sensorless induction welding of carbon fiber reinforced

Department of Automatic Control

Sensorless Induction Welding of Carbon Fiber Reinforced

Plastic Components

Johan Bladh

MSc Thesis TFRT-6113 ISSN 0280-5316

Department of Automatic Control Lund University Box 118 SE-221 00 LUND Sweden

© 2020 by Johan Bladh. All rights reserved. Printed in Sweden by Tryckeriet i E-huset Lund 2020

Abstract

In this work, a method to estimate the weld-zone temperature during inductionwelding of carbon fiber reinforced plastics (CFRP) is developed. Today severalmethods exist to measure, or in other ways ensure, the correct temperature in theweld. However, these methods all have some form of drawback ranging from leav-ing foreign material in the weld to complex sensor arrangements. The inductionwelding process is run by a frequency inverter. By using the current measured bythe frequency inverter the temperature in the weld-zone can be estimated and noexternal sensors are needed. The impedance the inductor is experiencing duringwelding is changing due to the changes to the electrical properties of the CFRP ma-terial. The changing resistance, in particular, results in a changing impedance whichcan be observed by measuring the phase angle. With the current measured by thefrequency inverter as input signal to a Wiener model, identified using a black-boxmodeling approach, the temperature in the weld-zone can be estimated. The modelconstructed in this work is to be used at the fixed operating frequency of 640 kHz,with the welding starting at room temperature and is validated on the specific CFRPmaterial and workpiece geometry used in this work.

i

Acknowledgements

I would like to start by expressing my gratitude to Magnus Hermodsson and Ken-neth Frogner at Corebon for the provided support, technical expertise and guidancethroughout this work. I would also like to thank my supervisor Anders Robertssonat LTH for the valuable input. Lastly, I would like to express my gratitude to theteam at Corebon for all the help during this work.

iii

Contents

1. Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Induction Heating . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Scope of Work and Delimitations . . . . . . . . . . . . . . . . . 6

2. Experimental Setup 72.1 Frequency Inverter . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Welding Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Carbon Fiber Reinforced Plastics . . . . . . . . . . . . . . . . . 112.5 Temperature Measurement . . . . . . . . . . . . . . . . . . . . 112.6 Weld Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3. Electrical Properties and Material Characterization 143.1 Power Loss Simulation . . . . . . . . . . . . . . . . . . . . . . 143.2 Equivalent Electrical Resistivity Computation . . . . . . . . . . 143.3 Temperature Dependence of Electrical Properties . . . . . . . . 16

4. Signal Analysis 184.1 Available Signals . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. System Identification 215.1 The System Identification Process . . . . . . . . . . . . . . . . . 225.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Model Estimation and Validation . . . . . . . . . . . . . . . . . 255.5 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . 26

6. Results 276.1 Equivalent Electrical Resistivity . . . . . . . . . . . . . . . . . . 276.2 Power Loss Simulation . . . . . . . . . . . . . . . . . . . . . . 286.3 Temperature Dependence of Electrical Properties . . . . . . . . 286.4 Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

Contents

6.5 System Identification . . . . . . . . . . . . . . . . . . . . . . . 366.6 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . 40

7. Discussion 427.1 Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . 427.2 Difference Between Samples . . . . . . . . . . . . . . . . . . . 437.3 Input Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.4 Effects of Welding . . . . . . . . . . . . . . . . . . . . . . . . . 44

8. Conclusions 469. Future Work 47Bibliography 48

vi

List of Figures

1.1 Current vs frequency in a series RLC circuit [electricalbaba, 2016]. . . 6

2.1 Simplified circuit overview of the frequency inverter MOSFET H-bridge on the transformer primary side and the series RLC load on thesecondary side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Illustrative figure displaying the typical waveform of voltage and cur-rent output from the frequency inverter. . . . . . . . . . . . . . . . . 8

2.3 Block diagram showing the adjustable parameters; frequency and dutycycle, along with the system output in the form of temperature. . . . . 8

2.4 Inductor coil in its housing. . . . . . . . . . . . . . . . . . . . . . . . 102.5 The inductor coil placed on top of two CFRP plates (black) with pres-

sure applied. Also seen is the thermocouple inserted between the twoCFRP plates, hoses for cooling, and the litz cable connecting the coil tothe frequency inverter. . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 The CFRP material used. . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Thermo image, taken with a slight angle, of two welded CFRP plates

heated to roughly 400°C. The inductor is situated beneath the plates,with no pressure applied. The thermocouple location is marked withthe black circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 The inductor and CFRP workpiece modelled in FEMM. The model isaxisymmetrical around the left edge of the image. The figure illustratesthe flux density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Black-box system identification workflow. Adapted from [Anderssonet al., n.d.] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Nonlinear ARX model structure [MathWorks, 2020d]. . . . . . . . . 245.3 Hammerstein-Wiener model structure [MathWorks, 2020c]. . . . . . . 25

vii

List of Figures

6.1 Measured (solid lines) and simulated (dash-dotted lines) resistance andinductance values for the frequencies 20-100 kHz. The simulated valueshave been compensated for with respect to the difference between thesimulated and measured non-loaded coil. . . . . . . . . . . . . . . . . 27

6.2 Cross-section of the workpiece illustrating the simulated power lossesin the workpiece with the inductor placed above the workpiece. Thehorizontal line marks the intersection between the two CFRP plates. . 28

6.3 Resistance and inductance temperature dependence of coil plus CFRP.Left: first welding. Right: reheating. . . . . . . . . . . . . . . . . . . 29

6.4 Temperature dependence of electrical resistivity using one measure-ment from each group seen in the left image in Figure 6.3. . . . . . . 30

6.5 Left: coil temperature during heating. Right: coil temperature 8 s afterstopping. The cursor marks the point on the coil where the temperatureis measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.6 Measured temperature during welding at 640 kHz. . . . . . . . . . . . 326.7 MA15 values for measured frequency, current RMS value, rail voltage

and phase value for the welding process in Figure 6.6. . . . . . . . . . 326.8 Temperature relative to room temperature for five different CFRP sam-

ples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.9 MA10 values for measured frequency, phase value, RMS current and

rail voltage for the five CFRP samples in Figure 6.8. . . . . . . . . . . 336.10 Multiple cycles at high temperatures. The first three cycles are run at

640 kHz, the following cycles are run at a variety of frequencies. . . . 346.11 Temperature curves from three cycles on the same CFRP sample. . . . 356.12 MA10 values for measured frequency, phase value, RMS current and

rail voltage of the three cycles shown in Figure 6.8. . . . . . . . . . . 356.13 Current vs temperature increase, starting from room temperature. . . . 366.14 Comparison between nonlinear ARX model and Wiener model. . . . 376.15 Pole-zero plot of the linear transfer function of the Wiener model. Poles

are marked "x", zeros are marked "o". . . . . . . . . . . . . . . . . . 386.16 Nonlinear estimator of the Wiener model. . . . . . . . . . . . . . . . 396.17 Model output comparison between the model running on the microcon-

troller and the equivalent simulation in Simulink. . . . . . . . . . . . 406.18 Images taken with microscope of two welded CFRP plates. Left: the

workpiece about 10 mm from the center, corresponding to locationmarked "A" in Figure 6.19. Right: cold spot in the center of the work-piece, corresponding to location marked "B" in Figure 6.19. Imageheight is about 4.5 mm, workpiece thickness is about 5 mm. . . . . . 41

6.19 Illustrative figure of cross-sections of coil and workpiece marking thelocations of images in Figure 6.18. . . . . . . . . . . . . . . . . . . . 41

viii

Symbols and Abbreviations

Abbreviations

AC Alternating currentARX Autoregressive with Exogenous VariablesCFRP Carbon fiber reinforced plasticIH Induction heatingMA Moving AveragePEEK PolyetheretherketoneRLC Resistance, Inductance, CapacitanceSISO Single-input, single-output

Symbols

C Capacitance [F]δ Skin depth [m]f Frequency [Hz]fr Resonance frequency [Hz]φ Phase angle [rad]I Current [A]J Current density [A/m2]L Inductance [H]R Resistance [Ω]ρ Resistivity [Ωm]µr Relative magnetic permeabilityV Voltage [V]XL Inductive reactance [Ω]XC Capacitive reactance [Ω]Z Impedance [Ω]

ix

1Introduction

1.1 Background

With the challenges the world is facing today regarding climate change and emis-sions from manufacturing and transport, there is an increasing demand on recyclingand the reduction of emissions from industries such as the transportation sector.There is also an economic incentive to lower fuel and production costs. This has ledto the search for lighter materials such as carbon fiber composites, which have cometo replace many metallic materials in mainly aeronautics due to its great mechani-cal properties in proportion to its low weight [Kane et al., 2020]. CFRP materialsare predicted to have an annual 10% usage increase in the next few years [Roberts,2019]. However, for this to become a reality more efficient manufacturing technolo-gies must be developed. Corebon AB has developed induction technology for fibercomposites manufacturing where the mold and/or the material itself can be heatedusing induction. By using this technology, shorter cycle times can be achieved com-pared to traditional methods while the result is a high-quality fiber composite. Thistechnology is applicable on most fiber composites but is particularly well suitedfor carbon fiber composites since the fibers can be characterized as semiconduc-tors, with a resistivity of typically 100-1000 times that of copper, and therefore bedirectly heated throughout the material thickness [Frogner et al., 2014].

Today it is becoming increasingly common to use thermoplastics as the matrixmaterial, i.e., the material that binds the fiber reinforcement. Using thermoplasticsas the matrix material has several advantages, one being the possibility to remelt thematerial. This means that components can be repaired, increasing the lifetime andrecyclability of products. However, there is another, and perhaps even greater ben-efit to use thermoplastics as the matrix material in a CFRP component as it allowsfor joining components by welding. Furthermore, when using traditional materialssome form of mechanical fastener, such as rivets, is often used. Eliminating thesefasteners and instead using welded CFRP parts would not only result in a weightreduction but would also result in lower production cost. There is the obvious costsaving of the fasteners themselves but also the elimination of production steps suchas the drilling and inspection of holes for mechanical fasteners. An alternative to us-

1

Chapter 1. Introduction

ing mechanical fasteners is to use some form of adhesive. However, the inspectionof the welded structures is easier compared to adhesively bonded parts [Gardiner,2018a].

