subsecref subsecname = thmref thmname = theorem lemref lemname = lemma

# A Class D Power Amplifier for Multi-Frequency Eddy Current Testing Based on
Multi-Simultaneous-Frequency Selective Harmonic Elimination Pulse Width
Modulation

Yang Tao, Christos Ktistis, Yifei Zhao, Wuliang Yin, and Anthony J. Peyton
Manuscript received March 22, 2019; revised July 17, 2019 and September 2,
2019; accepted October 8, 2019.This work was supported by the U.K. Engineering
and Physical Sciences Research Council via the University of Manchester Impact
Accelerator Account under Grant EP/R511626/1. (Corresponding author: Yang Tao;
Wuliang Yin.)The authors are with the Department of Electrical and Electronic
Engineering, School of Engineering, The University of Manchester, Manchester
M13 9PL, U.K. (e-mail<EMAIL_ADDRESS>christos.
<EMAIL_ADDRESS><EMAIL_ADDRESS>wuliang.yin@
manchester.ac.uk; a.peyton@manchester.ac.uk).Color versions of one or more of
the figures in this article are available online at
http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIE.2019.2947842

###### Abstract

Efficiency and multisimultaneous-frequency (MSF) output capability are two
major criteria characterizing the performance of a power amplifier in the
application of multifrequency eddy current testing (MECT). Switch-mode power
amplifiers are known to have a very high efficiency, yet they have rarely been
adopted in the instrumental development of MECT. In addition, switch-mode
power amplifiers themselves are lacking in the research literature for MSF
capability. In this article, a Class D power amplifier is designed so as to
address the two issues. An MSF selective harmonic elimination pulsewidth
modulation method is proposed to generate alternating magnetic fields, which
are rich in selected harmonics. A field-programmable-gate-array-based
experimental system has been developed to verify the design. Results show that
the proposed methodology is capable of generating high MSF currents in the
transmitting coil with a low distortion of signal.

###### Index Terms:

Class D power amplifier, eddy current, multifrequency, pulsewidth modulation
(PWM), selective harmonic elimination.

##  I Introduction

Multi-frequency eddy current testing (MECT) has attracted increasing attention
for non-destructive testing (NDT) in recent years. In comparison with the
single-frequency eddy current method, MECT takes advantage of wide-band
spectral signals, which is advantageous for applications such as detecting
cracks in conductive materials [1], measuring thickness of metal film [2],
monitoring microstructure in steel production [3] and imaging conductivity
profile of metal structures [4], etc. In these applications of MECT, a high-
current power amplifier with multi-simultaneous-frequency (MSF) output
capability is indispensable in the signal chain so as to achieve high overall
sensitivity [5].

Class D power amplifier has the virtue of high current handling capacity and
power efficiency [6, 7]. However, transmitting signals may suffer severe
distortion after amplification if they are not modulated properly. A notable
category of modulation methods is known as the programmed pulse width
modulation (PPWM) or selective harmonic elimination pulse width modulation
(SHEPWM) [8] which has drawn tremendous interests and been studied primarily
for high-power high-voltage converters [9]. Owing to the lack of motivation of
generating multi-simultaneous frequencies using SHEPWM, efforts have been
mainly devoted in a manner to achieve varying outputs of the fundamental
component which is characterised by the modulation index and to eliminate the
remaining harmonics. Yet in MECT applications, the varying modulation index is
not of interest while the MSF output capability is of importance. Therefore,
some harmonics may be well harnessed rather than eliminated.

This paper attempts, for the first time, to examine the MSF capability of the
SHEPWM technique and in addition, to introduce switch-mode power amplifier
into the instrumental development of MECT. A Class D power amplifier based
upon multi-simultaneous-frequency SHEPWM (MSF-SHEPWM) is designed. The
proposed design is capable of driving an excitation coil with a simultaneous
multi-frequency signal. Moreover, it is convenient to change the spectrum of
the transmitting signal. Experimental results have proved that the system
could transmit MSF alternating magnetic fields with low distortion. The
remainder of the paper is organised as follows. LABEL:Sec:MSFPWM-Principle
describes the principle of MSF-SHEPWM. The design of hardware and the results
of experiments are presented in LABEL:Sec:Hardware-Design. Finally,
conclusions are drawn in the last section.

##  II MSF-SHEPWM Principle

In eddy current testing, the load of the power amplifier is a coil that
generates alternating magnetic fields. In order to generate the magnetic
fields that are required for MECT, according to the
$\mathrm{Amp\grave{e}re^{\prime}s}$ law, we can directly design the current
through the coil. The waveform of coil current is a concatenation of a few
straight line segments of fixed gradients when the coil is connected to a
Class D power amplifier. This is due to the fact that in theory the voltage
across the coil can only have a fixed number of values or levels and, for a
perfect inductor, the coil current is an integral of the voltage.
LABEL:Fig:Topology-of-Class shows three example topologies for Class D power
amplifier. Enhancement-mode n-type MOSFETs are used for this illustration. The
components in blue colour constitute a conducting path. LABEL:Fig:Topology-of-
Class(a) shows a push-pull configuration where the voltage across the coil
have two levels, i.e. $+V_{DC}/2$ and $-V_{DC}/2$; LABEL:Fig:Topology-of-
Class(b) depicts a full H-bridge topology that is capable of driving the coil
with three-level voltage, i.e. $+V_{DC}$, $0$ and $-V_{DC}$. Multi-level
voltage can be achieved by utilising multiple full H-bridges. For example, a
five-level inverter is shown in LABEL:Fig:Topology-of-Class(c) that contains
two H-bridge cells. The five values of voltage are $+2V_{DC}$, $+V_{DC}$, $0$,
$-V_{DC}$ and $-2V_{DC}$.

