Inductive Charging of Electrical Vehicles
System Study
Mikael Cederlöf
Abstract
A prerequisite of a fast increasing market of the electrical vehicles is the access to charging and reliability/accessibility of the charging systems. Wireless charging i.e. charging without a cord, is an interesting alternative that has been put forward during recent years.
In this work a system study of topologies for inductively coupled power transfer over an air gap of tens of centimetres has been investigated. In order to obtain an effective power transfer compensation capacitors are used to achieve resonance circuits.
In the thesis four different compensation topologies for inductively coupled power transfer are examined. Expressions for the compensation capacitances and the output voltage or current are derived.
An example design for each of the four topologies capable of handling a transfer of 3 kW of power over an air gap of 20 cm with an efficiency of at least 96% is examined. These designs use an outer radius for both coils of 30 cm, and an operating frequency of 20 kHz. The efficiency only encompasses the windings, and does not take into account the efficiency of any power electronics before or after the coils.
Prototype housing for the primary coil has been designed and built using basalt fibre reinforced high performance concrete, and magnetic measurements on the materials used are included in the report.
Acknowledgements
First of all I would like to thank Sten Bergman at Elforsk for initiating this thesis work, for his enthusiasm and for contributing time and resources to this project. Then I would like to thank professor Göran Engdahl, my supervisor at KTH, for his support, feedback and proof reading. I also would like to thank Hector Zelaya de la Parra, Ener Salinas, Tomas Tegnér and Zichi Zhang at ABB for taking the time to help, guide, encourage and support me.
Further more I would like to thank Göte Lindfors and Günter Villman at Aquafence AB for building the concrete housing of the prototype, Stefano Bonetti and Johan Persson at NanOsc AB for performing the magnetic measurements. Also I would like to thank Johan Silfwerbrand at CBI for taking an interest in this thesis work, Niklas Thulin at Volvo, Thomas Bergfjord and Bjorn Karlstrom at ElectroEngine.
Finally I would like to thank my wife Malin for her constant care, support and encouragement.
Table of Contents
Abstract...iii
Acknowledgements...v
1: Background of wireless power transfer...1
1.1 History...1
1.2 Overview of wireless power transfer...1
2: Theory behind inductive power transfer...3
2.1 Maxwell's Equations...3
2.2 Magnetic field, magnetic flux and inductance...3
2.3 The Coils...4
2.4 Mutual inductance model...6
2.5 Leakage Flux...8
2.6 Power...9
2.7 Resonance...9
2.8 Compensation alternatives...10
2.9 The need for High Frequency...10
2.10 Skin Depth and Proximity Effect...11
2.11 Litz Wire...12
2.12 Field shaping materials...12
3: Modeling ...15
3.1 Field simulation program...15
3.2 Field simulation geometry...15
3.3 Interpreting obtained values...16
3.4 Circuit analysis program...16
3.5 Compensation Topologies...17
3.5.1 Secondary side impedance...17
3.5.2 Primary side impedance...19
3.5.3 Primary coil with constant amplitude current...21
3.6 Series-Series topology...23
3.7 Series-Parallel topology...24
3.8 Parallel-Series topology...25
3.9 Parallel-Parallel topology...26
4: Models...27
4.1 Model names...27
4.2 Model 230SS16A40...28
4.3 Model 230SP420V40...29
4.4 Model 230SP420V20...30
4.5 Model 230SS36A20...31
4.6 Model 230SP230V20...32
4.7 Model 230PS230V20...33
4.8 Model 230PP33A20...34
4.9 Summary of the introduced models...35
5: Prototype Design...37
5.1 Components and alternatives...37
5.2 Housing...37
5.3 Shielding material...37
5.4 Core...38
5.5 Coil...38
5.6 Power electronics...38
5.7 Output Constant Voltage or Constant Current...39
6: Concrete Housing...41
6.1 Basalt fibers...41
6.2 Measurements...41
6.3 Comments...42
7: Parameter study...43
7.1 Magnetic field calculations...43
7.1.1 Coil Radius...43
7.1.2 Coil Turns...44
7.1.3 Distance between primary and secondary coils...45
7.1.4 Frequency...45
7.2 Non-symmetrical sides...46
7.2.1 Coil Radius...46
7.2.2 Coil Turns...47
7.3 Simulink Studies...48
7.4 Coil Resistances...48
7.5 Compensation Capacitors...49
7.5.1 Series-Series...50
7.5.2 Series-Parallel...52
7.5.3 Parallel-Series...54
7.5.4 Parallel-Parallel...56
7.6 Distance between Coils...57
7.6.1 Series-Series...58
7.6.2 Series-Parallel...59
7.6.3 Parallel-Series...60
7.6.4 Parallel-Parallel...61
7.7 Features and comments regarding the studied topologies...62
8: Recommendations...63
8.1 Suggested compensation topologies for demonstration purposes...63
8.2 Suggested control alternatives...63
8.2.1 Adjusting the input voltage...63
8.2.2 Adjusting the distance between the two coils...64
8.2.3 Adjusting the frequency...64
8.3 Relative size of the two coils...64
8.4 Further work...64
9: Results and Conclusions...67
10: References...69
Appendices...71
1: Background of wireless power transfer
A pre-study on the feasibility of wireless power transfer was performed at KTH in the form of a Masters Thesis in 2009 [1]. That study gave good insight into the theory of inductive and resonant transfer of electric energy and how to design an inductive charging system.
