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Progress In Electromagnetics Research M, Vol. 28, 258–271, 2013

DESIGN OF A WIRELESS POWER TRANSFER SYSTEMFOR HIGH POWER MOVING APPLICATIONS

Saeed Hasanzadeh and Sadegh Vaez-Zadeh*

Center of Excellence on Applied Electromagnetic Systems andAdvanced Motion Systems Research Laboratory, School of Electricaland Computer Engineering, University of Tehran, Tehran, Iran

Abstract—In high power applications of wireless power transfersystems as Maglev, both a high transferred power and a high efficiencyare essential. However, these two requirements usually show dissimilarprofiles over a range of operating conditions. Magnetic and electricmodels for a capacitor compensated system are used to analyze theproblem. Using the analysis outcome, a compromise is made to cometo an acceptable design achieving both requirements. In particular,appropriate design parameters and resonance frequency are obtained.The analytical results are confirmed by 3D FEM analysis.

1. INTRODUCTION

High power moving objects can be supplied economically if appropriatewireless power transfer (WPT) sources exist. A maglev system, witha linear synchronous motor, for instance, requires primary windingsdistributed along the track, resulting in substantial increase in theconstruction and maintenance cost [1, 2]. Placing windings on themover plus a proper WPT system considerably reduces the cost. Asuitable structure for high power WPT systems should be designedto satisfy the best performance and meet the requirement of theapplication. The simplicity of the implementation is also essential inselecting the WPT structure.

In WPT for high power moving applications, the apparatususually includes a small air gap along the main flux path that linkstwo coils. A primary coil is located on the stationary base unit whilea secondary coil is located on the vehicle. The latter coil effectivelyreceives the power from the primary side through the air gap and

Received 22 October 2012, Accepted 25 January 2013, Scheduled 28 January 2013* Corresponding author: Sadegh Vaez-Zadeh ([email protected]).

Progress In Electromagnetics Research M, Vol. 28, 2013 259

delivers it to the vehicle. The power can be used immediately by atraction motors or can be stored for later use. Also, a WPT structurecan be constructed in conjunction with a magnetic levitation systemusing Hallbach arrays [3], permanent magnets (PMs) [4], passiveand self-controlled systems [5, 6]. Although the required power fora levitation system is usually lower than the power needed by apropulsion system, the combined system causes a reduction in totalweight of vehicle and improves its performance.

Different WPT structures for transferring high power form longprimary tracks to moving objects are considered. In particular, twotypes of WPT systems, i.e., with long bus bars and with long magneticmaterial cores are proposed [7]. Also, WPT structures for monorailsystems with U, S, E, Z and λ shapes are introduced [8]. Two WPTstructures are presented and analyzed for the linear servo motors [9].A WPT system consisting of a U shape pickup on the vehicle andthree wires as primary winding is proposed for supplying movablevehicles [10]. The mentioned works not thoroughly discuss the systemanalysis and design, considering practical limitations such as operatingfrequency when both high power transfer and high efficiency aredesirable. In order to attain high performances in WPT systems,capacitive compensations in both primary and secondary sides arerecommended to provide resonance conditions [11]. Resonance basedWPT systems save weight, space and cost of the system [12]. However,some special applications can be supplied by non-resonance basedWPT systems with proper designs [13].

Coaxial WPT systems including a straight primary wire passingthrough the center of a cylindrical secondary core with an air gaphave already been recognized. It can be used in Maglev as well as inwireless EV charging systems [14, 15] and power delivery system formining applications [16]. Also, a high power coaxial inductive powertransfer pickup is presented [17]. A partial design of the system isreported recently [18]. However, a systematic modeling, analysis anddesign procedure is not reported in the literature.

In this work, a coaxial WPT system as in Fig. 1 is considered.A mathematical model is used for the system analysis including thecompensating capacitors. The analysis includes the calculation oftransferred power, efficiency, coupling coefficient, etc.. Then, a designprocedure is proposed to achieve high efficiency and high transferredpower. The work extends the application of low power resonance-based inductive magnetic coupling to high power applications likeMaglev [19, 20]. The system parameters are obtained to meet thedesign specifications. Analytical results are verified by 3D FEMsimulations to confirm the design.

260 Hasanzadeh and Vaez-Zadeh

Figure 1. A schematic view ofthe gapped coaxial WPT system.

a

gc

d2γ

Secondary winding

Primary wire

Pickup core

P1 P2 P3 P4 P5 P7P6

Figure 2. Cross section of thegapped coaxial WPT system.

2. WPT MODEL

A physical model of the coaxial WPT system is presented in Fig. 1,where the primary side is a straight wire supplied by a high frequencysource on the ground. The secondary side consists of an open hollowcylindrical core of magnetic material and a pickup winding, all mountedon a maglev vehicle. A radial air gap takes the two sides apart.