Figure 1: Topology of Class D power amplifier using enhancement-mode n-type
MOSFETs. (a) push-pull half bridge. (b) full H-bridge. (c) two full H-bridges.

The objective of MSF-SHEPWM is to generate an output waveform that contains
the desired MSF spectral components, but eliminates the unwanted harmonics.
The principle of MSF-SHEPWM can be summarised in three stages. Firstly, the
objective signal is synthesised based on the fundamental sinusoidal signal and
a few harmonics in the time domain, and the waveform of the synthesised signal
is resembled by straight line segments in a morphological sense. Secondly, the
resultant resemblance signal is decomposed as Fourier series and analysed in
the frequency domain. Finally, the Fourier coefficients are adjusted by
solving an optimisation problem. These three stages will now be described in
detail in the following sub-sections. For clarity we use $f_{k}(t)$ to denote
the objective signal, $g_{k}(t)$ as the resemblance function of $f_{k}(t)$,
and $G_{k}(t)$ is the Fourier series of $g_{k}(t)$. The subscript $k$ refers
to a specific signal.

###  II-A Synthesis of the objective function $f_{k}(t)$ in the time domain

A periodic signal $f(t)$ that contains $n$ sinusoidal basic components is
expressed as (1) where $\omega_{p}$, $\theta_{p}$ and $w_{p}$ are the angular
frequency, phase angle and coefficient of the $p^{th}$ component respectively.
The function $f(t)$ serves as the objective coil current that is to be
generated and the coefficient $w_{p}$ forms the amplitude spectrum.

Additionally, three conditions are exerted on $f(t)$. Firstly, the lowest
frequency is set as the fundamental frequency $\omega$ and higher frequencies
are harmonics of $\omega$, i.e. $\omega_{p}=p\omega$. This is a natural
requirement since the signal generated by Class D power amplifier is rich in
harmonics. Secondly, the phase angle $\theta_{p}$ is zero implying all the
basic components are in-phase. In MECT applications, the relative phase values
are not critical factors. Thirdly, the coefficient $w_{p}$ is chosen as
$\frac{1}{p}$ for the selected frequencies and zero for absent harmonics. In
general, the magnitude of response signal tends to increase with frequency. In
order to achieve similar signal-to-noise ratio (SNR) in the whole spectrum of
interest, the coefficient $w_{p}$ is chosen to be inversely proportional to
the frequency $\omega_{p}$ or the order $p$. In fact, (1) is a special case of
the trigonometric form of Fourier series but only a few rather than an
infinite number of components are involved.

$f(t)=\sum_{p=1}^{n}w_{p}\sin\left(\omega_{p}t+\theta_{p}\right)$ (1)
$f_{1}(t)=\sin\left(\omega t\right)+\frac{1}{3}\sin\left(3\omega
t\right)+\frac{1}{9}\sin\left(9\omega t\right)\\\
+\frac{1}{27}\sin\left(27\omega t\right)+\frac{1}{81}\sin\left(81\omega
t\right)$ (2)

$f_{2}(t)=\sin\left(\omega t\right)+\frac{1}{3}\sin\left(3\omega
t\right)+\frac{1}{7}\sin\left(7\omega t\right)\\\
+\frac{1}{17}\sin\left(17\omega t\right)$ (3)

$f_{3}(t)=\sin\left(\omega t\right)+\frac{1}{2}\sin\left(2\omega
t\right)+\frac{1}{4}\sin\left(4\omega t\right)\\\
+\frac{1}{6}\sin\left(6\omega t\right)+\frac{1}{8}\sin\left(8\omega
t\right)+\frac{1}{10}\sin\left(10\omega t\right)\\\
+\frac{1}{12}\sin\left(12\omega t\right)+\frac{1}{14}\sin\left(14\omega
t\right)$ (4)

$f_{4}(t)=\sin\left(\omega t\right)+\frac{1}{2}\sin\left(2\omega
t\right)+\frac{1}{3}\sin\left(3\omega t\right)\\\
+\frac{1}{4}\sin\left(4\omega t\right)+\frac{1}{5}\sin\left(5\omega
t\right)+\frac{1}{6}\sin\left(6\omega t\right)\\\
+\frac{1}{7}\sin\left(7\omega t\right)+\frac{1}{8}\sin\left(8\omega t\right)$
(5)

$f_{5}(t)=\sin\left(\omega t\right)+\frac{1}{3}\sin\left(3\omega
t+\frac{\pi}{4}\right)+\frac{1}{9}\sin\left(9\omega t+\frac{\pi}{7}\right)\\\
+\frac{1}{27}\sin\left(27\omega
t+\frac{\pi}{5}\right)+\frac{1}{81}\sin\left(81\omega t+\frac{\pi}{6}\right)$
(6)

Figure 2: The objective function $f_{1}(t)$, the corresponding fundamental and
harmonic functions

As an example, a signal $f_{1}(t)$ that follows these principles is shown in
LABEL:Fig:Plots-of-signal. Here, $f_{1}(t)$ is a superposition of five basic
sinusoidal signals. The frequencies are chosen in a geometric scale as
$\omega$, $3\omega$, $9\omega$, $27\omega$ and $81\omega$. The weights are set
as $1$, $\frac{1}{3}$, $\frac{1}{9}$, $\frac{1}{27}$, and $\frac{1}{81}$.
Another three examples $f_{2}(t)$, $f_{3}(t)$ and $f_{4}(t)$ that have
different possible combinations of basic components are shown in
LABEL:Fig:Examples. The expressions of $f_{1}(t)$, $f_{2}(t)$, $f_{3}(t)$ and
$f_{4}(t)$ are shown in (2), (3), (4) and (5), respectively.

