Generiic RF passive device modeling

Generic Model Fitting of

Passive RF Devices

Tao-Yi Lee

Advisor: Yu-Jiu Wang

RFVLSI LAB @ NCTU

2014/4/18 Tao-Yi Lee @ RFVLSILAB 1

Outline

• The Model Fitting Design Flow

• Examples

– Model Fitting Of Inductors

– Model Fitting Of Center Tapped Inductors

– Model Fitting Of Transmission Lines

– Model Fitting Of Transformers

• Conclusion and Future Works

• References

2014/4/18 Tao-Yi Lee @ RFVLSILAB 2

Design Flow of Modeling Fitting

2014/4/18 Tao-Yi Lee @ RFVLSILAB 3

Start

Propose passive lumped equivalent model for an arbitrary high-frequency structure

Solve Y parameter matrix [Y] of the lumped equivalent network

Rum EM simulations of the desired structures, obtain [YEM]

Program the Ycost(R1, L1, C1)=[Y]-[YEM] matrix into MATLAB script as cost functions in numerical analysis

Solve values for lumped component, i.e. find R1, L1, C1,…, such that Ycost is minimized

Stop

Passive Lumped Equivalent Model For

Arbitrary High-frequency Structure

2014/4/18 Tao-Yi Lee @ RFVLSILAB 4

Propose passive lumped equivalent model for an arbitrary high-frequency structure

PORT1 PORT3

C13

C12 C23

C33C22C11

L12 L23R12 R23

Model #1

PORT2

Mi2i1

PORT1

PORT2

PORT3

Si Substrate

IMD

Cox Cox

• Main lumped elements• Skin effect• Loss

• Substrate• Eddy current

Modeling Skin Effect

• Skin effect: 𝑅𝑐𝑜𝑛𝑑 ∝ 𝑓; 𝐿𝑐𝑜𝑛𝑑 ≈ 𝑐𝑜𝑛𝑠𝑡.

– A non-linear effect

– Consider substrate coupling and proximity effect

2014/4/18 Tao-Yi Lee @ RFVLSILAB 5

• T. Kamgaing, T. Myers, M. Petras, And M. Miller, "Modeling Of Frequency Dependent Losses In Two-port And Three-port Inductors On Silicon," Radio Frequency Integrated Circuits Symposium, Pp. 307-310, 2002.

• C.-S. Yen, Z. Fazarinc, and R. L. Wheeler, “Time-Domain Skin-Effect Model for Transient Analysis of Lossy Transmission Lines,” Proceedings of the IEEE, vol. 70, pp. 750-757, 982• S. Kim and D. P..N eikirk, “Compact Equivalent Circuit Model for the Skin Effect”

Rm

Rf1

Lf1

Rf2

Lf2

Rf3

Lf3

Modeling Eddy Current

• Complex Image Method

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• D. Melendy and A. Weisshaar, “A New Scalable Model for Spiral Inductors on Lossy Silicon Substrate,” in 2003 MTT-S Symposium, June 2003, pp. 1007 – 1010• Melendy, D.; Francis, P.; Pichler, C.; Kyuwoon Hwang; Srinivasan, G.; Weisshaar, A.; , "A new wideband compact model for spiral inductors in RFICs," Electron Device Letters,

IEEE , vol.23, no.5, pp.273-275, May 2002• Kai Kang; Jinglin Shi; Wen-Yan Yin; Le-Wei Li; Zouhdi, S.; Rustagi, S.C.; Mouthaan, K.; , "Analysis of Frequency- and Temperature-Dependent Substrate Eddy Currents in On-Chip

Spiral Inductors Using the Complex Image Method ," Magnetics, IEEE Transactions on , vol.43, no.7, pp.3243-3253, July 2007

PORT1 PORT2Meddy

Rs,eddy

PORT1

PORT2

PORT3

Si Substrate

IMD

Image inductor on

lossy substrate

Modeling Oxide Capacitance and

Substrate Loss

• Model silicon substrate and IMD (oxides) as a 2D mesh

• Semi-empirical formula accounting for fringing and proximity effects (s: spacing, w: line width, ℎ𝑜𝑥: height above oxide)