2.1. Magnetic Modeling

Referring to Fig. 2, Ampere Law can be written as:

I1 =∮

H · dl = ϕ (Rm + Rg) (1)

where, Rm, Rg are the secondary core reluctance and the air gapreluctance respectively which can be calculated by a 3D model as shownin Fig. 1. The secondary flux, induced by I1, is then obtained as:

ϕ ' µ0

2π

ab

(d + a/2)µr

1 + γπ (µr − 1)

ln(

a + d

d

)I1 (2)

where, a, b, d, and γ are shown in Figs. 1 and 2. Also, γ can berepresented as:

γ ' sin−1

(gc

2d + a

)(3)

Comparing (2) to a linkage flux of a current carrying rectangularwire, an equivalent permeability is defined as:

µav =µr

1 + γπ (µr − 1)

(4)


Then, the secondary linkage flux is given by:

λav ' N2µ0µav

2πab

ln(

a+dd

)

(d + a/2)I1 (5)

As a result, the system mutual inductance is obtained as:

M ' N2µ0µav

2πab

ln(

a+dd

)

(d + a/2)(6)

Also, Applying Ampere Law, the primary and secondaryinductances are obtained respectively as:

L1 ' µ0lp8π

[1 +

12

ln

(√d(d + a)

r1

)](7)

L2 ' N22 µ0µrab

π(2d + a) + µrgc(8)

where, lp, N2, µr and r1 are the length of primary wire, the number ofsecondary turns, the relative permeability of the secondary core andthe primary wire radius respectively.

2.2. Electric Modeling

The system is represented by the equivalent circuit of Fig. 3. Theprimary wire and the secondary winding are modeled by two coupledRL circuits of impedances R1+jωL1 and R2+jωL2 respectively where:

R1 =ρcuJ1

I1lp (9)

R2 =ρcuJ2

I2(2N2(a + b)) (10)

and J1, J2 and ρcu are the primary and secondary winding currentdensities and the copper resistivity respectively.

The series compensating capacitors C1 and C2, are connected tothe primary and secondary sides respectively. The capacitances aredetermined such that a common resonance frequency of ω0 is providedin two sides as:

ω0 =1

L1C1=

1L2C2

(11)

The transferred power passing through the air gap is readilyobtained from the equivalent circuit as [21]:

P2 =ω2

0M2

RLI21 (12)


R1C1

L1 L2

R2 C2

RL

M

V1

Figure 3. Circuit model of the coaxial WPT system.

Also, the system efficiency can be calculated as follows:

η =RL

RL + R2

1

1 + R1(RL+R2)ω2

0M2

(13)

It is proved that the maximum efficiency occurs at the resonancefrequency, f0, and the maximum power is transferred at this frequencyor fL and fH at obtained as:

fL =f0√1 + k

, fH =f0√1− k

(14)

where, k is the magnetic coupling coefficient given by:

k =M√L1L2

(15)

If the system is compensated by series capacitors, it is possible tointroduce an overall coupling coefficient as [22]:

kov = k(ω0

ω

)√1 +

(ωL2

R2 + RL

)2

(16)

A desirable frequency range, over which acceptable transferred powerand efficiency are obtained, can be defined as:

BW = fH − fL (17)

3. SYSTEM ANALYSIS

Using the system models presented above, a WPT system for Maglevapplications is analyzed in this section where the system rated valuesand parameters are already known. The input and transfer powersof the system, plus the system efficiency are plotted in Fig. 4 by(12) and (13) respectively. It is seen that the input and transferpowers have similar shapes There are two frequencies, fL and fH , overwhich the input and transfer powers reach their maximum values. Themaximum transfer efficiency at the resonance frequency, f0, located in


fL f0 fH

Input power

120

100

80

60

40

20

0850 900 950 1000 1050 1100 1150

Frequency (Hz)

1

0.9

0.8

0.7

0.6

0.5

0.4

Eff

icie

ncy

Po

wer

(k

W)

Transferred Power

Efficiency

Figure 4. Input and transferpowers and efficiency versus fre-quency.

0

Input power

350

300

250

200

150

100

50

0 500 1000 1500 2000Resonance Frequency (Hz)

1

0.9

0.8

0.7

0.6

0.5

0.4

Eff

icie

ncy

Pow

er (

kW

)

Transferred Power

Efficiency

Figure 5. Input and transferpowers and efficiency versus reso-nance frequencies.

the frequency boundary of fL < f0 < fH , mentioned in the previoussection, is seen in Fig. 4.

The system efficiency and input and transferred power areplotted versus resonance frequencies in Fig. 5. It is seen that thetransfer efficiency drops gradually with the increasing transfer power.Therefore, there is a conflict between high efficiency and high transferpower. However, in high power applications like Maglev both highefficiency and high transfer power are essential. As a result, acompromise between high efficiency and high transfer power is needed.