The selection of involving harmonics has flexibility and is determined by the
fundamental frequency and the spectrum of interest. The associated
coefficients are determined by the requirement of power distribution for each
harmonics. It should be noted that the basic sinusoidal components do not
necessarily have to be in-phase relative to each other. For example,
$f_{5}(t)$ as expressed in (6), has the same weights for each basic components
as $f_{1}(t)$, but the phases are different which is shown in
LABEL:Fig:Examples. Although $f_{5}(t)$ has the same amplitude spectrum as
$f_{1}(t)$, some symmetric features are broken which makes the analysis in the
frequency domain more complicated.

Figure 3: Objective function: $f_{1}(t)$, $f_{2}(t)$, $f_{3}(t)$, $f_{4}(t)$
and $f_{5}(t)$

Once the waveform of the objective function is determined, such waveform can
be resembled using straight line segments as shown in LABEL:Fig:The-objective-
function. The resemblance function $g(t)$ is piece-wise linear with different
gradients and approximates the objective function $f(t)$. Accurate
approximation requires multiple levels of output voltage which, however, costs
multiple H-bridge cells. Therefore, the selection of voltage levels is a
trade-off between the fidelity of signal and feasibility in practice. The
number of time steps used for $g(t)$ is often determined by the clock
frequency used to drive the bridge circuit and limited by the dynamic
characteristics of the transistor.

Figure 4: An objective function $f(t)$ and its resemblance function $g(t)$

###  II-B Analysis of the resemblance function $g_{k}(t)$ in the frequency
domain

####  II-B1 Calculation of the Fourier coefficients for the resemblance
function $g_{k}(t)$

A resemblance function $g_{k}(t)$ can be generated by a multi-level Class D
power amplifier. The switching scheme in a quarter period is depicted in
LABEL:Fig:Switching-scheme-of. Switching angles satisfy
$0\leq\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{N}\leq\frac{\pi}{2}$ and
at each angle the voltage can raise or fall by $V_{0}$ until reaching the
limits. The coil current is essentially an integral of the pulse voltage which
is depicted as the area in the magenta colour in LABEL:Fig:Switching-scheme-
of. Therefore, the Fourier series of the coil current is calculated as (7).
The Fourier coefficient is shown in (8) which is a function of $N$ switching
angles $\alpha_{q}$, the coil inductance $L$, the order of harmonics $p$ and
the step of voltage $V_{0}$.

Figure 5: Switching scheme of multi-level Class D amplifier

$G(t)=\sum_{p=1,3,\cdots}^{\infty}b_{p}\sin\left(p\omega t\right)$ (7)

$\displaystyle b_{p}$ $\displaystyle=-\frac{4V_{{\it 0}}}{p^{2}\pi\omega
L}\sum_{q=1}^{N}(-1)^{s}\sin\left(p\alpha_{q}\right),$ $\displaystyle s$
$\displaystyle=\begin{cases}1&,\forall\,\alpha_{q}\,at\,which\,there\,is\,a\,falling\,edge\\\
0&,\forall\,\alpha_{q}\,at\,which\,there\,is\,a\,raising\,edge\end{cases}$ (8)

###  II-C Optimisation of the Fourier coefficients $b_{p}$

Once the waveform of the resemblance function $g_{k}(t)$ is obtained based on
the gradient of $f_{k}(t)$, the spectrum of $g_{k}(t)$ can be controlled by
adjusting the switching angles $\alpha_{q}$ which is essentially a problem of
solving simultaneous transcendental equations. Each equation corresponds to
the magnitude of a harmonic to be controlled.

Various methods have been developed in order to obtain one or multiple
solutions to the harmonic magnitude equations. One direction for developing a
solver is based on the elimination theory of polynomial and using the
$resultant$ [10]. Unfortunately, the order of polynomials increases with the
number of harmonics to be controlled which results in a significant
computational burden if the application entails a wide spectrum. Moreover,
completely solving the equations might be a stringent requirement and in fact
discrepancies could be tolerated within a threshold in a real application.
Another direction tends to recast the harmonic magnitude equations into an
optimisation problem [8]. Because of the trigonometric and non-convex nature
of the harmonic magnitude equations, it is difficult to guarantee that a
globally optimal solution can be achieved. Meta-heuristic algorithms such as
differential evolution [11], genetic algorithm [12], and particle swarm
optimisation [13], etc. have been applied to obtain a global solution through
modern stochastic search techniques. However, different results may be
returned for the same problem since these algorithms rely on random searches.
Many gradient based algorithms, on the other hand, guarantee that a local
optimal can be obtained in theory. If a good estimation of the solution
exists, the gradient based algorithms could converge to the optimal rapidly
[14].

