FEM Analysis of a PCB Integrated Resonant Wireless

Power Transfer

Žarko Martinović Danieli Systec d.o.o./Vinež 601, Labin, Croatia

e-mail: zmartinovic@systec.danieli.com

Roman Malarić Faculty of Electrical Engineering and Computing /

Department of Electrical Engineering Fundamentals and

Measurements, Unska 3, Zagreb, Croatia

e-mail: roman.malarić@fer.hr

Martin Dadić Faculty of Electrical Engineering and Computing Department

of Electrical Engineering Fundamentals and Measurements,

Unska 3, Zagreb, Croatia

e-mail: martin.dadic@fer.hr

Željko Martinović Combis d.o.o./Hektorovićeva 2, Zagreb, Croatia

e-mail: zeljko.martinovic@combis.hr

Abstract – The paper presents the results of the FEM analysis of

a printed-circuit-board (PCB) based wireless power transfer

(WPT) receiver/transmitter. The full-wave simulations were run

in HFSS. The design of the transmitter and the receiver was

based on the standard double-sided PCB FR4 fiberglass-epoxy

material. The WPT system was analyzed for different transmitter

and receiver distances (1mm – 100 mm), with and without

integrated capacitance. The influence of the additional

capacitance on resonant frequencies was analysed. Simulation

results were compared by extracting Z parameters (impedance

parameters) for different model scenarios.

Keywords – Wireless power transfer; Witricity; FEM, resonant

coupling; resonant frequency

I. INTRODUCTION

Wireless resonant power transfer was originally created by Nikola Tesla. His idea was to enable distribution of energy through the Earth's ionosphere [1][2][3][4]. The same concept gained a renewed interest in the last decade, mainly by the research team led by Marin Soljacic at MIT, who had described its physical behaviour and coined the term Witricity [5][6].

Wireless resonant power transfer from Fig. 1. consists of four main parts:

Power Source - Amplifier

Resonance Loops – Transmitter

Receiver Loops – Receiver

Load

Figure 1. Wireless power transfer components

Electromagnetic systems with the same specific frequency, can excite resonance due to electromagnetic coupling – to transfer energy at some distance. If those systems obtain the same resonant frequency – they can exchange energy. Transmitter side should have a constant supply, and receive side should have load perfectly matched [5]. In original paper [5] a separate single coil turn was introduced at the transmitter side as an impedance transformer, and later the same concept was used by other authors.

In order to design the entire transmitter system in the

double-sided PCB material, this concept is in this paper

replaced by a direct excitation of the transmitter coil, which is

made as a planar inductance at the one side of the board. In this

way, an easy integration of the transmitter coils and the

electronics is allowed, which further simplifies assembling of

the whole system. The resonant frequency of the planar coil is

determined by its inductance and the inter-turn capacitance.

The resonant frequency can be additionally tuned applying the

rectangular plates in the copper on the opposite side of the PCB

material, which serve as the additional parallel capacitances

and lowers the overall resonant frequency. Since the resonant

frequencies of the system depend of the distance of the

transmitter and receiver coils, while the accurate analytical

calculation of the overall lumped inductances and capacitances

of the coils is very difficult, the full-wave FEM simulation was

performed as an initial step in defining the demands for the

drive electronics of the system. The model was created in High

Frequency Structure Simulator (HFSS) according to the similar

models in papers [7][8][9]. The purpose of this paper is the

development of full wave Finite Element Method (FEM)

model of our electromagnetic system. Verified and simulated

model will enable us development of different optimization

technic and processes. By extracting of FEM model, it will be

created quadrupole network (Z - parameters).

II. COUPLED RESONATORS

Fig. 2 depicts a system of two coupled resonators. Here, sU

denotes voltage source, is angular frequency, sR is internal

resistance of the source, 1L is source inductance,

2L is device

inductance, 1,2M is mutual inductance,

1C is source

capacitance, 2C is device capacitance,

LR is load resistance, 1I

is source current and2I is load current.