A variety of welding methods for CFRP have been developed throughout theyears such as resistance welding, ultrasonic welding and laser welding to name afew [Gardiner, 2018b]. One very interesting method is induction welding which isthe method studied in this work. Since the carbon fibers can be characterized assemiconductors, they can be inductively heated [Frogner et al., 2014]. When in-ductively heating CFRP the material will be heated through the material thickness,and due to its low thermal mass the cycle time can be reduced compared to tradi-tional methods and materials [Lundström et al., 2017]. If the welding of CFRP isperformed with correct process conditions the resulting part will have no defineddividing line and can be viewed as a homogeneous, solid part. Such a joint will beaccepted by authorities with no mechanical fasteners [Gardiner, 2018b].

One major challenge of induction welding is to concentrate the heat in the weld-zone without (over)heating other parts of the workpiece. One way to overcome thischallenge is to use a susceptor material integrated between the parts to be welded.A susceptor material is a material that generates a substantial amount of heat whensubjected to a changing magnetic field, thus heating the adjacent workpieces at theinterface. This susceptor material could be a steel mesh or similar. However, it isnot always desirable to leave a foreign material in the weld which negatively affectsthe material properties of the weld. It would then not be considered a homogeneouspart. To solve this problem recent technology has been developed where a movingsusceptor is following the weld head and therefore not leaving any foreign materialin the weld [Gardiner, 2020], a seemingly complex method though. However, somemethods do not use a susceptor material but instead heat all the workpiece andnot only the weld. Due to effects discussed later in this study, most of the heatwill be generated at the surface closest to the induction coil. This brings us backto the problem of heat concentration in the surface layer rather than in the weld-zone. Technology has been developed where heat management system and toolingmaterial used to get rid of the heat generated in the surface layers, allowing theheat to be concentrated to the weld-zone without the use of a susceptor [Gardiner,2018b].

Another challenge is how to ensure the correct temperature in the weld. Todayseveral methods exist to measure the temperature, the most common being the us-age of pyrometers. By measuring the surface temperature of the workpiece one canestimate the temperature in the weld. However, this requires quite complex arrange-ments for many geometries. Another method is the integration of thermocouplesinto the weld, but again, this would leave behind foreign material resulting in anon-homogeneous part.

This study is an investigation into whether it is possible to eliminate a traditionaltemperature sensor and instead use the frequency inverter along with the inductoritself to monitor the temperature of the weld. If this would prove to be possible,

2

1.2 Induction Heating

the problem of leaving foreign material in the weld would be eliminated as well asany complex arrangement with external temperature sensors. A way to achieve thiscould be to develop a model describing how the various parameters measured bythe frequency inverter unit relate to the temperature in the weld. Previous work hasbeen done on the subject of modeling the electromagnetic and thermal properties ofCFRP, and it has been shown that one can do electric and thermal characterizationof CFRP structures using an inductor coil [Lundström et al., 2017]. By placingthe coil adjacent to the CFRP material it is possible to measure the in-plane bulkresistivity, which is an important property in induction heating (IH), and then fitthe measurements to a numerical simulation model. Other studies have been donewhere a multiscale approach has been applied [Wasselynck et al., 2011][Kane etal., 2020]. Good agreement has been shown between the simulated and measuredresults of the temperature at the surface layer. However, these models are validatedfor temperatures below 150°C, which is well below the melting point of most CFRPmaterials. The melting point of PEEK is about 340°C. Furthermore, the purposeof this study is to create a model to be used for real-time temperature estimation,implemented within the frequency inverter unit, using real-time data from the same.Thus, a model depending on FEM solving will not be applicable.

One can conclude that the research done on changes in mechanical propertiesof CFRP when subjected to elevated temperatures is quite extensive. However, thesame cannot be said for the research on the electrical properties. Studies like theones mentioned above are done at temperatures far below the melting point. Inthe aim to construct a model describing the relationship between the weld-zonetemperature and the properties measured by the frequency inverter, an investigationinto the electrical properties of CFRP at elevated temperatures is necessary.


When a current is flowing through a conductor a magnetic field is generated aroundsaid conductor. A change in this current will result in a change in the magneticfield. All conductive objects situated within this field will then induce currents inthe opposite direction to the conductor current direction to create a magnetic field oftheir own to counteract the change in the magnetic field. As can be seen in Equation(1.1), the rate of change in magnetic flux, ΦB, and the induced voltage, ε , haveopposite signs. Thus, the induced current, I, will oppose the change in flux. Theseinduced currents are known as eddy currents and will produce heat by the Jouleeffect in the object in which they are formed, see Equation (1.2). The resistanceand power are denoted R and P, respectively. However, when there is no changein current in the conductor, and thus no change in the magnetic field surroundingit, no currents will be induced in the parts within this field. Hence to generate acontinuously changing magnetic field an alternating current (AC) has to be flowingthrough the conductor.

3


ε =−∂ΦB

∂ t(1.1)

P = I2 ·R =ε2

R(1.2)

When using induction heating there are a few important electromagnetic effectsthat one has to be aware of. One such effect is the skin effect. When an AC cur-rent is flowing through a conductor the current distribution within the conductor’scross-section is not uniform. The current density will be highest at the surface anddecrease towards the center of the conductor. This principle applies to both the in-ductor coil and the workpiece. The current density through the workpiece thicknesscan be calculated by Equation (1.3) where J is the current density at distance dfrom the surface, J0 is the current density at the surface and δ is the skin depth,also known as penetration depth. This effect will cause most of the heat to be gen-erated in the part of the workpiece closest to the inductor coil within the distanceδ . Therefore, this effect has to be taken into account to not overheat this part of theworkpiece.

J = J0 · e−d/δ (1.3)

The skin depth δ is given by Equation (1.4) and is a function of electrical re-sistivity, ρ , frequency, f , and relative magnetic permeability, µr. Since CFRP isnonmagnetic the relative magnetic permeability is one, µr = 1. The skin depth has atemperature dependence since the electrical resistivity varies with temperature. Theelectrical resistivity can often be approximated as a linear equation, see (1.5), whereρ(T ) is the resistivity at temperature T, ρ0 is the resistivity at ambient temperatureT0 and α is the temperature coefficient [1/°C]. Such approximation is only valid inthe absence of phase transformation, [Rudnev et al., 2017].

δ ≈ 503√

ρ

µr f(1.4)

ρ(T ) = ρ0[1+α(T −T0)] (1.5)

The electrical circuit used in this induction heating application can be charac-terized as a series resonance circuit, RLC circuit. The resistance, R, and inductance,L, can mainly be attributed to the inductor. The capacitance, C, of the circuit is a re-sult of added capacitors to reduce the high inductive reactance of the inductor, thusmaking it an RLC circuit. Such a circuit will have an oscillating behavior whichwill be pronounced at a certain frequency, the resonance frequency. At the reso-nance frequency the reactance from the circuit inductance, see Equation (1.6), and

4


capacitance, see Equation (1.7), will cancel out each other making the impedance,Z, purely resistive, see Equation (1.8). The relationship between XL and XC will de-termine how much the current leads or lags the voltage. This can be described bythe phase angle, φ , see Equation (1.9). The phase angle will be positive in a mainlyinductive circuit, while in a mainly capacitive circuit it will be negative. The phaseangle will be zero at resonance.

XL = 2π f L (1.6)

XC =1

2π fC(1.7)

Z = R+ j(XL−XC) (1.8)

φ = arctanXL−XC

R(1.9)

The reactances have a frequency dependence as seen in Equations (1.6) and(1.7). Above the resonance frequency, the circuit is increasingly inductive with in-creasing frequency. Below this frequency, the circuit is increasingly capacitive withdecreasing frequency. This yields a relationship between current and frequency ac-cording to Equation (1.10), where V is the voltage. This relationship is visualizedin Figure 1.1, where the resonance frequency can be expressed according to Equa-tion (1.11). This relationship is of great importance in an induction heating processsince one can control the power output by changing the frequency. The shape ofthe curve can be described using the Q-factor, see Equation (1.12). A system with ahigh Q-factor will have a narrow curve, making small changes in frequency result-ing in large changes in current and thus large changes in power, while a system witha low Q-factor will have a wider curve and smaller changes.

I =VZ

(1.10)

fr =1

2π√

LC(1.11)

Q =XL

R(1.12)

The real part of the circuit impedance, i.e., the resistance, R, can be divided intothe resistance from the inductor coil and the resistance from the workpiece. Theresistance is subsequently a function of the electrical resistivity, ρ . The relationshipbetween resistance and electrical resistivity for a conductor is given by Equation(1.13), where l is the length of the conductor and A is the effective cross-sectionalarea of the conductor. As stated above, the electrical resistivity has a temperature

5


Figure 1.1 Current vs frequency in a series RLC circuit [electricalbaba, 2016].

dependence, thus making the impedance, see Equation (1.8), having a temperaturedependence. When the electrical resistivity in the material changes due to temper-ature changes, the eddy currents induced in the workpiece will vary in strength,making the circuit inductance, L, also change with temperature [Frogner, 2020]. Ascan be seen in Equation (1.11), this will alter the resonance frequency, moving thecurrent-frequency curve in Figure 1.1.

R =ρlA

(1.13)

1.3 Scope of Work and Delimitations

The aim of this work is to investigate the possibility of using the frequency inverterunit along with the inductor to measure the temperature in the weld-zone duringwelding. To make this study feasible, considering time and scope, the followingdelimitations are put into place.

• The welding is to be performed in stationary, meaning neither the weldinghead nor the workpiece will be moving during the welding process.

• The workpiece material will consist of one material only.

• The geometry of the workpieces to be welded will remain the same through-out this work.

• The effects of different amount of applied pressure will not be studied in thiswork and the pressure will remain constant.

6

2Experimental Setup

2.1 Frequency Inverter

The experimental setup consists of three major parts, all provided by Corebon; afrequency inverter, a transversal inductor and flat CFRP workpieces to be welded.The frequency inverter creates an AC current which is outputted to the inductor. Itachieves this by switching the MOSFETs in the H-bridge seen in Figure 2.1. Theoutput voltage becomes a modified square wave and the closer to resonance, themore the current will resemble a sinusoidal, see Figure 2.2.