1.1 History
Electromagnetic induction was discovered independently by Michael Faraday and Joseph Henry in 1831. Faraday published his results first and became the officially recognized discoverer of the phenomenon.
In Faraday's experimental demonstration of electromagnetic induction, he wrapped two wires around opposite sides of an iron torus. He plugged one wire into a galvanometer, watched it as he connected the other wire to a battery, and observed a transient current (which he called a "wave of electricity") both when he connected the wire to the battery and when he disconnected it [2]. The SI unit for capacitance is named after him.
Henry's experiment consisted of an electromagnet perched on a pole, rocking back and forth. The rocking motion was caused by one of the two leads on both ends of the magnet rocker touching one of two battery cells, causing a polarity change, and rocking the opposite direction until the other two leads hit the other battery. This apparatus allowed Henry to recognize the property of self inductance [3]. The SI unit for inductance is named after him.
James Clerk Maxwell later formulated the classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory [4]. This theory was later distilled down to the four famous equations known as Maxwell's equations by Oliver Heaviside [5].
Wireless transmission of energy was demonstrated in 1891 by Nikola Tesla. His many revolutionary developments in the field of electromagnetism were based on the theories of electromagnetic technology discovered by Michael Faraday [6]. The SI unit for the magnetic field was named after him.
1.2 Overview of wireless power transfer
Wireless power supply of an electric vehicle is considered by many as a ground breaking innovation, given that it is safe enough and can show an appropriate efficiency. No hassle with cords would be welcomed by many drivers.
Since a number of years back innovative concepts on electric vehicle charging has been going on, and wireless inductive charging has efficiently been used used in environments where conductive power transfer has been a problem.
In New Zealand, at the University of Auckland, the wireless technology for electric vehicle (EV) charging has been a key subject, and their technology has been patented and commercialized through the University by UniServices. Licenses have been sold and patents are now spread around the globe.
In Asia, Europe and US many attempts have been made to develop wireless power technology for both consumer applications and large power industrial use. Bombardier is e.g. using wireless inductive power transfer to feed continuous power to their trams. One successful demonstrator already runs in Berlin and development projects to feed heavy trucks at highways are going on.
Inductive power transmission is now getting into focus not only for personal vehicles but also for a number of mobile applications. The power transfer demonstrated so far include power levels up to about 7-10 kW but with ambitions to reach hundreds of kW within this decade. Efficiencies are reported to be between 80 – 95%.
Despite the fact that a potential customer only have to park the car at a given spot there are many things that must work and be synchronized. This includes performance, safety, interoperability, communication, frequency, and battery management issues.
International standardization efforts have been started within this field. Both in Japan, Europe and US standardization bodies are now forming task forces and defining areas of interest. The Society of Automotive Engineers (SAE) task force on wireless charging and positioning of electric vehicles (SAE J2954) has grown to form six sub teams.
Put together, there is a rapidly increasing interest of wireless charging of electric vehicles which is reflected trough the many ongoing university and industrial cooperation projects worldwide.
2: Theory behind inductive power transfer
Wireless transmission of energy using induction is based on the theory described by Maxwell's equations. A current through a conductor produces a magnetic field. Variations in the magnetic field creates an electric field. The electric field causes the electrical charges to move, producing a current.
