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RF INDUCTOR/TRANSFORMER

Enrollment No.: 10305179

Name: PALLAVIE TYAGI

Project Supervisor: PROF. A. B. BHATTACHARYYA

May 2012

Thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Technology

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY

(Deemed to be University)

NOIDA

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CERTIFICATE

This is to certify that the work titled “RF INDUCTOR/TRANSFORMER”

submitted by “Pallavie Tyagi” in partial fulfillment for the award of degree of

Master of Technology (Microelectronics and Embedded Technology) of Jaypee

Institute of Information Technology, Noida has been carried out under my supervi-

sion. This work has not been submitted partially or wholly to any other Universityor Institute for the award of this or any other degree or diploma.

Signature of Supervisor:

Name of supervisor:

Designation:

Date:

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ACKNOWLEDGMENT

I would like to express my deepest gratitude and appreciation for my supervisor

“Prof. A. B. Bhattacharyya”, without whose faith, continued interest, dedica-

tion and support, this work would not have been possible. Additionally, he deserves

my accolades for his patience and fairness during tough moments in my work. I

thank him for encouraging me at times when I needed it most. His energy and

enthusiasm for knowledge are boundless.

I would also like to thank “Mr. Kirmender Singh” for sharing his passion and

love for circuit design and device modeling. He encouraged me to work on a variety

of projects and thereby provided me with a well rounded perspective in engineering.

Above all, I thank him for his friendship, his natural charisma and great sense of

humour.

This project is a part of “NPMASS program” supported by Government of

India at MEMS Design Center at JIIT.

Signature of Student:

Name of Student:

Date:

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ABSTRACT

High performance Inductors and Transformers are playing an increasing role

in modern communication systems. Despite the superior performance offered by

discrete components, parasitic capacitances from bond pads, board traces and pack-

aging leads reduce the high frequency performance and contribute to the urgency

of an integrated solution. Embedded Inductors have the potential for significant

increase in reliability and performance of the IC. Due to the driving force of CMOS

integration and low costs of silicon-based IC fabrication, these Inductors and Trans-

formers lie on a low resistivity silicon substrate, which is a major source of energy

loss and limits the frequency response. Therefore, the quality factor of Inductors

and Transformers fabricated on silicon continues to be low.

In this thesis to improve the performance of the Inductors and Transformers

various approaches have been used. One approach is to fabricate the transformer

on a certain distance from the silicon substrate and then within this distance use

Air, High resistive silicon or Silicon nitride as a cavity. By doing so improvement

in quality factor is observed. Another approach is to increase the thickness of the

silicon dioxide. Fabrication of inductors or transformers on such a thick SiO2 layer

can be a good solution to achieve better performance. However, the thickness of such S iO2 layer is still limited by the process for further performance improvement.

One another approach is to use glass as a substrate and RF performance of the

glass-based transformer is improved compared to that of silicon-based transformer

highlighting a good prospect for the future 3-dimensional RF device application.

Meander structure is also investigated in this project. Perform the modeling

of meander inductor by decomposing it into individual straight segments and then

compute the self and mutual inductance. The obtained results are compared with

FASTHENRY.

Signature of Student: Signature of Supervisor:

Name: Name:

Date: Date:

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Contents

CERTIFICATE ii

ACKNOWLEDGEMENT iii

ABSTRACT iv

LIST OF TABLES vii

LIST OF FIGURES 1

1 INTRODUCTION 1

1.1 INTRODUCTION:- . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 MOTIVATION:- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 THESIS OUTLINE:- . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 LITERATURE REVIEW OF INDUCTOR AND TRANSFORMER 4

2.1 WHAT IS A INDUCTOR:- . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 CIRCUIT MODEL OF INDUCTOR:- . . . . . . . . . . . . . . . . . 4

2.3 WHAT IS A TRANSFORMER:- . . . . . . . . . . . . . . . . . . . . 4

2.4 TRANSFORMER APPLICATIONS:- . . . . . . . . . . . . . . . . . . 5

2.5 ELECTRICAL CHARACTERISTICS:- . . . . . . . . . . . . . . . . . 5

2.6 TRANSFORMER TOPOLOGIES:- . . . . . . . . . . . . . . . . . . . 7

2.6.1 ADVANTAGES OF INTEGRATED PASSIVE COMPONENTS:-

82.7 LOSS MECHANISMS IN TRANSFORMERS ON SILICON SUBSTRATES:-

9

2.7.1 CONDUCTOR LOSSES:- . . . . . . . . . . . . . . . . . . . . 9

2.7.2 SUBSTRATE LOSSES:- . . . . . . . . . . . . . . . . . . . . . 11

2.8 CIRCUIT MODEL OF THE TRANSFORMER:- . . . . . . . . . . . 11

3 QUALITY FACTOR ENHANCEMENT TECHNIQUES FOR IN-

DUCTOR AND TRANSFORMER 14

3.1 INTRODUCTION:- . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 PARASITIC EFFECTS IN THE SUBSTRATES:- . . . . . . . . . . . 14

