A Wide-Band 22 Equivalent-Circuit Model for Spiral

Inductors on Silicon

Xiang-zhan Wang, Jun Ren, Fan Yang, Wei Zheng, Qi Yu, Ning Ning, Mo-hua YangSchool of Microelectronics and Solid State Electronics

University of Electronic Science & Technology of ChinaChengdu, Sichuan, P. R. ChinaxiangzhanWpeoplemail.com.cn

Abstract-For monolithic RF spiral inductor on high-losssilicon substrate, a novel physical model is proposed, in whichfunctions of skin effect, proximity effect and eddy currentlosses in the substrate to frequency-dependent seriesparameters Ls and Rs are accounted in the light of modifiedpartial equivalent element circuit methodology and a full-coupled transformer loop and, in the meanwhile, distributedcharacteristics of parasitic capacitance are captured by 2irequivalent-circuit. Up to 15GHz, the model reveals quite goodaccuracy within 8% with data from full-wave electromagneticfiled simulator, including equivalent inductor Leff, resistor Reffand quality factor Q and, hopefully, it can be applied tofurther theory research and optimum design of RFIC spiralinductor on Si.

I. INTRODUCTION

Recently, the operational frequency of radio-frequencyintegrated circuits (RF IC's) continues to increase due to therapid growing of the wireless communication technology.On-chip spiral inductor is a crucial element of RF IC's, sothe accuracy of its model in higher frequency has receivedgreat attention[ 1]. Meanwhile, standard CMOS technology iswidely used in RF designs owing to their low cost and easeof process integration. On the other hand, losses in thesubstrate, which stem from its inherent low resistivity, bringa new challenge for on-chip spiral inductor modeling.

Most of current models which neglect proximity effect,substrate eddy current loss and capacitors distributedcharacteristics can not very well model silicon-base spiralconductors' high-frequency performance. An accuratesolution is to use full-wave electromagnetic simulators butwith an ultra low efficiency. As a result, an accurate and highefficient physical model for high frequency is required.

Based on partial equivalent element circuit (PEEC)methodology and a full-coupled transformer loop, ananalytical scalable equivalent-circuit model which isimplemented by MATLAB has been developed. Theefficiency of this scalable model approximates the simplecurrent model and the accuracy has great improved. It can be

widely and easily applied to design and optimization of on-chip inductors.

II. 2ri EQUIVALENT CIRCUIT MODEL FOR SPIRALINDUCTORS ON SILICON

In Fig.1 shows the accustomed structure of a spiralinductor on silicon. The transverse geometrical dimensionsinvolve le, w, s, and t, h, to,, tsub represent the verticalgeometrical dimensions.

Figure 1. The accustomed structure of a spiral inductor on silicon.

A c-

11C. C 111Cm CmII

L,(f)/2 R,(f)/2 L,(f)/2 R,(f)/2

: Cox C0 -I~~~~~~~~Rsi CSi Rsi CSi

Cox

Rs

Figure 2. The 2xi equivalent circuit model for spiral inductors.

0-7803-9584-0/06/$20.00(2006 IEEE.

csl

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As the structure of a spiral inductor isn't symmetrical andthe parasitical capacitances distribute throughout the inductor,the 221 equivalent circuit model is shown in Fig.2. Theterminals A and B represent the outer and inner ports of thespiral inductor. The series branch consists of L#(f) and R#(f),which are related to frequency due to the influence of skineffect, proximity effect and eddy current loss in the substrate.The parasitical capacitances between two adjacent parallelmetal lines are approximated by Cm, and Cp represents thecapacitance between the underpass line and the spiral. Theelectric loss in the substrate is modeled by COX, C,. and Rsiwhich are proportional to the area covered by the spiral.Most elements noticed above have the same implication inthe normal physical model[2]

Cm2 (N 1) ox

(1)Cp = (N-l) w h2 h

Cox = Ie w

c,i = le 'w Csub3

R, =le *w Gsub

N is the total turns of the inductor, le' is the averagediameter, 8ox is the permittivity of the oxidation layer, G,-1band C,-1b are conductance and capacitance per unite area ofthe spiral, and depend on process conditions.

A. Skin and Proximity Effects ModelingIn the higher frequency, issues of skin and proximity

effects become very important. In a spiral inductor, skin andproximity effects result in vertical and lateral nonuniformcurrent of a rectangular conductor. Also the currentdistribution changes with frequency, so the series parametersL, and R, are functions of frequency.