𝐶𝑜𝑥 = 1 −𝑠

𝑠 + 𝑤

1.16 𝜖0𝜖𝑜𝑥 ∙ 𝑤 ∙ 𝑙

ℎ𝑜𝑥

2014/4/18 Tao-Yi Lee @ RFVLSILAB 7

CoxCox Cox

RsubCsubRsubCsubRsubCsub

Rnon-uniform Rnon-uniform

OptionalOptional

• Kai Kang; Jinglin Shi; Wen-Yan Yin; Le-Wei Li; Zouhdi, S.; Rustagi, S.C.; Mouthaan, K.; , "Analysis of Frequency- and Temperature-Dependent Substrate Eddy Currents in On-Chip Spiral Inductors Using the Complex Image Method ," Magnetics, IEEE Transactions on , vol.43, no.7, pp.3243-3253, July 2007

Modeling Substrate

• Modeling of substrate extrinsic is generally difficult, but some closed form solution are found in micro-strip transmission line researches (ℎ𝑠𝑢𝑏:height of the substrate, 𝜖𝑠𝑢𝑏,𝑒𝑓𝑓: effective dielectric constant)

– 𝐶𝑠𝑢𝑏 =

𝑤

ℎ𝑠𝑢𝑏+1.393+0.667 ln

𝑤

ℎ𝑠𝑢𝑏+1.444

120𝜋𝑐∙

𝑙

2𝜖𝑠𝑢𝑏,𝑒𝑓𝑓

• In reference 2, shunt resistance 𝑅𝑠𝑢𝑏 in silicon can be determined using relaxation time constant

𝜖0𝜖𝑆𝑖

𝜎𝑆𝑖

– 𝑅𝑠𝑢𝑏 =𝜖0𝜖𝑆𝑖

𝐶𝑆𝑖𝜎𝑆𝑖

• Consider circuit optimization to look for practical design values

2014/4/18 Tao-Yi Lee @ RFVLSILAB 8

• Ref. 1 :M. Kirschning and R. H. Jansen, “Accurate wide-range design equations for the frequency-dependent characteristics of parallel coupled microstrip lines,” IEEE Trans. Microwave Theory and Tech., vol. MTT-32, pp. 83–90, Jan. 1984.

• Ref.2 :J. Zheng, Y.-C. Hahm, V. K. Tripathi, and A. Weisshaar, “CAD-oriented equivalent circuit modeling of on-chip interconnects on lossy silicon substrate,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1443–1451, Sept. 2000

Solve Y Parameter Matrix 𝑌 of The

Lumped Equivalent Network• Definition of Y parameters

– Yij = Ii

Vj Vk=0 for k≠j

–

𝐼1𝐼2𝐼3

𝑌11 𝑌12 𝑌13

𝑌21 𝑌22 𝑌23

𝑌31 𝑌32 𝑌33

𝑉1

𝑉2

𝑉3

– Short all other terminals to ground reference and write down 𝑌𝑖𝑗 as function of lumped elements

– Simple; Can be done by inspection

– Matrix symmetry of passive networks

2014/4/18 Tao-Yi Lee @ RFVLSILAB 9

Solve Y parameter matrix [Y] of the lumped equivalent network

Solve Y Parameter Matrix 𝑌 of The

Lumped Equivalent Network

• 2-port 𝜋 model

2014/4/18 Tao-Yi Lee @ RFVLSILAB 10

Y11+Y21

-Y12

Y21+Y22

PORT1 PORT2PORT1

C12

C22C11

L12 R12

PORT2i1

simple 2 port inductor model

Solve Y Parameter Matrix 𝑌 of The

Lumped Equivalent Network

• 2-port shunt model

2014/4/18 Tao-Yi Lee @ RFVLSILAB 11

Y11+Y21

-Y12

Y21+Y22

PORT1

Y11+Y21

PORT1

-Y12

Y11PORT1

-Y12

Solve Y Parameter Matrix 𝑌 of The

Lumped Equivalent Network

• 2-port differential model

2014/4/18 Tao-Yi Lee @ RFVLSILAB 12

Y11+Y21

-Y12

Y21+Y22

PORT1PORT2

-Y12PORT1 PORT2

Y11+Y21 Y21+Y22

-Y12

Y11//Y22+Y21/2 Y11//Y22-Y21/2

Run EM Simulations Of The Desired

Structures, Obtain 𝑌𝐸𝑀

• Convert S-parameters to

Y-parameters via post-

processing

2014/4/18 Tao-Yi Lee @ RFVLSILAB 13

Rum EM simulations of the desired structures, obtain [YEM]