Figure 2.1 Simplified circuit overview of the frequency inverter MOSFET H-bridge on the transformer primary side and the series RLC load on the secondaryside.

The user sets the desired values for frequency and duty cycle on the frequencyinverter affecting the power outputted to the inductor coil. The duty cycle is thefraction of one period in which the voltage across the load is non-zero. During eachperiod there is a fixed period where no MOSFET is conducting to avoid a short-circuit between the top and bottom MOSFETs in the H-bridge. This dead time willbe a larger fraction of each period with increasing frequency since the period timeis decreasing while the deadtime is constant. Thus, a low frequency allows for a

7

Chapter 2. Experimental Setup

high duty cycle and vice versa. Due to the properties of the system, the duty cycleshould be set as high as possible. This makes frequency the only variable parameter.A basic system overview can be seen in Figure 2.3.

To keep the switching losses of the inverter at an operable level, soft switchingneeds to be ensured. The inverter will be soft switching from frequencies slightlyabove resonance up to the point where the output current gets too low to turn theswitching node (the switching nodes are the points in Figure 2.1 where the transistoris connected to the H-bridge legs) during the deadtime. This yields a frequencyinterval in which the process can operate and thus limits the possible heating rates.

0 1 2 3 4 5

Time [s] 10-6

-Vdc

0

+Vdc

Voltage

-I

0

+I

Curr

ent

Frequency Inverter Output

Figure 2.2 Illustrative figure displaying the typical waveform of voltage and cur-rent output from the frequency inverter.

Figure 2.3 Block diagram showing the adjustable parameters; frequency and dutycycle, along with the system output in the form of temperature.

Load BalancingTo obtain a reasonable power output at a suitable frequency interval the RLC loadmust be balanced. This means to choose suitable capacitors and transformer to bemounted in the frequency inverter unit. The shape and placement of the current-frequency curve in Figure 1.1 is determined by the choice of transformer and ca-pacitors. The values of the capacitors are calculated such that the total capacitancein the circuit places the resonance frequency at the desired frequency using Equa-tion (1.11), where the inductance, L, is the measured inductance of the coil. The

8

2.2 Inductor

resonance frequency placement is a trade-off between efficiency and penetrationdepth. The transformer is chosen such that a suitable current is obtained on the pri-mary side to fit the specifications of the frequency inverter. The impedance of thesecondary side is determined according to Equation (1.8) using the measured in-ductance and resistance of the coil together with the chosen capacitor values. Therequired voltage on the secondary side is then determined using Equation (1.10).Since the voltage on the primary side is known a suitable transformer ratio, n, canbe obtained according to Equation (2.1), where Np and Ns are the number of turnsof the primary winding and secondary winding, respectively.

n =Np

Ns=

Vp

Vs(2.1)

Circuit CharacteristicsThe resonance frequency of the system used in this work was measured to beroughly 610 kHz, which was a bit lower than calculated since the inductance andresistance measurements were made on the coil when disconnected from the restof the system. The inductance of other parts of the system such as the cable con-necting the coil to the frequency inverter results in a lower resonance frequency. TheQ-factor of the system measured on the primary side of the transformer is about 1.2,which is quite low. This means that the current curve in Figure 1.1 is quite flat, thusmaking the changes in current relatively small when moving along the frequencyaxis.

2.2 Inductor

The inductor used in the experiments in this work is a transversal air core coil madefrom a copper tube. The coil can be seen in its housing in Figure 2.4. It is confined inan epoxy mold within a glass fiber housing to prevent it from moving or deformingduring the welding process. From an efficiency point of view, this is not an optimalcoil since all coil turns except the one closest to the workpiece will induce currentsin the adjacent coil parts resulting in lower efficiency. However, this is a coil designthat is easy to understand and simulate. A flux concentrator could be used to mini-mize this effect. However, adding a flux concentrator would result in nonlinearitiesadding complexity beyond the scope of this work. The flux concentrator would besubjected to temperature changes which would have to be either taken into accountin the model or some form of cooling mechanism would have to be implemented toensure a stable temperature. Both options result in unnecessary complexity addedto the system.

The inductor temperature will increase during welding due to the losses gener-ated in the coil itself along with the heat emitted from the workpieces being welded.An industrial chiller is used to circulate temperature-controlled cooling water in the

9


copper tube to counteract the heating of the coil. Furthermore, a ceramic plate issituated between the CFRP plates and the coil acting as an isolator.

Figure 2.4 Inductor coil in its housing.

2.3 Welding Rig

Pressure needs to be applied on the workpieces during the welding to ensure goodcontact between the two CFRP plates to be joined. The pressure in this setup isapplied using three pneumatic cylinders pushing an aluminum plate down upon theinductor that sits on the workpieces, see Figure 2.5. The total theoretical force ex-erted on the workpiece is about 480 N which corresponds to about 1.3 bars of pres-sure. To achieve an optimal weld, one would perhaps want to adjust the pressureapplied as the welding process proceeds and the properties of the material changeand deformation occurs. The pressure applied will remain constant to reduce thecomplexity and the number of changing parameters in these experiments and thefocus will be on electrical properties and temperature control. All conductive mate-rials within the magnetic field will be heated during the induction welding process.The welding rig is made of aluminum which is conductive, why the workpieces areplaced on a fixture which is not electrically conductive.

10

2.4 Carbon Fiber Reinforced Plastics

Figure 2.5 The inductor coil placed on top of two CFRP plates (black) with pres-sure applied. Also seen is the thermocouple inserted between the two CFRP plates,hoses for cooling, and the litz cable connecting the coil to the frequency inverter.

2.4 Carbon Fiber Reinforced Plastics

The material used in this work is TC1200 woven carbon fiber reinforced PEEK witha resin content by weight of 42%. The melting point of the PEEK is 343°C and theprocessing temperature 370-400°C. For more material specifications, see [Toray,2019]. The workpieces consist of plates with the dimensions 60x60x2.5 mm3 cutfrom a larger CFRP sheet to fit the size of the inductor, see Figure 2.6.

2.5 Temperature Measurement

A thermocouple is inserted between the CFRP plates to measure the temperature inthe weld-zone. A thermocouple is a temperature sensor that consists of two wires ofdifferent metals that are joined at two points; one junction at a reference temperatureand one junction where the temperature is to be measured. A difference in temper-ature between these two junctions gives rise to a voltage due to the thermoelectriceffect. This voltage is correlated to the temperature. The thermocouple is connectedto a transducer box, also provided by Corebon, which is connected to the frequencyinverter where the temperature is logged.

Due to the circular geometry of the coil, the heat pattern will take the form of aring with cold spots in the center and the corners, see Figure 2.7. The thermocoupleplacement is marked with a black circle about 15 mm from the center and 30 mmfrom the lower edge. Since the thermocouple is placed in an alternating magneticfield it is subjected to a lot of electromagnetic interference. These disturbances are

11


Figure 2.6 The CFRP material used.

filtered out by using a ferrite bead and by grounding the transducer box. The ferritebead functions as a passive low-pass common mode filter preventing the electro-magnetic interference from reaching the transducer.

2.6 Weld Inspection

To examine the quality of the weld, the joined pieces can be cut in half, the cross-section polished, and then viewed in a microscope. This will give an indication asto if the correct processing conditions are met. To further validate the quality of thewelded samples one would have to examine the material in more detail with betterequipment. Such examination is outside the scope of this work.

12

2.6 Weld Inspection

Figure 2.7 Thermo image, taken with a slight angle, of two welded CFRP platesheated to roughly 400°C. The inductor is situated beneath the plates, with no pressureapplied. The thermocouple location is marked with the black circle.

13

3Electrical Properties andMaterial Characterization

To simulate the electromagnetic behavior of the inductor the free software FiniteElement Method Magnetics (FEMM) [Meeker, 2018] is used. The FEMM model ofinductor and workpiece can be seen in Figure 3.1.

3.1 Power Loss Simulation

By dividing the workpiece into a number of smaller sections and integrate the powerlosses within these, an idea as to how the heat pattern will look can be established.The cross-section was divided into squares with the size of 0.25 mm2. In Figure 3.1,the cross-section can be seen divided into two rectangular sections with the size of75 mm2. In order to obtain results as close to the real process as possible the param-eters of the model, such as current and electrical resistivity, need to be determined.

3.2 Equivalent Electrical Resistivity Computation

The electrical resistivity is a parameter that has a key role in the induction heatingprocess. One can obtain an equivalent resistivity value of the CFRP material us-ing FEMM simulations together with measurements made with an LCR meter. Theequivalent electrical resisitivity is defined as the resistivity on a macroscopic level,i.e., the resistivity of a homogeneous and isotropic material. The CFRP materialis in reality non-homogeneous and anisotropic where the resistivity is direction-dependent. The method used in this work was suggested and validated in [Lund-ström et al., 2018]. By measuring the difference in resistance and inductance whenthe coil is loaded (CFRP is present), l, and when the coil is non-loaded (CFRP isabsent), nl, one can obtain a measured difference in impedance due to the pres-ence or absence of the CFRP material, see Equations (3.1) - (3.3). This is doneat room temperature for a number of frequencies, N f . Simulations are then done

14

3.2 Equivalent Electrical Resistivity Computation

Figure 3.1 The inductor and CFRP workpiece modelled in FEMM. The modelis axisymmetrical around the left edge of the image. The figure illustrates the fluxdensity.

at the corresponding frequencies using different resistivity values. By minimizingthe goal function (3.4) it is then possible to obtain a resistivity value of the CFRPmaterial such that the difference between simulation, s, and measurements, m, areminimized.

The goal function in this work was minimized for the frequencies 20, 40, 60,80 and 100 kHz. The accuracy of the LCR meter decreases with higher frequencieswhy 100 kHz was the maximum frequency used. The CFRP sheet the measurementswere made on is simulated as a circular sheet with a diameter of 100 mm, which inreality was a rectangular sheet of about double the size of the simulated CFRP sheet.