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3.2.1 MAGNETICALLY INDUCED PARASITIC EFFECTS IN THE

SUBSTRATE:- . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 ELECTRICALLY INDUCED PARASITIC EFFECTS IN THE

SUBSTRATE:- . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 QUALITY FACTOR:- . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 SELF RESONANCE FREQUENCY:- . . . . . . . . . . . . . . . . . . 16

3.5 PROCESS PARAMETERS AND DIMENSIONS:- . . . . . . . . . . . 17

3.6 Q-FACTOR ENHANCEMENT USING CAVITY:- . . . . . . . . . . 17

3.7 Q-FACTOR ENHANCEMENT USING COPPER:- . . . . . . . . . . 21

3.8 Q-FACTOR ENHANCEMENT USING HIGH RESISTIVE SUBSTRATES:-

22

3.9 LAYOUT OF INDUCTOR:- . . . . . . . . . . . . . . . . . . . . . . . 24

3.10 LAYOUT OF TRANSFORMER:- . . . . . . . . . . . . . . . . . . . . 25

3.11 EXPERIMENTAL RESULTS:- . . . . . . . . . . . . . . . . . . . . . 25

4 VERIFICATION OF SELF AND MUTUAL INDUCTANCE THROUGH

FASTHENRY 27

4.1 PARTIAL INDUCTANCE APPROACH:- . . . . . . . . . . . . . . . 27

4.2 FASTHENRY:- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 VERIFICATION OF PARTIAL INDUCTANCE APPROACH ON

MEANDER:- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 CIRCUIT DIAGRAM OF MEANDER ON SILICON SUBSTRATE:- 32

4.5 MODELING OF INDUCTANCE:- . . . . . . . . . . . . . . . . . . . 33

4.6 SELF INDUCTANCE:- . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.7 MUTUAL INDUCTANCE:- . . . . . . . . . . . . . . . . . . . . . . . 40

4.8 EXPERIMENTAL RESULTS:- . . . . . . . . . . . . . . . . . . . . . 43

5 CONCLUSIONS 44

APPENDIX 44

A FASTHENRY CODING OF SPIRAL INDUCTOR:- 45

B FASTHENRY CODING OF TRANSFORMER:- 47

C DIMENSIONS OF INDUCTOR LAYOUT:- 51

REFERENCES 51

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List of Tables

1.1 Effect of Q on typical RF circuits . . . . . . . . . . . . . . . . . . . . 2

3.1 Process Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Aluminium with Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Copper with Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Copper without Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Aluminium without Cavity . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Parameters of Ground Signal Ground structure . . . . . . . . . . . . 29

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List of Figures

2.1 Circuit diagram of Inductor . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Monolithic Transformer- Physical Layout . . . . . . . . . . . . . . . . 7

2.3 Illustration of magnetic fields in a Transformer . . . . . . . . . . . . . 10

2.4 Induced Eddy currents in a Transformer . . . . . . . . . . . . . . . . 11

2.5 Circuit diagram of Transformer by O.El.Gharniti[16] . . . . . . . . . 12

3.1 Magnetically induced substrate losses in a Transformer . . . . . . . . 15

3.2 Electrically Induced losses in a Transformer . . . . . . . . . . . . . . 16

3.3 Cavity(Air) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Cavity(Air) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Cavity(oxide) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Cavity(Oxide) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.7 Cavity(Silicon Nitride) . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.8 No Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.9 Layout of Inductor and Transformer in ADS . . . . . . . . . . . . . . 22

3.10 High Resistive Si Substrate . . . . . . . . . . . . . . . . . . . . . . . . 23

3.11 Inductor on Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.12 Transformer on Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.13 Layout of Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.14 Layout of Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Layout of Meander structure surrounded with Ground . . . . . . . . 28

4.2 Ground Signal Ground structure . . . . . . . . . . . . . . . . . . . . . 294.3 Ground Signal Structure . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Plot of self inductance by varying the dg . . . . . . . . . . . . . . . . 30

4.5 Plot of self inductance by Varying the length lw of the conductor . . . 30

4.6 Ground Signal Ground- Symmetric Structure . . . . . . . . . . . . . . 31

4.7 Ground Signal Ground- Asymmetric Structure . . . . . . . . . . . . . 31

4.8 Circuit diagram of meander . . . . . . . . . . . . . . . . . . . . . . . 32

4.9 Meander structure on Silicon substrate . . . . . . . . . . . . . . . . . 32

4.10 Return current through Capacitors . . . . . . . . . . . . . . . . . . . 33

4.11 Two magnetically coupled loops . . . . . . . . . . . . . . . . . . . . . 34

4.12 Circular conductor with radius b . . . . . . . . . . . . . . . . . . . . 35

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4.13 GSG: Symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.14 GSG: Asymmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.15 Coupling inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.1 Screen shot of INDUCTOR . . . . . . . . . . . . . . . . . . . . . . . 45

B.1 Screen shot of TRANSFORMER . . . . . . . . . . . . . . . . . . . . 47

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Chapter 1

INTRODUCTION

1.1 INTRODUCTION:-

Monolithic inductors/transformers have found extensive applications in radio-

frequency (RF) circuits in wireless communication systems, such as impedance

matching, signal coupling, phase splitting and formation of large inductances on

the order of tens of nano henries. Implemented in CMOS technology, the current

inductors/transformers generally have low quality factors (Q), low self-resonant fre-

quencies f res and large cross talk.