$\min_{0\leq\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{N}\leq\frac{\pi}{2},m_{p}}\sum_{p\in\Phi}\lambda_{p}\left(\frac{p\omega
L}{V_{0}}b_{p}-m_{p}\right)^{2},\\\ subject\,to\,\left|\frac{p\omega
L}{V_{0}}b_{p}\right|\leq\epsilon_{p},\,\forall p\in\Psi$ (9)

In this paper, the harmonic magnitude equations are calculated by solving a
constrained nonlinear optimisation problem as shown in (9). Following the
convention of multi-level inverter, we denote the modulation index for the
$p^{th}$ harmonic frequency as $m_{p}=\frac{V_{p}}{V_{0}}$, where $V_{p}$ is
the amplitude of the $p^{th}$ harmonic in the output voltage. The modulation
index characterises the relative amplitude of output voltage in the context of
a multi-level inverter, but there are two different considerations for MECT.
Firstly, $m_{p}$ is of interest for multiple harmonics, yet only the
fundamental frequency component is relevant for a multi-level inverter.
Secondly, $m_{p}$ is treated as an optimising parameter, while a multi-level
inverter tends to be able to output a range of values of the modulation index.
In (9), $\Phi$ is a set whose elements are the orders of selected harmonics
while the elements of $\Psi$ are the orders of harmonics to be eliminated. The
objective function in (9) represents the sum of weighted ($\lambda_{p}$)
squared residuals of the modulation index for selected harmonics subject to
such constrains that the modulation index of undesirable harmonics is bounded
with threshold $\epsilon_{p}$. The optimising parameters are $N$ switching
angles $\alpha_{1},\alpha_{2},\cdots,\alpha_{N}$ and the modulation index
$m_{p}$.

The problem (9) could provide many flexibilities in practice. For instance,
the adjacent harmonics of an objective frequency will cause interference at
that frequency and hence the corresponding threshold $\epsilon_{p}$ for the
adjacent harmonics should be smaller than those of other less critical
harmonics. Additionally, if annulation of the objective function is not
possible, the objective function could be relaxed by putting higher weights
$\lambda_{p}$ for more important harmonics and lower weights for other
harmonics.

The interior-point algorithm is adopted to solve the problem (9). This choice
of algorithm is due to the fact that the interior-point algorithm has been
proved to be an effective solver for constrained nonlinear optimisation
problems [15]. Moreover, both the objective function and constrain function
have well-defined Gradient and Hessian. Besides, the initial value of
switching angles can be obtained based on the gradient of objective function
$f(t)$ which turns out to be a good estimation. The $MATLAB^{\circledR}$
$Optimization\,Toolbox^{\circledR}$ provides a solver $\mathrm{fmincon}$ that
implements the interior-point algorithm.

##  III Hardware and Experimental Results

###  III-A Hardware Implementation

In order to test the proposed MSF-SHEPWM method, an FPGA based experimental
system was designed which is shown in LABEL:Fig:Experimental-system. The
system schematic diagram is shown in LABEL:Fig:Schematic-diagram-of. The
system mainly comprises three parts, namely a $Spartan^{\circledR}-6$ FPGA,
power supply and H-bridge circuit. An oscillator on the experiment board
generates a clock signal of a fixed frequency, and a digital clock manager
(DCM) inside the FPGA is used to generate a specified baseline clock signal
for control sequence. The FPGA periodically reads the switching sequence from
in-chip memory and outputs pulse signals through a PWM logic block. The pulse
signals are then fed into the bridge driver. Four enhancement-mode n-type
MOSFETs are utilised in the system configured in a full H-bridge. Therefore,
three levels of voltage can be generated by the H-bridge cell. According to
the analysis of two cases of application that will be shown in the following
paragraphs, even a bipolar pulse voltage is sufficient in generating the
desired magnetic fields. Although the proposed MSF-SHEPWM method applies to
the design of multiple H-bridge cells, only a single full H-bridge is adopted
in the system. In order to make the enhancement-mode n-type MOSFETs working in
the ohmic region, the gate-to-source voltage must be higher than not only the
threshold voltage, but the addition of the drain-to-source voltage and the
threshold voltage. Therefore, a bootstrap circuit is required to drive the
high side MOSFETs. The dc voltage is selected as $24\,V$ supplied externally.

Figure 6: Experimental system Figure 7: Schematic diagram of the experimental
system

###  III-B Power and Efficiency Metrics

The drain efficiency $\eta$ [16] as expressed in (10) is adopted as an
efficiency metric, in which the input dc power $P_{DC}$ is the product of the
dc power supply voltage $V_{DC}$ and current $I_{DC}$. The real output power
$P_{out}$ is defined in (11), in which $u_{out}$, $i_{out}$ and $T$ are the
load voltage, current and integration interval, respectively. As the objective
is generating ac magnetic field rather than dissipating dc power in the load,
we define an energy conversion factor $\zeta$ as (12), in which the output
reactive power $Q_{out}$ [17] is defined as (13) where $U_{i}$ and $I_{i}$ are
the root-mean-squared values of the output voltage and current harmonics of
order $i$. $\theta_{i}$ is the phase angle difference between them, and $\Psi$
is the set containing the orders of transmitting harmonics.

$\eta=\frac{P_{out}}{P_{DC}}$ (10)

$P_{out}=\frac{1}{T}\int_{T}u_{out}i_{out}\;\mathrm{d}t$ (11)

$\zeta=\frac{Q_{out}}{P_{DC}}$ (12)

$Q_{out}=\sum_{i\in\Psi}U_{i}I_{i}\sin\theta_{i}$ (13)

###  III-C Two Experiments

The selection of excitation frequencies is of high importance and it can be
expected that different spectrum is required for different applications. For
example, the $\beta-dispersion$ frequency range from hundreds of kilohertz to
megahertz is often harnessed to detect the property which relates to the
cellular structure of biological tissues [18], while $20\,kHz$ to $100\,kHz$
is a suitable range for measuring copper film with thickness of
$10\,\text{\textendash}\,400\,\mu m$ [19].

Two experiments are carried out based on the experimental system. In the first
experiment, the full design procedures are illustrated and the questions
needing attentions are discussed. In the second experiment, we show a fast
design without the procedure of analysing in the frequency domain and the
optimisation process. The setups of two experiments can be found in
LABEL:Tab:Details-of-setup.

