Modeling of Multi-Port Inductor ConsideringMutual Components

Takeshi Ito, Kenichi Okada, and Kazuya MasuIntegrated Research Institute, Tokyo Institute of Technology

4259-R2-17 Nagatsuta, Midori-ku, Yokohama, 226-8503 Japan.Tel: +81-45-924-5031, Fax: +81-45-924-5166, E-mail : paper@lsi.pi.titech.ac.jp

ABSTRACT

This paper proposes a modeling method of multi-port on­chip inductors considering mutual components. Mutual in­ductances in a multi-port inductor influence each other. Inthis paper, self and mutual inductances are derived from S­parameters by using a matrix-decomposition technique . Anequivalent circuit model for multi-port inductors is presented,and extracted results using measured S-parameters are demon­strated. The average errors of Land Q are 2.1% and 10.0%,respectively.

2-port inductors

(a) four z-port inductors

(b) one s-port inductor

Fig. 1. Decreasing number of inductors and layout area.

where admittance matrix Ymeas is converted from measuredS-parameter. To decompose each part of multi-port inductor

parameters of each segment are individually extracted from thematrices. In this paper, the proposed method is demonstratedwith measurements.

II. MATRIX-DECOMPOSITION TECHNIQUE

In this section, the matrix-decomposition technique is ex­plained, and 5-port inductor is utilized as an example. Figure 2shows layout structure of the 5-port inductor, which has asimilar structure to a general 2-port inductor with 3 additionalterminals .

A. Derivation of matrix Y c

Figure 3 shows an equivalent circuit of the 5-port inductor,which consists of core, shunt, and lead parts[3] , [4]. Thecore part expresses self and mutual inductances with parasiticresistance and capacitance, which are characterized by Znin Fig. 3. The core part is also expressed by a matrix Yc'

The shunt part characterizes ILD and Si substrate, and it isexpressed by a matrix Y;;ub. Y;;ub consists of admittancesY;,ub n in Fig. 3. The lead part characterizes lead lines, and it isalso expressed by matrix Yo p en and Zshort . These matricescan be combined by the following equations.

(1)

(2)

(Ym eas - Yo p e n )- l

- (Zshort -1 - Yo p en )- l ,

Y m eas/ - Y;;ub ,

Zmeas/

I. INTRODUCTION

Recently, many kinds of commercial radio ICs using SiGeand CMOS process integrate on-chip inductors . It is not easyto characterize on-chip inductors because of eddy-current losscaused by Si substrate, and use of inaccurate models mightinvolve degradation of circuit performance. Thus, accuratemodeling of on-chip inductors is a very important issue inhigh-frequency circuit design.

This paper proposes a modeling method of multi-port in­ductors . Multi-port inductors are utilized in switched inductor,distributed amplifier, VCO, etc [I], [2]. For example, a center­tapped symmetric inductor, i.e. 3-port inductor, is a kind ofmulti-port inductors. A multi-port inductor can also be utilizedto save layout area, e.g., 5-port inductor can substitute four 2­port inductors as shown in Fig. I [I] .

In modeling of multi-port inductors, it is important toconsider mutual components. The state-of-art simulators candeal with S-parameter components directly. On the otherhand, an equivalent circuit model is indispensable to figureout self and mutual inductances for circuit design . In thiscase, each parameter of equivalent circuit has to reflect actualelectromagnetical properties. A parameter extraction methodusing numerical optimization easily lose such the physicalintegrity even if the total characteristic expressed by the ex­tracted parameters meets the original S-parameter numerically.Especially, equivalent circuit models of multi-port inductorshave many circuit components, which also cause degradationof the physical integrity. Thus, a novel modeling methodusing a matrix-decomposition technique is proposed in thispaper. The proposed method has two steps. As the first step,multi-port S-parameter is decomposed into several matricesby an analytical method, which expresses self- and mutual­effects of each segment. As the second step, equivalent-circuit

0-7803-9763-0/07/$20.00 ©2007 IEEE

1

5Fig. 4. Equivalent circuit of core.

Fig. 2. Structure of the 5-port inductor.

