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Thikness Dependance Investigation of the MutualInductance Link in Concentric Planar Transformers

Hala Ghadhab, Mohamed Hadj Said, Fares Tounsi,Brahim Mezghani

EMC research Group, National Engineering School of Sfax,University of Sfax, BP 1173, 3038 Sfax, Tunisia

Tel: +216-74274409; Fax: +216-74275595Correspondence e-mail: fares.tounsi@isimsf.rnu.tn

1Sandeep G. Surya, 2V. Ramgopal Rao1Centre for Research in Nanotechnology and Science

2Department of Electrical EngineeringIndian Institute of Technology, Bombay 400076, India

Tel: +91-222 576 7456; Fax: +91-222 572 3707

Abstract In this paper , we present a new analytic approachfor the calculation of the mutual inductance between twoconcentric planar conductors. From this study, we aim to provethat coil thickness has a minor impact on mutual inductancevalue. This study will help to determine a simplified adaptedexpression of the induced voltage output at our electrodynamicmicrophone ends. This micromachined new sensor exploits theconcentric transformer principle which relies on the use of twoplanar inductors: an outer fixed one and an inner suspended one.If the outer is biased, a mutual inductance link will be generated.This link strongly depends on inductors coil design geometry andthe induced output voltage. The present work proves that forlong conductors (large number of turns), coil thicknesses above6m will have no significant influence on mutual inductancevalues. This has been proven by an increase of only 0.01% if coilthickness reaches as high as 20 m. Moreover, for thicknessesunder 2m, mutual inductance value is constant arround 7.2H.

This allows the use of much simpler expressions, with nothickness dependence, to evaluate mutual inductance values.

Index Terms Analytic modeling, Mutual inductance, MEMS sensor, electrodynamic transduction, Planar Transformer.

I. I NTRODUCTION The microelectronics revolution has led to Microsystems

technologies or MEMS (Micro-Electro-Mechanical Systems)which combines semiconductor microelectronic processeswith bulk and/or surface micromachining techniques [1]. Thisemerging technology permits the realization of completesystems on a single chip for microsensors and microactuatorsapplications [2,3]. Integrated planar transformers are quiteuseful for MEMS applications mainly in order to transferenergy between two circuits, such us pacemakers, softswitching and resonant power converters [4,5]. Moreover, itcan be used also as a transduction technique to convertmechanical force (or pressure) into an electrical signal [6].Thus, in planar transformers, the mutual inductance coefficient

between primary and secondary coils is a key parameter thathas to be optimized in order to enhance the inductive linkcoefficient between both inductor coils. The targetedapplication is a microsensor which uses this electrodynamic

transduction technique to detect the acoustic pressure. Theidea consists on the use of two planar inductors: fixed outerone on the substrate and suspended inner one on top of asuspended membrane on a micromachined cavity. When the

primary inductor is biased using a DC or AC current, a mutualinductance linkage will be generated between the primary andsecondary coils. The vibration of the suspended membrane,along with the inner inductor, will lead to a variation in themutual inductance. This phenomenon gives rise to an inducedoutput voltage at the ends of the secondary. This voltage will

be proportional to the membrane fluctuation amplitude caused by the incident acoustic wave [7,8]. The objective of this paperis firstly to determine the mutual inductance linkage between

both planar inductors forming our microsensor. Then,secondly investigate the inductors coil thickness effect on themutual inductance. We will clearly show that coil thicknesshas no significant effect on mutual inductance values. Thiswill be done using different methods described in literatureand adapted to our DUT and then a comparison will be given

between different evaluated results.The paper is organized as follows: we will start by

presenting the inductors arrangement in the proposedmicrosensor and intended fabrication process. Thereafter, wewill evaluate the mutual inductance in halt position usingdifferent methods. The coil thickness will be the maincomparison parameter in our developed theory. Finally, resultswill be discussed and a comparison will be given.