TABLE I: Detailed setups of two experiments

|  Experiment No. 1 |  Experiment No. 2
---|---|---
Selected Harmonics |  $1^{st}$, $3^{rd}$, $7^{th}$, $17^{th}$ |  $1^{st}$, $3^{rd}$, $9^{th}$, $27^{th}$, $81^{st}$
Clock Frequency |  $24\,MHz$ |  $6.25\,MHz$
Lookup Table Length |  $476\,bits$ |  $3888\,bits$
Transmitting Frequency |  $50.42\,kHz$, $151.26\,kHz$, $352.94\,kHz$, $857.14\,kHz$ |  $1.61\,kHz$, $4.82\,kHz$, $14.47\,kHz$, $43.40\,kHz$, $130.21\,kHz$
Coil Inductance |  $1.4\,\mu H$ |  $662\,\mu H$
Wire Diameter (AWG) |  $13\,AWG$ |  $26\,AWG$

####  III-C1 Experiment No. 1

The objective of this experiment is to generate high currents for a low-
impedance coil in the frequency range between $10\,kHz$ and $1\,MHz$. Test
samples can be transferred through the aperture of the coil. This
configuration could be applied to inline discrimination of metal contaminants,
grading of food product and monitoring of meat ageing process, etc.

##### Excitation frequencies

Deciding the number of transmitting frequencies is a trade-off between power
distribution and spectral resolution. If too many frequencies are selected,
it’s inevitable that the power will be relatively low for an individual
frequency. Moreover, if two selected frequencies are too close, not enough
distinction of the response signal could be drawn from them. As a result, four
frequencies are chosen for this experiment as the fundamental, $3^{rd}$,
$7^{th}$ and $17^{th}$ harmonics which results in the earlier mentioned
objective function $f_{2}(t)$ in LABEL:Subsec:Synthesis-of-harmonics. The
clock frequency is configured as $24\,MHz$ and the highest frequency is
$\frac{1}{28}$ of the clock frequency. Therefore, the four transmitting
frequencies are about $50.42\,kHz$, $151.26\,kHz$, $352.94\,kHz$ and
$857.14\,kHz$. It is noted that these frequencies well span the objective
spectrum but are relatively arbitrary and the transmitting frequencies could
be altered by either changing the clock frequency or the ratio of the highest
frequency over the clock frequency.

##### Levels of output voltage

Determining the number of output voltage levels of the amplifier is another
trade-off. As shown in LABEL:Fig:Plots-of-f2, the resemblance function
$g_{2}(t)$ generated by the bipolar voltage across the coil is able to
approximate the objective function $f_{2}(t)$ with a tolerable error in
practice. It will be justified through practical test that the bipolar output
voltage is capable of generating the coil currents with low distortion. In
addition, there are only six switching angles $\alpha_{1}$, $\alpha_{2}$,
$\alpha_{3}$, $\alpha_{4}$, $\alpha_{5}$ and $\alpha_{6}$ in a quarter period.
The highest switching frequency is close to the $17^{th}$ harmonic frequency
implying that the magnitude of the harmonic frequencies higher than the
highest transmitting frequency will be low. Therefore, a low-pass filter is no
longer needed.

Figure 8: $f_{2}(t)$, $g_{2}(t)$ and the pulse voltage that generates
$g_{2}(t)$

##### Setup of optimisation problem

According to (8) the Fourier coefficients $b_{p}$ of $g_{2}(t)$ can be
calculated as (14). Cautions should be made that there are two falling edges,
i.e. $+V_{DC}$ to $0$ and $0$ to $-V_{DC}$ at $\alpha_{1}$, $\alpha_{3}$,
$\alpha_{5}$, two raising edges, i.e. $-V_{DC}$ to $0$ and $0$ to $+V_{DC}$ at
$\alpha_{2}$, $\alpha_{4}$, $\alpha_{6}$, and one falling edge, i.e. $+V_{DC}$
to $0$ at $\frac{\pi}{2}$. Substituting (14) in (9) gives rise to the
optimisation problem (15) where the dc voltage $V_{DC}$ and coil inductance
$L$ have been cancelled. Additionally, because we would like $b_{p}$ to be
inversely proportional to $p$, (15) is in fact normalised where each residual
in the objective function should have similar level of magnitude.

$b_{p}=-\frac{4V_{{\it DC}}}{p^{2}\pi\omega
L}\left(2\sum_{q=1}^{6}(-1)^{q}\sin\left(p\alpha_{q}\right)-\sin\left(\frac{p\pi}{2}\right)\right),\\\
p=1,3,5,\cdots$ (14)

$\min_{0\leq\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{6}\leq\frac{\pi}{2},m_{p}}\sum_{p=1,3,7,17}\lambda_{p}\left(\frac{8}{p\pi}\sum_{q=1}^{6}(-1)^{q}\sin\left(p\alpha_{q}\right)-\frac{4}{p\pi}\sin\left(\frac{p\pi}{2}\right)+m_{p}\right)^{2},\\\
subject\,to\,\left|\frac{8}{p\pi}\sum_{q=1}^{6}(-1)^{q}\sin\left(p\alpha_{q}\right)-\frac{4}{p\pi}\sin\left(\frac{p\pi}{2}\right)\right|\leq\epsilon_{p},\,\forall\,p\in\left\\{5,9,11,13,15\right\\}$
(15)

