Analytical Sizing Methodology for Inductive Power Transfer Systems

Iker Porras Arteaga CEIT, San Sebastian, Spain.

Electronics & Communications Department iporras@ceit.es

Miguel Martinez-Iturralde TECNUN, University of Navarra, San Sebastian, Spain.

Electrical, Electronic and Automation department

Federico Martin Ibañez CEIT, San Sebastian, Spain.

Electronics & Communications Department

Ibon EloseguiCEIT, San Sebastian, Spain.

Electronics & Communications Department

Abstract—In this paper an analytical sizing methodology for inductive power transfer (IPT) systems is proposed. The methodology is based on the inductance coupling principle. It allows to obtain the geometry (dimension of the coil, number of turns, according to the geometry type: circular, rectangular, square...) and the performance of an IPT system from just a few parameters: nominal power, nominal voltage in primary and secondary winding, air gap and operation frequency. This procedure has been implemented using Matlab, where different prototypes can be obtained in a very short time, reducing significantly the effort during the design process. The validation of the methodology has been carried out through finite element method (FEM) software and the construction and experimentation on a prototype. This paper includes a sensitivity analysis regarding coil sizing methodology, with the introduction of misalignment between windings.

Keywords—Inductive Power Transfer Systems; sizing; analytical methodology; coils; mutual inductance; contactless energy transmission

I. INTRODUCTION

Global pollution, fossil fuels reduction and green care policies are motivating an increase on the electric car research. One of the main drawbacks of these vehicles is their reduced driving range due to the low energy density of the available electric batteries and the infrastructure required to charge them. Among the existing proposals, such as improved battery systems [1] or the use of refueling places where car batteries are changed in minutes [2], the inductive power transfer system is one of the most investigated in the last years. These systems are based on the energy transmission from a source to a receiver, without any electrical connection between them. Therefore they have several advantages: they work well in hostile environments because of the electric isolation, require less maintenance costs than traditional charging posts and are more user-friendly. Nowadays, there are many applications where this technology is applied such as biomedical devices[3], consumer electronic appliances [4] and electric vehicles [5].

Several authors propose design methodologies [6, 7] for IPT systems, all of them relying on finite element method

(FEM) software to calculate the system inductances. In [8, 9] an analytical calculation for a given IPT system that includes expressions for the mutual inductance is developed, but no global sizing methodology is offered. This paper proposes a analytical sizing methodology that allows obtaining the complete geometry, the winding characteristics and the performance of an IPT system, by introducing only five parameters. This methodology allows to perform a wide range of sensitivity analysis for different compensation topologies in both primary and secondary coils such as series-series, series-parallel, parallel-series and parallel-parallel. In addition, it allows to study the effects of the misalignment between coils.

II. IPT ANALYTICAL SIZING METHODOLOGY

The analytical sequence of the proposed methodology is shown in Fig. 1. In this section, every part of the flowchart will be described.

Fig. 1. Analytical Sizing Methodology flowchart

A. IPT Specifications and system constraintsThe first design step is to define the “IPT system

specifications”: nominal power (Pn ), nominal voltage in the primary (input, Vin) and secondary windings (load, VL), air gap length (hagap) and operation frequency (f). Furthermore, the “Design constraints” must be defined. These constraints are: compensated power factor (cos ) in the load, minimum expected efficiency ( ), maximum current densities for the primary and secondary windings (Jp and Js) and maximum IPT physical dimensions (restricted by the application).

B. IPT analitycal sizing of the geometry By defining the power, the required operating voltages and

the power factor, the primary and secondary winding currents are implicitly specified (Ip and Is).

Initially, both primary (Ep) and secondary (Es) windings electro motive force values (EMF) are approximated to Vin and VL. Once the EMF and the currents are calculated, a first approximation to the system’s mutual inductance is obtained in (1). This approximation will be used as an objective for the initial geometry sizing process.

fIjE

fIjE

Mp

s

s

p

22 (1)

Among the different possible combinations of geometries and coil turn numbers, only a few of them fulfill the calculated mutual inductance. The objective is to define the relation between the size and the total number of turns of the primary and secondary windings, in order to obtain this mutual inductance.

First, the wire cross section is sized according to the calculated current and the settled current density in the design constraints. The wire diameter is selected as small as possible in order to reduce the high frequency effects (Proximity effect and skin effect) and the number of parallel wires per conductor (strands) is selected according to the desired current density.

Afer that, the primary and secondary winding dimensions are imposed, with a small number of turns. With the preliminary geometry and turn number, the coil inductances are calculated using the Neumann expression (2).