2.1 Maxwell's Equations
Maxwell's equations is a collective name for four famous equations. These equations were discovered and refined by Gauss, Ampère, Faraday and Maxwell.
∮SE⋅ds=Qϵ0 (2.1)
∮SB⋅ds=0 (2.2)
∮CE⋅dl=−d Φdt B (2.3)
∮CB⋅dl=μ0(I+ϵ0d ΦdtE) (2.4)
The first equation relates the electrical field with electrical charges. The second equation states that there are no sources or sinks for magnetic fields, all field lines are closed loops. The third equation states that the electrical field is related to the change in the magnetic flux. The fourth equation relates the magnetic field with a current and the change in the electric flux.
2.2 Magnetic field, magnetic flux and inductance
Figure 2.1: A closed loop and magnetic flux lines.
According to equation (2.4), running a current through a closed loop will create a magnetic field, also known as magnetic flux density, B. This closed loop encloses a surface S. Through this surface there flows a magnetic flux ΦB, henceforth called Φ.
ΦB=∫SB⋅dS (2.5)
The magnetic flux density is proportional to the applied current, meaning the magnetic flux is also proportional to the current. This proportionality coefficient is the inductance L.
Φ=LI (2.6)
Figure 2.2: Two closed loops with mutual flux Φ12.
Placing a second closed loop in the vicinity of the first will cause some of the magnetic flux from the first loop, due to B1, to pass through the surface of the second loop, S2. This is designated as the mutual flux Φ12.
Φ12=∫S2B1⋅dS2 (2.7)
Running a current through the second loop instead, will produce another mutual flux.
Φ21=∫S1B2⋅dS1 (2.8)
The proportionality coefficient between the current and the mutual flux is the mutual inductance.
Φ12=L12I1 (2.9)
Φ21=L21I2 (2.10)
It turns out that the mutual inductances are the same [7].
M=L12=L21 (2.11)
2.3 The Coils
To run a current through a closed loop a coil is used. If a coil has N coil turns, the current in each turn will add to the magnetic flux density. In other words, B is proportional to the number of turns carrying the current I.
B ∝NI (2.12)
The number of turns the magnetic flux links with determines the flux linkage, Ψ.
Ψ=N Φ (2.13)
For a coil with more than one turn the flux linkage is used to calculate the inductance.
L= ΨI (2.14)
Now, L is proportional to NΦ, and Φ is proportional to B, and B is proportional to NI. This means that the self inductance is proportional to the square of the number of turns of the coil.
L∝N2 (2.15)
The same reasoning is done with the mutual inductance. A current I1 runs through a coil with N1
turns, causing a magnetic flux density B1.
B1∝N1I1 (2.16)
The flux density B1 will cause a mutual flux Φ12 which will link to the second coil.
Φ12=∫S2B1⋅dS2 (2.17)
If the mutual flux links with N2 turns of the second coil, the flux linkage becomes N2Φ12.
Ψ12=N2Φ12 (2.18)
From the flux linkage the mutual inductance is calculated.
M=Ψ12
I1 (2.19)
Now, M is proportional to N2Φ12, and Φ12 is proportional to B1, and B1 is proportional to N1I1. This means that the mutual inductance is proportional to the product of the number of turns in the coils.
M ∝N1N2 (2.20)
The inductance of a coil is determined by the geometrical shape and the physical arrangement of the conductor as well as the permeability of the medium [7]. The mutual inductance between two coils is further dependent on the distance and relative position of the two coils.
The ratio between the mutual inductance and the square root of the product of the self inductances is the coupling coefficient, k.
k = M
√(L1L2) (2.21)
0⩽k⩽1 (2.22)
This coefficient measures the magnetic coupling between the coils and is independent of the number of turns in the coils. It only depends on the relative positions of the two coils and the physical properties of the media in the vicinity of these coils.
2.4 Mutual inductance model
According to Faraday's law, equation (2.3), an alternating flux linkage will cause an induced electromotive force, or voltage.
u2(t)=d ψ12(t)
dt (2.23)
Here the negative sign has been compensated for by changing the direction of the winding of the second coil. From the current I1 through the first coil a voltage U2 will be induced in the second coil.
Figure 2.3: Two magnetically coupled coils with good coupling.