Figure 2. Equivalent circuit for coupled resonator system

If we denote overall resistance at the transmitting side as 'SR ,

the input power can be expressed as [10]:

2122

221221

22

22

212

22

22

''

'

LRRLRMLLRRR

URRMLRRRP

LSLS

SLLSIN

(1)

while the output power can be expressed as:

2122

221221

22

2212

'' LRRLRMLLRRR

RUMP

LSLS

LSO

(2)

Based on the equations (1) and (2), the system efficiency is

[10]:

LLS

L

RRMLRRR

RM

22

122

22

2

212

' (3)

The transferred power splits into two peak values when the

coupling coefficient of the coils is large, where the lower

frequency corresponds to the odd mode. The higher frequency

corresponds to the even mode. At splitting frequencies, the

transferred power reaches its maxims. The odd mode allows

larger efficiency, and therefore it is mostly chosen as the

operating frequency of the system. The system efficiency

reaches its maximum at the natural resonant frequency [10].

III.PLANAR COILS

The inductance of the printed planar spiral inductors can be

estimated using several analytical expressions. Fig. 3 depicts

the square realization of the planar inductor.

The planar inductor is specified by the number of turns n, the

turn width w, the turn spacing s, the outer diameter dout , the

inner diameter din. Furthermore, the average diameter

inoutavg ddd 5.0 , and the fill ratio is

inoutinout dddd / .

Fig. 3. Square realization of the spiral inductor

Based on the work of Mohan et al. [11], the modified Wheeler

formula for the inductance of the square realization of the

spiral inductor is

75.2134.2

2

0avg

mw

dnL . (4)

There is also an expression based on current sheet

approximation and mean distances [11]:

22

013.018.0/07.2ln

2

27.1

avggmd

dnL (5)

The natural resonant frequency of a RLC system is determined by Thompson equation where L represents lumped inductance and C is lumped capacitance:

r

1f =

2π LC (6)

Capacitance value [9] is dependent on conductor dimension area (A), insulator thickness (d) and permittivity of the substrate ( ε ) and includes both inter-turn capacitance and

capacitances between turns and additional plates on the opposite side of the substrate. Inductance value is dependent on loop characteristics (e.g. lengths, width, and thickness). For a large number of turns it is increasingly difficult to calculate accurately the parasitic capacitance, due to the nonlinear adjacent winding capacitance [13]. This capacitance is typically on the order of a few pF. An estimation of the parasitic capacitance of the circular spiral inductors can be deduced from [14]:

off

i

S K

R

R

hKC

10

2

0

ln

2 (7)

where h denotes height of the traces, 0 is permittivity of

vaccum, iR2 is the inner radius of the second winding, iR2 is

the outer radius of the first winding. K is a constant of 0.3 and Koff is an offset of about 2.6 pF. Due to the very small height of the traces compared to their distance the results of this

analytical estimation are very inaccurate. Analytical expressions is especially inaccurate with the added rectangular elements, and the capacitance calculations should rely on the numerical models.

IV. Z-PARAMETERS

Fig. 4. Two-port

Fig. 4 presents a two-port connected to a source with the

source impedance Z1, terminated with load impedance Z2. The

input and output voltages can be generally expressed as

2121111 IzIzU (6)

2221212 IzIzU (7)

where z11 and z22 are open-circuit driving point impedances

and z12 and z21 are open circuit transfer impedances of the two-

port [12]. The impedance parameters (Z parameters) can be

easily extracted from the FEM solver for a fixed distance of

the receiving and transmitting coil, and further applied in the

calculation of the transferred power and system efficiency

using standard method of two-port analysis.

V. WPT MODEL OVERVIEW

Developed model is based on two identical PCB coils. At the top side is the receiver, and the bottom one is transmitter (Fig. 5). Transmitter and receiver plate has dimensions 80 x 80 mm with thickness 1 mm. They consist of spiral turns with 80 loops with 1 mm distance between them. The loops are made of cooper with thickness 0.035 mm (Fig. 6.). The capacitor is presented with four rectangular elements, which electrically breaks the path for the induced currents in the ground plane and allows easy implementation of the return conductor in the PCB. Strip length is 35 mm and width is 21 mm. Capacitor layers were made of copper. The insulator is FR-4 type. Advantages of FR-4 material are low cost and good integration properties related to PCB design development in electronic applications. Two external separate copper conductors are connected for the feed and the load of the system. Based on the equations (4) and (5), the inductance of the coils was estimated as Lmw=207 μH and Lgmd=218 μH.

