An Approach to Calculate the Efficiency for an N-Receiver Wireless Power Transfer System

A wireless power transfer system with more than one receiver is a realistic proposition for charging multiple devices such as phones and a tablets. Therefore, it is necessary to consider systems with single transmitters and multiple receivers in terms of efficiency. Current offerings only consider single device charging systems. A problem encountered is the efficiency of one receiver can be affected by another because of the mutual inductance between them. In this paper, an efficiency calculation method is presented for a wireless power transfer system with one to N-receivers. The mutual inductance between coils is implicitly calculated for different spatial positions and verified by practical experimentation. The effect of changing parameters, such as resonant frequency, coil size and distance between coils, on the efficiency has been studied. A clarification of the special performance of a wireless power transfer system at a specific point has been presented. Keywords—wireless power transfer; multiple receivers


INTRODUCTION
Wireless power transfer by magnetic resonant frequency coupling is a new version of the classic theory presented by [1].Wireless charging is more convenient and has a number of advantages over wired connections.In wireless charging there is no physical connection between the load and the source encouraging its use for no spark applications.Wireless power transfer depends on the near field and is convenient, safe, and simple [2].Due to its dependence on magnetic resonance, Witricity is also safe from a biological point of view i.e. environmentally friendly [3].The seminal work of this technique was conducted by Nikola Tesla in the late 19th century, he was one of the wireless power transmission pioneers [1].
In 2007 scientists at Massachusetts Institute of Technology (MIT) revisited the concept of wireless power transfer by announcing a testbed which wirelessly transferred tens of watts over a distance of greater than two meters.Following this, research in this area has received increased attention by many large multinational companies such as Apple, Duracell and Qualcomm [4].Researchers have tended to study wireless power transfer systems with a single transmitter and a single receiver.It is clear that the performance of wireless power transfer changes according to factors such as the distance between the transmitter and the receiver [5][6][7].
In [8] the authors explained how one can express inductive coupling or magnetic resonance with the same analytic form as used in coupled mode theory.However, according to [9] this method is undesirable and inconvenient.Therefore researchers tend to use the equivalent circuit for a magnetic coupled system.For example, in [5] the researchers analysed an equivalent circuit wireless system and found its efficiency at different distances between the transmitter and the receiver demonstrating optimum performance at a specific point.In their analysis, s-parameters have been employed because they are more convenient for practical experiments at high frequency.In spite of the wealth of research to study efficiency for single receiver wireless systems, there is still a shortage of similar studies about multiple receivers.
In this paper, the equivalent circuit of the magnetic coupling system is proposed with some modification to study the efficiency of wireless power transfer with multiple receivers.An essential part of this study is the calculation of the mutual inductance between coils.Resonant frequency, coil size and distance between coils, and the effect of these parameters on efficiency has been considered.The optimum performance of the wireless power transfer system has been demonstrated for various cases.
The findings of our study can be used to form the basis of any multiple receiver system.The proposed method can also be applied analytically at different frequencies.To investigate the results practically, experiments have been conducted inside a Faraday cage at frequencies of 2.07MHz and 2.5MHz.Choosing these frequencies is due to practical considerations such as a.c.signal sources and practical dimensions of the coils.

II. WIRELESS POWER TRANSMISSION USING MAGNETIC RESONANCE
The principle of the mutual induction phenomena, which happens between two coils, is the passing of a continuous time varying current through one of the coils called the primary coil leading to the production of a magnetic field around it.The interaction of this magnetic field with the second coil called secondary, leads to an induced current in it as shown in Fig. 1.The maximum interaction happens when the two coils work at the same frequency and the mutual induction here is called magnetic resonance [3,10].www.ijacsa.thesai.org

A. One load system representation
A simple wireless power transmission system consists of a transmitter and a receiver as shown in Fig. 2. The transmitter is represented by the source Vs and its internal resistor Rs, the capacitor C 1 and the inductor L 1 .While C 2 and L 2 are the capacitor and the inductor that achieve the resonance to transfer the power to the load resistor R 2 .Due to the inductance value of each coil, capacitor values are chosen to achieve the resonance condition X L =X C , through the resonance frequency equation [11]:

B. Two load system representation
The same principle can be used to represent a system with two receivers or more.Fig. 3 shows a schematic circuit of a resonant coupled system which consists of a transmitter and two receivers.It shows the basic variables which are used in the calculation of mutual inductance.The variables A and a are the radius of the transmitter and each receiver, respectively.The distance between the transmitter and each receiver or between the two receivers are represented by b and D, respectively.Finally, p is the pitch between any two turns in any of the coils.It is clear from this circuit that the calculation method of mutual inductance between the transmitter and each receiver is different for the two receiver case due to variation in location.Different methods are proposed to calculate the mutual inductance for each case [12,13].The equivalent circuit of a wireless power transfer system with two receivers is shown in Fig. 4. In the figure, the source V s drives a transmitter coil and its internal resistor is R s .L 1 and C 1 are the inductor and the capacitor of the transmitter.On the other side of the circuit, the first receiver is represented by L 2 , C 2 , and R 2 as an inductor, capacitor and load resistor, respectively.Similarly, L 3 , C 3 , and R 3 form the second receiver.The second receiver should be under the resonance condition with the same resonance frequency.
IV. BASICS OF THEORETICAL ANALYSIS Applying Kirchhoff's law of voltage on each loop [14] (the transmitter and the receiver) is the start of the analysis.M 12 or M 21 is the mutual inductance between the transmitter and the receiver.
Applying Z-parameters on this set of equations leads to: The previous equation can be written as: Here, the output and input power can be calculated: Under the condition of maximum transfer of power, the efficiency can be calculated as: (10) V. THEORETICAL ANALYSIS OF MULTIPLE RECEIVERS Following the same theoretical analysis as for a single receiver system we extend the method for a number of the receivers; this is the main contribution of this paper.

A. Two Receivers
For the circuit shown in Fig. 3, Kirchhoff's law of voltage is applied on each loop (the transmitter and the two receivers) with regard to the mutual inductance between each pair of coils [14].The mutual inductance between the transmitter and the first, second receiver are M 12 and M 13 , respectively.Both of them follow the same calculation method because of their relation to the transmitter.As a special case, M 12 is equal to M 13 if the two receivers are identical.While M 23 , the mutual inductance between the two receivers has to be calculated using another method due to their parallel nature.
Applying Z-parameters on this set of equations leads to: The previous equation can be written as: So (14-a) (14-b) (14-c) It is concluded from (14-a) that: (15) Here, the output power for each receiver can be calculated: (16-a) (16-b) And the input power is: Under the condition of maximum transfer of power, the efficiency for each receiver can be calculated as: (18-a) (18-b)

B. Three Receivers
Similar representation for the two receivers system can be applied on the three receivers system: Applying Z-parameters on this set of equations leads to: The previous equation can be written as: Here, the output power for each receiver can be calculated: (24-a) (24-b) (24-c) And the input power is: Under the condition of maximum transfer of power, the efficiency for each receiver can be calculated as:

C. N-Receivers
The method can be applied in the general case for an Nreceiver system and the general equation for any system is:

A. Concentrical Coils
According to [5], The Neumunn formula can be applied to find the mutual inductance between two concentric turns: Where D: the distance between dl 1 and dl 2 The same solution for this integral can be achieved by the Maxwell equation [12]: Where A, a: the radius of what is regarded as two concentric turns as shown in Fig. 2 √ √ (30) K and E: the first and second kind of the complete elliptic integral and they are defined as: As a modification to deal with a coil of more than one turn, the following expression was used instead of D in (30): (31) Where N 1 and N 2 : the number of turns for each coil.

P: the pitch between turns
Finally, the total mutual inductance between two concentric coils with multiple turns can be found by [12]:

B. Parallel Coils
Mutual inductance calculation between two parallel coils needs another formula instead of the Maxwell equation.In [13], the researchers presented two formulas to express the mutual inductance between two turns at the same level and two turns with lateral misalignment.

1) Two turns at the same level
The following formula can be applied to calculate the mutual inductance between two turns at the same level: (33) Where

√
In this case C equals zero because the two turns are in the same level

2) Lateral Misalignment
If the two turns are allocated at different levels, the mutual inductance between them can be calculated by: Again the mutual inductance between each of the two turns of the two parallel coils can be summed by (32) to find the total mutual inductance.

VII. RESULTS AND DISCUSSION
The representation of a wireless power transfer system with one to n-receivers is programmed in Matlab to calculate the mutual inductance between coils and then the efficiency.