The low performance of current inductors/transformers is in part due to the par-

asitics between the devices and the lossy silicon substrate. First, the eddy currents

induced in the silicon substrate beneath a inductor/transformer by the magnetic field

generated in the device cause energy loss and reduce Q, this effect is especially seri-

ous at high frequencies, because the intensity of the eddy currents is proportional to

the rate of change in the magnetic field. Second, the parasitic capacitances between

the inductor/transformer traces and the substrate are in shunt with the inductance

obtained from the spiral traces, thus lowering f res of the inductor/transformer, these

parasitic capacitances also enable electric coupling between the device and the sub-strate, causing further energy loss. Lastly, but not the least important,adjacent

devices are coupled through these parasitics and their ambient (including both the

substrate and air), hence large cross talk.

Because spiral inductors are typically used to form transformers, the very nature

of coupling entails even greater parasitic capacitances and eddy currents among the

spirals, resulting in more energy loss. Therefore, to improve the performance of

the on-chip monolithic transformers, the parasitics due to the substrate should be

suppressed as much as possible. One efficient approach is to drastically increase

the thickness of the isolation layer, typically silicon dioxide or by using air as a

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cavity, between the inductor/transformer and the silicon substrate. In doing so,

the magnetic and electric coupling between the device and the silicon substrate and

the coupling among devices through the substrate are greatly weakened. Another

approach to reduce these effects is the use of glass substrate instead of silicon sub-

strate. By using glass as a substrate quality factor is increased up to 40 percent.

1.2 MOTIVATION:-

Modern RF communication systems require stringent specifications for RF com-

ponents. Despite much work , which has been done on the integration of RF systems

into a single chip, many components remain off-chip. This is because the RF pas-

sives with the required performance are not available in the standard silicon process.

Many recent studies have shown that two aspects are extremely important in order

to obtain high performance integrated passive components when the signal frequency

is in the GHz range. One is the metal thickness and the other is how far the device is

isolated from the lossy substrate. For these two reasons, research has been performed

to investigate other technological options among available fabrication processes.

Recently, Transformers/Inductors have been required in many RFIC applications

for impedance matching/transforming, signal coupling, phase splitting (balun).Important

specifications for inductors/transformers are their Q and self-resonance frequency.High-Q inductors are essential for many different passive and active circuits and

can substantially reduce the phase noise or power consumption of oscillators and

amplifiers. Also, they result in low-loss matching networks and filters as shown in

Table 1.1.

Table 1.1: Effect of Q on typical RF circuits

Circuit Parameter Effect of Q

Oscillator Phase Noise 1Q2

Amplifier Gain Q

Oscillator Power Consumption 1

Q

Amplifier Power consumption 1

Q

Matching System Loss 1

Q

Filter Loss 1

Q

System Noise Figure 1Q

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MEMS based products combine both mechanical and electronic devices on a

monolithic microchip to produce superior performance over solid state components,

especially for Wireless Applications. Micromachined versions of off chip components,

including Vibrating Resonators , Switches, Capacitors, Inductors and Transformers

could maintain or shrink the size of future Wireless phones. Next generation hand-

sets need multiband reconfigurability and even larger number of high Q components.

However, on-chip Transformers acquired from the conventional silicon IC technolo-

gies do not meet the required performance of circuit designers. In order to address

this issue, non-standard substrates such as high resistivity silicon or glass substrates,

or sometimes substrates with insulation layers have been used to reduce the sub-

strate loss.

1.3 THESIS OUTLINE:-

This thesis consists of six chapters. Chapter 1 provides an introduction to the

research. This mainly consists of the motivation and importance of RF passive com-

ponents in modern communication system.

Chapter 2 presents literature review of inductors and transformers. This chapter

discusses various topologies proposed by previous researchers as well as applicationsof RF transformers. A physics based compact model of inductor and transformer

along with the loss mechanisms, responsible for the degradation of Q factor when

placed above the silicon substrate is discussed.

In chapter 3 Quality factor enhancement techniques for both the inductors and

transformers is presented. Conventional monolithic inductors can achieve a maxi-

mum Q-factor of 10, It poses a limitation for narrow band circuits but by fabricating

the inductor/transformer farther away the silicon substrate using the cavity or byfabricating the inductor/transformer on high resistive substrate, higher value of

Quality factor and self resonance frequency is obtained.

Chapter 4 presents the modeling of meander inductor surrounded by ground on

both the sides, decomposing the structure into individual straight segments and then

perform the computation of self and mutual inductance.