Figure 5. Developed Wireless Power Transfer model in HFSS

Figure 6. Wireless power transfer model – spiral turns

Figure 7. Wireles power transfer model in HFSS with air box around

the model

On transmitter and receiver sides the appropriate lumped ports

were assigned, and the whole system was placed into an air

box with radiating boundaries (Fig. 7). First 30 simulations

were run for transmitter and receiver distances between 1 – 30

mm and after that we continued simulations with distances

equal to 40, 60, 80 and 100 mm. Analysis of resonant

frequency was the main output of simulations which were

conducted between 1 and 100 MHz. Maximum number of

passes was 6. The resonant frequency of the system was

targeted to satisfy the European regulations for the inductive

loop systems in the frequency range 9 kHz to 30 MHz, i.e.

ETSI EN300 330 [15].

VI. RESONANT FREQUENCY ANALYSIS

a) WPT model without additional capacitance

WPT model without capacitance (Fig. 8.) consists of two coils

and insulator boards with the identical dimensions. Simulation

was run with distances between coils equal to 1, 20, 40, 60,

80 and 100 mm, for all simulations presented in the paper.

Figure 8. WPT model without capacitance

Fig. 9 depicts the frequency response of the Z21 parameter for all examined distances. From Fig. 9. we can conclude that the main resonant frequencies is in the band 10 - 30 MHz, for all examined distances, with the biggest peak at 25 MHz and 40 mm.

Figure 9. Z21-parameters of model without capacitance

Since the wavelength was much greater than the physiscal dimensions of the system, the adaptive meshing of all models was based on the element length based refinment, with the restricted maximum lemgth of the elements in the whole domain (including air box) equal to 17.82 mm (wavelength was always below 3 m), and using the unrestricted number of elements.

b) WPT model with additional capacitance

WPT coils with additional capacitance elements (Fig. 10.)

have additional rectangular elements at the reverse side of the

PCB board.

Figure 10. WPT model with capacitive elements

Figure 11. Z21-parameters of model with additional capacitance

From Fig. 11. we can notice decreasing in resonant frequency by adding capacitance, which shifts main resonances below 20 MHz. The biggest resonant peak is now at 13 MHz and at a distance of 60 mm.

c) WPT model with additional capacitors and lateral

misalignment of the coils

In this subsection the influence of the lateral misalignment of

the coils is examined, Fig. 12. depicts the examined Witricity

model with lateral misalignment of the coils. Receiver and

transmitter boards are shifted lateraly for 5 mm.

Figure 12. WPT model with lateral shift of the coils

From Fig. 13. we can see that the frequency of the strongest peak is again 13 MHz with distance between plates equal to 60 mm. As we can notice, the frequency responses is similar to the model with the aligned coils, both in the shape and in the magnitudes. It may be concluded that the small lateral shifts does not influence the response of the system significantly.

Figure 13. Z21-parameters of the WPT model with the lateral shift

d) WPT conductor model

In the last subsection, the influence of the permittivity on the increase of the resonant frequencies of the substrate was examined for the initial model without additional rectangular capacitive elements. In this model the PCB substrate is replaced by the air. Such a conductor model represents simplified model, which is consisted of two loops with air gap distance (Fig. 14.).

For model without any insulators and additional capacitances, the results are presented in Figure 15. The decrease of the permittivity increased the frequencies of the resonance peaks for approximatelly 4 MHz, which is the absolute upper bound for possible change in the resonant frequencies due to different substrate materials (FR-3, Rogers), or the frequency dependance of the permittivity.

Figure 14. Conductor model of WPT

Figure 15. Z21-parameters of the WPT model consisted of coils only (without substrate)

VI. CONCLUSION

In order to properly design wireless power transfer system, it is necessary to carefully determine the resonant frequency of the system, as well as its frequency response.

Previous papers [7]-[9] shows a similar design where resonant frequency was adjusted according to the number of turns from the receiver and transmitter side, thickness of the substrate and dimensions of capacitors. In this paper approach was on adjusting the resonant frequency but with different parameters. In order to reach specific resonance frequency limits we changed shape of the possible additional capacitive elements.

As it can be seen from the FEM analysis, the main resonant peak of the system with additional capacitors is near 13 MHz, while without them it has been near to 20 MHz. Another set of simulations were run with transmitter and receiver coil axes shifted laterally for 5 mm, with no significant change in the response. For all presented scenarios, the impedance parameters (Z-parameters) were extracted, which allows easy calculation of the system response for different loads, as well as their matching regarding to the efficiency and the maximum power transfer.

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[15] ETSI EN 300 330 (draft), v.2.1.0 (2016-05), Short Range Devices (SRD), Radio equipment in the frequency range 9 kHz to 25 MHz and inductive loop systems in the frequency range 9 kHz to 30 MHz;