YEMSEM

Define The Minimization Problem In

MATLAB

∀i, j, minimizeR1,R2,…,L1,L2,…,C1,C2,…

ΔYij

= minimizeR1,R2,…,L1,L2,…,C1,C2,…

𝐘 − 𝐘𝐄𝐌

subject to all passive elements ≥ 0

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Program the Ycost(R1, L1, C1)=[Y]-[YEM] matrix into MATLAB script as cost functions in numerical analysis

Solve Component Values Using Non-linear

Least-square Solvers

• “lsqnolin” function in

MATLAB

– trust-region-reflective

– levenberg-marquardt

• Computational intensive

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Solve values for lumped component, i.e. find R1, L1, C1,…, such that Ycost is minimized

MODEL FITTING OF INDUCTORS

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Inductor 1 Port Model #1

Y11 = 𝑠𝐶𝑠1 +1

𝑠𝐿1 + 𝑅1

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Model#11R1L

1SC

Inductor 1 Port Model #2

𝑌11 =𝐶𝑆1𝐶𝑆5

(𝐶𝑆1 + 𝐶𝑆5)⋅

𝑠 ⋅ (𝑠 +1

𝑅𝑆5𝐶𝑆5)

𝑠 +1

𝑅𝑆5 𝐶𝑆1 + 𝐶𝑆5

+1

𝑠𝐿1 + 𝑅1

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Model#2

1SC

5SC

5SR

1R1L

Inductor 1 Port Model #3

• 𝑌11 = 𝑠𝐶𝑓 +1

𝐿0⋅

𝑠+𝑅0+𝑅1

𝐿1

𝑠2+𝑠 𝑅0𝐿0+𝑅1𝐿0+𝑅0𝐿1

𝐿0𝐿1+

𝑅0𝑅1𝐿0𝐿1

+𝐶𝑆1𝐶𝑆2

(𝐶𝑆1+𝐶𝑆2)⋅

𝑠⋅(𝑠+1

𝑅𝑆2𝐶𝑆2)

𝑠+1

𝑅𝑆2 𝐶𝑆1+𝐶𝑆2

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2R2LModel#3

1SC

5SC

5SR

1R1L

EM Setup – Symmetrical Inductor

• Inductor@M9, UTM = 3.4𝜇𝑚, 2 turns

• IMD Simplification

• Localized Excitation

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Name Thickness (nm) Rel Permittivity Z

FOXEQ 300 3.9 1

ILDEQ 310 4.2 2

IMD_1aEQ 4100 3.523395 3

IMD_9aEQ 725 4.2 4

IMD_9bEQ 110 8.1 5

IMD_9cEQ 3230 4.2 6

PASS1EQ 1800 5.254054 7

MODEL FITTING OF CENTER-TAPPED

INDUCTORS

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Center Tapped Inductor 3 Port Model #1