It is wise to consider the knee of the efficiency locus of Fig. 5 as anacceptable operating region. This is shown in Fig. 5 by a rectangularzone. Choosing a desired frequency on the acceptable region, theprimary and secondary capacitances are obtained from (11) where L1

and L2 are known.

4. WPT SYSTEM DESIGN

The design specifications of a system are given in Table 1. using thisdata, one would find RL = 1.25Ω, and I2 = 200. Also, having used (12)yields:

M =√

PoutRL

ω0I1(18)

Choosing an appropriate current density, R1 is given by (9).Assuming that there are series resonance circuits in both sides of thesystem, the following relationship is held:

V1 = R1I1 + jMωI2 (19)


Table 1. Given data for system design.

Characteristics:Operating Frequency, f = 1 kHz

Output Power, Pout = 50 kWLoad Voltage, V2 = 300 V

Primary Current, I1 = 500ACopper Resistivity, ρcu = 2.05 ∗ 10−8 Ω-mPrimary Length lp = 200m

Pickup Core Material:Maximum Flux Density, Bm = 1.4TRelative Permeability, µr = 105

Considering the system structure of Fig. 2, the air gap length, gc,is chosen four times the radius of the primary wire, r1, in order toprevent undesirable physical contact. As a result, r1 is obtained as:

r1 =√

I1

πJ1(20)

gc = 4r1 (21)Considering physical limitation of secondary winding, the values

of a and d are obtained. The values of γ and µav are then calculatedfrom (3) and (4).

Deciding on proper values of b and N2, M is calculated from (18).An appropriate value for a can be regarded as a = b/3. An examinationof M versus N2 and a is carried out as presented by Fig. 6 before N2 isdecided. Fig. 6 shows that for a fixed length of the secondary windingwire, the less the turn number, N2, and the higher the value of a, thelarger the mutual inductance, M .

Having, J2, it is possible to find out R2 by using (10). The valueof r2 is then calculated by (20). Then L1 and L2 are calculated. Also,the compensating capacitors are determined as:

C1 =1

4π2f20 L1

, C2 =1

4π2f20 L2

(22)

Finally, the system efficiency is obtained by using (13). The fluxdensity is then examined in order not to saturate so that:

Bm ≤ MωI1

2√

2πf(ab)(23)

It is possible to get close to a system design with good efficiencywithout saturation. It is achieved by choosing appropriate dimensions,


0

0. 5

1 5

10

15

20

25

0

50

100

150

Turnswidth (m)

Mu

tual

ind

ucta

nce (µ

H)

Figure 6. Mutual inductance versus turns and width of secondarycoil.

operating frequency and current densities. Such a design presented inTable 2. It is wise to mention if the designed specifications are notmet, some of the parameters are to be reconsidered. For instance, theprimary and secondary current densities must be revised. This in turnchanges many other parameters.

5. DESIGN EVALUATION

A 3D FEM analysis is carried out on a certain resonance frequencyto evaluate the design. A 2D FEM cannot be used for the systemanalysis since the primary side is much longer than the secondary sideand consists of two parts, i.e., a part coupled to the secondary sideand an uncoupled part. The results of 3D FEM are illustrated inFigs. 7(a) and (b) by two and three dimensional graphs respectively.It is seen in Fig. 7(a) that high flux densities appear in and around theair gap, while the flux density in the secondary core remains limited.The magnetic flux vectors are depicted in Fig. 7(b). It is seen that thevector values are independent from the secondary core length. Also,regardless of a relative speed between the primary and secondary,end effects are not visible. This is in contrast to a linear inductionmachine since in this case the flux produced by the primary wire isfixed throughout the wire in every instance. While in an inductionmachine the flux distribution is sinusoidal.

An evaluation of flux density distribution along with a line passingthrough the center of primary wire and the middle of the air gap ispresented in Fig. 8. The flux density, in particular, is considered at


(a) (b)

Figure 7. 3D FEM results. (a) Two dimensional graph, (b) threedimensional graph.

P1 P2 P3 P4 P5 P6 P7

Flu

x D

ensi

ty (

T)

0.6

0.5

0.4

0.3

0.2

0.1

0

Path

Figure 8. Flux density alongwith a path through the pickupcenter.