If the second coil is closed, a current I2 will flow in the second coil. This current will cause a magnetic flux density B2, which in turn will cause a magnetic flux. This flux will oppose the mutual flux from the first coil. Using superposition the total flux linkage of the second coil will be described as a combination of the self-flux linkage Ψ22 and the mutual flux linkage Ψ12.
ψ2(t)=ψ12(t)−ψ22(t) (2.24)
Some flux from the current in the second coil will link with the first coil, and the total flux linkage of the first coil is defined in the same way.
ψ1(t)=ψ11(t)−ψ21(t) (2.25)
These equations are now described in terms of inductances.
ψ1(t)=L1i1(t)−M i2(t) (2.26)
ψ2(t)=M i1(t )−L2i2(t) (2.27)
Using Faraday's law on the total flux linkages of each coil, assuming the inductances are time invariant leads to the relationship between input voltage and current and output voltage and current [8].
u1(t)=L1di1(t)
dt −Mdi2(t)
dt (2.28)
u2(t)=Mdi1(t)
dt −L2di2(t)
dt (2.29)
Assuming all currents are sinusoidal and steady-state these equations can be written on phasor form.
U1=jω L1I1−j ω M I2 (2.30)
U2=jω M I1−j ω L2I2 (2.31)
A non-ideal coil consist of a resistance in series with an inductance. The equations are modified to include these resistances.
U1=R1I1+j ω L1I1−j ω M I2 (2.32)
U2=jω M I1−R2I2−jω L2I2 (2.33)
Figure 2.4 illustrates equations (2.32) and (2.33) in a circuit diagram.
Figure 2.4: Circuit diagram of mutual inductance model.
Equation (2.33) can be seen as an induced voltage Uind due to the current in the primary coil, and a voltage drop Udrop due to the current in the secondary coil.
U2=Uind−Udrop (2.34)
Where
Uind=jω M I1 , (2.35)
Udrop=(R2+j ω L2)I2 . (2.36)
Connecting the load resistance Rload to the secondary coil will create a closed circuit.
j ω M I1=R2I2+j ω L2I2+RloadI2 (2.37)
The current through the second coil is given by the induced voltage divided by the total impedance.
I2= j ω M I1
R2+j ω L2+Rload (2.38)
And can be rewritten as I2= j ω M I1
Z2 (2.39)
Where Z2 is the total impedance of the second coil and load. This is inserted into the expression for the first coil, equation (2.32).
U1=R1I1+j ω L1I1+ω2M2 Z2
I1 (2.40)
It is now clear that from the point of view of the first coil, the second coil is seen as a transformed impedance from the secondary side, ZS.
Zs=ω2M2
Z2 (2.41)
2.5 Leakage Flux
The self-flux linkages are composed of two parts. The main flux linkage that connects with other coils and the leakage flux linkage that does not connect with any other coil. The name main flux linkage comes from the fact that in transformers the main flux linkage is the dominant part of the flux linkage. This is not the case with inductive power transfer.
Ψ11=Ψ11m+Ψ11l (2.42)
The main flux linkage is the mutual flux linkage linked with the coil that produced it.
Ψ11m=N1Φ12 (2.43)
The main flux linkage is scaled by a factor N2/N2,
N1Φ12=N1 N2
N2Φ12 , (2.44)
and is expressed as a scaling of the mutual inductance.
Ψ11m=N1 N2
M I1 (2.45)
Dividing by the current gives the main inductance.
L1m=N1
N2M (2.46)
This gives the leakage inductance expressed in inductances.
L1l=L1−L1m (2.47)
It is easy to see that in the case of no secondary coil, the mutual inductance is zero, and the self inductance consists only of the leakage inductance. If the coupling coefficient k goes to one, then the leakage inductance goes to zero. This can be seen in equation (2.43), if Φ12 is the total flux produced by the coil, then this is the definition of the self inductance.
2.6 Power
Due to the large leakage flux most of the magnetic energy will only link with the coil itself, and return the energy. This energy is what causes the current in the coil to lag the voltage. The product of the voltage and current is called the power.
p (t)=u(t )i(t) (2.48)
When dealing with complex voltages and currents on phasor form, the power will be complex and denoted S. Complex power, S, consists of Active power, P, and Reactive power, Q.
