Alternative Expressions for Mutual Inductance and Coupling Coefficient Applied in Wireless Power Transfer
 Author: Kim Gunyoung, Lee Bomson
 Publish: Journal of electromagnetic engineering and science Volume 16, Issue2, p112~118, 30 Apr 2016

ABSTRACT
Alternative analytic expressions for the mutual inductance (
L_{m} ) and coupling coefficient (k ) between circular loops are presented using more familiar and convenient expressions that represent the property of reciprocity clearly. In particular, the coupling coefficients are expressed in terms of structural dimensions normalized to a geometric mean of radii of two loops. Based on the presented expressions, various aspects of the mutual inductances and coupling coefficients, including the regions of positive, zero, and negative value, are examined with respect to their impacts on the efficiency of wireless power transmission.

KEYWORD
Coupling Coefficient , Magnetic Flux , Mutual Inductance , Transfer Efficiency , Wireless Power Transfer

I. INTRODUCTION
We live in a world of wireless communication, where huge amounts of information are transferred between mobile terminals. With the growing demand for wireless transfer of electric power, wireless power transmission (WPT) technology has become increasingly important. In 2007, a magnetically coupled WPT was first investigated by Soljacic and his research group at Massachusetts Institute of Technology [1]. They successfully lit a 60W bulb at a distance of 7 feet—more than 2 m using helical coils of high Qfactor (=
ω _{0}L/R ). Since then, numerous papers have investigated theoretical [2,3] and practical [4,5] evolutions of WPT systems. The transfer efficiency of WPT systems has been well known to depend on the figure of merit, defined as the product of the Qfactor (the geometric mean ofQ _{1} andQ _{2}) and the coupling coefficient (k ), and the load resistance. The Qfactors of the coils are determined by the dimensions and structures of two resonators. Therefore, the coupling coefficient (k ) remains to be the most important parameter to be determined carefully and accurately.The mutual inductance (
L_{m} ), expressed as the product of the coupling coefficient (k ) and selfinductance (L ) (actually, the geometric mean ofL _{1} andL _{2}) have been usually calculated based on the Neumann integral, which can be expressed as an integral of the Bessel function or complete elliptic integrals [6–9], given byand
where
R _{1} andR _{2} are radii of two loops.d is the distance between two loops. The calculation of the mutual inductance between inclined circular coils is also provided in [6,9]. A method of extracting the coupling coefficients (or mutual inductance) was suggested in [10]. These expressions have been found to agree well with the results extracted with [10].In this paper, alternative analytic expressions for the mutual inductance and the coupling coefficient are presented. The new expressions consist of geometric parameters of the loops. Specifically, the coupling coefficient in this work is more explicitly expressed as a function of structure dimensions normalized to the radius of a transmitting loop, which is indeed the nature of the coupling coefficient. Based on the presented expressions, various aspects of the coupling coefficients, including the regions of positive
k , zerok , and negativek , are examined with their impacts on WPT transfer efficiency. Comparisons with other expressions are also performed.II. CALCULATION OF MUTUAL INDUCTANCE AND COUPLING COEFFICIENT FOR CIRCULAR LOOPS
Fig. 1(a) and (b) show a WPT system consisting of two resonant loops magnetically coupled with each other and its equivalent circuit. When each loop is resonant at a resonant angular frequency (
ω _{0}), the current on each loop can be easily obtained. Details can be found in [10]. One compact but meaningful expression for the power transfer efficiency was obtained aswhere
F_{m} andb are the figure of merit of the WPT system and a load deviation factor, defined by andb =R_{L} /R_{L,opt} , respectively. Whenb = 1 in (4) [10], maximum efficiency is achieved. Whenb > 1, the WPT system is in the undercoupled region. Whenb < 1, the WPT system is in the overcoupled region. It is also notable thatη_{L} (b ) =η_{L} (1/b ) as shown in (4).In Fig. 2, the magnetic flux Φ_{21} crossing the opened secondary loop (
I _{2} = 0) due to the currentI _{1} on the primary loop is readily obtained by converting the surface integral to a line integral, and is given by [7]where is the magnetic flux density and is the magnetic vector potential, is the differential surface vector on
S _{2}, is the differential line vector on the secondary loop, and the other symbols are similarly defined. The mutual inductance (L_{m} ) is defined byThis is the wellknown Neumann integral for the mutual inductance. The selfinductance of the primary loop (
L _{1}) is given bywhere
l _{1}' is the closed circle with radiusr _{1}' (slightly smaller thanr _{1}) and the selfinductance of the secondary loop (L _{2}) is similarly defined [7]. The theoretical coupling coefficient between the two loops is then given byWe attempt to obtain a more tractable expression in terms of
r _{1},r _{2},d , andc . The distance between the source on the primary loop and field on the secondary loop is given byThe differential line element is given by
with similarly expressed. The dot product of the and is expressed as
where
r is the geometric mean ofr _{1} andL_{m} and theL _{1} can be expressed asand
and
L _{2} can be expressed similarly. If two loops are coaxially aligned (c = 0),L_{m} is simplified toThis expression is quite exact as others since it started from the Neumann integral.
For the very special case in which