(7)

(8)

(9)

(10)

jwM13

jwM23

Z3

jwM43

jwM12

Z2

jwM32jwM42

(VZl) (iZl)Vz2 i-a - Z .= Zeore . - eore~zVz3 Zz3

Vz4 i z4

vz =

(

ZI

jwM21Zeore = . M

JW 31

jWM41

where vectors v, i, v z , and i z are defined as shown in Fig. 4.Each element of the matrix Zeore expresses self and mutualcomponents directly. The matrix ~ has the same numericalinformation as Zeore, which is explained later. The self andmutual inductances in Zeore can be derived from Ye .

Basically, rank of the matrix Zeore is 4, and the matrix Ye

consists of the same components as shown in Fig. 4. Thus,rank of ~ is also 4 although Ye is a 5 x 5 matrix. Thematrix Ye is not a regular matrix, and it does not have aninverse matrix. In this work, converting matrices A and Bare utilized, which are also not a regular matrix.

The converting matrices A and B are derived by thefollowing procedure . Vectors i and v are converted into vectorsi z and Vz by the following equations.

B. Conversion of matrix Y e to Zeore

Figure 4 shows the core part of the entire equivalent circuitin Fig. 3, which is expressed by the matrix Ye • In this case,we need each parameter of Zn and M n m , so the matrix Ye isconverted into a matrix Zeore . When Ye is a n x n matrix,Zeore is a (n - 1) x (n - 1) matrix. The matrix Zeore isdefined by the following equations.

Therefore, each Y;;ubn in ~ub can be calculated by thefollowing equation.

sYsu bn = LYmeas'ni

i= 1

(5)

(6)

(Y.."0 0 0

JJY..b ~ ~Y sub2 0 0

0 Y sub3 0 (3)0 0 Y sub4

0 0 0

For the sum of the matrices Ye and ~ub, the followingequation can be defined.

Fig. 3. Equivalent circuit of the 5-port inductor.

When VI = V2 = V3 = V4 = Vs, no current flows into Zn

shown in Fig. 3, which is described by the following equation.

in Fig. 3, first the matrix ~ub is calculated. The matrix ~ub

can be expressed by admittances Y;;ubn as follows.

The following equation is derived from Eqs.(4)(5)

Fig. 5. 1r-type equivalent circuit.

Fig. 6. Microphotograph of the 5-port inductor.

(14)

(15)

(13)

(11)

(12)

=Av

oo-11

o-11o

-11oo

(VZI) (VI - V2)Vz2 _ V2 - V3

Vz3 V3 - V4

Vz4 V4 - V5

V z =Av

i z = Bi

Matrices A and B are not regular matrix. Matrix Zcore is4 x4 matrix . ~ is shrunk to Zcore by pseudo-inverse matrixA +. For example, A + can be defined as follows .

According to Eq.(16), i l + i2 + i3 + i 4 is also expressed by- i 5 as an example. Thus, matrix B has several solutions asfollows.

B~ (10 0 0

~), (10 0 0

~)1 0 0 1 0 01 1 0 1 1 01 1 1 0 0 0 -1

(l0 0 0

J}(~0 0 0

~1)1 0 0 0 -1 -10 0 -1 0 0 -1 -1 '0 0 0 -1 0 0 0 0 -1

(l-1 -1 -1 -1)0 -1 -1 -10 0 -1 -1 , ...

0 0 0 -1

(17)

Eqs. (8)(11)(12) are substituted into Eq. (9), and the fol-lowing equations are obtained.

A m~ (~~) ~ Z~r e (1~1) ~ Z~~Bm(18)

A = ZcoreB~ (19)

(22)

(23)

= I (21)(

1 -1 0 0o 1 -1 0o 0 1 -1000 1

I = ZcoreBYcA+

Zcore = (BYcA+)-1

AA+=

III. EXPERIMENTAL RESULTS

In this section, the proposed method is performed formeasurement result. Figure 6 shows microphotograph of the5-port inductors. The configuration of the 5-port inductor issymmetric, 3 turns , width of 15 /Lm, line space of 1.2/Lm, andouter diameter of 250 /Lm. 5-port S-parameter is obtained fromtwo TEGs shown in Fig. 6 because the vector network analyzerwe used has four ports for measurement. Port 3 of inductor(a) is terminated by 50n resistor as indicated in Fig. 6. Port4 of inductor (b) is also terminated by 50n resistor.

Finally, the following equations are derived . Zcore is ex­pressed by Eq. (23). The self and mutual inductances arecalculated from S-parameter.