An improved PEEC method is presented, which caneffectively capture skin and proximity effects. A rectangular

conductor cross section is usually divided into mxn parts byaccustomed PEEC methods, suppose a uniform current ofevery part, and then, solve the problem with electromagneticformulas. This method leads to precise results but costsmuch time.

Based on accustomed PEEC methods, in the light ofcharacteristics of geometrical dimensions of a typical spiralinductor (w>>t), a rectangular conductor cross-section isdivided laterally into m parts by the improved PEEC method,which have equal lateral width. Every part is equivalent to aself-inductor L in series with a resistance R. M representedmutual inductances among different parts. L and M arecalculated with Greenhouse formulas. Due to proximityeffect causing the lateral non-uniform current of crosssection in a rectangular conductor and skin effect causing thevertical non-uniform current, R is only dependent on skineffect

(2)woff (I - e-t' )

and , 2C 1 0Ho

where 6 is the skin depth, pto vacuum magnetic conductance,a electricity conductance and co radian frequency.

Therefore, the I-V characteristics of a spiral inductorwhich takes account into skin and proximity effects can besolved with matrix equation(3). As shown in Fig.3, thevoltage V, represents the voltage of the ith segment conductorin x direction., and the current im represents the current of themth part of the ith segment conductor in x direction. Onaccount of Dij= I (i = ...m), it can be obtained byMATLAB that the relationship equation between the totalcurrent I and the voltage Vi in x direction. In the same way,the relationship equation between the total current I and thevoltage Vi in y direction can be acquired. Finally, L,,(f) andR,,(f) which calculates skin and proximity effects can besolved by equations(4)(5)

R±I+ joLll io°M1,12 ... ... froMMl,ili°M li2i°OM12,11 RI2 + i°tL12 ... i°CtM12,ilI i°cM1,i22

jiM,il j°Mil,2

ro)M12,nm

... ... ±+ jwLil jwM ,,rI2o°Mi2,11 fo°M2,12 ..M.. j 2,il Ri2 +±oLi2 j)M2,im ... .. jct)M2,nm

.....

i°oMim, joMi ,12 i°.Mim il i°Mimi2

i°joMnn,ll joMm.12 ncm.il i°Mnm.i2

Rim + jo)Lim foAMz,n

*-- j.coMm,..± ... ...--LR + jot)L,

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VI

Vi

Vi

L.,

il

ill

1i2

(3)

(4)R, (f) =realV +z,realt

X,= ( I )YL I

IIA12-1-tm bin-b "I El 42-hwX .. ...i,UfIC.Zr. tm

Figure 3. Modified PEEC method calculates skin and proximity effects

B. Substrate Eddy-Current Losses ModelingIn standard CMOS technology, electromagnetic loss in

silicon substrate due to its low resistivity is much higher thanthat in GaAs substrate which has high resistivity. Most ofcurrent simple models just take into account the electric losswithout the significant eddy-current loss which is generatedby the magnetic field penetration in the substrate for higherfrequency.

Eddy current creates its own magnetic field that opposestime-varying magnetic field. This decreases the inductanceof the spiral and induces the power loss in substrate. In thismodel, a RL loop coupling with spiral inductor is employedto capture above two effects accurately.

It is complicated to solve the magnetic field distributionof a rectangle inductor. Thus, the substrate RL loop ofcircular inductor is calculated firstly and then, the RL loop ofthe rectangle inductor is obtained according to theirgeometric dimensions relationship.

In order to investigate the magnetic field distribution ofthe circular inductor, the space s is neglected (w>>s) and anassumption of a cirque with uniform current density isemployed. The magnetic induction intensity at the sameplane of the cirque

__ r rcosW).dQB=f R, poiR dR f (R-rco) =B' (6)Br JRdRJj (r2 + R2 _2Rrcos()23 r

where R1 and R2 represent the inside radius and externalradius of the circular inductor, R the radius of the circularcurrent element in the cirque, r the distance of arbitrary pointin the same plane to the center of the circle and i the currentdensity in the circular inductor.