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600

700

gap (mm)

Eq

uiv

ale

nt R

elat

ive P

erm

eab

ilit

y µ

rAnalytical

FEM

0

10

20

30

40

50

60

70

Lea

kag

e F

lux

φl (

mW

eb

)

µr

φ leakage

Figure 9. Equivalent relativepermeability and leakage fluxversus crack gap.

seven points P1–P7, in Fig. 2 and Fig. 8. It is seen in Fig. 8 thatthe flux density inside the secondary core, from P1 to P2, increasesrather linearly and stopped down to zero at the core border. It is startincreasing rather exponentially towards the primary wire up to pointP3. The flux density distribution is also shown inside and outside thewire from P3 to P4 and from P4 to P7 respectively. At P5 the leakageflux is substantial; while, the flux density in the middle of the air gap,point P6, reaches a local maximum.

The equivalent permeability, µav, is shown in Fig. 9 versus air gapof the secondary coil. It is seen that µav tends towards a certain valueof π/γ when the air gap gets increased. The leakage flux in air gap isalso depicted in Fig. 9 showing that it keeps increasing exponentiallywith the increasing air gap. The FEM results for µav are also depictedin Fig. 9 confirming the analytical results.

The system analysis on two resonance frequencies is done by bothanalytical method and FEM. The corresponding efficiencies at thesefrequencies are plotted in Figs. 10 and 11 for the sake of comparison.


Table 2. Calculated parameters of the WPT system.

Symbol Quantity ValueRL Load Resistance 1.8 ΩI2 Secondary Current 167AM Mutual Inductance 95.5µHJ1 Primary Current Density 2 ∗ 106 A/m2

R1 Primary Wire Resistance 0.0164ΩS1 Primary Wire Area 250mm2

r1 Radius of Primary Wire 8.92 mmgc Gapped Distance 3.57 cmd Distance Between Two Sides 20 cma Width of Secondary Winding 15 cmγ Gapped Corresponding Angle 3.72 deg

µav Equivalent Relative Permeability 48.3667b Length of Secondary Winding 45 cm

N2 Turn Number of Secondary Winding 72J2 Secondary Current Density 4 ∗ 106 A/m2

R2 Secondary Resistance 0.0424Ωr2 Radius of Secondary Wire 3.64 mmS2 Secondary Wire Area 42 mm2

Req2 Reflected Secondary Impedance 0.1954 ΩV1 Source Voltage 106 VL1 Primary Inductance 27µHL2 Secondary Inductance 12.3 mHk Coupling Coefficient 0.167C1 Primary Capacitor 940µFC2 Secondary Capacitor 2µFη System Efficiency 90%

Bm Maximum Flux Density 0.5 T

It also shows that the system efficiency is improved with an increasingresonance frequency. The figures confirm the validity of the analyticalresults as they are close to the FEM results. It is seen that the FEMresults adopt the analytical results in the high power ratings as well.The exerted parameters in the analytical model are imported from theFEM analysis of the WPT model.


0 40 80 120 1600.1

0.3

0.5

0.7

0.9f0 = 400 Hz

Transferred Power (kW)

Analytical

FEM

Efficiency (η)

Overall Coupling (kov

)

Figure 10. Efficiency andoverall coupling coefficient versustransferred power at resonancefrequency of 400Hz.

0 40 80 120 1600.1

0.3

0.5

0.7

0.9

1f0 = 1000 Hz

Transferred Power (kW)

Analytical

FEM

Overall Coup ling (kov

)

Efficiency (η)

Figure 11. Efficiency andoverall coupling coefficient versustransferred power at resonancefrequency of 1000 Hz.

Figure 10 depicts the system efficiency and overall couplingcoefficient versus the transferred power at a resonance frequency of400Hz. It is seen while the overall coupling coefficient decreasesmonotonically by increasing the transferred power; the systemefficiency reaches a maximum of 82% before decreasing with theincreasing transferred power. The same situation is repeated inFig. 11 for a resonance frequency of 1000 Hz. In Figs. 10 and 11,a compromise between kov and transferred power can be found onthe knee of kov curve where both kov and transferred power are notlow. It is interesting to notice that the maximum efficiency in bothfrequencies almost coincides with the knee of kov. Therefore, anoptimal operating point can be found for every resonance frequencyat which the transferred power, system efficiency and overall couplingcoefficient are acceptable.

6. CONCLUSIONS

Presenting magnetic and electric models for a capacitor compensatedwireless power transfer system, the transfer power, the efficiency andthe coupling coefficient are analyzed for a high power application. It isobserved that the transferred power and efficiency versus the resonancefrequency follow different trends. In fact, these two requirementscouldn’t be achieved at the same frequency at the first glance. Inother hand, the high transferred power is obtained at a low resonancefrequency but the high efficiency is achieved at a high resonancefrequency. Fortunately, a compromise can be reached to achieveacceptable transferred power and high efficiency at the knee of the


coupling coefficient curve. This is done by a graphical determinationof an acceptable operating region on the knee. The analytical resultsare verified by 3D FEM analysis. The proposed analysis and design rulecan be used as a tool for more accurate system design and optimization.

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