The initial values of the six switching angles are the points where the
gradient of $f_{2}(t)$ is zero as shown in LABEL:Fig:Plots-of-f2. The angles
can only have discrete values and the resolution is determined by the product
of the order of the highest harmonic and the ratio of the clock frequency over
the highest transmitting frequency. As a result, the length of the lookup
table which stores the control sequence is $476$ as a product of $17$ and $28$
and hence the resolution of angle is $\frac{2\pi}{476}\,rads$. Since every
switching angle is an integer multiplied by the resolution, it is convenient
to use the number of clock cycle to represent the angles. A set of initial
values is listed in the first row of LABEL:Tab:Initial-values-and. These
initial values are substituted into (14) in order to calculate the modulation
index $m_{p}$. The objective $b_{p}$ of selected harmonics is inversely
proportional to $p$, so the corresponding modulation index $m_{p}$ should have
the same value. As a result, the average value of $m_{1}$, $m_{3}$, $m_{7}$
and $m_{17}$ when substituting the initial angles is used as the initial value
for $m_{1}$, $m_{3}$, $m_{7}$ and $m_{17}$.

The constrain threshold $\epsilon_{p}$ is set up in a way that the scaled
modulation index $\frac{m_{p}}{p}$ for those undesirable harmonics has the
same bound. The modulation index $m_{p}$ is scaled by $p$ because we would
like $b_{p}$ to have the same bound for undesirable harmonics. All the weights
$\lambda_{p}$ are configured as one for the first setup and then the residual
of the $17^{th}$ harmonic is relaxed by setting $\lambda_{17}$ as zero which
is the second setup. The reason of such relaxation will be discussed in
following paragraphs. Details of the initial condition and two setups for the
parameters are listed in LABEL:Tab:Initial-values-and.

##### Results

The optimisation problem (15) given the initial condition under two setups of
parameters is solved using the interior-point algorithm. The resultant
switching angles, total harmonic distortion (THD) and scaled modulation index
$\frac{m_{p}}{p}$ are listed in LABEL:Tab:Initial-values-and where the THD is
defined as (16). Here we use $\frac{m_{p}}{p}$ to represent $b_{p}$ because
the former is independent of the coil inductance $L$ while the latter is not.
An Fast Fourier Transform (FFT) is conducted to the resemblance functions when
the initial switching angles, the optimal angles of the first setup and of the
second setup are applied. The results are plotted in LABEL:Fig:Plot-of-m.

$THD=\frac{\sqrt{\sum_{p\in\Psi}b_{p}^{2}}}{\sqrt{\sum_{p\in\Phi}b_{p}^{2}}}\times
100\,\%$ (16)

TABLE II: Residual weight $\lambda_{p}$, constrain threshold $\epsilon_{p}$,
switching angle $\alpha_{q}$, scaled modulation index $\frac{m_{p}}{p}$ and
the total harmonic distortion for the initial and two optimisation setups of
experiment No.1

| $\lambda_{1}$ | $\lambda_{3}$ | $\lambda_{7}$ | $\lambda_{17}$ | $\epsilon_{5}$ | $\epsilon_{9}$ | $\epsilon_{11}$ | $\epsilon_{13}$ | $\epsilon_{15}$ | Switching angle (clock cycle) | $\frac{m_{p}}{p}$ | $THD\,(\%)$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{5}$ | $\alpha_{6}$ | $50.42\,kHz$ | $151.26\,kHz$ | $352.94\,kHz$ | $857.14\,kHz$
Initial | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $36$ | $49$ | $68$ | $77$ | $88$ | $111$ | $0.534$ | $0.145$ | $0.066$ | $0.043$ | $7.33$
No. 1 | $1$ | $1$ | $1$ | $1$ | $0.10$ | $0.18$ | $0.22$ | $0.26$ | $0.30$ | $39$ | $51$ | $70$ | $82$ | $90$ | $111$ | $0.553$ | $0.171$ | $0.047$ | $0.032$ | $9.04$
No. 2 | $1$ | $1$ | $1$ | $0$ | $0.10$ | $0.18$ | $0.22$ | $0.26$ | $0.30$ | $35$ | $48$ | $65$ | $74$ | $86$ | $111$ | $0.493$ | $0.157$ | $0.068$ | $0.043$ | $7.80$

Figure 9: Scaled modulation index $\frac{m_{p}}{p}$ with respect to frequency
for the initial switching angles, optimal angles of the first setup and the
second setup of experiment No.1

It can been seen from LABEL:Tab:Initial-values-and that the initial switching
angles already lead to a low THD of $7.33\,\%$. However, the power
distribution among the four selected frequencies is not the same as desired.
The fourth component is higher while the second and third component are lower
than they should be. This situation is improved by the optimisation under the
first setup. Equal weights are assigned to the four residuals resulting in a
more desirable power distribution. However, the THD has been increased to
$9.04\,\%$ which is mainly caused by the increase of the adjacent harmonic
components of $857.14\,kHz$ that can be seen in LABEL:Fig:Plot-of-m. The
reason behind this is that initially the frequency of the highest-frequency
ripple is close to $857.14\,kHz$ which in turn gives rise to the large
component of $857.14\,kHz$ and a clean spectrum higher than $857.14\,kHz$. The
optimisation process tends to penalise the residual of $857.14\,kHz$ by,
unfortunately, increasing its adjacent harmonic components rather than the
lower desirable harmonics. Consequently, it is reasonable to always relax the
penalty of the ’$benign\,ripple$’ as has been done in the second setup. As
shown in LABEL:Tab:Initial-values-and, the scaled modulation index of the
first three frequencies have a better ratio comparing to those of the first
optimisation while the highest-frequency $\frac{m_{p}}{p}$ and THD are
similar. The waveforms of the associated resemblance functions for the three
sets of switching angles are shown in LABEL:Fig:Plots-of-the.