Finally in chapter 5 conclusions are highlighted and some future work in order

to continue this project are suggested.

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Chapter 2

LITERATURE REVIEW OF

INDUCTOR AND

TRANSFORMER

2.1 WHAT IS A INDUCTOR:-

An inductor is a component that stores energy in the form of a magnetic field.

Maxwells equations imply that current moving through a conductor induces a mag-

netic field. Similarly, changes in a magnetic field near a conductor induce changes in

the current flowing inside that conductor. Key parameters in integrated inductors

are the quality factor Q and the self-resonance frequency where Q would peak as

the device transforms from inductive to capacitive characteristics. Resistive metal

lines and dielectric losses in the substrate are the main contributing factors to the

degradation of the Q factor.

2.2 CIRCUIT MODEL OF INDUCTOR:-

A general model that describes the performance of a planar inductor ( square coil) is

shown in Figure 2.1. Ls is the low-frequency inductance, Rs is the series resistance of

the coil, C s is the capacitance between the different windings of the inductor and in-

cludes the fields in air and in the supporting dielectric layers, C ox is the capacitance

in the oxide layer between the coil and the silicon substrate, C p is the capacitance

between the coil and the ground through the silicon substrate, and R p is the eddy

current losses in the substrate[20].

2.3 WHAT IS A TRANSFORMER:-

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Figure 2.1: Circuit diagram of Inductor

A Transformer is a passive device that “Transforms” or converts a given impedance,

voltage or current to another desired value. In addition, it can also provide DC isola-

tion, common mode rejection, and conversion of balanced impedance to unbalanced

or vice versa. Transformers come in a variety of types, our focus is on Transform-

ers used in RF and Microwave signal applications. Essentially, an RF Transformer

consists of two or more windings linked by a mutual magnetic field. When one wind-

ing, the primary has an ac voltage applied to it, a varying flux is developed, the

amplitude of the flux is dependent on the applied current and number of turns in

the winding. Mutual flux linked to the secondary winding induces a voltage whose

amplitude depends on the number of turns in the secondary winding.

2.4 TRANSFORMER APPLICATIONS:-

Transformers are used for:-

• Impedance matching to achieve maximum power transfer between two devices.

• Voltage/current step-up or step-down.• DC isolation between circuits while affording efficient AC transmission.

• Interfacing between balanced and unbalanced circuits, example: push-pull am-plifiers, ICs with balanced input such as A to D converters.

• Common mode rejection in balanced architectures.

2.5 ELECTRICAL CHARACTERISTICS:-

A monolithic integrated planar Transformer comprises two windings. Each wind-

ing consists of an integer number of turns N 1 ≥ 1,N 2 ≥ 1 where N 1 is the number

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of primary turns and N 2 is the number of Secondary turns. The two windings are

arranged in a single plane with conductors crossing one another .

The Turn Ratio (n) of a Transformer is one of the main electrical parameters

of interest and is defined as

n = N 1/N 2

In the case of an ideal transformer, the Turn Ratio (n) is also equivalent to

n = V 1/V 2 = I 2/I 1

where V 1 is the voltage between the primary ports, V 2 is the voltage between the

secondary ports. I 1 and I 2 is the current-flow into the related ports. Each wind-ing, the Primary and the Secondary, has a self-inductance LP and LS and they are

inductively coupled denoted by the mutual-inductance M. The self inductance of

a given winding is the inductance measured at the Transformer terminals with all

other windings open circuited.

The strength of the magnetic coupling between the Primary and Secondary winding

is indicated by the Coupling Coefficient k (k-factor) as

k = M P /√

LP LS

A typical range for the k-factor achieved in monolithic transformer designs is 0.6 ≤k ≤ 0.95. The phase of the voltage induced at the secondary of the Transformerdepends upon the selection of the reference terminal and basis of this Transformer

connections are of two types:-

• Inverting Connection• Noninverting connection

For an ac signal source with the output and ground applied between terminals

P and P̄ , there is minimal phase shift of the signal at the secondary if the load is

connected to terminal S(with S̄ grounded). This is the Noninverting connection. In

the Inverting connection, terminal S is grounded and S̄ is connected to the load so

that the secondary output is antiphase to the signal applied to the primary[13].

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2.6 TRANSFORMER TOPOLOGIES:-

There are various types of Transformer topologies:-

• parallel(Shibata) Transformer• Interwound(Frlan) Transformer

• OverlayFinlay Transformer

• Concentric Spiral transformer

A microstrip line is the simplest on-chip element for monolithic implementa-

tion of a transmission line inductor, and the strip is normally wound into a spiral

to reduce chip area of the component. Interwinding microstrip spiral inductors tomagnetically couple independent conductors is a logical extension of this concept,

and results in a monolithic transformer, as shown in Figure 2.2.