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PORT1 PORT3

C13

C12 C23

C33C22C11

L12 L23R12 R23

Model #1

PORT2

Mi2i1

Center Tapped Inductor 3 Port Model #1

• 𝜇 =𝑠𝑀

𝑠𝐿12+𝑅12 𝑠𝐿23+𝑅23 −𝑠2𝑀2

• A =𝑠𝐿23+𝑅23

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿23𝑅12 +𝑅12𝑅23

• 𝐵 =𝑠𝐿12+𝑅12

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿23𝑅12 +𝑅12𝑅23

• 𝐴′ = −𝑠𝐿23+𝑅23+𝑠𝑀

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿23𝑅12 +𝑅12𝑅23

• 𝐵′ = −𝑠𝐿12+𝑅12+𝑠𝑀

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿23𝑅12 +𝑅12𝑅23

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Self Mutual

Y11 = sC11 + sC12 + sC13 + A 𝑌13 = 𝑌31 = −𝑠𝐶13 + 𝜇

Y22 = sC22 + sC12 + sC23 − A′ − B′ 𝑌12 = 𝑌21 = −𝑠𝐶12 + 𝐴′

Y33 = sC33 + sC23 + sC13 + B 𝑌23 = 𝑌32 = −𝑠𝐶23 − 𝐵′

Center Tapped Inductor 3 Port Model #2

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PORT1

PORT2

PORT3

C13

C12 C23

C33C22C11

L12 L23R12 R23

Model #2

CS2RS2 CS3RS3CS1RS1

Mi2i1

Center Tapped Inductor 3 Port Model #2

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Ys1

=sC11 sRS1CS1 + 1

sRS1 C11 + CS1 + 1

Ys2

=sC22 sRS2CS2 + 1

sRS2 C22 + CS2 + 1

Ys3

=sC33 sRS3CS3 + 1

sRS3 C33 + CS3 + 1

Self Mutual

Y11 = sC12 + sC13 + Ys1 + A 𝑌13 = 𝑌31 = −𝑠𝐶13 + 𝜇

Y22 = sC13 + sC23 + Ys2 − A′ − B′ 𝑌12 = 𝑌21 = −𝑠𝐶12 + 𝐴′

Y33 = sC13 + sC23 + Ys3 + B 𝑌23 = 𝑌32 = −𝑠𝐶23 + 𝐵′

Center Tapped Inductor 3 Port Model #3

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PORT1

PORT2

PORT3

C13

C12 C23

C33C22C11

L12 L23R12 R23

Model #3

CS2RS2 CS3RS3CS1RS1

L12i R12i L23iR23i

M

i2i1

Center Tapped Inductor 3 Port Model #3

• C =sL23+R23

′

s2L12L23−s2M2+s L12R23′ +L23R12

′ +R12′ R23

′

• D =sL12+R12

′

s2L12L23−s2M2+s L12R23′ +L23R12

′ +R12′ R23

′

• 𝐶′ = −𝑠𝐿23+𝑅23

′ +𝑠𝑀

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23′ +𝐿23𝑅12

′ +𝑅12′ 𝑅23

• 𝐷′ = −𝑠𝐿12+𝑅12

′ +𝑠𝑀

𝑠2𝐿12𝐿23−𝑠2𝑀2+𝑠 𝐿12𝑅23′ +𝐿23𝑅12

′ +𝑅12′ 𝑅23

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Self Mutual

Y11 = sC12 + sC13 + Ys1 + C 𝑌13 = 𝑌31 = −𝑠𝐶13 + 𝜇′

Y22 = sC12 + sC23 + Ys2 − C′ − D′ 𝑌12 = 𝑌21 = −𝑠𝐶12 + 𝐶′

Y33 = sC13 + sC23 + Ys3 + D 𝑌23 = 𝑌32 = −𝑠𝐶23 + 𝐷′

R12′ =

R12 sL12i + R12i

R12 + R12i + sL12iR23

′ =R23 sL23i + R23i

R23 + R23i + sL23i

Center Tapped Inductor 3 Port Model #4

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PORT1

PORT2

PORT3

C13

C12 C23

C33C22

C11

L12 L23R12 R23

Model #4

CS2RS2 CS3RS3CS1RS1

L12i R12i L23iR23i

RS4 RS4

M

Quality of Fitting

• Good from 1GHz thru 30 GHz

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EM Setup –

Symmetrical Center-Tapped Inductor• Inductor@M9, UTM = 3.4𝜇𝑚, 2 turns

• IMD Simplification

• Localized Excitation

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 30

Name Thickness (nm) Rel Permittivity Z

FOXEQ 300 3.9 1

ILDEQ 310 4.2 2

IMD_1aEQ 4100 3.523395 3

IMD_9aEQ 725 4.2 4

IMD_9bEQ 110 8.1 5

IMD_9cEQ 3230 4.2 6

PASS1EQ 1800 5.254054 7

MODEL FITTING OF 4 PORT CENTER-

TAPPED INDUCTORS

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Center Tapped Inductor 4 Port Model #1

