Optimal Design and Analysis of the Stepped Core for Wireless Power Transfer Systems

The key of wireless power transfer technology rests on finding the most suitable means to improve the efficiency of the system. The wireless power transfer system applied in implantable medical devices can reduce the patients’ physical and economic burden because it will achieve charging in vitro. For a deep brain stimulator, in this paper, the transmitter coil is designed and optimized. According to the previous research results, the coils with ferrite core can improve the performance of the wireless power transfer system. Compared with the normal ferrite core, the stepped core can produce more uniform magnetic flux density. In this paper, the finite element method (FEM) is used to analyze the system. The simulation results indicate that the core loss generated in the optimal stepped ferrite core can reduce about 10% compared with the normal ferrite core, and the efficiency of the wireless power transfer system can be increased significantly.


Introduction
Wireless power transfer technology has a history of more than one hundred years.Since the nineteenth century, Nikola Tesla proposed two coils which were named as Tesla coil to transfer the power between the two coils wirelessly [1].Over the decades, after Tesla, the wireless power transfer system developed very slowly because of the limit of the electronic technology [2].With the rapid development of modern technology in the new century, wireless power transfer system has great application potentials especially in implantable medical devices.
With the advent of power electronics, various implanted medical devices, such as deep brain stimulator, nerve stimulator, and cardiac pacemaker, are widely used in the medical field [3][4][5].These devices can effectively prolong the patients' lives.In order to alleviate the patients' financial burden and their suffering, the technology of wireless power transfer has become popular in biomedical engineering.
Before the magnetic resonant coupling method is proposed, the wireless power transfer technology mainly has two types: one is far field method which focuses on microwave power transfer system; the other one is near field method which is applied in electrical toothbrush with very small distance.The far field method can transfer power over a long distance but the power transfer efficiency is very small.In addition, the systems using the far field mode need complicated detection devices to realize the power transfer from point to point.In the millimeter distance, the near field method can achieve high power transfer efficiency.When the power transfer distance is up to centimeters, the power transfer efficiency of near field systems is only about 1% [6][7][8].
In 2007, the MIT research team published their findings in Science, and since then, the magnetic resonant coupling method has become the hottest methodology to realize wireless power transfer system.In order to distinguish from the above two methods, this method is named as midfield method.In this method, the power transfer distance can 2

Transmitter coil
Receiver coil Figure 1: (a) Wireless power transfer system without a core.(b) Wireless power transfer system with a core.be prolonged significantly with relative high power transfer efficiency [9,10].
In this paper, the ferrite material is employed as the core to be applied into the wireless power transfer system.According to the application of the system, a new shape of the transmitter coil is proposed to improve the performance of the system.In addition, the stepped ferrite core is optimized to reduce the core loss and increase the efficiency of the wireless power transfer system.

Structure of Wireless Power Transfer Systems
Figure 1(a) presents the studied analyzing common wireless power transfer system.It consists of two coils with the same resonant frequency, and the coils are connected with compensated capacitors.In order to stretch the power transfer distance, ferrite cores are inserted into the two coils as shown in Figure 1(b).Previous research results [11] indicate that the receiver coil with the ferrite core can attain more magnetic flux linkages which are generated by the transmitter coil with the ferrite cores.In this paper, the wireless power transfer system is applied to the deep brain stimulator which is implanted inside the skull of patients.Transmitter coil with a novel shape as shown in Figure 2 is proposed.In this system, the transmitter coil with a ferrite core has an arc along with the shape of the human head.
During the analysis, the four-layer concentric sphere head model [12][13][14] being used is modeled and simulated as shown in Figure 3.The four layers are the scalp, skull, cerebrospinal fluid (CSF), and brain.The properties of each layer in the human head model are listed in Table 1.
For deep brain stimulator, the receiver coil should be as small as possible.In order to obtain the maximum power transfer efficiency, the ferrite core inserted into the transmitter coil needs to be optimized.As shown in Figure 2, the angle  represents the length of the ferrite core, while the angle  represents the relative position of the transmitter coil and the receiver coil.In addition, the angle  represents the chord of the transmitter coil.The specifications of the system are listed in Table 2.
In this paper, the resonant frequency of the coils is 3 MHz.The Nickel-zinc (NiZn) ferrite core is chosen as the core material which is inserted into the coils.NiZn ferrite has higher resistivity which is more suitable for applications with operating frequencies above 1 MHz.