II. PLANAR TRANSFORMER INTRODUCTION Fig. 1 shows the inductors arrangement in our acoustic

microsensor [8]. Since only the primary inductor is actuated by a DC current, the mutual inductance will be constant andthe flux caused by the outer inductor through the inner one isgiven by 12 = M I 1, where I 1 is the biasing current and M denotes the constant magnetic link between both inductorscorresponding to a separation of (see Fig. 1). In a previouswork [7], we did demonstrate that the mutual inductance

produced between two planar square inductors, with n spiralseach, is the same obtained from the superposition of n single

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spirals having the inductor's average diameter with an averagespacing a (see Fig. 1). So, all proposed calculation methodswill simply use the above assumption to evaluate the mutualinductance and multiply the result by n2. Moreover, since asquare inductance design is symmetric on both x and y axis,we can simply evaluate the mutual between two unequalstraight conductors as show in Fig. 2. Finally, the resultingmutual expression will be multiplied by 4, which leads to atotal coefficient of 4 n1 n2, where n1 and n2 represent thenumber of turns of the primary and secondary inductors,respectively. For mutual inductance evaluation, the useddimensions applied to the stacked transformer are shown inTABLE I. Both coils width is set to the smallest value possiblein order to put them as close as possible to each other.

Fig. 1. Magnetic interaction between two inductors in the concentric planartransformers.

Fig. 2. Magnetic interaction between two conductors with different lengths.

TABLE I. A DOPTED DIMENSIONS OF BOTH MICROSENSOR INDUCTORS .

Inductors design parameters ValueAverage inductors separation ( a) 113 mOuter inductor average length ( a) 1648 m

Number of turns ( n1 and n2) 50 turnsCoils width ( w) 2 mCoils thickness ( t ) variable

In the next section, we will present the technologyavailable at the Indian Institute of Technology, Bombay(IITB), which was used for the fabrication process of ourinductors. The actual fabrication was carried out in the Centreof Excellence in Nanoelectronics within the IITB.

A. Fabrication process of the planar transformer

The planar inductors were fabricated by the Electron BeamLithography technique using Raith150 equipement. Due to therequired minimum feature for inductors (~2 m), the shape

paterning using lazer writer (lithography laser) was not possible. Therefore, EBL was used in our case. We started bythe deposition, using spin coating (3000 RPM for 50 seconds),of 250 nm of PMMA (Polymethyl Methacrylate) on top of ourwafer which already contains 200 nm-film of SiO 2. It is then

placed on a hot plate (180 C for 5 minutes) to soft bake.Thereafter, the wafer was loaded on the Raith150 using 90A/cm 2 current dose; 22.844 mm/s beam speed, 40 nmresolution, 1000 m write field and 10kV voltage. Finally, thePMMA was developed by placing the wafer in a beaker withMethyl Isobutyl Ketone MIBK:IPA 1:3 for 30 seconds. Theused ratio offers an optimal developing speed. Next, the waferwas placed in a different breaker containing IPA propanol forapproximately 15 minutes with periodic agitation or until thesmallest features became visible. In Fig. 3, we show a SEM

photograph of both inductors after development (Fig. 3a) andafter lift-off operation using acetone and IPA propanol andmetal deposition (Fig. 3b).

(a)

(b)

Fig. 3. SEM photograph of inductors (a) after development, (b) after lift-off process and metal deposition.

The caracterization of our prototype will be done at a laterstage. This will enable us to measure not only output inducedvoltage under different biasing modes but also under variableincident acoustic pressures. These experimental values will beexploited to validate the developed model expressions of themutual inductance values.

B. Induced voltage evaluation

The estimated induced voltage expressions will be derivedin this section. The study will be done when the outer fixedinductor is biased by a DC current and using a mutualinductance analytic approach. The coupling design between

both inductors occupying distinct regions, as shown in Fig. 1,represents a concentric planar transformer structure (Fig. 4).

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Fig. 4. Magnetic interaction in a concentric planar transformer structure.

Induced voltage conponents are strongly time dependenton the current passing through the external inductor.Generally, the induced voltage expression is given by:

dt dM

I dt dI

M dt

d v 1

1122

(1)

III. MUTUAL INDUCTANCE CALCULATION The mutual inductance will be calculated between two

parallel segments as shown in Fig. 2. It has a positive value forthe currents flowing in the same direction and negative in theopposite case. The sum of mutual inductances calculated(positive and negative) between conductors forming inner andouter inductors gives the value of the total mutual inductance.Several approaches have been gathered to compute the mutualinductance in various practical cases. The coils thicknessdependence is present only in some of these expressions. Forothers, the used methods depend only on conductor length andassume that all other parameters, as width and thickness, arenegligible compared to the length. Therefore, an accuratecomparison should be performed in order to validate thederived simplified formula.