1 2

'

4 rdldlNNM sp

(2)

Since Neumann's formula depends on coil positioning and geometry, expression (2) can take into account misalignments in the three axes. Therefore, the Neumann formula adapts very well to this methodology. If the obtained mutual inductance does not match the one calculated in (1), loop 1 in Fig. 1 adjusts the geometry and turn number until the convergence is reached. Once the system is sized and every geometric parameter is obtained, the systems performance is evaluated.

C. IPT system CalculationThe first step to calculate the system performance is to

calculate the resistances of the primary (Rp) and the secondary (Rs) windings. As the working frequency is in the kHz range, in order to correctly estimate the copper resistance, Dowell’s formula is applied (3) [10].

2cos2cosh2sin2sinh1

32

2cos2cosh2sin2sinh

22

2

mdlNR

cu

(3)

Where N is the number of turns, dcu is the diameter of the wire, m is the number of layers, l is the length of the coil per turn, is the conductivity of the material and (4) is the penetration ratio:

cud (4)

where (5) is the skin depth and it's calculated as:

2 (5)

where is the pulsating frequency of the waveform

If Litz wire is used, more precise results can be obtained by using expression (6) [11].

22

2

4

2 24116

461

1921

2m

pN

dN

lNR

fst

cust

(6)

2

cud (7)

where Nst is the number of strands per conductor and pf is the packing factor of the strands in the coil section.

The IPT systems work with large air gaps, resulting in a high leakage flux and a small coupling between the primary and secondary windings. In order to compensate for the excessive reactive power due to the leakage and to improve the efficiency of the system, capacitors are usually introduced. There are several compensation topologies: the Series-Series (SS) compensation (Fig. 2), the Parallel-Series (PS) compensation (Fig. 3), the Series-Parallel (SP) compensation (Fig. 4) and the Parallel-Parallel (PP) compensation (Fig. 5). A precise description of the calculation of the capacitors can be found in [12].

There are different topologies for the compensation of inductive charging systems, each with their own advantages and drawbacks. The choice of the compensation topology depends mainly on the application; being a major issue to be addressed when designing an IPT system. If the secondary is

compensated in series, at the nominal resonance frequency there is no reflected reactance. Owing to this reason, the primary can be compensated in series independently of the mutual coupling and the load value. If the secondary is compensated by a parallel capacitor, a capacitive reactance is reflected. In this case if the primary is compensated in series, the compensation is dependent on the coupling but not on the load. But if the compensation of the primary is in parallel the compensation depends on the coupling and the load.

As a conclusion of the previous paragraph, the series-series compensation has clear advantages. The secondary reflected impedance has only resistive part and in consequence, all the secondary power is active power, resulting in a unity power factor in the secondary. Additionally, the compensation capacitor values are independent of the mutual coupling and the load. This is perfect for systems where the load is variable and the mutual coupling may vary due to misalignment in the three axes.

Fig. 2. Circuit with SS compensation

Fig. 3. Circuit with PS compensation

Fig. 4. Circuit with SP compensation

Fig. 5. Circuit with PP compensation

Afterwards, the voltages and currents of each part of the circuit (Fig. 2, Fig. 3, Fig. 4 or Fig. 5) are calculated to obtain the system performance. Those parameters will depend on the different compensation type. At this point, the second convergence loop (Convergence in Fig. 1) is activated in order to achieve currents and voltages convergence.

For the Fig. 2 circuit (System with SS compensation) the Kirchhoff's second law is applied in the input and the output of the circuit, resulting in the next equations (8, 9).

sp

pppin IMjI

CLjRV 1

ps

ssLs IMjI

CLjRR 10

Where inV is the input current of the circuit, Cp and Cs are the compensation capacitors of the primary and the secondary windings respectively, Lp and Ls are the inductances of the primary and the secondary windings respectively, Rp and Rsare the resistances of the primary and the secondary windings and RL is the load that will be feeded.

In the case of a PS compensation as can be seen in Fig. 3, the resulting equations are described by (10, 11):

p

ssLs

ppin I

CLjRR

MLjRV1

22

ps

ssLs IMjI

CLjRR 10

In the case of the SP compensation shown by Fig. 4, the resulting equations are (12, 13):

sp

pppin IMjI

CLjRV 1

p

L

Lsss

LssLs

IMj

IRCRLj

RCLRR 2

0

Where LI is the current through the load.

