose according to
(11) and form the modulated signal (right).
The modulated signal 𝐴(𝑡) is mathematically obtained by application of (11):
𝐴(𝑡) = [�̂�C + 𝐴inf(𝑡)] ∙ cos(𝜔𝑡) = �̂�(𝑡) ∙ cos(𝜔𝑡) (11)
where 𝐴C(𝑡) = �̂�C ∙ cos(𝜔𝑡) is the carrier signal and 𝐴inf(𝑡) the periodic modulating
(information) signal. Details about 𝐴inf(𝑡) are discussed in the next chapter. However, it
should be evident that 𝐴inf(𝑡) is simply the additional amplitude at time t, added to the
amplitude of the carrier signal. This results in a time dependency of the amplitude �̂�(𝑡) of the
measured signal 𝐴(𝑡). To point it out: The information signal 𝐴inf(𝑡) is a direct consequence
of the applied alternating magnetic field 𝐻(𝑡). They are related by the materials physics, but
are still completely different quantities and may generally differ a lot regarding their
M214: ∆𝐸 Effect Sensors
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respective appearance in time and frequency domain. According to (10) it is at the working
point:
𝑑𝐴inf(𝑡) = 𝑑�̂�(𝑡) = 𝑆 ∙ 𝑑𝐻(𝑡) (12)
For small magnetic fields at the working point, the sensitivity 𝑆 is approximately constant,
thus integrating yields:
𝐴inf(𝑡) = Δ�̂�(𝑡) ≈ 𝑆 ∙ 𝐻(𝑡) (13)
If S is independent of 𝐻(𝑡), then 𝐴inf(𝑡) is a linear function of 𝐻(𝑡). Referring to (13), this is
approximately valid for small fields, if the course of 𝑓R(𝐻) and �̂�(𝑓R) can be approximated as
linear functions around the respective working point. The influence of the linearity of the
information signal in 𝐻(𝑡) on the output signal 𝐴(𝑡) is discussed in the subsequent section.
2.4.1 Linearity
Consider the frequency domain of 𝐴(𝑡) by taking (11) and assuming 𝐴inf(𝑡) to be a pure
cosine function:
𝐴(𝑡) = [�̂�C + �̂�inf ∙ cos(𝜃𝑡)] ∙ cos(𝜔𝑡) = �̂�C cos(𝜔𝑡) + �̂�inf ∙ cos(𝜃𝑡) cos(𝜔𝑡) (14)
Using
cos(𝜃𝑡) cos(𝜔𝑡) =1
2(cos([𝜔 − 𝜃]𝑡) + cos([𝜔 + 𝜃]𝑡)) (15)
it follows:
𝐴(𝑡) = �̂�C cos(𝜔𝑡) +�̂�inf
2∙ (cos([𝜔 − 𝜃]𝑡) + cos([𝜔 + 𝜃]𝑡)) (16)
According to (16) the signal in frequency domain has to be composed of at least three peaks
shown in Fig. 7 (left). One center peak at the frequency 𝜔 corresponding to the carrier signal
and two side peaks at [𝜔 − 𝜃] and [𝜔 + 𝜃] from the modulating signal 𝐴inf(𝑡). In the general
case, however, 𝐴inf(𝑡) is not a pure sine function due to nonlinearities in S. Expressing it as a
Fourier series will result in sinusoidal contribution of higher order. Thus, additional peaks
occur in the spectrum as pictured in Fig. 7 (right).
Fig. 7: Power density spectrum of a carrier signal with frequency 𝜔 modulated by an
information signal with frequency 𝜃 (left) and additional harmonics from nonlinearities of the
modulating signals in H (right)
M214: ∆𝐸 Effect Sensors
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2.5 Other Sensor Parameters
2.5.1 Total harmonic distortion (THD)
The linearity is an important sensor characteristic. As an appropriate measure the (total)
harmonic distortion (THD) provides an appropriate quantification for nonlinearities. There are
many different definitions, depending on the field of application and area of expertise. For the
purpose of this lab course, the harmonic distortion is used and defined as
𝑇𝐻𝐷dB,𝑖 = 10 ∙ 𝑙𝑜𝑔10 (𝑃𝑖
𝑃1) = 𝑃𝑖,dB − 𝑃1,dB (17)
𝑃1 is the peak height at the basis frequency and 𝑃𝑖 the peak height at the 𝑖𝑡ℎ harmonic in the
power density spectrum (Fig. 7). During the lab course this will be determined and discussed
with respect to applications.