Figure 2.2: Monolithic Transformer- Physical Layout

An early example of this type of structure is the compact spiral directional

coupler reported by Shibata[2] in 1981. The parallel conductor (Shibata)[2] config-

uration is made up of two conductors which are inter-wound and lie in the same

plane. This is done to promote edge coupling between the primary and secondary

windings which increases the coupling coefficient, k. The differences in primary

and secondary winding lengths make the Shibata Transformer layout physically and

electrically asymmetric, and a transformer ratio of 1:1 is not achievable. This non-

symmetric property subsequently produces coupled coils with different inductance

values.

An improvement on the Shibata Transformer topology is the planar inter woundtransformer. This layout was introduced in 1989 by E. Frlan and is referred as the

‘Frlan Transformer’ [11]. In Frlan configuration primary and secondary windings

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are identical and lie in the same plane. This configuration ensures that when both

windings have the same number of turns, they are electrically identical. In addition,

this configuration has the advantage of placing the terminals of the Transformer at

opposite ends which makes it easier to connect the transformer to other components

in the system. The Frlan[11] winding configuration was selected as the most suitable

configuration for the resonant tank of the oscillator.

Multiple conductor layers are used to fabricate an overlay or broadside coupled

Transformer. This implementation was first introduced by Finlay[12]. In this con-

figuration two windings are on different metal layers and are staked one on top of

the other. This has the advantage of high coupling coefficient, k, because magnetic

coupling is achieved both from the edges and the flat surfaces of the metal traces.

Thus, coupling coefficients close to 0.9 are easy to achieve with this configuration. In

addition, a relatively smaller area is required to achieve the same inductance as theother layouts. The disadvantage of this winding configuration lies in the fact that

the primary and secondary windings are constructed with different metals which

have different sheet resistances. Thus, the Q-factors of the two windings will be

different.

A Transformer can also be implemented using concentrically wound planer spi-

rals. The common periphery between the two windings is limited to just a single

turn. Therefore, mutual coupling between adjacent conductors contributes mainlyto the self-inductance of each winding and not to mutual inductance between the

windings. As a result, the concentric spiral transformer has less mutual inductance

and more self-inductance than the Frlan and Finlay configurations, giving it a lower

k-factor. Also there is no symmetry between windings in this configuration[11][12].

2.6.1 ADVANTAGES OF INTEGRATED PASSIVE COMPONENTS:-

• Improved Performance

– Very high Q from 40 to 70 at 4 GHZ Inductance

– Reduce of capacitive effects for Inductances

– Less noise

– Low consumption

• Integration: Light, small, compatibility

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• Packaging: Substantially improved packaging efficiency

2.7 LOSS MECHANISMS IN TRANSFORMERS

ON SILICON SUBSTRATES:-

Losses arise in transformers because of the finite conductivity of the metal wind-

ings, the finite resistivity of the silicon substrate, and eddy currents in the wind-

ings and substrate. These losses lead to reduction in quality factor, self resonant

frequency. Hence, minimization of losses is important. To reduce these losses,

transformers fabricated on different substrates with different dimensions have been

studied. The following sections explain in detail about the various losses that occur

in transformers.

2.7.1 CONDUCTOR LOSSES:-

Conductor losses in monolithic transformers arise because of the finite conduc-

tivity of the metal. At low frequencies, mainly dc resistance losses due to finite

conductivity of the metal contribute to the conductor losses[14]. At high frequen-

cies, eddy currents resulting in conductor skin and proximity effects also contributeto the conductor losses.

METAL LOSSES:-

The metal losses arise because of the finite resistivity of the windings. This loss

can be modeled as series resistance as illustrated in Figure 2.3. The dc resistance is

directly related to the resistivity of the windings and is given as

R = ρL

wt (2.7.1)

where R is the resistance of the winding of length L, width w, thickness t, and ρ is

the resistivity of the winding.

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Figure 2.3: Illustration of magnetic fields in a Transformer

EDDY CURRENT LOSSES:-The time-varying currents of the primary and secondary windings give rise to

time-varying magnetic fields. Because of the close proximity of the windings, these

magnetic fields pass through the windings and generate eddy currents as shown in

Figure 2.4. According to Faraday-Lenzs law, these eddy currents in turn produce

magnetic fields which oppose the applied magnetic field [5]. This decreases the ef-

fective magnetic field. These opposing magnetic fields are strong at the center of

the conductor at high frequencies and so the current density in the center of the

conductor decreases. This results in non-uniform current distributions in the metalwindings, where most of the current flows at the surface of the conductor at high

frequencies [6]. These effects are commonly called skin effect and proximity effect.

The depth of current penetration in a conductor depends on the frequency and

also on the properties of the conductive material, its conductivity σ or resistivity ρ

and its permeability µ. The depth of penetration is defined as the depth at which

the current density is attenuated by 1e

[21]. The depth of penetration δ , is:

δ = 1√

πfσµ (2.7.2)

The proximity effect is another form of eddy currents, in which nearby conductive

segments experiences induced eddy currents due to the magnetic field of a separate

conductor. Proximity effect is considered as a important loss mechanism when the

distance between adjacent conductive segments in a spiral inductor is made verysmall.