As le>>tox, the distribution of magnetic field on thesurface of the substrate can be assumed the same with that onthe plane of circular inductor. Thus, on the surface of thesubstrate the magnetic flux in circle with r radius is

O0r f r2I-2TBrdr = if r2IrBr'dr (7)

As the current density of the spiral inductor changes, theinductive electromotive force is given by eq. (8)

dtdi f r21B ddt

(8)

Compared with the resistance of the spiral inductor, the(5) resistance of substrate is much bigger so that the substrate

eddy current is much smaller than the current of the spiralinductor. Also, magnetic induction intensity produced by thesubstrate eddy current is much smaller than that produced bythe spiral inductor. So the impact of the substrate eddycurrent on its distribution could be neglected, that is, thesubstrate eddy current density distribution is only determinedby the inductive electromotive force produced by the spiralinductor.

'eddyrYrteff

(9)

Where Psub is the substrate resistivity and teff denotes the eddycurrent effective thickness that includes the skin effect.

i eddyrBr oeoe ieddyr

- Br

I < Rl R2 reff ee. 1

Figure 4. The distributions of the magnetic inductionintensity B on the surface of substrate and eddy

Supposing the constant items which are independent on r ineq. (6) - (9) and the distributed characteristic of ieddy. iSillustrated in Fig. 4.

Based on the inductance of substrate eddy currentdetermined by effective radius [2], in the light of therelationship between ieddy. and r in eq. (9), the effectiveradius reff is given

(10)Id 1edrrreff o I dr

eddy

and eddy =l o 'eddyrdr

Where radius d is usually 1-2 times length of le and leddy thetotal substrate eddy current within radius d.

Substrate eddy current is supposed to be cylindricalcurrent with radius reff, the self-inductance of Substrate eddycurrent is given by eq. (11)

ksubc lief rjyAr.ff.drJf 7 (rff -rcos )- dy(r2+rejff I2rrcosq 2

(1 1)

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Substrate resistance R,-1b, is calculated by loss equivalentmethod[6].

subc 1 2 1 2I lo reddy eddy eff

(12)

and teff=l -e-tsub )

The general conversion rule among the inductors ofdifferent shapes is that the inductances ratio is approximatelysquare root of areas ratio and the resistances ratioapproximately perimeters ratio. Making use of Rsubc and Lsubcof the circular inductor, Rs-1b and Ls-1b of square inductor isgiven by eq.(13) and (14).

(13)LsUb = ( LSUbC

RSIT

700 13

500 - typical model le=226 iLm 12n model w,¢/ w= tLm

500 - x x xx HFSS t 0 s=2W - 9

400 ¢ f a7

300 5',i' !;100

-200 - £0 5 10 15 20

Frequency (GHz)

(a)450 i- -1'7 A5d

400

350

300

250

t 200

150

100

(14)

Coupling degree between the RL loop of substrate eddycurrent and spiral inductors is related to geometricaldimensions of the spiral inductor. In the light of le>>toxcharacteristic of typical spiral inductors, mutual inductancebetween L,-1b and LSC is supposed to be 1 [7].

Therefore, the resistance R#(f) and the inductance L#(f) of221 equivalent circuit in Fig.2 are computed using formulas offull-coupled transformers

Rs(f) =Rs,(f)+ 2/M2RubRsub2 + (2Lsub )2

Ls(f) = Ls, (f)-2T(2fV)2Ls,bRsub2 + (2;tLs,b)2

(15)

(16)

and M = K L cLmubc

III. VERFICATION AND ANALYSE OF THE MODEL

Two three-turn inductors which have different geometricdimensions are chosen to verify the accuracy of the 221equivalent circuit model. And parameters of CMOStechnology are shown in Table I. The data from full-waveelectromagnetic filed simulator (HFSS), typical model [2]and the 221 equivalent circuit model are compared.

TABLE I. PARAMETERS OF CMOS TECHNOLOGY

Parameter Value Parameter ValuePsub 5Q cm h I1pmPmetal 0.03Q.cm t Iltmtsub 500ptm T tox 55pm

0 5 10 15 20Frequency (GHz)

6.4

5.4

4.4

3.4

2.4

1.4 -

0.4

-0.6

-1.6

-2.6

(b)

Figure 5. (a) & (b) The effective resistance Reff andeffective inductance L,ff of the two different inductors

-2 U 2 4 b E 1U0 12 14 lb6Frequency (GHz)

(a)1 | r r T 1~~~~

Frequency (GHz)

(b)Figure 6. (a) & (b) The quality factors of the two

different inductors

Connecting the inner ports of the inductors to the ground,the effective resistances and effective inductances derived

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3

1

9

7

5 sr-

3 0)I

1

3

5

-50

from the outer ports are shown in Fig.5 (a) & (b), and thequality factors Q= WLeff/Reff in Fig.6 (a) & (b).