For a PP compensation (Fig. 5), the resulting equations are (14, 15):

sLssL

psLpppin CRjLjRR

IMCRjILjRV

11 22

pLLsss IMjIRILjR0

According to the calculated values of the electric magnitudes, the system’s winding copper losses and iron losses are calculated. The core losses can be calculated by the Steinmetz equation described in [13]. Sometimes the manufacturer provides a curve fitting constants for one

sinusoidal voltage excitation. For non sinusoidal cases the Improved Generalized Steinmetz Equation (IGSE) was introduced [14].

Finally, the system efficiency is calculated. If the system does not meet specifications, loop 2 (Fig. 1) is enabled in order to modify some design constrictions such as the current densities. Other modifications may be the number of strands of the conductor, the diameter of the strands or the insulation of the strand. Changing those variables, the coil resistance will be modified and the system can be optimized to have the needed specifications. Other possible changes to apply are the material and the geometry of the core or to use a coreless system. The use of ferromagnetic elements add the advantage of improving the coupling between coils, but the disadvantage of higher losses. In these cases, a compromise has to be reached according to the importance of these parameters in the final application. As this method is generic it may be considered the use of ferromagnetic elements, even though the presented prototype is coreless due to its major advantages for the final application.

All these changes can be made to optimize the system’s behavior and achieve the needed specifications.

III. METHODOLOGY VALIDATION

In order to validate the methodology, an IPT prototype has been designed and built. The IPT design specifications and constraints are collected in Tables I and II. The obtained geometrical dimensions and performance characteristics according to the sizing methodology are shown in Tables III and IV. Because of the final application, the system has been made of rectangular shape. As is focused on the electric vehicle application, the relationship between the width and length of the receptor coil has been set to 2.6, because the ratio of vehicles, i.e. the Toyota Prius, have this relationship between the length and width of the vehicle.

Different options of coil area sizes and number of turns have been evaluated, as can be seen in Fig. 6 and Fig. 7. In Fig. 6 can be seen that the third system presents a higher efficiency value than the other systems. Moreover Fig. 7 shows that the lowest value of required copper volume for the coils is also the third system’s. For these reasons this is the one selected for manufacturing. The complete system specifications are in Table III.

Fig. 6. Efficiency vs system area

Fig. 7. Cooper volume vs system area

The validation of the proposed methodology has been carried out by comparing the analytical results to results from electromagnetic FEM simulations, to results from an electric simulation software (Cadence) and to measurements from the constructed prototype.

TABLE I. SYSTEM SPECIFICATIONS

Specification Values

Output power (Pn) 3300 W

Output voltage (VL) 144 V

Input voltage (Vin) 240 V

Air gap (hagap) 200 mm

Frequency (f) 100 kHz

TABLE II. SYSTEM CONSTRAINTS

Constraint Values

Max. length of the primary and secondary windings 1200 mm

Max. width of the primary and secondary windings 1200 mm

Max. primary winding current density 5 A/mm2

Max. secondary winding current density 5 A/mm2

cos (winding compensated in the load) 1

Min. efficiency 90%

TABLE III. PROTOTYPE CHARACTERISTICS

Characteristic Values

Length of the primary winding 775 mm

Width of the primary winding 450 mm

Length of the secondary winding 775 mm

Width of the secondary winding 300 mm

Air gap (hagap) 200 mm

Number of turns on the primary winding (Np) 10

Number of turns on the secondary winding (Ns) 6

TABLE IV. PROTOTYPE PERFORMANCE

Characteristic Values

Output power 3300 W

Frequency 100 kHz

Input current (Iin) 15.7 A

Output current (IL) 22.91 A

Input voltage (Vin) 227 V

Output voltage (VL) 144 V

Primary winding self-inductance (Lp) 194.21 μH

Secondary winding self-inductance (Ls) 61.202 μH

Mutual inductance (M) 14.338 μH

Efficiency 92.6%

FEM simulations using the commercial CST Studio Suite have been conducted (Fig. 8 and Fig. 9). Table V shows the results for the calculation of the inductances, both with the analytical and the FEM methods. As it can be seen the error values are below 4%, which is deemed acceptable for validation purposes.

Fig. 8. Flux density lines between the primary and the secondary windings

Fig. 9. Cut view of the flux density lines between the primary and the secondary windings

TABLE V. INDUCTANCE CALCULATION

Magnitude Analytical FEM Error

Primary winding self- inductance (Lp) 194.2 μH 202.0 μH 4.0%

Secondary winding self- inductance (Ls) 61.20 μH 63.23 μH 3.2%

Mutual inductance (M) 14.34 μH 13.85 μH 3.5%

In order to check the validity of the considered electrical circuit, Cadence software has been used. In Table VI the results obtained analytically and with the electric simulation software can be compared. Its graphic representation is in Fig. 10.