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Figure 2.4: Induced Eddy currents in a Transformer

2.7.2 SUBSTRATE LOSSES:-

Substrate losses arise in monolithic transformers because of the finite conductivity

of the silicon substrates. The substrate losses can be divided into two categories,

namely the electric losses, which arise because of the electrically induced conduction

currents, and the magnetically induced eddy current losses called the magnetic losses

[5][7][17].

2.8 CIRCUIT MODEL OF THE TRANSFORMER:-

• The transformer turns are modeled by ideal inductances L p and Ls. Theinductances, L p and Ls are the sum of all self inductances Li and mutual in-

ductances M i,k of primary conductors and respectively secondary conductors.

L p =

Li + 2

M i,k (2.8.1)

Ls =

Li + 2

M i,k (2.8.2)

The primary and secondary inductances can be extracted by using the tool

FASTHENRY and by the following equations:

L p = imgZ 11

ω (2.8.3)

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Ls = imgZ 22

ω (2.8.4)

Figure 2.5: Circuit diagram of Transformer by O.El.Gharniti[16]

• The ohmic losses in the conductors are represented by the series resistances

R p and Rs.

• CL1, CL2, C pp, and C ss are used to model the parasitic capacitive couplingbetween different winding turns. C pp and C ss can be determined using the

following equations:

C pp = W.l p.C m1m2 (2.8.5)

C ss = W.ls.C m1m2 (2.8.6)

C m1m2 is the capacitance per unit area between metal layers 1 and 2, l p and ls

are the lengths of primary and secondary turns, W is width of metal traces.

• C sub1 and C sub2 are used to model the parasitic capacitive coupling into thesubstrate.

C sub1 = C sub2 = 1

2W.l p.C sub (2.8.7)

C sub3 = C sub4 = 1

2W.ls.C sub (2.8.8)

C sub is the substrate capacitance per unit area.

• In order to model the parasitic capacitive coupling into the oxide C ox1 to C ox4

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is used.

C ox1 = C ox2 = W.l p.C ox (2.8.9)

C ox3 = C ox4 = W.ls.C ox (2.8.10)

C ox is the oxide capacitance per unit area.

• To model the substrate losses Rsub1 and Rsub1 is used. To determine Rsub1 andRsub1 the formula presented in[7] is used:

Rsub1 = Rsub2 = 1

πσsublln

2Coth

π

8

W + 6H ox + T

H sub

(2.8.11)

W p is the width of the primary turns, T is the thickness of metal layer, H sub

is the thickness of the substrate. Similarly Rsub3 and Rsub4 can be calculated.

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Chapter 3

QUALITY FACTOR

ENHANCEMENT

TECHNIQUES FOR INDUCTOR

AND TRANSFORMER

3.1 INTRODUCTION:-

In this chapter we have presented two approaches for the quality factor en-

hancement of Inductor and Transformer: (i) By fabricating the Inductor on a high

resistive glass substrate or high resistive Si substrate and this is equivalent to elim-

inate the substrate underneath the spiral (ii) By fabricating the Inductor farther

away the silicon substrate and within this distance use air, high resistive silicon and

silicon nitride as a cavity. The simulation results are obtained by using the tool

ASITIC. Inductors are key devices in RF circuits that, when fabricated on tradi-

tional semiconductor substrates like silicon, suffer from poor RF performances due

to substrate related RF losses and capacitive parasitics due to inter-spiral tracks/-

substrate coupling. Inductor and Transformer on a glass substrate results in a high

quality factor(Q) as well as high self resonance frequency(SRF) which show a goodprospect in various RFIC’s applications.

3.2 PARASITIC EFFECTS IN THE SUBSTRATES:-

When an inductor is integrated on a Silicon based technology, some undesirable

induced effects show up. The reason being that the metallic layers are separated

from the semiconductor substrate by a layer of silicon oxide. These effects can be

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classified in two types, magnetically induced and electrically induced.

3.2.1 MAGNETICALLY INDUCED PARASITIC EFFECTS

IN THE SUBSTRATE:-One of the fundamental properties of an inductor is that generates a magnetic

field. This alternating field penetrates into the conductive substrate and induces

a voltage difference, which in turn generates a current as shown in Figure 3.1[18].

This phenomenon diminishes the energy in the coil, decreasing at the same time the

quality of the inductor.

Figure 3.1: Magnetically induced substrate losses in a Transformer

3.2.2 ELECTRICALLY INDUCED PARASITIC EFFECTS

IN THE SUBSTRATE:-

Other parasitic effect that shows up in Silicon integrated inductors is the capacita-

tive coupling between the inductor and the substrate. In addition, ohmic losses are

produced since there are displacement currents induced in the substrate as shownin Figure 3.2[18].

For substrates with lossless dielectrics, the higher the dielectric constant, the

higher the electric field concentration; therefore higher the current density at the

edges of the transmission line(each segment is treated as a Transmission line) and

higher the attenuation of the signal[1]. On the other hand if the dielectric constant

of the substrate is low, then capacitive parasitics will be reduced because

C = Ad

if is lower then C will be reduced.