∆R = Rl−Rnl (3.1)

∆L = Ll−Lnl (3.2)

15

Chapter 3. Electrical Properties and Material Characterization

∆Z = ∆R+ j2π f ∆L (3.3)

N f

∑n=1

(|∆Zm( fn)−∆Zs( fn)|)2 (3.4)

3.3 Temperature Dependence of Electrical Properties

Resistance and InductanceAs mentioned in Section 1.2, the resistance along with the inductance were sus-pected to change during the welding process. Therefore, LCR measurements weremade to investigate the temperature dependence of those properties. The fact thatthe inductor is used both for heating and measurement is somewhat of a problem.The measurements cannot be made while the coil is in use. Furthermore, to performthe measurements the coil needs to be isolated from the rest of the system, i.e., thecable and frequency inverter. The measurements were therefore carried out in stepswhere the temperature was increased to a certain level followed by the disconnec-tion of the inductor. The LCR meter probes were then attached for measurementsto be made. The procedure described was repeated seven times at different tem-peratures with four measurement points taken each time. After the last temperaturestep the LCR meter was left on and measurements were made until the temperaturehad reached room temperature. This experiment was carried out twice on the samesample to study the effect of rewelding the material.

The cable is normally connected to the inductor with screws. A plug connectorwas attached to the cable and the inductor for the purpose of this experiment to re-duce the time it would take to disconnect the cable to make the LCR measurements.

The LCR meter is quite sensitive to how it is connected to the coil and howcables are placed etc. The LCR meter was therefore connected and disconnectedfive times at room temperature before the reheating experiment was performed tosee how much the readings would vary.

Electrical ResistivityThe measured impedance values from the initial welding experiment were used todetermine how the electrical resistivity is changing with temperature. This was doneby minimizing the difference between measured and simulated impedance, similarto the process described in Section 3.2, except measurements were taken at onefrequency and with the coil being loaded only.

Coil Temperature InfluenceSince the resistance and inductance measurements are made on the coil, the temper-ature dependence of the inductor itself must be taken into account. The change in

16


coil temperature will alter the resistivity of the coil. Copper has a temperature coeffi-cient, α , of 0.004 [1/°C], and a resistivity value at room temperature of 0.017 µΩm[Rudnev et al., 2017]. The temperature of the coil was measured using a thermalcamera. Due to the low emissivity of the copper the camera will measure the re-flected temperature instead of the temperature of the object of interest. Therefore,a part of the coil was painted with a graphite coating (black) to increase the emis-sivity in order to obtain a correct temperature reading. The frequency inverter wasrun until a stationary coil temperature was reached. The time it took for the temper-ature to return to room temperature was then measured. The possible effects of thecoil temperature on the results of the LCR measurements could then be taken intoaccount by comparing the time it takes for the coil to reach room temperature withthe time it takes to disconnect the inductor and attach the measurement probes.

17

4Signal Analysis

There are several electrical properties measured and logged by the frequency in-verter unit along with the temperature measured by the thermocouple. Since thetemperature is the signal to be estimated it will be chosen as the model output sig-nal. Which signal(s) to be used as the input signal(s) in the model to be constructedis not obvious though. As mentioned in Section 1.2, many of the electrical proper-ties are temperature dependent why these might provide information useful in esti-mating the temperature. The increasing temperature during the welding process isexpected to result in a dislocation of the current-frequency curve seen in Figure 1.1.When running the process on a fixed frequency this displacement could be detectedin the signals of the measured electrical properties. Previous work suggests that theCFRP resistivity temperature coefficient, α in Equation (1.5), is slightly negative inthe linear region, i.e., at temperatures below that when phase transformation occurs[Lin, 2018]. This indicates that up to the point of phase transformation no significantchanges will occur as a result of changing material properties of the CFRP.

What happens to the electrical signals at the melting point temperature andabove is not known. However, more dramatic changes are expected due to the phasetransformation altering the electrical resistivity of the CFRP. As mentioned in Sec-tion 1.2, a change in electrical resistivity will affect the resistance and inductancein the system which will alter the impedance. A change in current is to be expectedprovided that the voltage stays constant according to Equation (1.10), along with achange in phase value due to the altered resistance and inductive reactance, accord-ing to Equation (1.9).

4.1 Available Signals

The electrical properties available that might be of interest are the measured fre-quency, current RMS value, rail voltage and phase value. The current RMS valuewas measured at the primary side of the transformer, seen in Figure 2.1, the railvoltage is the voltage source seen in Figure 2.1 and determines the maximum am-plitude of the voltage as seen in Figure 2.2. The phase value is a value given by the

18

4.2 Experiments

frequency inverter to monitor the phase shift, or phase angle, between the voltageand current. The phase value sensor is known to be nonlinear in its region and alsohas an initial period of about 100 ms when the output is not usable. Furthermore, itis not intended for any exact phase angle readings but is rather used as an indicationof how close to resonance the system is. The frequency inverter logs a value be-tween 0 and 4095 where 0 indicates that current and voltage are in phase meaningthat the system is at the resonance frequency. A flag shows whether the system isabove or below resonance. When the flag is positive a higher value means a moreinductive circuit and likewise when the flag is negative a higher value indicates amore capacitive circuit. Since the system is to be run at frequencies above resonancethe circuit will always be mainly inductive, thus, a higher phase value will alwayscorrespond to a more inductive circuit in this work. The default logging frequencyis 1 Hz which was increased to 10 Hz for the purpose of this study in order to gatherenough data when running the system at frequencies closer to resonance.

4.2 Experiments

To gain some knowledge about the system characteristics and the behavior of thedifferent electrical signals during the welding process a number of experiments werecarried out. The processing temperature of the CFRP material is 370-400°C whymost experiments will not exceed 400°C by any large extent.

Step ResponseStep response experiments were performed to investigate how the temperature andthe electrical signals measured by the frequency inverter vary over time. These ex-periments were performed for a number of frequencies. As previously mentioned,experiments run with a fixed frequency should make the possible dislocation of thecurrent-frequency curve (Figure 1.1) observable by a change in current along witha changing phase value.

Multiple Samples ComparisonA series of tests were performed on a few different samples to investigate whetherthe welding process would be consistent for all samples. A fixed frequency of640 kHz had shown to give a suitable heating rate and was used in these experi-ments.

Multiple Cycles on SampleSince the electrical properties are to be used as input signals in a model it is of in-terest to see whether these properties change and if the constructed model can beexpected to be valid when reheating a previously welded part. An experiment was

19

Chapter 4. Signal Analysis

conducted consisting of several high-temperature cycles to investigate the correla-tion between the phase value and temperature. The temperature in this experimentwas repeatedly brought up to around 400°C before dropping to below 300°C.

Another experiment was conducted where a sample was welded and reheatedtwice to make an investigation as to how the welding process alters the electricalsignals when the temperature is brought down to room temperature between eachcycle. This experiment was performed again on a different sample to investigatewhether the results were repeatable.

20

5System Identification

To achieve the goal of eliminating any traditional temperature sensor from the in-duction welding process, such as pyrometers or thermocouples, a model used fortemperature estimation will be constructed. This can either be done by deriving themathematical differential equations from physical relationships, also known as awhite-box model approach, or by observation of input-output data from the system.When constructing a system model using observations of input-output data there aretwo types of approaches: gray-box models and black-box models. Gray-box mod-els are used when the system can be described by a set of differential equations andinput-output data is used to estimate the equation coefficients. Black-box modelson the other hand, are models created solely from the observation of input-outputdata. A generic model structure is chosen and data from experiments is then used toestimate the model structure parameters to fit the measurement data.

As mentioned in Section 1.1, not much is known about the electrical proper-ties at elevated temperatures. Therefore, along with the complexity of the system awhite-box model is disregarded. A gray-box model would perhaps be viable sinceprevious work has shown that models can be constructed from a set of equationssolved using FEM. As previously mentioned, the validity of those at elevated tem-perature is uncertain. The most crucial part of the welding process occurs at elevatedtemperatures, more specifically, when the temperature approaches the melting pointand changes to the known system properties will occur. Some parts of the systemssuch as the enthalpy of fusion could perhaps be described though. Regarding black-box models, they require no previous knowledge about the system and the sys-tem model is created by the observation of collected experimental input-output dataonly. This is the easiest model structure of the three to use and will be the startingpoint in the system identification of this system. MATLAB® System IdentificationToolbox™ [MathWorks, 2020b] will be used in the system identification part of thiswork.

21

Chapter 5. System Identification

5.1 The System Identification Process

The system identification process approach is to a large extent an iterative trial-and-error process, see Figure 5.1. After stating the purpose of the model, which inthis case is to construct a model to be used for real-time temperature estimation,several steps are performed which often have to be revisited multiple times duringthe process. The first step in black-box identification is to plan and conduct exper-iments. The data from these experiments is then examined and if the data is goodthe next step is to select a model structure, otherwise new experiments will haveto be performed. Using the selected model structure, a model is estimated and thenvalidated. If the constructed model is sufficient the system identification process iscompleted, if not, some or all steps of the system identification process will have tobe performed again.

Figure 5.1 Black-box system identification workflow. Adapted from [Anderssonet al., n.d.]

22

5.2 Data Acquisition

5.2 Data Acquisition

As mentioned in Section 5.1 the approach taken in this work is the black-box mod-elling approach which means the identification process is data driven. The frequencyinverter unit logs the data on an SD card including current, voltage, phase value andtemperature values which will be used as input and output data. However, the fre-quency inverter unit is not designed with the intent of high-frequency logging ofdata. To save memory, only changes to parameters are logged, resulting in a possi-ble uncertainty whether "missing" data is due to an error with the data logging orbecause no change of the parameter in question occurred. Furthermore, the sampletime is at times drifting somewhat according to the data log. The data used in theSystem Identification Toolbox™ is required to be uniformly sampled why the datais interpolated to get data points with uniform time interval according to the sampletime used.

5.3 System Models

There are several types of black-box models available in the System Identifica-tion Toolbox™. They can be divided into linear and nonlinear models. Althoughthe system is expected to show a quite nonlinear behavior it is generally a goodidea to try a few different linear models to see if any linear model will suffice. Ifthe linear models turn out to not being able to describe the system in a satisfy-ing way a nonlinear model will have to be constructed. The System IdentificationToolbox™ provides two types of nonlinear black-box models: nonlinear ARX mod-els and Hammerstein-Wiener models.