S =P+ jQ (2.49)
A current and voltage in phase will always have a positive product. This is active power, and it is the power that conducts work. When the voltage and current are out of phase with each other, part of the time the power will be positive and part of the time the power will be negative. This is reactive power, and it will flow back and forth in the system without performing any net work. It will however give cause to resistive losses in the system.
2.7 Resonance
Inductances and capacitances cause phase shifts. The phase shift is expressed by the equation of the impedance.
ZL=j ω L (2.50)
ZC= 1 j ω C
=−j ωC
(2.51)
Since one of the components cause a phase shift of +90 degrees, and the other a phase shift of -90 degrees, they can be used to cancel each other out.
ωL= 1
ωC or L= 1
ω2C or C= 1
ω2L or ω= 1
√ LC (2.52)
This is called resonance. When resonance occurs the current caused by the collapsing magnetic field in the inductor charges the capacitor. The discharging capacitor then drives a current through the inductor that produces a magnetic field. The energy in the system shifts between magnetic energy and electric energy. In the ideal case there are no losses and the system will continue to resonate. In reality however there are resistances in the inductor and capacitor that will lead to losses and the ringing will die out.
2.8 Compensation alternatives
Resonance can be achieved with series compensation or with parallel compensation. The following equations expresses the impedance of the series case and the parallel case.
ZSeries=ZL+ZC (2.53)
ZParallel= ZLZC
ZL+ZC (2.54)
These can be rewritten as ZSeries= j(ω2L C −1)
ωC , (2.55)
and
ZParallel= ωL
j (ω2LC −1) . (2.56)
During resonance the impedance of the series compensated case goes to zero, and the impedance of the parallel compensated case goes to infinity.
lim
ω→ 1
√LC
ZSeries=0
(2.57) lim
ω→ 1
√LC
ZParallel=∞
(2.58) In both these cases the resonance is used to preserve the energy from the leakage flux of the coil, and use it to maintain the magnetic coupling of the system. Without this preservation only a very small amount of the energy entered into the system would arrive at the other end.
2.9 The need for High Frequency
The equation for the induced voltage in the secondary coil is
Uind=jω M I1 . (2.59)
If the distance between the two coils is increased, only the value of the mutual inductance M will
change in this equation. As the distance increases, the mutual inductance decreases. To keep the induced voltage constant without running a larger current through the first coil the frequency needs to be increased.
As the frequency increases there are some things that needs to be taken into consideration.
2.10 Skin Depth and Proximity Effect
Conductivity, σ, is the ability of a material to conduct electric currents. Metals are very good conductors, for example copper has a conductivity of 5.8x107 S/m, iron has a conductivity of 107 S/m, and rubber which is an isolator has a conductivity of about 10-15 S/m.
In the case of a direct current, the current density distribution is constant over the cross section area.
An alternating current however will change the current density distribution. Reducing it at the center of the conductor and concentrating it at the surface.
Figure 2.5: Current density distribution for low, medium and high frequency.
The skin depth, δ, is defined as the depth at which the current density falls to 1/e, (about 0.37), of the current density at the surface. This can be approximated with
δ=√ω μ2ρ , (2.60)
where ρ is the resistivity of the material and is related to the conductivity as 1/σ, and μ is the permeability of the material.
When conductors are placed in parallel near each other, as with the the windings of a coil, they will be affected by the proximity effect. Figure 2.6 shows how this further affects the current density distribution.
Figure 2.6: The influence of the proximity effect on the current density distribution for a medium frequency
These effects increases the AC resistance of a conductor or wire dramatically, especially at higher frequencies.
2.11 Litz Wire
Since the skin depth determines the effective current carrying area of a conductor, a conductor thicker than the skin depth will not be an effective use of material. A Litz wire consists of many thin strands, each individually insulated, wound into a wire. Each strand is no thicker than the skin depth, this ensures an effective use of the conductive area. The strands are then woven together so that the location of each strand alternates between the center of the wire and the edge of the wire.
This ensures that the proximity effect will affect each strand the same, and thus carry the same current.
This will not remove the influence of the proximity effect and skin depth, but it will significantly decrease the AC resistance compared to normal wires.
2.12 Field shaping materials
A magnetic field created by a coil will extend equally above and below the coil. For inductive power transfer this is undesirable. There are however ways to shape the field.
Figure 2.7: A conductor without field shaping materials.