Figure 10: Waveform of the resemblance function for the initial switching
angles, optimal angles of the first setup and the second setup of experiment
No.1

The optimal switching angles of the second optimisation setup are implemented
in the experimental system to drive a coil with an inductance of about
$1.4\,\mu H$. The screenshot of an oscilloscope displaying the coil current
and bipolar output voltage is shown in LABEL:Fig:Coil-current-(green. The
peak-to-peak value of the current through the coil is higher than $60\,A$. An
FFT is done to the current and the largest ten root-mean-square (RMS) values
and the corresponding frequencies are listed in LABEL:Tab:FFT-results-of.

Figure 11: Coil current (green trace), output voltage of one half bridge
(yellow trace) and of the other half bridge (red trace) for optimisation No. 2
of experiment No. 1 TABLE III: FFT results of the coil current for
optimisation No. 2 of experiment No. 1

No. | Frequency | Current (RMS) | No. | Frequency | Current (RMS)
---|---|---|---|---|---
$1$ | $50.42\,kHz$ | $16.97\,A$ | $6$ | $554.62\,kHz$ | $0.76\,A$
$2$ | $151.26\,kHz$ | $6.15\,A$ | $7$ | $252.10\,kHz$ | $0.49\,A$
$3$ | $352.94\,kHz$ | $2.08\,A$ | $8$ | $1462.18\,kHz$ | $0.49\,A$
$4$ | $857.14\,kHz$ | $1.49\,A$ | $9$ | $756.30\,kHz$ | $0.32\,A$
$5$ | $655.46\,kHz$ | $0.81\,A$ | $10$ | $453.78\,kHz$ | $0.31\,A$

####  III-C2 Experiment No. 2

Compared to the first experiment, the objective of the second experiment is to
drive a higher-impedance coil in a lower spectrum from about $1\,kHz$ to
$200\,kHz$. The axis of the coil aperture would point to the test samples.
Application of this spectrum can be found in the inspection of welding
process, crack detection of metal samples and thickness measurement of metal
film, etc.

The objective function $f_{1}(t)$, as shown in LABEL:Fig:Plots-of-f1, is
implemented in this experiment where the fundamental, $3^{rd}$, $9^{th}$,
$27^{th}$ and $81^{st}$ harmonics are involved. The resemblance function
$g_{1}(t)$ is also plotted in LABEL:Fig:Plots-of-f1 and there are $38$
switching angles in a quarter period of $g_{1}(t)$. Because the highest order
harmonic is the $81^{st}$ harmonic, $g_{1}(t)$ requires more switching angles
than $g_{2}(t)$. The initial values of the switching angles are determined in
the same way as it is in the first experiment, i.e. examining the gradient of
objective function. The clock frequency is configured as $6.25\,MHz$ and the
five transmitting frequencies are chosen as about $1.61\,kHz$, $4.82\,kHz$,
$14.47\,kHz$, $43.40\,kHz$ and $130.21\,kHz$. As a result, the memory needs
$3888\,bits$ to store the control sequence.

In order to show that a fast design could be done without the optimisation
process discussed in the first experiment, the initial values of the $38$
switching angles are directly implemented in the experimental system. The
screenshot of an oscilloscope displaying the coil current and bipolar output
voltage is shown in LABEL:Fig:Coil-current-(green-1. The peak-to-peak value of
the current through the coil is about $2.5\,A$. An FFT is done to the current
and the largest ten root-mean-square (RMS) values and the corresponding
frequencies are listed in LABEL:Tab:FFT-results-of-1.

Figure 12: $f_{1}(t)$, $g_{1}(t)$ and the pulse voltage that generates
$g_{1}(t)$ Figure 13: Coil current (green trace), output voltage of one half
bridge (yellow trace) and of the other half bridge (red trace) for experiment
No. 2 TABLE IV: FFT results of the coil current for experiment No. 2

No. | Frequency | Current (RMS) | No. | Frequency | Current (RMS)
---|---|---|---|---|---
$1$ | $1.61\,kHz$ | $751.46\,mA$ | $6$ | $8.04\,kHz$ | $10.53\,mA$
$2$ | $4.82\,kHz$ | $284.49\,mA$ | $7$ | $20.90\,kHz$ | $10.30\,mA$
$3$ | $14.47\,kHz$ | $89.05\,mA$ | $8$ | $11.25\,kHz$ | $7.75\,mA$
$4$ | $43.40\,kHz$ | $24.60\,mA$ | $9$ | $17.68\,kHz$ | $7.52\,mA$
$5$ | $130.21\,kHz$ | $22.80\,mA$ | $10$ | $27.33\,kHz$ | $5.49\,mA$

####  III-C3 Power and efficiency comparison between two experiments

According to (10), (11), (12) and (13), the power and efficiency metrics of
the two experiments are calculated and listed in LABEL:Tab:Efficiency. In
terms of the efficiency $\eta$, the first experiment has a lower value of
20.02%, which is due to the fact that the coil has a high quality factor and
much of the power is dissipated by the MOSFETs. In contrast, the efficiency
$\eta$ is 76.92% in the second experiment, because the coil is lossy. The
energy conversion factor $\zeta$ is larger than unity for both experiments.
The reactive power $Q_{out}$ for the first experiment is as large as 224.81
var with a total dissipated power of 34.32 W. The second experiment has a
lower $\zeta$ mainly because of its higher impedance of the coil. If possible,
we would pursue a low efficiency $\eta$, because power dissipation in the load
would be a waste of energy. On the other hand, a large energy conversion
factor $\zeta$ is preferred, which implies that an ac magnetic field would be
generated while consuming low dc power.