A. Single Receiver System
As a starting point to investigate the previous analysis, a wireless power transfer system with two identical coils is practically implemented.Each coil, with 3cm radius, 21 turns and 3mm pitch between the turns, has an inductance equal to 18.6µH.One of them is the transmitter and the other is the receiver.Choosing specific parameters such as resonance www.ijacsa.thesai.orgfrequency at 2.07 MHz, load resistor 50Ω, C 1 = C 2 = 330pF (which includes the natural capacitance of the coil) and the distance between the two coils is 3cm.Next, finding the efficiency over a range of frequencies was achieved as shown in Fig. 5.The theoretical results are compared with the experimental in the figure.It is clear from Fig. 5 that the higher efficiency of 46% for wireless transfer happens at the resonance frequency and there is a good match between the theoretical and practical results.However, the efficiency here is relatively low, that is because of the small value of mutual inductance between the two coils.It is equal to 1.5468µH.
Replacing the two coils with another set of coils, which have higher inductance and higher mutual inductance between them, gives higher efficiency over larger distances at the same frequency as shown in Fig. 6.In this experiment, each coil has 12.25cm radius, 8 turns and 3mm pitch between the turns, and 31.3 µH inductance.Working at the same frequency of 2.07MHz, 188pF capacitors are required for the transmitter and receiver.The system has an efficiency of 72% at 15.5 cm.This is as a result of the 2.1583 µH mutual inductance, which is larger compared to the previous experiment.It was explained in section 6 that mutual inductance increases when the two coils approach.Fig. 7 shows the effect of changing the distance between these two coils on the mutual inductance.Using the larger coils as in Fig. 6, at smaller distances, the efficiency at the same resonant frequency gradually increases until it approaches the maximum value of 100% at a distance of 10cm as shown in Fig. 8.A further decrease in the distance surprisingly leads to a decrease in efficiency of the system.As shown in Fig. 9, there is a splitting in the efficiency of the system at a distance of 3.5cm.This splitting starts to appear as the mutual inductance increases between the two coils (the transmitter and the receiver), increasing when the two coils are approaching.This effect shifts the maximum efficiency to two distinct frequencies around the resonance frequency.This means that there is a perfect point to transfer the maximum power in the wireless system as shown in Fig. 10 from the relation between the efficiency and the distance at the resonant frequency.The reason for this point is shown in Fig. 11, where the input resistance of the system, which changes with the distance, equals 50Ω at this point.The 50Ω value is equal to the input resistance of the source Z in which leads to maximum transfer of power.
Working at a different frequency will also affect the efficiency of the wireless power transfer system.To show this effect, the capacitors of the transmitter and the receiver are changed from 188pF to 128pF.The results are shown in Fig. 12.The efficiency of the system is shown at the resonant frequency of 2.5MHz with 15.5cm distance between coils.Compared to Fig. 6 and Fig. 8, the higher frequency enables the system to transfer more power through a larger distance.It can be concluded that the distance of the perfect point increases at higher frequency but is ultimately limited by the frequency and mutual inductance in both coils.Before studying multiple receivers it is necessary to study the efficiency of a small receiver with a large transmitter.The efficiency of a system with the large coil of 12.25cm and the small coil of 3cm is shown in Fig. 13 at a resonant frequency of 2.07MHz and a distance of 3cm.

A. Two Receiver System
Using two identical receivers of 3cm radius and a transmitter of 12cm the efficiency of a two coil system is studied.The efficiency of a receiver can be affected by the presence of another receiver in the same field.Fig. 14 shows the efficiency of each one of two closed receivers in the same transmitter field.The distance between their centres is 7cm.It appears from the figure that the efficiency here is less than it is in case of just one receiver as shown in Fig. 13.

B. Three Receiver System
It is possible to apply our modelling and simulation approach for the case of more than two receivers, i.e. three to N. Fig. 15 shows the efficiency of one of the three receivers if all three are together inside the field of the same transmitter.In this part the three identical receivers form a triangular shape at the same level.This means that the mutual inductance between any two of them are equal, which is 0.21076µH.Moreover, the mutual inductance between the transmitter and any one of the receivers are also equal which is 1.7358µH.Moving a receiver to another location in the same field requires a new calculation for the mutual inductance between coils.The mutual inductance between any two coils depends on their shape, their location and the distance between them.This study presented a method for the calculation of mutual inductance between any solenoid coils with a different location and distance.
The efficiency of a wireless power transfer system is dependent on factors such as: the resonance frequency; the size of the coils; the mutual inductance between them; the distance between coils and the presence of more than one receiver in the transmitter area.The following observations can be made:  There is a specific distance, where the input resistance of the system equals the internal resistor of the source, at which point the system has the maximum efficiency and can transfer maximum power to the load.
 Higher frequency leads to an increase in the distance at which the system has the maximum efficiency subject to frequency and mutual inductance limitations.
 A Larger transmitter size, produces more flux around it, and also leads to an increase in the distance at which the system has the maximum efficiency.
 The Presence of more than one receiver gives decreased efficiency for each individual (compared to the larger coil), but the sum is greater than the efficiency of the same individual coil  There are a number of practical limitations observed at higher frequencies regarding measurements

Fig. 2 .
Fig. 2. Equivalent Circuit of a Wireless Power Transfer System

Fig. 3 .
Fig. 3. Schematic Circuit of a Resonant coupled system

Fig. 4 .
Fig. 4. Equivalent Circuit of a Wireless Power Transfer System with Two Receivers

:
The loops currents.