Methods and Analysis of the New Shape of the Transmitter Coil
3.1.Finite Element Method.In order to reduce the AC resistance of the coils, the transmitter coil and the receiver coil are fabricated using Litz wires.According to the Maxwell equations, the two-dimensional (2D) magnetic field equation can be expressed as where  is the magnetic vector potential; ] and  are the reluctivity and the conductivity of the material, respectively;   is the given current density.
Field equation is as follows: Additional equation is as follows: Electric circuit branch equation is as follows: The induced voltage  2 in the receiver coil is where  is the angular frequency;  2 is the cross section area of the receiver coil;  2 is the number of turns of the receiver coil;  2 () is the average magnetic flux density in the receiver coil determined by where   is the length of the receiver coil.The power transfer efficiency of the system can be calculated by where  1 and  2 are the input power in the transmitter coil and the output power induced in the receiver coil, respectively;  1 and  2 represent the resistance of the transmitter coil and the receiver coil, respectively;  1 is the input current in the transmitter coil.Magnetic flux lines in the wireless power transfer system are shown in Figure 4.It can be seen that the receiver coil can pick up the major bulk of the magnetic fluxes generated by the transmitter coil.

Analysis.
The angle  represents the positional relationship between the transmitter coil and receiver coil.When the line connecting the centers of the transmitter coil and the receiver coil is parallel to -axis, the value of angle  is equal to zero.The curve of power transfer efficiency varying with different  values is shown in Figure 5.It can be seen that the power transfer efficiency is maximum when the angle  is equal to −10 degrees.
The angle  represents the length of the core inserted into the transmitter coil.The initial value of the angle  is equal to 70 degrees when the lengths of the core and the transmitter coil are the same.The curve of power transfer efficiency varying with the different angle  is shown in Figure 6.It can be seen that the power transfer efficiency is maximum when the angle  is equal to 120 degrees.

Analysis of the Stepped Core
According to the theory of the electromagnetic field, if the thickness of the ferrite core is even, the magnetic flux density in the ferrite core is not uniform as shown in Figure 7.The thickness of the ferrite core should be optimized to make the magnetic flux density uniform and reduce the amount of the ferrite material.The magnetic flux density in the ferrite core of the transmitter coil, as shown in Figure 7, can be numerically modeled as follows: According to the numerical model of Figure 7, the ferrite core can be divided into the wound section and the unwound section.In order to make the magnetic flux density uniform, the unwound section of the ferrite core will be fabricated as a stepped shape [11].The magnetic flux density of the stepped core can be obtained by (8b).As shown in Figure 8, the wound section is expressed as  0 () which is the same as (8a).Each step of the unwound ferrite core is expressed as   () ( = 1, 2, . . ., ), where  is the number of the total steps of the ferrite core.  () can be described as The magnetic flux at the stepped junction of the core should be continuous which agrees with the following description: where   is the cross section area of the th step core.The magnetic flux density increases by the ratio of cross section area  0 / 1 = /( − 1).From (11), the relationship of the magnetic flux density between the stepped cores can be obtained such as Then  1 can be calculated as follows: In this case, the unwound section of the ferrite core is divided into stepped core. 1 is equal to /2 which is 35 degrees from Table 2.In this condition, the number of step core  should be 3.
From Figure 8, the magnetic flux density is about 10 T when  is equal to 60 degrees.Applying these values to (10b),  1 can be determined and its value is about −3.6.Similarly,  2 can be calculated as −8.2.Combined with (10a), (10b), and (11),  2 can be determined by the following: Substituting the values of the coefficients into (14),  2 is calculated as 49 degrees.
Based on the above calculated parameters, the comparison of the calculated result with the simulated result is shown   As shown in Figure 4, the ferrite core can make the magnetic flux density concentrated, but the core loss generated in the ferrite core will weaken the performance of the wireless power transfer system.After the core loss in the stepped core will decrease about 20% compared with the normal core.