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 32

PORT1 PORT3

C14

C12

C34

C44C33C11

L12 L34R12 R34

Model #1

PORT2

M i2i1

PORT3C22

C23PORT4

C24C13

Center Tapped Inductor 3 Port Model #1

• 𝜇 =𝑠𝑀

𝑠𝐿12+𝑅12 𝑠𝐿34+𝑅34 −𝑠2𝑀2

• A =𝑠𝐿34+𝑅34

𝑠2𝐿12𝐿34−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿34𝑅12 +𝑅12𝑅34

• 𝐵 =𝑠𝐿12+𝑅12

𝑠2𝐿12𝐿34−𝑠2𝑀2+𝑠 𝐿12𝑅34+𝐿34𝑅12 +𝑅12𝑅34

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Self

𝑌11 = 𝑠𝐶11 + 𝑠𝐶12 + 𝑠𝐶13 + 𝑠𝐶14 + 𝐴

𝑌22 = 𝑠𝐶12 + 𝑠𝐶22 + 𝑠𝐶23 + 𝑠𝐶24 + 𝐴

𝑌33 = 𝑠𝐶14 + 𝑠𝐶23 + 𝑠𝐶33 + 𝑠𝐶34 + 𝐵

𝑌44 = 𝑠𝐶14 + 𝑠𝐶24 + 𝑠𝐶34 + 𝑠𝐶44 + 𝐵

Mutual

𝑌12 = 𝑌21 = −𝑠𝐶13 − A

𝑌13 = 𝑌31 = −𝑠𝐶12 − 𝜇

𝑌14 = 𝑌41 = −𝑠𝐶23 + 𝜇

𝑌23 = 𝑌32 = −𝑠𝐶13 + 𝜇

𝑌24 = 𝑌42 = −𝑠𝐶12 − 𝜇

𝑌34 = 𝑌43 = −𝑠𝐶23 − 𝐵

Center Tapped Inductor 4 Port Model #2

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PORT1

PORT2

PORT4

C14

C12 C34

C44C33C11

L12 L34R12 R34

Model #2

CS2

RS2 CS4RS4CS1RS1

Mi2i1

CS3

RS3

C22

C23

PORT3

Center Tapped Inductor 3 Port Model #2

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Ys1

=sC11 sRS1CS1 + 1

sRS1 C11 + CS1 + 1

Ys2

=sC22 sRS2CS2 + 1

sRS2 C22 + CS2 + 1

Ys3

=sC33 sRS3CS3 + 1

sRS3 C33 + CS3 + 1

Ys𝟒

=sC𝟒𝟒 sRS𝟒CS𝟒 + 1

sRS𝟒 C𝟒𝟒 + CS𝟒 + 1

Self

𝑌11 = Ys1 + 𝑠𝐶12 + 𝑠𝐶13 + 𝑠𝐶14 + 𝐴

𝑌22 = Ys2 + 𝑠𝐶12 + 𝑠𝐶23 + 𝑠𝐶24 + 𝐴

𝑌33 = Ys3 + 𝑠𝐶13 + 𝑠𝐶23 + 𝑠𝐶34 + 𝐵

𝑌44 = Ys1 + 𝑠𝐶14 + 𝑠𝐶24 + 𝑠𝐶34 + 𝐵

Mutual

𝑌12 = 𝑌21 = −𝑠𝐶13 − A

𝑌13 = 𝑌31 = −𝑠𝐶12 − 𝜇

𝑌14 = 𝑌41 = −𝑠𝐶23 + 𝜇

𝑌23 = 𝑌32 = −𝑠𝐶13 + 𝜇

𝑌24 = 𝑌42 = −𝑠𝐶12 − 𝜇

𝑌34 = 𝑌43 = −𝑠𝐶23 − 𝐵

Center Tapped Inductor 4 Port Model #3

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PORT1

PORT2

PORT4

C14

C12 C34

C44

C22C11

L12 L34R12 R34

Model #3

CS3RS2

CS4RS4CS1RS1

L12i R12i L34iR34i

M

i2i1

C33

RS3CS2

C23

PORT3

Center Tapped Inductor 3 Port Model #3

• 𝜇′ =𝑠𝑀

𝑠𝐿12+R12′ 𝑠𝐿34+R𝟑𝟒

′ −𝑠2𝑀2

• C =𝑠𝐿34+R𝟑𝟒

′

𝑠2𝐿12𝐿34−𝑠2𝑀2+𝑠 𝐿12𝑅23+𝐿34R12′ +R12

′ R𝟑𝟒′

• 𝐷 =𝑠𝐿12+R12

′

𝑠2𝐿12𝐿34−𝑠2𝑀2+𝑠 𝐿12R𝟑𝟒′ +𝐿34R12

′ +R12′ R𝟑𝟒

′

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 37

R12′ =

R12 sL12i + R12i

R12 + R12i + sL12iR𝟑𝟒

′ =R𝟑𝟒 sL𝟑𝟒𝐢 + R𝟑𝟒𝐢

R𝟑𝟒 + R𝟑𝟒𝐢 + sL𝟑𝟒𝐢

Self

𝑌11 = Ys1 + 𝑠𝐶12 + 𝑠𝐶13 + 𝑠𝐶14 + 𝐶

𝑌22 = Ys2 + 𝑠𝐶12 + 𝑠𝐶23 + 𝑠𝐶24 + 𝐶