A. Formulas without thickness dependance

In this section, the used method depend only on the lengthof the conductors assuming that all other parameters arenegligible compared to the length. To adapt expressions fromliterature to our DUT, original parameters were replaced withour set of parameters throughout calculations.

1) Grovers mutual inductance formulaGrover method is useful for the mutual inductance

calculation between two parallel conductors with no thicknesseffect parameter. Thus, the mutual inductance M a,a-2 a betweentwo parallel segments with lengths a and ( a-2 a) can be

determined by the following [9]:

2 a a aa a a M M

(2)

where,

Q M aa

2

0 (3)

with Q a purely geometrical parameter that can be written as:

aa

aa

aGMD

a

GMD

GMD

aGMDa

Q

2

2

2

2

1

1ln (4)

Here, GMD (Geometric Mean Distance) is the geometricmean of the distance between surfaces of both conductors (seeFig. 2). The exact value of GMD can be obtained from theGrover table [10] and can be estimated with an error less than3% by [11]:

2

2

12lnln

aa

ww

wGMD

6

6

4

4

16860 aa ww

ww

...

66036010

10

8

8

aa w

w

w

w

(5)

2) Neumanns mutual inductance formula

The mutual inductance M f between two parallel conductorsseparated by a distance d , with lengths l 1 and l 2 and withnegligible width and thickness ( Fig. 5) is given by [12]:

z z z z z M l l l l l l l l f 2313 3123,

,2222ln001.0

(6)

where,

4

1

1,, 1

31

42k

k k s s

s s s f z z f (7)

The result M f is expressed in micro henrys for dimensionsin centimeters. To be noted that Eq. (6) covers all possible

positions of the two parallel filaments. It is given in itsexpanded form for the special case elsewhere in literature[13,14]. In TABLE II. , we identify all the parameters used inEq. 6 and Eq. 7 but that have been adapted to our case of two

parallel segments with respective lengths a and ( a-2 a) asshown in Fig. 2. This will enable the use of Neumannsformula to evaluate the mutual inductance value produced atour microphone ends.

Fig. 5. Two parallel conductors arrangement for Neumann case study [12].

l 1

l 2l 3

z

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TABLE II. P ARAMETERS FROM EQ . (6) AND RESPECTIVE ADAPTED ONES USEDTO EVALUATE MUTUAL INDUCTANCE BETWEEN TWO PARALLEL CONDUCTORS .

Parameters in Eq. (6) l 1 l 2 l 3 Adapted parameters a a (a - 2 a) a

3) Ruehlis mutual inductance formulaThe Ruehli's method can be used to express the mutual

inductance formula without a dependence on coils thicknessand width. The mutual inductance presented by Ruehli for two parallel conductors, as drawn in Fig. 6, is given by [15]:

1 ,2 2 2 2

1 ,

1

ln( ) ( )

4

P P v

r P v P

l z z z r z r z

(8)

where,

2

1

l l

v ,2l

D P z and

2 2

2

x y D Dr

l (9)

Fig. 6. Parallel conductors arrangement for the Ruehli case study.

Table III identifies parameters used in Eq. 8 and thoserespective ones which have been modified accordingly to fit

our case study.

TABLE III. P ARAMETERS FROM EQ . (8) AND RESPECTIVE ADAPTED ONESUSED IN OUR CASE .

Parameters in Eq. (8) l 1 l 2 D x D y D z

Adapted parameters a (a - 2 a) 0 a a

B. Method with thickness dependance (Hoers formula)In this section, we focus on mutual inductance expressions

involving coil thickness parameter. In our study, originaldesign factors used in basic formulas are to be replaced withsuitable parameters which are not only less complex but alsoadapted to our specific design geometry. The exact expressionof the mutual inductance between two rectangular conductorsis given by the Hoer formula of Eq; (11) [10]. The parametersof Eq. 11 are shown in Fig. 6 using as length a=l 1, as widthw=e=d and thickness as t=b=c . The first conductor is

positioned so that its lower part is located at the origin of the(x, y, z) coordinate system. The second one is placed in anarbitrary position. Erreur ! Source du renvoi introuvable. identifies the parameters used in Eq. 11 which are adapted toour microsensor case. In Eq. 11, the mutual inductance has

been left intentionally in its original complicated form to beable to evaluate it while dimensions are varied considerably.The positive parts can cancel their negative counterparts.