TABLE V: Power and efficiency metrics of two experiments.

Experiment | $V_{DC}$ | $I_{DC}$ | $P_{DC}$ | $P_{out}$ | $Q_{out}$ | $\eta$ | $\zeta$
---|---|---|---|---|---|---|---
No. 1 | 24.00 V | 1.43 A | 34.32 W | 6.87 W | 224.81 var | 20.02% | 6.55
No. 2 | 24.00 V | 0.13 A | 3.12 W | 2.40 W | 8.62 var | 76.92% | 2.76

##  IV Conclusions

The instrumental development of MECT entailed a highly efficient amplifier in
the transmitting power stage. Switch-mode power amplifiers exhibited
compelling advantages in terms of efficiency. This article attempted to serve
as a bridge that introduced a switch-mode power amplifier into the design of
MECT instrumentation. Being able to transmit MSF magnetic fields was critical
in MECT but was not a primary goal in the research of modulation strategy of
switch-mode amplifiers or converters. Hence, this article also examined the
MSF capability of the SHEPWM technique. A Class D power amplifier was
developed in this article aiming at generating high MSF currents in the
transmitting coil. An MSF-SHEPWM method was proposed to modulate the pulse
voltage across the coil. Three concise steps could summarize the new
modulation strategy, i.e., time-domain synthesis of harmonic signals,
frequency-domain analysis of the resemblance signal, and optimisation of the
resultant Fourier coefficients. An FPGA-based experimental system was
developed, and the proposed methodology was implemented in two experiments.
The first experiment transmitted magnetic fields exciting at four simultaneous
frequencies with a THD as low as 8%. The peak-to-peak value of the coil
current was as high as 60A while the total power dissipation was low.The size
of the circuit board was small and no large heat dissipating facilities were
needed. The second experiment tries to illustrate a fast design process
showing that the proposed method was still effective even with the first step
only. Five frequencies were transmitted in a smaller span of spectrum
comparing to the first experiment. The two experiments have shown that the
design is versatile and could be applied to different electromagnetic
inductive sensing scenarios.

## Acknowledgement

All research data supporting this publication are directly available within
this publication.

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|  Yang Tao (M’19) received the B.Sc. degree in automation and the M.Sc.
degree in control science and engineering from Tianjin University, Tianjin,
China, in 2010 and 2013, respectively, and the Ph.D. degree in electrical and
electronic engineering from the University of Manchester, Manchester, U.K., in
2018. He is currently an Research Associate with the Department of Electrical
and Electronic Engineering, School of Engineering, University of Manchester.
His research interests include electromagnetic instrumentation, metal
detection, multifrequency power inverter, electromagnetic tomography, inverse
problem, sparse representation, and deep learning.
---|---
|  Christos Ktistis received the B.Sc. degree in mechanical engineering from
the Technological Educational Institution, Serres, Greece, in 2000, the M.Sc.
degree in mechatronics from Lancaster University, Lancaster, U.K., in 2002,
and the Ph.D. degree in electrical and electronic engineering from The
University of Manchester, Manchester, U.K., in 2007. He is currently a Visitor
with the Department of Electrical and Electronic Engineering, School of
Engineering, The University of Manchester.
---|---
|  Yifei Zhao received the B.Eng. (Hons.) degree and the Ph.D. degree in
electrical and electronic engineering from The University of Manchester,
Manchester, U.K., in 2010 and 2013, respectively. He is currently a Visitor
with the Department of Electrical and Electronic Engineering, School of
Engineering, The University of Manchester.
---|---
|  Wuliang Yin (M’05–SM’06) received the B.Sc. and the M.Sc. degrees in
electronic measurement and instrumentation from Tianjin University, Tianjin,
China, in 1992 and 1995, respectively, and the Ph.D. degree in automotive
electronics from Tsinghua University, Beijing, China, in 1999. He was
appointed as a Mettler Toledo (MT) Sponsored Lecturer with the Department of
Electrical and Electronic Engineering, School of Engineering, The University
of Manchester, Manchester, U.K., in 2012, and was promoted to a Senior
Lecturer in 2016. He has authored 1 book, more than 230 papers, and was
granted more than 10 patents in the area of electromagnetic sensing and
imaging. Dr. Yin was a recipient of the 2014 and 2015 Williams Award from the
Institute of Materials, Minerals and Mining and the Science and Technology
Award from the Chinese Ministry of Education in 2000.
---|---
|  Anthony J. Peyton received the B.Sc. degree in electrical engineering and
electronics and the Ph.D. degree in medical instrumentation from The
University of Manchester Institute of Science and Technology (UMIST),
Manchester, U.K., in 1983 and 1986, respectively. He was appointed as a
Principal Engineer with Kratos Analytical Ltd. in 1989, developing precision
electronic instrumentation systems for magnetic sector and quadrupole mass
spectrometers, from which an interest in electromagnetic instrumentation was
developed. He returned to UMIST as a Lecturer and worked with the Process
Tomography Group. He moved to Lancaster University in 1996 taking up the post
of Senior Lecturer and promoted to a Reader in Electronic Instrumentation in
July 2001 and a Professor in May 2004. Since December 2004, he has been a
Professor of Electromagnetic Tomography Engineering with the University of
Manchester, Manchester. He has been a Principal Investigator of numerous
national- and industry-funded projects and a partner of ten previous EU
projects, one as a coordinator. He has been a coauthor on almost 140
international journal papers, two books, several hundred conference papers,
and 12 patents in areas related to electromagnetics and tomography. His main
research interests include instrumentation, applied sensor systems, and
electromagnetics.
---|---