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

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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.

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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

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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.

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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:

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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

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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

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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)

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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)

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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.

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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.

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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

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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,

vol. 37, Supplement, pp. 117 – 122. 2004

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