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Figure 3.2: Electrically Induced losses in a Transformer

3.3 QUALITY FACTOR:-

Quality (Q) factor and self-resonant frequency (SRF) originate from resonant

circuits. They are relevant to loop inductors as the inductors exhibit finite parasitic

capacitances.

It is defined as the ratio of the energy stored to the total dissipation per cycle for a

sinusoidal excitation:

Q = 2π Energy stored

Energy lost per cycle (3.3.1)

Q = ω. Energy stored

Average power loss (3.3.2)

Q = −ImagY 11

RealY 11(3.3.3)

Ideally, the quality factor should be infinite for a lossless transformer/inductor.But due to losses in the monolithic transformer/inductor, the quality factor is finite

and low. It also depends on the frequency of operation of the transformer/inductor.

3.4 SELF RESONANCE FREQUENCY:-

When the inductor starts behaving like a capacitor rather than as an inductor.

The value of the frequency where this occurs is called Self Resonance Frequency(F sr)

of the inductor. Transformer and inductor can not be used beyond this frequency.At

self-resonance, the magnitude of the imaginary impedance is zero (the inductive and

capacitive parts cancel), yielding a Q of zero

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3.5 PROCESS PARAMETERS AND DIMENSIONS:-

The key parameters for the design of Inductors and Transformers involve the

outer dimensions or inner dimensions, width and spacing of metal tracks, number of

turns, thickness of the metal and the substrate material. The inductance as well as

its quality factor can be fine tuned by the proper selection of the above parameters.

Process related parameters are given in Table 3.1.

Table 3.1: Process Parameters

PROCESS PARAMETERS VALUE UNITSUBSTRATE RESISTIVITY(ρsub) 10 Ω− cm

OXIDE RESISTIVITY(ρ) 10E 10 Ω− cmSUBSTRATE PERMITTIVITY() 11.7 F/m

OXIDE PERMITTIVITY() 3.9 F/mMETAL (ALUMINIUM) SHEET RESISTANCE(Rsh) 30 mΩ/sq

METAL (COPPER) SHEET RESISTANCE(Rsh) 30 mΩ/sq DIMENSIONS:

OXIDE THICKNESS 1 µmOXIDE THICKNESS WITHOUT CAVITY 6 µm

SUBSTRATE THICKNESS 230 µm

CAVITY(High Resistive Silicon,Air,Glass)DEPTH 5 µmMETAL THICKNESS(t) 1 µm

INNER HOLE LENGTH(Lin) 40 µmNUMBER OF TURNS(N) 4WIDTH OF SPIRAL(w) variable µm

SPACING BETWEEN TURNS(s) variable µmVIA LENGTH 0.5 µm

3.6 Q-FACTOR ENHANCEMENT USING CAVITY:-

In this technique improvement in Quality Factor is observed by fabricating the

inductor farther away the Silicon substrate and then within this distance(between

the substrate and inductor) use high resistive silicon, air and silicon nitride as a cav-

ity or by increasing the thickness of the silicon dioxide. By using these techniques

capacitive parasitics in the substrate will be reduced.

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Figure 3.3: Cavity(Air)

Figure 3.4: Cavity(Air)

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Figure 3.5: Cavity(oxide)

Figure 3.6: Cavity(Oxide)

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Figure 3.7: Cavity(Silicon Nitride)

Figure 3.8: No Cavity

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3.7 Q-FACTOR ENHANCEMENT USING COPPER:-

One another approach to improve the Q-factor of inductor and transformer is

the use of Copper metal instead of conventional Aluminium metal. Through these

Tables 3.2-3.5 we have presented a comparative analysis of Inductor, fabricated us-

ing Aluminium as a metal and Copper as a metal. When Copper is used as a metal

provide an overall higher Qmax than Aluminium metallization, due to a two fold

lower metal resistivity and a higher aspect ratio of the conductor for a given metal

pitch[3][4].

Table 3.2: Aluminium with Cavity

SR.NO.

Width/Spacing QualityFactor(Qmax)

Self ResonanceFrequency(F sr)GHZ

Inductance(L)nH

1 W=S=8 4.540 16GHZ 10.0402 W=S=12 4.110 14GHZ 8.1953 W=S=16 3.611 13GHZ 8.0274 W=S=20 3.356 12GHZ 8.7785 W=S=25 2.957 10GHZ 10.3586 W=S=30 2.733 10GHZ 86.907

Table 3.3: Copper with Cavity

SR.NO.



Inductance(L)nH

1 W=S=8 6.185 16GHZ 9.6182 W=S=12 5.504 14GHZ 8.0173 W=S=16 5.044 13GHZ 7.8994 W=S=20 4.580 12GHZ 8.6885 W=S=25 3.663 10GHZ 10.2196 W=S=30 2.763 10GHZ 91.789

Table 3.4: Copper without Cavity

SR.NO.