Nonlinear ARX ModelThe nonlinear ARX model is an extension of the linear ARX model. The linearARX model of a SISO system with zero input delay (nk = 0) has the structure seenin Equation (5.1), where t is the time index, y is the output signal, u is the inputsignal, e is the noise, na and nb is the number of past output and input, respectively,used to predict the current output. The current output, y(t), is thus computed as aweighted sum of the regressors, i.e., current input, past inputs and past outputs. Bycollecting the weighting coefficients in one vector and the regressors in another,Equation (5.1) can be rewritten as seen in Equation (5.2).

y(t)+a1y(t−1)+a2y(t−2)+ · · ·+anay(t−na) =

b1u(t)+b2u(t−1)+ · · ·+bnbu(t−nb+1)+ e(t) (5.1)

y(t) = [−a1,−a2, ...,−ana,b1,b2, ...bnb]·[y(t−1),y(t−2), ...,y(t−na),u(t),u(t−1), ...,u(t−nb+1)]T (5.2)

23


In a nonlinear ARX structure the linear mapping is replaced by a more flexiblenonlinear mapping function, F, see Equation (5.3). The structure of the nonlinearARX model is visualized as a block diagram in Figure 5.2. The output is determinedby first computing the regressor values from current input values as well as pastinput and output values. It then maps the regressors to the model output using thenonlinearity estimator block containing a linear function in parallel with a nonlinearfunction. The nonlinearity estimator block is given by Equation (5.4), where LT (x−r)+d is the linear function output and g(Q(x− r)) is the output from the nonlinearfunction. x is the regressor vector, r is the mean value of the regressors and d isa scalar offset. The form of g(x) is determined by the nonlinearity estimator used.Q is a projection matrix making the calculations well conditioned [MathWorks,2020d]. The nonlinear estimators used in this work are wavelet network and sigmoidnetwork.

y(t) = F(y(t−1),y(t−2), ...,y(t−na),u(t),u(t−1), ...,u(t−nb+1)) (5.3)

F(x) = LT (x− r)+d +g(Q(x− r)) (5.4)

Figure 5.2 Nonlinear ARX model structure [MathWorks, 2020d].

Hammerstein-Wiener ModelThe Hammerstein-Wiener model consists of a discrete-time linear transfer functionwith a nonlinearity on its input (Hammerstein) and another on its output (Wiener),see Figure 5.3. Different types of input and output nonlinearity estimators can beused. Nonlinearity estimators used in this work are wavelet network, sigmoid net-work and piecewise linear function. The nonlinear blocks are static, i.e., memory-less, while the linear block represents the dynamic part of the system. The Systemidentification Toolbox™ computes the output signal, y(t) in three steps. First the in-ternal variable w(t) is calculated from the input data according to Equation (5.5).This variable is then used as input to the linear transfer function B/F with the out-put calculated according to Equation (5.6). The model output signal y(t) is thencalculated as shown in Equation (5.7) [MathWorks, 2020c].

24

5.4 Model Estimation and Validation

Figure 5.3 Hammerstein-Wiener model structure [MathWorks, 2020c].

w(t) = f (u(t)) (5.5)

x(t) = (B/F)w(t) (5.6)

y(t) = h(x(t)) (5.7)

5.4 Model Estimation and Validation

The data used in the system identification consists of data from experiments per-formed on a number of different CFRP samples presented in Section 6.4. Figure 6.6and Figure 6.7 show how the raw data from an experiment run at 640 kHz typicallylooks. The data was then low-passed filtered to prevent aliasing before downsam-pling to achieve a sampling frequency of 0.5 Hz. This was done to achieve a suitablesampling frequency in relation to the dominating time constant of the system in thefollowing system identification process. An example of how the input-output datalooks can be seen in Figure 6.13.

The estimation data used consists of data from experiments on Samples 1 and3 seen in Figure 6.8 with corresponding input signals which are seen filtered usingmoving average in Figure 6.9. The validation data consists of data from other ex-periments including experiments on Samples 2,4 and 5 also seen in Figure 6.8 andFigure 6.9. All four input signals could potentially be used in the system identifica-tion process though the current is perhaps the most suitable signal to look at sinceit is easily measured and has a clear relationship with the output power and thus thetemperature in the workpiece.

With the current as the input signal and temperature as the output signal, differ-ent model structures were selected, estimated and validated beginning with linearmodels. An error of±20°C in the process temperature region of 370 - 400°C is con-sidered good enough in the identification process of this work, that is in line withother methods. For a real application ±10°C or better would be preferred [Frogner,2020].

The initialization of the model can have a great impact on the data fit when com-paring the model output against the measured temperature. The initial conditions ofthe model can be estimated to present the model’s best possible fit to the workingdata. The model to be implemented in the process will have to be functioning inreal-time. That means that an estimation of the initial conditions to improve the fit

25


to the measured data cannot be done. There are a few possible ways to go aboutthis issue. One option is to set the initial conditions to zero and start all experimentsand simulations from rest, i.e., from room temperature. Other ways would be to useinput-output data measured immediately prior to the start of the simulation to es-timate the initial conditions. That is not possible in this case since no output datawill be available for measurement in a real scenario. One could also use state in-formation from previous simulations for initialization [MathWorks, 2020a]. It canbe assumed that the welding for this process will be carried out starting at roomtemperature why the initial conditions can be set to zero. That is the method thatwill be used in this work.

Regarding the overall strategy, there are two possible options. The first one be-ing the use of a comprehensive model that is valid for all operating frequencies.Another strategy that could prove successful would be to create a set of less com-plex models, each valid for a smaller frequency interval, and then change betweenthese depending on the frequency used. With the strategy to construct several lesscomplex models there might be simplifications to be made of the model structureused. The second option is the strategy used in this work.

5.5 Model Implementation

To test if the output of a model within the Simulink environment is reproducibleon the real process, C code was generated and implemented on the microcontrollerrunning on the system. The code was generated using MATLAB Embedded Coder®

for use on an embedded target. Tests were then carried out where the temperatureestimated by the constructed model was added to the frequency inverter log whichwas then compared to the logged temperature measured by the thermocouple.

26

6Results

6.1 Equivalent Electrical Resistivity

Using the method described in Section 3.2, a resistivity value of about 100 µΩmwas obtained (equivalent to a conductivity value of 10 kS/m). The difference be-tween simulated and measured resistance and inductance values can be seen in Fig-ure 6.1. A decent fit is achieved although there is an increasing discrepancy betweenmeasured and simulated resistance with increasing frequency.

20 30 40 50 60 70 80 90 100

Frequency [kHz]

0

0.01

0.02

0.03

0.04

0.05

0.06

Resis

tance [

]

0

0.2

0.4

0.6

0.8

1

1.2

1.4In

ducta

nce [H

]10

-6e = 100m

Rl,s

- (Rnl,s

- Rln,m

)

Rl,m

Ll,s

- (Lnl,s

- Lln,m

)

Ll,m

Figure 6.1 Measured (solid lines) and simulated (dash-dotted lines) resistance andinductance values for the frequencies 20-100 kHz. The simulated values have beencompensated for with respect to the difference between the simulated and measurednon-loaded coil.

27

Chapter 6. Results

6.2 Power Loss Simulation

Initial experiments showed that a frequency of 640 kHz resulted in a suitable heatingrate. The current flowing through the inductor at this frequency has a peak value ofabout 80 A. Using this value, along with the frequency as input to the FEMM soft-ware, the losses in the workpiece was simulated according to the method presentedin Section 3.1. As seen in Figure 6.2 the majority of the losses is concentrated in thelayers closest to the inductor at a point roughly 10-15 mm from the center. This cor-responds well with the circular pattern seen in the thermal image Figure 2.7 (note:the inductor is placed beneath the workpiece in Figure 2.7 and above the workpiecein Figure 6.2).

Figure 6.2 Cross-section of the workpiece illustrating the simulated power lossesin the workpiece with the inductor placed above the workpiece. The horizontal linemarks the intersection between the two CFRP plates.

.


Resistance and InductanceThe method described in Section 3.3 was used to measure the resistance and induc-tance at different temperatures. Even with the plug connector in place, the tempera-ture dropped about 50°C before measurements could be made after each increase intemperature. The maximum temperature reached before each measurement groupwas taken is seen in Table 6.1. The results are marked with circles in the left imagein Figure 6.3. Measurements in Groups 1-5 are made when the material has not yetreached the melting point, Group 6 has reached the melting point (343°C), but notthe processing temperature (370-400°C). Group 7 has reached a temperature justabove the processing temperature. After the CFRP had been heated to 404°C (thelast group) the LCR meter was left on and measurements were taken during coolingto around 26°C. These measurements are marked with the dash-dotted lines.

It should be noted that the LCR meter can be quite sensitive regarding wireplacement and contact between the measurement probes and the surface of the cop-per tube. This might result in reading discrepancies when reconnecting the LCRmeter. The range of the resistance values when connecting and reconnecting themeasurement probes at room temperature, as described in Section 3.3, was roughly4 mΩ. The corresponding value for the inductance was 0.0164 µH. However, theresults show that the resistance is clearly decreasing with increasing temperature.

28


Table 6.1 Maximum temperature reached before measurement.

Group Max. temp [°C]1st welding

Max. temp [°C]reheating

1 132 1242 178 1703 219 2074 264 2605 304 3006 360 3607 404 400

0 50 100 150 200 250 300 350

Temperature [°C]

190

195

200

205

210

215

Re

sis

tan

ce

[m

]

1.13

1.14

1.15

1.16

1.17

1.18

1.19

Ind

ucta

nce

[H

]

Resistance and Inductance Temperature Dependence

0 50 100 150 200 250 300 350

Temperature [°C]

190

195

200

205

210

215

Re

sis

tan

ce

[m

]

1.13

1.14

1.15

1.16

1.17

1.18

1.19

Ind

ucta

nce

[H

]

Resistance and Inductance Temperature Dependence

Figure 6.3 Resistance and inductance temperature dependence of coil plus CFRP.Left: first welding. Right: reheating.

The inductance stays more or less constant initially, though Group 5 is somewhat ofan outlier, but seems to be increasing at higher temperatures. Furthermore, it can beseen that the resistance after welding is lower than before welding, while the induc-tance after welding is higher than before welding. The results indicate that the phaseangle is increasing during welding, see Equation (1.9), and the resonance frequencymight be decreasing somewhat, see Equation (1.11).