Fig. 7. Parallel conductors arrangement for Hoer case study, which are boththikness depending.

2 2 22 2 4 4

2 2

2 2 22 2 4 4

2 2

2 2 22 2 4 4

2 22 2

4 4 4 2 2 2 2 2 2 2 2 2

ln4 2 4 2 4

ln4 2 4 2 4

ln0 .001 4 2 4 2 4

1( 3 3 3 )6 0

b

x x y z y z y z x

y z

y x y z x z x z y

x z

z x y z x y x y z

M x yw t

x y z x y y z z x x y z

x

,,

3 31 1

2 2 2 2 2 2

31

2 2 2

, ,

6 6

6

P b P E e E d

E d e E P c b P

x

yz xy xy z xz T a n T a n

z x y z y x y z

x yz yz T a n

x x y z

3 1, 3 2

2 3 1, 3

( ) ( )

l l l l c

l l l l

y z

(10)

where,

3 3 31 1 12 4 2 4 2 4

4 4 4i+j+k+1

i=1 j=1 k=1

( , , ) ( ) ( ) (z) = (-1) ( , , )q r sq r sq q r r s s i j k f x y z x y f q r s (11)

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TABLE IV. P ARAMETERS FROM EQ . (11) AND RESPECTIVE ADAPTED ONESUSED IN OUR CASE .

Parameters Eq. (11) l 1 l 2 l 3 e d b c E P

Adapted parameters a (a-2 a) a w w t t (a+w ) 0

IV. R ESULTS AND DISCUSSION

Mutual inductance expressions have been estimated in thecase of two conductors. Now, its total value can beapproximated through the multiplication of the previouslyderived expressions by the factor 4n1n 2 [16]. Aftercalculation, the results obtained show that the Ruehli, Groverand Neumanns formulas give nearly the same values of7.184H for mutual inductance. To prove that thickness

parameter has a small or no effect on mutual inductance, we plot this latter as a function of thickness using Hoer formula(Fig. 8). We can clearly notice that the effect of thickness onmutual inductance is almost constant up to 6m and barelyincreases for up to 20m thickness. The variation of mutualinductance value is as low as 0.01%.

2 4 6 8 10 12 14 16 18 207.152

7.153

7.154

7.155

7.156

7.157

7.158

7.159

7.16

7.161

7.162

Thickness [m]

M u

t u a

l i n d u c

t a n c e

f r o m

H o e r ' s

f o r m u

l a [ H ]

Fig. 8. Mutual inductance obtained from Hoers formula vs inductors coil

thickness.

As a result, we can state that inductors coil thickness has aminor influence on the mutual inductance value. This says thatwe can use much simpler mutual inductance expressions toevaluate output induced voltage of our sensor.

Finally, for our case, the inductors thickness depends onthe available fabrication technology at the CEN in IITBombaywhich does not exceed 1 m.

V. CONCLUSION In this paper, the mutual inductance between two planar

square inductors coil was evaluated aiming an electrodynamicmicrophone application. This micromachined microsensor was

partly fabricated at the CEN in IITBombay using an EBL process. This paper highlights that the conductor thicknessaffects on the mutual inductance calculation was too small andwas arround 0.01% when using different calculation methodsand when the thickness reachs 20 m. For a small thicknessless than 2m, we can confirm that the mutual inductance was

constant and was evaluated to be arround 7.2H between thetwo inductors. The mutual inductance calculation for ourmicrosensor is very important since it can be used for theoutput induced voltage determination when an incidentacoustic pressure is applied on the internal inductor.

ACKNOWLEDGMENT

This work was carried out with support from the TunisianMinistry of Higher Education and Scientific Research and theDepartment of Science & Technology, India in the frameworkof the Tunisian-Indian joint research cooperation in the fieldof scientific and technological research.

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