C. Parameter extraction

Next, each parameter of equivalent circuit is extracted. Fig­ure 5 shows a 1T-ladder equivalent circuit, which is transformedfrom the circuit model shown in Fig. 3. Zn and M nm candirectly be obtained from Zcore' Ysubm is the sum of YSm-1and YSm when m=2, 3,..., n - 1. Ysubi is YSI' and Ysubn isYSn-I' Each parameter in Fig. 5 is fitted to Zn and YSn by anumerical optimization.

(16)

Here, the follwing equation can be derived from Fig. 4.

5

L in = i l + i2 + i3 + i4 + i5 = 0n=1

(20)

0.1 1 10Frequency [GHz]

Fig. 7. Inductance of the 5-port inductor. Fig. 9. Coupling coefficient of the 5-port inductor.

1(25)Zl + v-; [II

Im(~J(26)

[21

w

Im(~J[31

(27)

Re(~J [41

Fig. 8. Quality factor of the 5-port inductor.

(28)

REFERENCES

T. Ito, D. Kawazoe, K. Okada, and K. Masu, "A DC-7GHz small-areadistributed amplifier using 5-port inductors in a 180nm Si CMOS tech­nology," in Proceedings of IEEE Asian Solid-State Circuits ConferenceDigest of Technical Papers, Nov. 2006, pp. 363-366.W. Khalil, B. Bakkaloglu, andS. Kiaei, "Aself-calibrated on-chip phase­noise-measurement circuit with -75dBc single-tone sensitivity at 100kHzoffset," in IEEE International Solid-State Circuits Conference Digest ofTechnical Papers, Feb. 2007, pp. 546,547,621.J. R. Long and M. A. Copeland, "The modeling, characterization, anddesign of monolithic inductors for silicon RF Ie's:' IEEE Journal ofSolid-State Circuits, vol. 32, no. 3, pp. 357-369, Mar. 1997.A. M. Niknejad and R. G. Meyer, "Analysis, design and optimizationof spiral inductors and transformers for Si RF IC's," IEEE Journal ofSolid-State Circuits, vol. 33, no. 10, pp. 1470-1481, Oct. 1998.

ACKNOWLEDGMENT

This work was partially supported by MEXT. KAKENHI,JSPS. KAKENHI, STARC, MIC. SCOPE, Intel and VDEC in collab­oration with Cadence Design Systems, Inc., and Agilent TechnologiesJapan, Ltd.

IV. CONCLUSIONS

The modeling method using the matrix-decomposition tech­nique is proposed in this paper. S-parameter of multi-portinductors can be analyzed as self and mutual inductances.Parameter extraction using the proposed modeling methodis demonstrated. The average errors between measured andmodeled values are 2.1% in L, and 10.0% in Q.

Mn mkn m = --===

VLnLm

knm is coupling coefficient between Ln and Lm. Couplingcoefficients kn m has various values from -0.00 to 0.50 becauseline to line coupling intensity is different depending on topol­ogy of each segment. In this experiment, coupling coefficientk23 is larger than the others because L2 and L3 are arrangedparallelly. Coupling coefficient k 14 is almost zero because L 1

and L 4 are arranged orthogonally.

errors of Land Q are 2.1% and 10.0% from 0.5 GHz to 2 GHz,respestively.

Figure 9 shows coupling coefficient of the 5-port inductorcalculated by Eq. (28).

(24)

I~~~~,-,-----~~J

0.1 1 10Frequency [GHz]

Figures 7 and 8 show measured and modeled inductancesand quality factors of the 5-port inductor, respectively. Themodelded Land k agree with the measurement. The average

Yu

Ymeas is obtained by the following equation.

(

a u a12 b31 a14 a15)a 21 a22 b32 a24 a25

Ymeas = b31 b32 b33 b32 b35 ,

a41 a42 b23 a44 a45

a 51 a52 b53 a 54 a 55

where a ij and bi j are measured S-parameter elements ofinductors (a) and (b), respectively. In this case, Ymeas34 andYme asas cannot be obtained, so these components are substi­tuted by Ymeas 32 and Ymeas 23' respectively. The matrix Zcore

is calculated from Y m eas , Y o pen, and Zshort as explained inSec. II . In this experiment, Y o p en and Zshort are obtainedfrom electromagnetic field simulation (Ansoft HFSS).

To evaluate Land Q of Zn, Ynn is utilzed. For example,Yu is derived from Zl in the matrix Zcore, and the followingequations are used.