The same experiment was carried out again on the same sample to examine theeffect on the resistance and inductance. This experiment was performed on anotheroccasion when the cable placement and inductor-workpiece placement might havebeen slightly different compared to the first experiment. This makes comparisons ofabsolute values between the experiments problematic. However, the overall patternof the two experiments can still be compared. The maximum temperature reachedduring this experiment before each measurement group was taken can be seen inthe rightmost column in Table 6.1. The result of this experiment can be seen in theimage to the right in Figure 6.3. The result is quite different compared to the re-sult obtained during the first experiment. The initial values of both the resistance

29

Chapter 6. Results

and inductance are lower than the values from the first experiment. Furthermore,the overall pattern in the measurements made for Groups 1 to 7 is not the same forthe first welding and the reheating. The resistance seems to be decreasing at highertemperatures in both cases though. In the first experiment, the resistance seems tobe decreasing across the whole temperature range, with a greater rate at higher tem-peratures. However, in the second experiment, the resistance seems to initially beincreasing before decreasing. These changes are not very large though, comparedto the first experiment. The inductance values do not show much similarity betweenthe two experiments other than that there seems to be an increase at higher temper-atures.

What these experiments show is that sometime during the welding there seemsto be a point where the resistance will show a clear decline regardless if it is thefirst or second cycle. If it is the first welding the decline can be expected to begreater with a long duration whereas if the workpiece has been previously weldedthe decline in resistance will be smaller with a shorter duration.

Electrical ResistivityBy minimizing the difference between measured and simulated impedance as de-scribed in Section 3.3, it is shown that the electrical resistivity is increasing at highertemperatures, see Figure 6.4.

50 100 150 200 250 300 350

Temperature [°C]

-5

0

5

10

15

20

25

30

35

Resis

tivity C

hange [%

]

Resistivity Temperature Dependence

Figure 6.4 Temperature dependence of electrical resistivity using one measure-ment from each group seen in the left image in Figure 6.3.

Coil Temperature InfluenceThe influence of the coil temperature was investigated according to the methoddescribed in Section 3.3. The frequency inverter has been running for about 10 s

30

6.4 Signal Analysis

and the coil has reached a stationary temperature level of 33°C in the image to theleft in Figure 6.5. The electrical resistivity of copper at 33°C is about 0.018 µΩmaccording to Equation (1.5) compared to 0.017 µΩm at 22°C. This increase in coiltemperature will push down the phase angle during the first 10 s of operation andthen stabilize according to Equations (1.13) and (1.9).

The image to the right in Figure 6.5 shows the coil temperature about 8 s afterthe frequency inverter has been then stopped and the coil temperature has droppedfrom 33°C and has reached the room temperature of 22°C. 8 s is shorter than theminimum time it takes from stopping the frequency inverter until the first resistanceand inductance measurements can be taken according to the experimental proceduredescribed in Section 3.3. The results presented in Figure 6.3 are thus not affectedby any temperature dependencies of the coil.

Figure 6.5 Left: coil temperature during heating. Right: coil temperature 8 s afterstopping. The cursor marks the point on the coil where the temperature is measured.

6.4 Signal Analysis

Step ResponseThe temperature in the weld-zone during a step response experiment can be seenin Figure 6.6 and the corresponding electrical signals can be seen in Figure 6.7filtered using a moving average (MA) filter with a window of 15 data points. Thecurrent which was expected to show changes during the welding process stays quiteconstant, there is perhaps a small decline after 10 s. The phase value on the otherhand shows an interesting pattern. Since the phase value is increasing there is eithera decreasing resistance in the system and/or an increasing inductance. However, itshould be noted that the changes are very small.

31

Chapter 6. Results

0 5 10 15 20 25 30 35

Time [s]

0

50

100

150

200

250

300

350

400

450

Tem

pera

ture

[°C

]

Measured Temperature

Figure 6.6 Measured temperature during welding at 640 kHz.

0 5 10 15 20 25 30 35

Time [s]

639.2

639.3

639.4

639.5

639.6

639.7

Fre

quency [kH

z]

Measured Frequency

logged data

MA15

0 5 10 15 20 25 30 35

Time [s]

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

Cu

rre

nt

[A]

Current RMS Value

logged data

MA15

0 5 10 15 20 25 30 35

Time [s]

525

525.5

526

526.5

527

527.5

528

528.5

529

Vo

lta

ge

[V

]

Rail Voltage

logged data

MA15

0 5 10 15 20 25 30 35

Time [s]

1690

1695

1700

1705

1710

1715

1720

1725

1730

Phase V

alu

e

Phase Value

logged data

MA15

Figure 6.7 MA15 values for measured frequency, current RMS value, rail voltageand phase value for the welding process in Figure 6.6.

Multiple Samples ComparisonThe temperature responses of a series of step response experiments on multiple sam-ples can be seen in Figure 6.8 with corresponding electrical signals in Figure 6.9.The time it takes for the different samples to reach 400°C varies somewhat, as can be

32

6.4 Signal Analysis

0 5 10 15 20 25 30 35

Time [s]

0

50

100

150

200

250

300

350

400

Te

mp

era

ture

in

cre

ase

[°C

]

Temperature Comparison of Different Samples

sample 1

sample 2

sample 3

sample 4

sample 5

Figure 6.8 Temperature relative to room temperature for five different CFRP sam-ples.

0 5 10 15 20 25 30 35

Time [s]

639.4

639.6

639.8

640

640.2

Fre

qu

en

cy [

kH

z]

Frequency Comparison of Different Samples

sample 1

sample 2

sample 3

sample 4

sample 5

0 5 10 15 20 25 30 35

Time [s]

1690

1695

1700

1705

1710

1715

1720

1725

1730

Ph

ase

va

lue

Phase Value Comparison of Different Samples

sample 1

sample 2

sample 3

sample 4

sample 5

0 5 10 15 20 25 30 35

Time [s]

8.5

9

9.5

10

10.5

11

Cu

rre

nt

[A]

Current Comparison of Different Samples

sample 1

sample 2

sample 3

sample 4

sample 5

0 5 10 15 20 25 30 35

Time [s]

520

522

524

526

528

530

532

534

Vo

lta

ge

[V

]

Rail Voltage Comparison of Different Samples

sample 1

sample 2

sample 3

sample 4

sample 5

Figure 6.9 MA10 values for measured frequency, phase value, RMS current andrail voltage for the five CFRP samples in Figure 6.8.

seen in Figure 6.8, even though the same frequency and duty cycle reference valuesare set on the frequency inverter. The overall pattern of the phase value displayed inFigure 6.7 can be seen in all five samples, though the values are not the same. The

33

Chapter 6. Results

data indicates that there might be a noticeable difference between samples.

Multiple Cycles on SampleTwo types of experiments were performed containing multiple cycles, as describedin Section 4.2. The temperature response of a number of welding cycles at hightemperature on the same sample can be seen in Figure 6.10. The results from thistest showed that the phase value behavior found in previous tests is present only inthe first ramp up, thereafter it changes.

0 200 400 600 800 1000 1200 1400

Time [s]

0

50

100

150

200

250

300

350

400

450

Tem

pera

ture

[°C

]

Multiple High Temperature Cycles

Figure 6.10 Multiple cycles at high temperatures. The first three cycles are run at640 kHz, the following cycles are run at a variety of frequencies.

The results of the other experiment type where the temperature is starting fromroom temperature each cycle can be seen in Figure 6.11 and Figure 6.12. The initialwelding (blue line) has a shorter temperature rise time than cycles 2 (red line) and 3(yellow line) where the sample is being reheated. The phase value, see Figure 6.12,displays a change in behavior with a higher starting value for each cycle along witha different pattern between the first welding and the following cycles.

This experiment was performed again on another sample which also showed afaster heating rate for the initial welding compared to the following cycles althoughit did not display the same increase in initial phase value between cycles. How-ever, the difference in the shape of the phase value curves between initial weldingand reheating seen in Figure 6.12 could be seen also in this experiment. The phasevalue during the first cycle covers a wider range, while the reheating shows smallerchanges.

34

6.4 Signal Analysis

0 5 10 15 20 25 30 35 40

Time [s]

0

50

100

150

200

250

300

350

400

Tem

pera

ture

Incre

ase [°C

]

Temperature Multiple Cycles

cycle 1

cycle 2

cycle 3

Figure 6.11 Temperature curves from three cycles on the same CFRP sample.

0 5 10 15 20 25 30 35 40

Time [s]

639.5

639.55

639.6

639.65

639.7

639.75

639.8

639.85

Fre

qu

en

cy [

kH

z]

Measured Frequency Multiple Cycles

cycle 1

cycle 2

cycle 3

0 5 10 15 20 25 30 35 40

Time [s]

1695

1700

1705

1710

1715

1720

Ph

ase

Va

lue

Phase Value Multiple Cycles

cycle 1

cycle 2

cycle 3

5 10 15 20 25 30 35 40

Time [s]

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

Cu

rre

nt

[A]

Current Multiple Cycles

cycle 1

cycle 2

cycle 3

0 5 10 15 20 25 30 35 40

Time [s]

520

521

522

523

524

525

526

527

Vo

lta

ge

[V

]

Rail Voltage Multiple Cycles

cycle 1

cycle 2

cycle 3

Figure 6.12 MA10 values for measured frequency, phase value, RMS current andrail voltage of the three cycles shown in Figure 6.8.

35

Chapter 6. Results

Analysis ConclusionThe signal analysis has shown that there is a noticeable difference between samplesand that reheating of a sample will give different results compared to the first weld-ing. The signal analysis also showed that the phase value might be a useful signal asit indicates how the impedance is changing with temperature. The fact that a certainphase value does not correspond to a certain temperature and has a varying behav-ior between welding cycles was not unexpected but worth noting. The phase valuecould still be of interest since the initial welding is of interest and not so much anyfollowing cycles, and the phase value pattern displayed might provide informationthat could help estimate the weld-zone temperature.

6.5 System Identification

The system identification was performed according to the method described in Sec-tion 5.4. An example of how the input-output data looks can be seen in Figure 6.13.As expected, no linear model gave results that were good enough why these werediscarded and focus shifted to nonlinear models. The linear models struggled athigher temperatures when a temperature increase of about 300°C was reached. Withthe linear models proving insufficient the nonlinear ARX model structure and theHammerstein-Wiener model structure were tried.