d is very large (d ≫r _{1},r _{2}), the closedform formulas for the mutual inductance of a coaxial case were given by [2,7]and
Under the assumption that
r _{1}' ≈r _{1} andr _{2}' ≈r _{2} (filamentary loops) andr _{1}'/r _{1} =r _{2}'/r _{2}, coupling coefficient (8) can be finally expressed aswhere
r is the geometric mean ofr _{1} and (17) has been found to be exactly the same as the coupling coefficients in [8,9] but can serve as a convenient alternative expression written in terms of structural dimensions normalized to the radius of the transmitting loopr _{1} (r _{2}/r _{1},d/r , andc/r ). This indicates that if the ratiosr _{2}/r _{1},d/r , andc/r remain to be the same, the coupling coefficients are the same irrespective of the real sizes ofr _{2},d , andc . Of course, this holds true whenr _{2},d , andc are very small compared with the wavelength, which is usually very large at WPT frequencies. For example, at 6.78 MHz, the wavelength is approximately 44 m.III. DISCUSSION OF MUTUAL INDUCTANCE AND COUPLING COEFFICIENT
We compared the normalized mutual inductance (
L_{m} /r ) given by (14) with (15) and (16) for the coaxial cases ofr _{2}/r _{1} = 1 in Fig. 3(a) andr _{2}/r _{1} = 0.5 in Fig. 3(b), respectively. The expressions in [8] and [9] were also included for comparisons. The results based on (14) (or (12)), [8] and [9] are shown to be in excellent agreement (and exact). The results based on (15) and (16) show large discrepancies whend/r is small. Thus, (15) and (16) are not recommended for nearfield WPT problems.Fig. 4(a) and (b) show the normalized mutual inductances (
L_{m} /r ) between two circular loops as a function ofc/r for fixedd/r = 0.1, 0.5, and 1. The cases ofr _{2}/r _{1} = 1 and 0.5 are shown in Fig. 4(a) and (b), respectively. Asc/r increases, the mutual inductances are all shown to converge to zero as expected. It is also observed that whenc/r is about 2, negative mutual inductance becomes more pronounced asd/r decreases. The negative mutual inductance simply means that the net magnetic flux in loop 2 is crossing downward. The peak whenc/r is about 0.75 for the case ofr _{2}/r _{1} = 0.5 in Fig. 4(b) is due to the fact that the magnetic flux from loop 1 is the strongest near the loop itself not at the center. The sign of the mutual inductance is not important for the twoloop WPT system, because even though the direction of the current on loop 2 may change, the efficiency remains the same. However, it does matter for the multipleinput and multipleoutput (MIMO) systems.Fig. 5 shows
k (17) as a function ofr _{2}/r _{1} andd/r withr _{2}/r _{1} ≤ 1 assumingc = 0. Again,r is the geometric mean ofr _{1} andr _{2}. Based on the property due to reciprocity theorem, Fig. 5 covers all possible combinations ofr _{1},r _{2}, andd if a rule is made thatr _{1} is the larger ofr _{1} andr _{2}. Eq. (17) holds true as long asr _{1},r _{2}, andd are much smaller than the wavelength. The coupling coefficients are shown to monotonically decrease asr _{2}/r _{1} decreases andd/r increases.Fig. 6(a) and (b) show the coupling coefficients (17) for the cases of
r _{2}/r _{1} = 1 and 0.5, respectively, as a function ofd/r andc/r . As explained in Fig. 5(a) and (b), the negative coupling coefficients are shown to be most pronounced whend/r is very small andc/r is about 2. For the case ofr _{2}/r _{1} = 0.5 shown in Fig. 6(b), the maximum coupling coefficient occurs whend/r approaches zero andc/r is about 0.5.We examine the coupling coefficients and the transfer efficiencies of the WPT systems consisting of two resonant loops as examples. Three kinds of loop structures are employed. The radii of the loops are 5 cm, 10 cm, and 20 cm, respectively. In Table 1, the values of
R, L, C , and Qfactor are summarized for the three loops. Each loop is made of copper withσ = 5.8 × 10^{7} S/m. The chip capacitors are loaded on the loops for a resonance at 6.78 MHz. The Qfactors of the loops have been calculated to be 334.2 (r = 5 cm), 668.3 (r = 10 cm), and 1,331.9 (r = 20 cm).Fig. 7 shows the coupling coefficients and transfer efficiencies as a function of
d/r for the cases ofr _{2}/r _{1} = 0.5, 1, and 2. The radius of the primary loop (r _{1}) is fixed at 10 cm. The coupling coefficients for the case ofr _{2}/r _{1} = 1 are shown to be larger than those of the other cases. The higher efficiency for the case ofr _{2}/r _{1} = 2 compared to the case ofr _{2}/r _{1} = 0.5 is due to the larger Q factor, because the coupling coefficients remain the same.Fig. 8(a) and (b) show the coupling coefficients and transfer efficiencies as a function of
c/r with differentd/r ’s of 0.1 and 1 for the cases ofr _{2}/r _{1} = 1 and 0.5. For the case ofr _{2}/r _{1} = 1 andd/r = 0.1, the coupling coefficient changes sign from positive to negative whenc/r is about 1.5. Near this position, the 80% efficiency is shown to drop abruptly to zero but then to quickly recover to a value above 70%. A similar phenomenon is observed for the case ofr _{2}/r _{1} = 1 andd/r = 1 but the efficiency recovers to about 50%. For the case ofr _{2}/r _{1} = 0.5, similar effects are demonstrated, but the levels of the results are smaller than those for the case ofr _{2}/r _{1} = 1.IV. CONCLUSION
This paper presents alternative analytic expressions for the mutual inductance and coupling coefficient between circular loops. The mutual inductance and coupling coefficient have been expressed in more familiar forms and explicitly written in terms of structure ratios and the placement of loops. Various aspects of the coupling coefficients, including the regions of positive
k , zerok , and negativek , have been examined respect to with their impacts on WPT transfer efficiency.