The results indicate that up to 15GHz, the 22 equivalentcircuit model reveals quite good accuracy within 8% withdata from HFSS, including equivalent inductor Leff, resistorReff and quality factor Q. And Q is estimated too high by thetypical model above 3GHz for the ignorance of the proximityeffect, substrate eddy current loss and the distribution ofparasitical capacitances.

As the 221 equivalent circuit model presented is sufficientaccurate and much more efficient than full-waveelectromagnetic filed simulation, it is competent for thedesign and optimization of spiral inductors.

The impacts of skin effect, proximity effect and thesubstrate loss on L, and R, are compared in Fig.7 (eg. thedimensions of the spiral inductor are le 190ptm, w 13pm ands 1.5pm). Skin effect has no impact on L, and makes R, toincrease in square root of frequency. After adding proximityeffect, the non-uniform current in conductors becomesterrible, which causes Rs increasing faster as frequency.Confessedly, the proximity effect induce the currents whichflow in the same direction in conductors apart from eachother and the currents which flow in the opposite directionclose to each other [5], resulting in Ls decreasing. Thesubstrate eddy current produces magnetic field in theopposite of time-varying magnetic field, thereby decreasingLs. Also, the substrate eddy current brings power loss in thesubstrate, increasing Rsb too. The latter is the main factorimpacting on the characteristics of spiral inductors.

Fig.7 demonstrates that neither the substrate eddy currentabove 3GHz nor proximity effect can be neglected instandard CMOS process.

Verification with the data from full-wave electro-magnetic filed simulator demonstrates that the model showsexcellent agreement within 8% up to 15GHz.

This model can be applied to design and optimization forspiral inductors on silicon substrate. The idea of this modelalso has some referenced value for the study of spiralinductors.

REFERENCES

[1] D. Pukneva, G. Dodeva, and M. Hristov, "Design, analysis andoptimisation of monolithic inductors for RF applications" in Proc. ofthe 2002 First International IEEE Symposium on Intelligent Systerms,Vol.3, pp.63-68, Sept. 10-12, 2002.

[2] C. Patrick Yue, S. Simon Wong, "Physical modeling of spiralinductors on silicon", IEEE Trans. Electron Devices, Vol.47, pp.560-568, 2000.

[3] Li Fuhua, Zhao Jixiang, and Li Zhengfan, "Modeling for spiralinductors on-chip with partial element equivalent circuit method",Chinese Journal of Semiconductors, Vol.26, pp.770-774, 2005.

[4] H. M. Greenhouse, "Design of planar rectangular microelectronicinductors", IEEE Trans. Parts, Hybrids, Packaging, Vol.10, pp.101-109,1974.

[5] Yu Cao, Robert A. Groves, and Huang Xuejue, et. al., "Frequency-independent equivalent-circuit model for on-chip spiral inductors",IEEE JSSC, Vol.38, pp.419-426, 2003.

[6] K. T. Tong, C. Tsui. "A physical analytical model of multilayer on-chipinductors", IEEE Trans. Microwave Theory and Techniques, Vol.53,pp. 1 143-1 149, 2005.

[7] P. Arcioni, R. Castello, and L. Perregrini, et. al., "An innovativemodelization of loss mechanism in silicon integrated inductors",IEEE Trans. Circuits and Syst. II, Vol.46, pp. 1453-1460, 1999.

2.21

2.19

217

2.15

213T'

2.11 -S:1

2.09

2.07

2.05

2.03

0 5 10 15 20

Frequency (GHz)

Figure 7. Effects on Ls and Rs

IV. CONCLUSIONSA novel 22 equivalent circuit model has been developed,

which accounts for skin and proximity effects in the metalconductors, eddy current loss in the substrate as well as

capacitors distributed characteristics for on-chip spiralinductor on high-loss silicon substrate.

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22 2ooskin effect

20 skin & proirnfity effect 2

18 - skin proxsmity eddy current effect 2

16 2

114 2

12 -2

10 2

6 2.

4 -2.