0 5 10 15 20 25 30 35 40

0

200

400

Tem

p. in

cre

ase [°C

] Input and output signals

0 5 10 15 20 25 30 35 40

Time [s]

0

5

10

Curr

ent [A

]

Figure 6.13 Current vs temperature increase, starting from room temperature.

Since the model is to be run at a fixed frequency, the changes to the input sig-nal will be relatively small why the input signals was approximated to be linear.Therefore, the input nonlinearity of the Hammerstein-Wiener model was removed,making the model a Wiener model. Two different types were created that showed

36


promising results; a nonlinear ARX model and a Wiener model. Their configura-tions can be seen in table Table 6.2 and Table 6.3. na and nb is the number of pastoutput and inputs respectively used in the ARX model. In the Hammerstein-Wienermodel nb is the number of zeros plus one of the linear transfer function, i.e., thelength of the numerator polynomial (B) and nf is the number of poles, i.e., the orderof the denominator polynomial (F). nk is the input delay expressed as the numberof samples for both model types.

Table 6.2 Constructed nonlinear ARX model.

current Temperaturena - 1nb 3 -nk 1 -

nonlinearity wavelet 6 units

Table 6.3 Constructed Wiener model.

current Temperaturenb 1 -nf 3 -nk 1 -

nonlinearity - wavelet 7 units

0 5 10 15 20 25 30 35 40

Time [s]

0

50

100

150

200

250

300

350

400

Tem

pera

ture

incre

ase [°C

]

Measured and simulated model output

Nonlinear ARX

Wiener

Measured temperature

Figure 6.14 Comparison between nonlinear ARX model and Wiener model.

37

Chapter 6. Results

As can be seen in Figure 6.14 both models give good results, with the Wienermodel giving a slightly better overall fit to the measured data. Six unique sets ofvalidation data were used to validate the models. The two models perform similarresults although the Wiener model gives the better fit in the majority of the tests.The chosen model is therefore the Wiener model.

The constructed model’s transfer functions can be seen in Equation (6.1). z isthe time shift operator such that z−1yk = yk−1. Corresponding pole-zero plot can beseen in Figure 6.15. The nonlinear estimator is visualized in Figure 6.16.

HI(z) =z

z3−1.1256z2 +0.3503z−0.0666(6.1)

Figure 6.15 Pole-zero plot of the linear transfer function of the Wiener model.Poles are marked "x", zeros are marked "o".

38


Figure 6.16 Nonlinear estimator of the Wiener model.

39

Chapter 6. Results


So far, all work has been done within the MATLAB® environment, using datalogged on the SD card for both estimation and validation. The implementation onthe process was done according to Section 5.5. The output from the implementedmodel on the microcontroller shows good agreement with the simulated model asseen in Figure 6.17. This confirms that the model implementation is done correctlyand the results obtained from simulations can be assumed to be equivalent to theresults obtained from the real system. Note that the model in Figure 6.17 is not thechosen model presented in Section 6.5.

5 10 15 20 25 30 35

Time [s]

0

50

100

150

200

250

300

350

400

Te

mp

era

ture

in

cre

ase

[°C

]

Model Output Comparison Microcontroller and Simulink

Model output, Microcontroller

Model output, Simulink

Figure 6.17 Model output comparison between the model running on the micro-controller and the equivalent simulation in Simulink.

Weld InspectionAs mentioned in Section 2.6, the quality of the weld can be examined when thecross-section is viewed in a microscope. The left image in Figure 6.18 shows theresulting weld about 10 mm from the center. No distinct weld line can be seen,which indicates that the weld is good. The right image in Figure 6.18 is taken at thecenter of the workpiece, i.e., the area beneath the center of the inductor. The resultof too low temperature can be seen since the interface between the two CFRP platesis clearly visible. This is the part of the workpiece that is located beneath the centerof the coil where there is a cold spot, as seen in Figure 2.7 and Figure 6.2. Theparticular sample seen in Figure 6.18 reached a weld-zone temperature of 408°C,which is a bit high.

40


Figure 6.18 Images taken with microscope of two welded CFRP plates. Left: theworkpiece about 10 mm from the center, corresponding to location marked "A" inFigure 6.19. Right: cold spot in the center of the workpiece, corresponding to loca-tion marked "B" in Figure 6.19. Image height is about 4.5 mm, workpiece thicknessis about 5 mm.

Figure 6.19 Illustrative figure of cross-sections of coil and workpiece marking thelocations of images in Figure 6.18.

41

7Discussion

7.1 Electrical Properties

The experiments where the temperature dependence of the resistance and induc-tance were measured, see Figure 6.3, were not performed in the most sophisticatedmanner. The time it takes to disconnect the cable to perform the LCR measurementswould ideally have been shorter to prevent the temperature from dropping to the ex-tent it did. To collect more data points over a larger temperature region a methodthat would not include any manual disconnection of the cable would be advisable,e.g., a relay switching the connection of the coil between a measurement circuit andthe frequency inverter. Furthermore, the connecting and reconnecting of the LCRmeter affected the results to some extent. However, the overall changes to the re-sistance, in particular, were large enough to be caused by actual changes and notmeasurement errors. Thus, the results obtained from the experiment conducted inthis work are still sufficient to draw conclusions regarding the overall change of theresistance.

It has been found that the resistance and inductance of the CFRP change withtemperature. The results indicate that the resistance is decreasing with increasingtemperature while the inductance seems to be increasing, as seen in Figure 6.3.This suggests that the resistivity is increasing according to the FEMM simulationsmade, see Figure 6.4. The fact that the resistance is decreasing while the resistivityis increasing can seem unintuitive when looking at Equation (1.13). However, theeffective cross-sectional area of the workpiece is both temperature and frequencydependent, why the resistivity is not proportional to the resistance.

The increasing inductance, also seen in Figure 6.3, indicates that there mostlikely is a dislocation of the current-frequency curve, but this change is extremelysmall. The resonance frequency is decreasing with an increasing inductance, seeEquation (1.11), which could yield a decreasing current. However, the decreasingresistance counteracts this. When looking at the current in Figure 6.7, for example,the current is not displaying any major changes. After about 10 seconds, the currentseems to be decreasing slightly which would correspond with the current-frequencycurve moving to the left. Again, these changes are very small.

42

7.2 Difference Between Samples

7.2 Difference Between Samples

The results presented in Section 6.4 showed that the time it takes for the differ-ent samples to reach 400°C varies somewhat. That could be the result of a numberof things; the thermocouples might have been displaced during the set up, the fre-quency output power might have varied, or the material properties or thickness ofthe CFRP plates are slightly different causing different heating rates.

The overall pattern of the phase value displayed in Figure 6.9 can be seen inall five samples, though the values are not the same. The phase value is very muchaffected by the frequency which could explain some of the differences in phasevalue between the different samples. By looking at Samples 2 and 4, which have asimilar measured frequency, there is a clear offset between the phase value curves.If all system properties of these two samples were the same the phase values shouldideally be the same at the same frequency. The results show that this is not the case.Thus, the phase values measured during welding of the different samples indicatethat the impedance of the different CFRP samples is not the same. This variationbetween samples is the reason why feedback is necessary to ensure the correct weld-zone temperature.

As previously mentioned, the phase value logged by the frequency inverter is notintended for any exact phase angle measurements. This makes it hard to interpretthe significance of the difference in phase value between samples. The possibilitythat the difference is a result of altercations of other parts of the system such as ca-ble placement etc., cannot be ruled out. Important to consider is the uncertainty inthe accuracy and precision in the measurement of the phase value. Regardless, thephase value has proven to be a useful signal to measure. In future work, a methodfor sensing the phase angle with greater accuracy and precision would be of greatinterest. An alternative to measuring the phase angle, which can be hard to accu-rately measure, would be to measure the other electrical properties that give rise tothe phase shift.

7.3 Input Signal

The chosen signals to use as input to the constructed models have to a large extentbeen limited by the current design of the frequency inverter. As mentioned, the fre-quency inverter was not designed with the intent of being used for high-frequencylogging for the detection of minor changes to the electrical properties. Although itis proven that the electrical properties measured by the frequency inverter can beused for temperature estimation, there are likely other properties that would im-prove the performance of the model. The active power would perhaps yield a bettermodel since that would not only take into account the current but also the voltageand phase angle which give the useful power outputted from the frequency inverter.Included in the active power would be other losses in the system in addition to the

43

Chapter 7. Discussion

losses in the workpiece, such as coil losses, but it would likely still be closely cor-related to the temperature in the workpiece. The frequency inverter unit does notpresently support such measurements to be made though. An option that was con-sidered and to some extent tried in the early steps of this work was to use externalmeasurement equipment to measure the current and voltage over the load and deter-mine the zero-crossing of these curves to calculate the active power. However, thisis not trivial to do accurately. The voltage has the form of a modified square wavewith ripples where even small errors in the measurement would result in large errorsin the calculated active power why this method was not used in this work.

7.4 Effects of Welding

In Section 6.4 it was found that the phase value behavior changes when comparinga number of consecutive cycles at high temperature. However, this altered behaviorof the phase value between different cycles might be because the temperature mea-sured by the thermocouple does not reflect the overall temperature in the workpiece.The overall workpiece temperature will increase with an increasing number of cy-cles due to thermal conduction. This makes it difficult to draw any conclusions re-garding the phase value-temperature relation from this experiment. The most likelyexplanation is instead that the welding alters the properties of the material thus mak-ing the following cycles different from the first, which is supported by the findingspresented in Section 6.3, see Figure 6.3.

The suspicion that the welding process does have an impact on the materialproperties such that a reheating of the material will not give the same results as theinitial welding was confirmed in Section 6.4, see Figure 6.11 and Figure 6.12. Theinitial welding process is faster and there is a difference in the phase value. Thefact that the two CFRP plates are already joined might be a contributing factor tothe difference in measured heating rate due to better thermal conduction betweenthe upper and lower plate. Regarding the phase value, it can be seen in Figure 6.12that the measured frequency during Cycle 1 is not closer to resonance than duringCycle 2, which indicates that the change in phase value is either due to the resis-tance and inductance changes seen in Figure 6.3, or the precision of the phase valuemeasurement is not good enough and the difference seen is a measurement error.