The Relative Permeability, μr, defines the ability of a material to conduct magnetic fields. In vacuum the relative permeability is equal to one, whereas iron has a relative permeability in the range of 4x103. Magnetic materials can be roughly classified into three groups.
Diamagnetic: μr is slightly smaller than one.
Paramagnetic: μr is slightly larger than one.
Ferromagnetic: μr is much larger than one.
When a magnetic field encounters a paramagnetic material such as copper, the relative permeability is almost equal to one, and the field will only slightly prefer the copper over free space. However, copper is conductive and there will be eddy currents induced in the copper. These eddy currents will create their own magnetic field, a field in the opposite direction of the external field. This will cause the external field to bend off instead of entering the material. This effect is called repulsion.
When a magnetic field encounters a ferromagnetic material, the high permeability causes the field to prefer the material over open space. The magnetic field will be captured by the material.
Iron is a ferromagnetic material that is also conductive, in this case the relative permeability of the iron is high enough to overcome the repulsive effects of the eddy currents, but there will be a lot of eddy current losses due to the captured magnetic field. These losses will heat the iron.
In transformers an iron core is often used to conduct the magnetic flux between the coils. In order to limit the eddy current losses the transformer core consists of thin insulated sheets of iron. This laminated core will prevent much of the eddy currents, and substantially reduces the losses.
The magnetic field can be shaped by placing a ferromagnetic material with low conductivity on a paramagnetic material with high conductivity. The ferromagnetic material will conduct the magnetic field while the conductive material will repel the magnetic field.
Figure 2.8: A conductor with field shaping materials.
3: Modeling
Calculating the inductance can be done by solving differential equations. Adding field shaping materials will significantly increase the complexity of these equations. A finite element program can handle complex geometries with different materials and produces a numerical solution to the problem.
In this work the program FEMM was used for finite element calculations. FEMM stands for Finite Element Method Magnetics. The results were then used in Matlab to perform circuit analysis.
3.1 Field simulation program
FEMM [9] is a free program that can be controlled from within Matlab or Octave. A 2-dimensional geometry is defined either in a in a matlab script or in the program itself, and after the simulation information such as the inductances, resistances and losses can be extracted.
In order for FEMM to be able to use a material in its simulations, the material must be defined.
When defining a material the two most important parameters are its conductivity σ and its relative permeability μr. Values for these parameters can be found in text books [7] or Wikipedia [10][11].
FEMM accepts the conductivity in mega Siemens per meter. That is 106 S/m.
Table 3.1: FEMM material definitions.
Material Conductivity σ [S/m] FEMM σ [MS/m] Relative Permeability μr
Air 5 x 10-15 0 1
Aluminium 3.54 x 107 35.4 1
Copper 5.80 x 107 58 1
Ferrite 1 10-6 1000
When it comes to the relative permeability of the ferromagnetic material it comes down to what the manufacturer can supply. For example Mumetal is a special alloy that has a relative permeability of about 100000. In comparison pure Iron has a relative permeability of 4000. In this work the relative permeability of ferrite will be set to 1000.
3.2 Field simulation geometry
FEMM can solve both planar problems or axisymmetric problems. A planar problem takes a 2- dimensional problem definition and then applies a depth. The axisymmetric problem takes a 2- dimensional problem definition and then rotates it around a central axis. This work uses the axisymmetric definition and will therefore correspond to a circular geometry.
The primary side of the Inductively Coupled Power Transfer (ICPT) will consist of a copper coil on top of a ferrite core and an aluminium shield. The secondary side will be a mirror image of the primary side. This design will shield the area above and below the ICPT and focus the field in the area between the two coils. Surrounding the geometry a large volume of air is defined. Figure 3.1 illustrates the geometry used in the field calculations.
Figure 3.1: FEMM image of two coils with shields and cores.
3.3 Interpreting obtained values
Circuit definitions are used to define the parts of the geometry that conduct defined currents, and are used to run a current through a coil. Coil turns can be individually modeled and included in the same circuit.
FEMM will calculate the voltage drop and the flux linkage for each circuit definition. From this the resistance of the coil, R, is calculated by taking the real part of the ratio between the voltage and the current. The inductance of the coil, L, is calculated by dividing the flux linkage, Ψ, with the current, I.