TABLE VI. ANALYTIC VS SIMULATIONS

Magnitude Analytical Cadence Error

Power 3300 W 3273 W 0.80% Frequency 100 kHz 100 kHz (fixed) Primary winding current 15.7 A 15.6 A 0.36% Secondary winding current 22.91 A 22.82 A 0.38% Primary winding voltage 227.0 V 226.3 V 0.25% Secondary winding voltage 144.0 V 143.4 V 0.41% Efficiency 92. 6% 92.4% 0.20%

0 0.5 1 1.5 2 2.5 3

x 10-5

-400

-200

0

200

400

Vp(

V)

Time (s)

Input voltage and current

0 0.5 1 1.5 2 2.5 3

x 10-5

-40

-20

0

20

40

Ip(A

)

IpVp

0 0.5 1 1.5 2 2.5 3

x 10-5

-400

-200

0

200

400

Vs(

V)

Time (s)

Output voltage and current

0 0.5 1 1.5 2 2.5 3

x 10-5

-40

-20

0

20

40

Is(A

)

IsVs

Fig. 10. Primary and secondary operation graphics

Finally, results for the built prototype (Fig. 11) are given. The converter used to feed the system is a DC-AC resonant converter.

In Table VII results for the measurement of the system inductances are compared to the results given by the analytical calculation method, giving an error of less than 4%.

Fig. 11. DC-AC converter (left) and the IPT system (right).

TABLE VII. PROTOTYPE CHARACTERISTICS

Magnitude Analytical Measurements Error

Primary winding self- inductance (Lp) 194.2 μH 199.7 μH 2.8%

Secondary winding self- inductance (Ls) 61.20 μH 62.67 μH 2.4%

Mutual inductance (M) 14.34 μH 14.85 μH 3.5%

In Table VIII results for the system performance are compared to the results given by the analytical calculation method, giving an error of less than 9%.

TABLE VIII. SYSTEM PERFORMANCE

Magnitude Analytical Measurement Error

Power 3300 W 3501 W 6 % Frequency 100 kHz 100 kHz (fixed) Primary winding current 15.7 A 16.42 A 5.6% Secondary winding current 22.91 A 24.9 A 8.6% Primary winding voltage 227.0 V 230.8 V 1.6% Secondary winding voltage 144.0 V 140.6 V 2.4% Efficiency 92. 6% 92.4% 0.20%

In Fig. 12 experimental results of the system inductances for different misalignments are compared to the results given by the analytical calculation method, giving an error of less than 8%.

Fig. 12. Error in mutual inductance calculation for X and Y misalignments

The values of efficiency in the next figures are the comparison between the systems misaligned and the system without misalignments.

In Fig. 13 the efficiency calculated in the analytical methodology is compared with the efficiency measured in the prototype with x-axis misalignments (The misalignment percentage in x-axis is related to the wide of the designed coil), giving an error of less than 7% between them with misalignments. As can be seen the system efficiency is reduced around a 24% in the values calculated in the analytical methodology with misalignments and around the 30% in the final prototype.

Fig. 13. Analytical vs measured system efficiency regarding X axis misalignment

The same comparison is done with the y-axis misalignments (The misalignment percentage in y-axis is related to the large of the designed coil). Fig. 14 shows error values lower than 3%. The calculated system efficiency is around 12.6% lower with a 50% misalignment while the measured values drops a 16%.

Fig. 14. Analytical vs measured system efficiency with Y axis misalignment

Finally, in Fig. 15 an efficiency graph of the prototype is shown, taking into account the misalignments in x and y axes (X and Y at the same time). Comparing with the results given by the analytical calculation method the error is lower than 7%. In the case of the analytical methodology the efficiency drop with a 50% misalignment in x and y axes is around 55% and in the prototype around the 60%.

Fig. 15. Measured system efficiency with X and Y misalignment

IV. CONCLUSIONS

An analytical sizing methodology for an IPT system is proposed, which can size a complete IPT system in a few seconds from only five input parameters and some design constraints. The obtained results have been validated through FEM simulations, electrical circuit simulations and an IPT prototype. The error values in inductances calculation with reference to measured values are below 4% without misalignments. In the case of the efficiency calculation with misalignments the highest error is around 7%. This method permits a simple and quick sizing and optimization of IPT systems.

ACKNOWLEDGMENT

Iker Porras thanks to the Asociación de Amigos of the University of Navarra for its financial support.

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