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[Fig. 1.] Magnetically coupled wireless power transmission between two resonant loops. (a) Geometry, (b) equivalent circuit.

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[Fig. 2.] Two circular loops with radii of loops (r1 and r2) and distances (c, d) for derivation of mutual inductance (Lm) and coupling coefficient (k).

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[Fig. 3.] Normalized mutual inductances (Lm/r) as a function of d/r (rring1/r1 = rring2/r2 = 0.02 assumed). (a) r2/r1 = 1, (b) r2/r1 = 0.5.

[Fig. 4.] Normalized mutual inductances (Lm/r) as a function of c/r1 (rring1/r1 = rring2/r2 = 0.02). (a) r2/r1 = 1, (b) r2/r1 = 0.5.

[Fig. 5.] Coupling coefficients (k) as a function of r2/r1 and d/r (rring1/r1 = rring2/r2 = 0.02, c = 0).

[Fig. 6.] Coupling coefficients (k) as a function of d/r and c/r (rring1/r1 = rring2/r2 = 0.02). (a) r2/r1 = 1, (b) r2/r1 = 0.5.

[Table 1.] Circuit element values of three loops at 6.78 MHz

[Fig. 7.] Coupling coefficients (k) and transfer efficiencies (ηL) as a function of d/r for different r2/r1 (r1 = 10 cm, rring1/r1 = rring2/r2 = 0.02, c/r = 0).

[Fig. 8.] Coupling coefficients (k) and transfer efficiencies (ηL) as a function of c/r1 for different d/r1 (r1 = 10 cm, rring1/r1 = rring2/r2 = 0.02). (a) r2/r1 = 1, (b) r2/r1 = 0.5.