2.5.2 Quality factor and losses
The Q-factor is a measure for the energy loss of a resonator system at resonance frequency. It
is defined by the energy 𝐸max stored per cycle by the energy loss 𝐸loss per cycle and is
defined as:
𝑄 =2𝜋 ∙ 𝐸max
𝐸loss≈
𝑓c
𝐹𝑊𝐻𝑀 (18)
The approximation using the peak center frequency 𝑓0 and the full width at half maximum is
only valid for small losses. Since the decay time of an oscillation is a function of the losses,
the quality factor can be related to the decay constant 𝜏 and the number 𝑛cycles of periods to
decay
𝑄 = 𝜋 ∙ 𝑛cycles = 𝜋 ∙𝜏
𝑇= 𝜋𝑓0𝜏 (19)
2.5.3 Signal to Noise Ration (SNR) and Limit of detection (LOD)
The SNR is a measure for the relative strength of desired signal and defined as the ratio of
signal power to mean noise power. Since we measure all the power in dB (20) serves as an
appropriate definition.
𝑆𝑁𝑅 dB = 10 ∙ 𝑙𝑜𝑔10
𝑃signal
𝑃noise= 𝑃signal,dB − 𝑃noise,dB (20)
The magnetic field at which 𝑆𝑁𝑅 = 1 is referred to as the limit of detection (LOD), since
magnetic fields with lower amplitudes cannot be distinguished from the measured background
noise.
2.5.4 Bandwidth
For this sensor, the bandwidth ∆𝑓 is defined by the frequency, at which the measured
amplitude is decreased by 3 dB, called the 3dB cut-off frequency 𝑓3dB.
∆𝑓 = 𝑓3dB (21)
M214: ∆𝐸 Effect Sensors
10
3 Experiment
3.1 Sensor
The ∆𝐸 effect sensor used during the lab course is based on a polycrystalline silicon
cantilever with piezoelectric aluminum nitride (AlN) deposited on top. As sketched in Fig. 8 it
is connected to the circuit by two gold/platinum electrodes. For the magnetoelastic
component, deposited on the bottom, amorphous FeCoSiB is used. As a metallic glass it
exhibits a large quality factor, due to less eddy current losses and enhanced elastic properties.
Compared with other highly magnetostrictive materials the lack of crystalline anisotropy
energy results in a low total anisotropy energy and therefore in a strong ∆𝐸 effect [9].
Fig. 8: Structure and composition of the ∆𝐸 effect thin film sensors. The cantilever is excited
by an alternating external voltage 𝑈ex to oscillate. A magnetic easy axis (EA) is induced along
the short axis. The mechanical behavior is dominated by the substrate with about 50µm
thickness. The thickness of all other layers is in the range of a few µm. This is advantageous
due to the large quality factor of the substrate. Because the electric and magnetic response of
the sensor originate from the piezoelectric and magnetostrictive layer, their volume fraction is
a decisive factor contributing to the frequency sensitivity and the magnetic sensitivity. [6]
3.2 Setup
During the experiment, the sensor is placed in the center of two concentrically arranged
solenoids, which are used to apply an alternating and a static magnetic field, respectively.
Conversion factors between solenoid current and magnetic field are listed in following chart.
The complete setup is magnetically shielded with Permalloy foil.
Table 1: Conversion factors for solenoids between current and flux density
Solenoid Conversion factor /mTA−1
DC 7
AC 0.89
As signal source and readout the inputs and outputs of an interface commonly used for audio
recording can be triggered by a MatLab script. Details will be provided by the supervisor. As
sketched in Fig. 9 output out1 provides the excitation voltage for the piezoelectric layer,
whereas out2 is connected to the solenoid to produce the alternating magnetic field.