𝑌33 = Ys3 + 𝑠𝐶13 + 𝑠𝐶23 + 𝑠𝐶34 + 𝐷

𝑌44 = Ys1 + 𝑠𝐶14 + 𝑠𝐶24 + 𝑠𝐶34 + 𝐷

Mutual

𝑌12 = 𝑌21 = −𝑠𝐶13 − C

𝑌13 = 𝑌31 = −𝑠𝐶12 − 𝜇′

𝑌14 = 𝑌41 = −𝑠𝐶23 + 𝜇

𝑌23 = 𝑌32 = −𝑠𝐶13 + 𝜇

𝑌24 = 𝑌42 = −𝑠𝐶12 − 𝜇

𝑌34 = 𝑌43 = −𝑠𝐶23 − 𝐷

Center Tapped Inductor 3 Port Model #4

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 38

PORT1

PORT2

PORT4

C14

C12 C34

C44

C22

C11

L12 L34R12 R34

Model #4

CS3RS2

CS4RS4CS1RS1

L12i R12i L34iR34i

M

i2i1

C33

RS3CS2

C23

PORT3

Rs12 Rs34

EM Setup –

Symmetrical Center-Tapped Inductor• Inductor@M9, UTM = 3.4𝜇𝑚, 2 turns

• IMD Simplification

• Localized Excitation

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 39

Name Thickness (nm) Rel Permittivity Z

FOXEQ 300 3.9 1

ILDEQ 310 4.2 2

IMD_1aEQ 4100 3.523395 3

IMD_9aEQ 725 4.2 4

IMD_9bEQ 110 8.1 5

IMD_9cEQ 3230 4.2 6

PASS1EQ 1800 5.254054 7

Quality of Fitting

• Good from 1GHz thru 30 GHz

2014/4/18 (C) RFVLSI LAB Confidential TYLEE 40

Future Works

• Accuracy of transformer models

• Accuracy in higher frequencies

2014/4/18 Tao-Yi Lee @ RFVLSILAB 41

2014/4/18 Tao-Yi Lee @ RFVLSILAB 42

Thank you for listening!

References

• Sung-gi Yang, Gi-hyon Ryu, And Kwang-seok Seo, "Fully Symmetrical, Diff Erential-pair Type Floating Active Inductors," International Symposium On Circuits And Systems, Pp. 93-96, Jun. 1997.

• Kenichi Okada And Kazuya Masu, "Modeling Of Spiral Inductors," In Advanced Microwave Circuits And Systems, April 1, 2010, P. 291.

• C. Patrick Yue, Changsup Ryu, Jack Lau, Thomas H. Lee, And S. Simon Wong, "A PHYSICAL MODEL FOR PLANAR SPIRAL INDUCTORS ON SILICON".

• T. Kamgaing, T. Myers, M. Petras, And M. Miller, "Modeling Of Frequency Dependent Losses In Two-port And Three-port Inductors On Silicon," Radio Frequency Integrated Circuits Symposium, Pp. 307-310, 2002.

• J. R. Long And M. A. Copeland, "Modeling, Characterization And Design Of Monolithic Inductors For Silicon Rfics.," Custom Integrated Circuits Conference, 1996.

• Sunderarajan S. Mohan, Maria Del Mar Hershenson, Stephen P. Boyd, And Thomas H. Lee, "Simple Accurate Expressions For Planar Spiral Inductances," JOURNAL OF SOLID-STATE CIRCUITS, Vol. 34, No. 10, Oct. 1999.

2014/4/18 Tao-Yi Lee @ RFVLSILAB 43