The slower heating rate for Cycle 2 and 3 compared to Cycle 1 was reproducedwhen the same experiment was performed on another sample while the phase angledid not show the same increase between cycles. However, the measured frequencyduring the reheating was quite a bit lower than during the initial welding whichresults in a lower phase value. Since a certain phase value does not correspond to aspecific phase angle it is not possible to compensate for the frequency offset. It isthus not clear whether the phase value during the reheating would have been lowerif the measured frequency during the initial weld and the reheating were the same.

The fact that the electrical properties are affected by the welding such that a

44

7.4 Effects of Welding

rewelding does not give the same result as the initial welding is perhaps not of greatimportance in most cases but should be considered if previously welded materialwere to be reheated in some application.

45

8Conclusions

In this work, it has been shown that it is possible to estimate the temperature in theweld-zone during induction welding using only the inductor itself along with thefrequency inverter, and by doing so eliminating the need for any external sensorssuch as pyrometers or thermocouples. This is accomplished by using the currentmeasured by the frequency inverter as an input signal to a Wiener model. The sys-tem identification was done using a black-box modeling approach, meaning thatthe model was constructed by estimating the model coefficients of a generic modelstructure by the observation of input and output data only. Both linear and nonlin-ear models were tested with different combinations of input signals. Furthermore,it has been shown that the electrical properties of the CFRP material have a tem-perature dependence which was to be expected. More specifically, the resistanceis decreasing with increasing temperature. The electrical bulk resistance is shownto be increasing at higher temperatures. These temperature dependencies are pos-sible to measure during welding using the phase value measured by the frequencyinverter. Furthermore, the time it takes for different samples to reach the processtemperature varies, why feedback would be necessary to ensure the correct weld-zone temperature.

The model proposed in this work is valid for operation using the fixed frequencyof 640 kHz with the welding process starting at room temperature using the sameworkpiece material and dimensions used in this work. The same approach would beapplicable on other frequencies as well.

46

9Future Work

The model constructed in this work is capable of temperature estimation when thesystem is run at a fixed frequency of 640 kHz. Since the model is constructed forone frequency only it cannot be used if the frequency is changed. That means thatcurrently no temperature control can be done since that would include changing thefrequency. There are other ways to control the process though. The process couldbe controlled with an on/off controller on a fixed frequency for example. The pro-cess would most likely be controlled with the frequency as the only control signalin future work. To make control possible on the model the multi-model strategywould perhaps be viable, where a set of models like the one constructed in thiswork are made for different frequencies and then switched between during control.A model using measured electrical signals as input provides the opportunity forfast temperature control. In an induction heating system, two processes are inter-acting; electromagnetic processes which are very fast, and far slower heat transferprocesses. The difference in time constants between the two processes can be uti-lized by using a cascade control structure where an outer control loop is controllingthe temperature and an inner loop is controlling the electrical properties. With sucha strategy, changes in the measured electrical signals can be managed before theyhave any major impact on the temperature. It also simplifies the dynamics of thesystem the primary controller is tasked with.

Something that has been touched upon but not further examined is the temper-ature variations within the workpiece as both theory and simulations showed exist.This work has focused on the weld-zone temperature where the constructed modelcan estimate the temperature. For a model like the one developed in this work tobe used in a real process the temperature profile of the workpiece has to be takeninto account to avoid thermal degradation of the polymer in the layers closest to theinductor.

47

Bibliography

Andersson, L., U. Jönsson, K. H. Johansson, and J. Bengtsson (n.d.). A manualfor system identification. [Part of course literature in Course on System Iden-tification (FRTN35), 2019]. Department of Automatic Control, Lund Univer-sity. URL: http://www.control.lth.se/education/engineering-program/frtn35-system-identification/. (accessed: 26.10.2020).

electricalbaba (2016). Why Series Resonance Circuit Called an Acceptor Cir-cuit? URL: https://electricalbaba.com/why- series- resonance-circuit-called-an-acceptor-circuit/. (accessed: 09.11.2020).

Frogner, K. (2020). Personal communication.Frogner, K., O. Wiberg, V. Akujärvi, T. Cedell, and M. Andersson (2014). “In-

duction heating of carbon fiber structures - properties and challenges”. In:Johansson, B. (Ed.). Proceedings of the 6th Swedish Production Symposium.Conference date: 16.09.2014 - 18.09.2014. Chalmers Tekniska Högskola. URL:http://conferences.chalmers.se/index.php/SPS/SPS14/paper/viewFile/1748/420. (accessed: 31.07.2020).

Gardiner, G. (2018a). New horizons in welding thermoplastic composites. URL:https : / / www . compositesworld . com / blog / post / new - horizons -in-welding-thermoplastic-composites. (accessed: 27.03.2020).

Gardiner, G. (2018b). Welding thermoplastic composites. URL: https : / /www . compositesworld . com / articles / welding - thermoplastic -composites. (accessed: 27.03.2020).

Gardiner, G. (2020). Using mobile susceptors to innovate thermoplastic inductionwelding. URL: https://www.compositesworld.com/blog/post/using-mobile - susceptors - to - innovate - thermoplastic - induction -welding. (accessed: 27.03.2020).

Kane, B., A. Pierquin, G. Wasselynck, and D. Trichet (2020). “Coupled Numer-ical and Experimental Identification of Geometrical Parameters for Predictingthe Electrical Conductivity of CFRP Layers”. IEEE Transactions on Magnetics56:2, pp. 1–4.

48

http://www.control.lth.se/education/engineering-program/frtn35-system-identification/

http://www.control.lth.se/education/engineering-program/frtn35-system-identification/

https://electricalbaba.com/why-series-resonance-circuit-called-an-acceptor-circuit/

https://electricalbaba.com/why-series-resonance-circuit-called-an-acceptor-circuit/

http://conferences.chalmers.se/index.php/SPS/SPS14/paper/viewFile/1748/420

http://conferences.chalmers.se/index.php/SPS/SPS14/paper/viewFile/1748/420

https://www.compositesworld.com/blog/post/new-horizons-in-welding-thermoplastic-composites

https://www.compositesworld.com/blog/post/new-horizons-in-welding-thermoplastic-composites

https://www.compositesworld.com/articles/welding-thermoplastic-composites



https://www.compositesworld.com/blog/post/using-mobile-susceptors-to-innovate-thermoplastic-induction-welding



Bibliography

Lin, Y. (2018). “Analysis Model and Numerical Simulation of Thermoelectric Re-sponse of CFRP Composites”. Applied Composite Materials 26. DOI: 10 .1007/s10443-018-9700-6.

Lundström, F., K. Frogner, and M. Andersson (2018). “A method for inductive mea-surement of equivalent electrical conductivity in thin non-consolidated multi-layer carbon fibre fabrics”. Composites Part B:Engineering 140, pp. 204–213.

Lundström, F., K. Frogner, O. Wiberg, T. Cedell, and M. Andersson (2017). “Induc-tion heating of carbon fiber composites: investigation of electrical and thermalproperties”. International Journal of Applied Electromagnetics and Mechanics53:S1, pp. 21–30. ISSN: 1383-5416.

MathWorks (2020a). Estimate Initial Conditions for Simulating Identified Mod-els. URL: https://se.mathworks.com/help/ident/ug/estimate-initial-conditions.html. (accessed: 24.06.2020).

MathWorks (2020b). System Identification Toolbox. URL: https : / / se .mathworks.com/products/sysid.html. (accessed: 09.11.2020).

MathWorks (2020c). What are Hammerstein-Wiener Models? URL: https://se.mathworks.com/help/ident/ug/what- are- hammerstein- wiener-models.html. (accessed: 26.05.2020).

MathWorks (2020d). What are Nonlinear ARX Models? URL: https : / / se .mathworks.com/help/ident/ug/what-are-nonlinear-arx-models.html. (accessed: 26.05.2020).

Meeker, D. C. (28, 2018). Finite element method magnetics. Version 4.2. URL:http://www.femm.info. (accessed: 24.08.2020).

Roberts, A. J. (2019). “Carbon fiber commercial update 2020 - 2025”. In: CarbonFiber 2019 (Knoxville, Tenn., U.S, Nov. 19, 2019). AJR Consultancy.

Rudnev, V., D. Loveless, and R. L. Cook (2017). Handbook of Induction Heating.2nd ed. CRC Press, Boca Raton. ISBN: 978-1-1387-4874-3.

Toray (2019). Toray Cetex® TC1200 PEEK. Toray Industries. URL: https://www.toraytac.com/media/7765d981- 1f9f- 472d- bf24- 69a647412e38/8InI2A / TAC / Documents / Data _ sheets / Thermoplastic / UD % 5C %20tapes,%5C%20prepregs%5C%20and%5C%20laminates/Toray-Cetex-TC1200_PEEK_PDS.pdf. (accessed: 09.11.2020).

Wasselynck, G., D. Trichet, B. Ramdane, and J. Fouladgar (2011). “Micro-scopic and Macroscopic Electromagnetic and Thermal Modeling of CarbonFiber Reinforced Polymer Composites”. IEEE Transactions on Magnetics 47:5,pp. 1114–1117.

49

https://doi.org/10.1007/s10443-018-9700-6

https://doi.org/10.1007/s10443-018-9700-6

https://se.mathworks.com/help/ident/ug/estimate-initial-conditions.html

https://se.mathworks.com/help/ident/ug/estimate-initial-conditions.html

https://se.mathworks.com/products/sysid.html

https://se.mathworks.com/products/sysid.html

https://se.mathworks.com/help/ident/ug/what-are-hammerstein-wiener-models.html



https://se.mathworks.com/help/ident/ug/what-are-nonlinear-arx-models.html



http://www.femm.info

https://www.toraytac.com/media/7765d981-1f9f-472d-bf24-69a647412e38/8InI2A/TAC/Documents/Data_sheets/Thermoplastic/UD%5C%20tapes,%5C%20prepregs%5C%20and%5C%20laminates/Toray-Cetex-TC1200_PEEK_PDS.pdf





Document name

Date of issue

Document Number

Author(s) Supervisor

Title and subtitle

Abstract

Keywords

Classification system and/or index terms (if any)

Supplementary bibliographical information

ISSN and key title ISBN

Language Number of pages Recipient’s notes

Security classification