M214: ∆𝐸 Effect Sensors
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Fig. 9: Schematic representation of the experimental setup. Coils and
Input/Outputs are connected to the same ground.
To apply a magnetic bias field 𝐻bias (or 𝐻w at the optimum WP) an external power supply is
used connected to the second solenoid. Both enclose a tube in which the sensor is placed. The
sensors response is recorded as a voltage at Input1, which is measured across the input
resistance and therefore it is proportional to the current or the total admittance of the circuit.
In the equivalent circuit (Fig. 10), the sensor is represented as a capacitance with admittance
adjustable by an external magnetic field.
Fig. 10: Equivalent circuit of the sensor - interface connection via output
channel 1 (bottom) and input channel 1 (top). The voltage is measured
across the input resistance 𝑅In1at the input.
M214: ∆𝐸 Effect Sensors
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3.3 Tasks
3.3.1 General remarks
All graphs have to be scaled appropriately. All results should be discussed. Plots including
magnetic field amplitudes should be plotted in mT. All voltages in MatLab are denoted as
voltages 𝑈rel relative to the maximum output voltage. Therefore amplitudes > 1 must not be
applied to avoid overdriving the system! The necessary conversion factors will be provided by
the supervisor, as well as all necessary factors for calculating the magnetic field amplitudes.
All amplitudes without units are relative amplitudes.
3.3.2 Working point
Record the resonance curve for different applied 𝐻bias fields. Therefore, apply an excitation
voltage of 𝑈exc = 0.2 and increase the bias voltage 𝑈bias from 0 V up to 12 V.
a) Measure the resonance curve for each voltage step and denote the approximated resonance
frequency.
b) Plot 𝑓r vs. 𝑈bias during the measurement to determine the working point
c) Now apply 𝐻ac frequency 𝑓ac = 10 𝐻𝑧 and an amplitude 𝐴ac = 0.02. Record a spectrum
and adjust 𝑈bias until the spectrum is mirror symmetric around the center peak.
Perform all subsequent measurements at the working point you determined above.
3.3.3 SNR
a) Apply 𝑈exc = 0.01 and 𝑈exc = 0.05 and measure the resonance curve for both excitation
amplitudes.
b) Record spectra with 𝑈exc increased from 0.1 up to 0.9. Measure the peak height of the first
harmonics relative to the noise level.
Evaluation:
(a) Calculate the frequency sensitivities and compare.
(b) Plot SNR over excitation amplitude and determine the optimum excitation amplitude
3.3.4 THD
Increase 𝐴ac from 0.1 up to 0.9 and measure the spectra. Record the spectra and peak values
of all occurring harmonics and the respective noise level.
Evaluation:
Calculate the THD for each measurement and plot it vs. the magnetic field.
3.3.5 Bandwidth
Start with a 𝐻ac frequency of 1 Hz and increase it stepwise to 200 Hz. Determine the peak
height of the first sidebands and respective noise level from the spectrum for each frequency.
Evaluation:
Plot the signal amplitude vs. 𝑓ac and determine the bandwidth. Use an appropriate scale!
3.3.6 LOD
Measure the signal amplitude for decreasing amplitude of 𝐻ac by recording frequency spectra
for each amplitude value.
Evaluation:
Plot the SNR vs. amplitude of 𝐻ac and determine the LOD
M214: ∆𝐸 Effect Sensors
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4 References
[1] P. W. Macfarlane, A. van Oosterom, O. Pahlm, P.Kligfield, M. Janse, and J. Camm, Eds.,
Comprehensive Electrocardiology. Springer, 2011.
[2] H. Koch, “Recent advances in magnetocardiography”, Journal of Electrocardiology,
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[3] J. Clarke and A. I. Braginski, The SQUID Handbook Vol II: Applications of SQUIDs and
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