R=Real{UI } (3.1)
L= ΨI (3.2)
The mutual inductance is calculated by first running the current I1 through the first coil and saving the flux linkage Ψ11, then running the current I2 through the second coil and saving the flux linkage Ψ22, then finally running the current I1 through the first coil while simultaneously running the current I2 through the second coil and obtaining the total flux linkage for each coil, Ψ1 and Ψ2.
M=(Ψ1−Ψ11)
I2 =(Ψ2−Ψ22)
I1 (3.3)
Which is the same as the definition of the mutual inductance.
M=Ψ21 I2 =Ψ12
I1 (3.4)
It does not matter which mutual flux linkage is used to define the mutual inductance, since FEMM produces the same result for both cases. This is in accordance to the theory in chapter 2.
3.4 Circuit analysis program
Figure 3.2 shows the basic circuit model used to simulate the inductively coupled power transfer. It is simulated in the Matlab toolbox Simulink using the library SimPowerSystems [12], and consists of a voltage source connected to a mutual inductance model of a transformer. On the other side of the mutual inductance block, the load will be simulated using a resistance.
A small line resistance is included to prevent errors when a capacitor is connected in parallel to the mutual inductance block.
Figure 3.2: Circuit model without compensation.
The mutual inductance block uses six variables to model a transformer. Primary Winding Impedance [R1, L1], Secondary Winding Impedance [R2, L2] and Mutual Impedance [Rm, Lm].
Figure 3.3: SimPowerSystems mutual inductance block circuit diagram.
Figure 3.3 shows the circuit diagram of the mutual inductance block. If the mutual resistance, Rm, is set to zero, this circuit diagram is equivalent to the mutual inductance model in figure 2.4. The mutual resistance is used to model eddy current losses and hysteresis losses in a transformer core.
These losses will be disregarded in this work.
3.5 Compensation Topologies
Both sides of the inductively coupled power transfer needs a resonant capacitance. Since the capacitor can be connected either in series or in parallel with the coil, this gives a total of four different compensation topologies.
3.5.1 Secondary side impedance
Figure 3.4: Secondary impedance with series compensation and with parallel compensation.
Figure 3.4 illustrates the total impedance of the secondary side, Z2, as seen by the induced voltage in the secondary coil. The total impedance of the secondary side when using series compensation is given by equation (3.5)
Z2=ZL2+ZC2+RLOAD , (3.5)
and the total impedance of the secondary side when using parallel compensation is given by equation (3.6)
Z2=ZL2+ 1 1
ZC2+ 1 RLOAD
. (3.6)
Figure 3.5 illustrates how the voltage over the load resistance varies with the compensation capacitance when the secondary side is series compensated and is connected to a voltage source with a constant amplitude.
4 5 6 7 8 9
x 10-7 60
80 100 120
Compensation capacitance [F]
Load voltage [V]
Load resistance = 300 Ohm Load resistance = 25 Ohm Load resistance = 10 Ohm Load resistance = 5 Ohm Resonance occurs
for C2 = 6.3e-7 [F]
Figure 3.5: Series connected compensation capacitance.
Figure 3.6 illustrates how the current through the load resistance varies with the compensation capacitance when the secondary side is parallel compensated and is connected to a voltage source with a constant amplitude.
4 5 6 7 8 9
x 10-7 6
7 8 9 10
Compensation capacitance [F]
Load current [A]
Load resistance = 0 Ohm Load resistance = 10 Ohm Load resistance = 20 Ohm Load resistance = 30 Ohm Resonance occurs
for C2 = 6.3e-7 [F]
Figure 3.6: Parallel connected compensation capacitance.
The value of the resonant capacitance, C2, is the same in the parallel compensated case as in the series compensated case.
C2= 1
ω02L2 (3.7)
Applying equation (3.7) to equation (3.5) and equation (3.6) gives the total impedance of the secondary side, Z2, when the secondary side is in resonance. The impedance of the secondary side series compensated by C2 is given by equation (3.8)
Z2=RLOAD , (3.8)
and the impedance of the secondary side parallel compensated by C2 is given by equation (3.9)
Z2= ω02L22
RLOAD−j ω0L2 . (3.9)
Combining equation (3.8) and equation (3.9) with equation (2.41) results in ZS, which is the total impedance of the secondary side as seen by the primary coil. When the secondary side is series compensated, ZS is given by equation (3.10)
ZS=ω02M2
RLOAD , (3.10)
and when the secondary side is parallel compensated, ZS is given by equation (3.11)
ZS=ω02M2(RLOAD−j ω0L2)
ω02L22 =M2RLOAD
L22 −jω0M2
L2 . (3.11)
This means that when the secondary side is parallel compensated, a phase shift is introduced into the system. That phase shift can only be compensated on the primary side.