Results
Practically, the core loss in the ferrite material will increase the temperature of the coil; meanwhile, the higher temperature will decrease the saturation flux density of the ferrite material.The NiZn ferrite material used in this paper has a saturation flux density of about 400 mT at room temperature.Considering the temperature caused by the core loss, the flux density in the core should not exceed 250 mT.For the deep brain stimulator, however, the security of the system to the patients is the most important problem that should be considered.According to the standard established by the Minister of Public Works and Government Services of Canada, the maximum magnetic field strength should not exceed 2.05 T [21].
In order to guarantee the security of the system, the distribution of the magnetic field strength in the human head is given in Figure 10.The curve of the magnetic field strength from left to right is shown in Figure 11.It can be seen that the maximum magnetic field strength in the human head is under 2 T which meets the above-mentioned standard.In this case, the input current in the transmitter coil is about 5 mA.Under this excitation, the current induced in the receiver coil is shown in Figure 12.It indicates that the induced voltage in the receiver coil is above 3 V when the input current in the transmitter coil is about 5 mA.
According to the above analysis, the wireless power transfer system will operate with the best performance when the angle  is equal to −10 degrees and the angle  is equal to 120 degrees.The power transfer efficiency of the system achieves the maximum value when the frequency of the excitation is 3 MHz as shown in Figure 13.It can be seen that the power transfer efficiency of the system with the normal core is about 60%, while the power transfer efficiency of the system with stepped core achieves larger than 70%.The power transfer efficiency increases significantly.When the operating frequency deviates from the resonant frequency, the power transfer efficiency of the system will have a sharp decline.

Conclusion
In this paper, a transmitter coil with a novel shape is proposed for wireless power transfer system in implanted medical devices.A ferrite core which is suitable for high frequency application is inserted into both the transmitter coil and the receiver coil.The simulation results indicate that the receiver coil with the core can pick up more magnetic fluxes when compared with receive coils without the core.In addition, the relative positional relationship of the transmitter coil and the receiver coil and the length of the ferrite core inserted into the transmitter coil are both quantitatively analyzed.In order to reduce the core loss induced in the ferrite core, moreover, the normal core is replaced by the optimized stepped core.The simulation result indicates that this stepped core can reduce the core loss generated by the ferrite core and increase the power transfer efficiency of the wireless power transfer system significantly.The power transfer efficiency of the system can reach as high as 70%.Such findings are instrumental for building a solid theoretical basis for future work in wireless power transfer systems.

Figure 2 :Figure 3 :
Figure 2: Novel shape of the wireless power transfer system with core applied to a deep brain stimulator.

Figure 4 :
Figure 4: Simulation result of the magnetic flux lines of the proposed coil configuration: (a) without core; (b) with core.

Figure 5 :
Figure 5: The efficiency of the system varies with .

)Figure 6 :
Figure 6: The efficiency of the system varies with .

Figure 7 :
Figure 7: Simulated magnetic flux density for the transmitter coil with NiZn ferrite core.(a) Magnetic flux density in the ferrite core; (b) structure of the transmitter coil.

Figure 8 :
Figure 8: (a) Structure of the stepped core in the transmitter coil.(b) Assumed magnetic flux density in the stepped core.

Figure 9 :
Figure 9: Comparison of the simulated magnetic flux density with the calculated result for the proposed stepped core.

Figure 10 :
Figure 10: Distribution of the magnetic field strength in the human head.

Figure 11 : 4 Figure 12 :Figure 13 :
Figure 11: Curve of the magnetic field strength from left to right of the human head.

Table 1 :
The properties of each layer in human head model.

Table 2 :
Specifications of the wireless power transfer system.