3.5.2 Primary side impedance
Figure 3.7: Primary impedance with series compensation and with parallel compensation.
Figure 3.7 shows the total impedance of the primary side, Z1, as seen by the voltage source. When using series compensation the total impedance of the primary side is given by equation (3.12)
Z1=ZC1+ZL1+ZS , (3.12) and when using parallel compensation the total impedance of the primary side is given by equation (3.13)
Z1=ZC1(ZL1+ZS)
ZC1+ZL1+ZS . (3.13)
Resonance is achieved on the primary side if the total impedance on the primary side, Z1, is purely resistive. If the secondary side is series compensated, ZS is purely resistive as per equation (3.10), and the resonant capacitance on the primary side, C1 is given by equation (3.14)
C1= 1
ω02L1 . (3.14)
If the secondary side is parallel compensated, ZS has a reactive part as per equation (3.11), and the resonant capacitance on the primary side, C1 is given by equation (3.15)
C1= 1
ω02(L1−ML22) . (3.15)
The resonant capacitance on the primary side, C1, has the same value for the parallel compensated case as for the series compensated case. Applying equation (3.14) to equation (3.13) gives the total impedance of the primary side, Z1, when the primary side is parallel compensated and the secondary side is series compensated
Z1=ω02L12
ZS −j ω0L1 . (3.16)
Combining equation (3.16) with equation (2.41) gives the total impedance on the primary side, Z1, as a function of the total impedance on the secondary side, Z2,
Z1=Z2L12
M2 −j ω0L1 . (3.17)
Equation (3.17) puts some constraints on Z2 in order for the system to be in resonance. Z2 is divided into its resistive part RZ2 and its reactive part XZ2,
Z2=RZ2+j XZ2 . (3.18)
The criteria for resonance is now that the reactance of the secondary side compensates for the inductance of the primary coil, L1. The expression for this is given by equation (3.19),
XZ2=ω0M2
L1 . (3.19)
The reactance XZ2 is introduced into Z2 by adjusting the secondary compensation capacitance C2. C2= 1
ω02(L2−ML12) (3.20)
When the primary side is parallel compensated a phase shift is introduced to the the secondary coil, that brings the secondary side out of resonance. Using the compensation capacitance defined by equation (3.20) brings the secondary side back into resonance again.
Parallel compensation introduces phase shifts into the system that must be compensated for on the opposite side.
3.5.3 Primary coil with constant amplitude current
Figure 3.5 shows that the series compensated secondary side behaves as a voltage source, and figure 3.6 shows that the parallel compensated secondary side behaves as a current source. This is only true if the secondary side is fed by an induced voltage with a constant amplitude.
The voltage induced in the secondary coil, Uind, is proportional to the current through the primary coil, IL1,
Uind=jω M IL1 . (3.21)
The current through the primary coil, IL1, is determined by the input voltage, U1, and by the the total impedance of the secondary side as seen by the primary coil, ZS. When the primary side is series compensated the current through the primary coil is
IL1=U1 ZS
, (3.22)
and when the primary side is parallel compensated the current through the primary coil is IL1= U1
ZL1+ZS
. (3.23)
The current through the primary coil can be kept constant in amplitude if the input voltage is varied as a function of ZS. That would result in an induced voltage in the secondary coil, Uind, with a constant amplitude. Which in turn would result in the series compensated secondary side behaving as a voltage source, and the parallel compensated secondary side behaving as a current source.
The amplitude of the load voltage, ULOAD, when the secondary side is series compensated is given by equation (3.24)
ULOAD=ω0M IL1 , (3.24)
and the amplitude of the load current, ILOAD, when the secondary side is parallel compensated is given by equation (3.25)
ILOAD=M
L2 IL1 . (3.25)
However, this is only the case when the input voltage is regulated to keep the amplitude of the current through the primary coil constant. The following pages will show the behavior of the different compensation topologies when the inductively coupled power transfer is supplied by a constant amplitude voltage source.