Inductance(L)nH

1 W=S=8 6.519 11GHZ 12.9032 W=S=12 6.161 10GHZ 10.0793 W=S=16 6.000 9GHZ 9.1894 W=S=20 5.782 8GHZ 9.1385 W=S=25 4.800 7GHZ 10.157

6 W=S=30 3.585 6GHZ 11.492

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Table 3.5: Aluminium without Cavity

SR.NO.



Inductance(L)nH

1 W=S=8 4.642 11GHZ 13.5052 W=S=12 4.414 10GHZ 10.471

3 W=S=16 3.853 9GHZ 9.4374 W=S=20 3.934 8GHZ 9.3585 W=S=25 3.610 7GHZ 10.1226 W=S=30 2.943 6GHZ 11.596

3.8 Q-FACTOR ENHANCEMENT USING HIGH

RESISTIVE SUBSTRATES:-

Many techniques have been used to improve the Q-factor of inductors by reduc-

ing substrate loss, especially capacitive loss. In these techniques use high resistivity

silicon substrates, or glass or quartz to enhance the Q factor of inductor. High sub-

strate resistivities result high Q values, whereas Q degrades quickly when resistivity

decreases between 10 and 0.1 Ω cm, to reach values close to zero. This degradation

is due to the energy dissipation into the substrate which can be explained by ana-

lyzing both electric E and magnetic H fields in the substrate. These fields induce

leakage(E) and eddy(H) currents into the substrate which densities depend on the

resistivity. The layout of Inductor and transformer over the substrate is shown in

Figure 3.9.

Figure 3.9: Layout of Inductor and Transformer in ADS

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Figure 3.10: High Resistive Si Substrate

Figure 3.11: Inductor on Glass

Figure 3.12: Transformer on Glass

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3.9 LAYOUT OF INDUCTOR:-

A typical spiral inductor surrounded with ground is shown in Figure 3.11. Quali-

tatively, the spiral inductor consists of a number of series-connected metal segments.

In each segment, time varying conductive current will flow due to a time-varyingvoltage impressed on the segment. Spiral inductor generally makes use of one or

more metal layers. In the most conventional design, the spiral is build with several

turns in the one of the metal layers and the end of the inductor connects to the out

of the inductor with other layer.

Figure 3.13: Layout of Inductor

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3.10 LAYOUT OF TRANSFORMER:-

In this structure, two parallel coils on the same metal layer are symmetrically

interwound side by side[11][16]. Lower metal layer conductors are used to connect

the inner terminals to other circuitry. Layout of Transformer surrounded by ground

is shown in Figure 3.14.

Figure 3.14: Layout of Transformer

3.11 EXPERIMENTAL RESULTS:-

These simulation results are obtained by using the tool ASITIC(Analysis and

simulation of Inductors and Transformers in Integrated Circuits). Through ASITIC

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we can model the electrical and magnetic behavior of passive metal structures re-

siding above a lossy conductive substrate. In all these plots the Q-factor initially

rises with frequency as the reactive component of the impedance increases, peaks,

and then decreases due to the increasing energy dissipation at higher frequencies.

The estimated quality factor for inductor and transformer, when fabricated on High

resistive glass substrates is shown in Figure 3.9 and Figure 3.10. Inductors made on

a glass substrate have shown a Q of 14.75 at 14 GHz with an inductance of 1.974

nH and Transformers made on a glass substrate have shown a Q of 13.89 at 11 GHz

with an inductance of 2.957 nH.

The thickness of the oxide layer is an important parameter which influences the

inductor/transformer performance. The measured effect of changes in the oxide

thickness upon the component factor is illustrated in Figure 3.5 and Figure 3.6. A

thicker oxide layer reduces the parasitic capacitance of the structure, which improvesthe inductor self-resonant frequency.

A spiral inductor suspended approximately 5µm above the silicon substrate is

found to reduce the effect of substrate proximity on the performance of inductor.

Figure 3.4 and Figure 3.5 presents the effect of air cavity on the quality factor of the

inductor. Similarly the plot of the Q factor and inductance, when Silicon nitride is

used as a cavity is shown in Figure 3.7. The inductors with cavity have higher Q

and self-resonance frequency than one without any cavity as shown in Figure 3.8.

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Chapter 4

VERIFICATION OF SELF AND

MUTUAL INDUCTANCE

THROUGH FASTHENRY

4.1 PARTIAL INDUCTANCE APPROACH:-

There are several approaches for inductance calculation. Traditionally, the con-

cept of partial inductance is used. Partial inductances are useful when the induced

current return paths are unknown. In the partial inductance approach, the signal

lines, ground and supply lines are treated equivalently, resulting in a large, densely

coupled network representation[9]. When a multi conductor problem is described

by a set of partial inductances, the sum of the partial self and partial mutual in-

ductances along any closed loop path will yield the total loop inductance of the

path. The most accurate way of extracting inductance is to divide the structure

into smaller regions and numerically solve the magnetic field within each region to

find the magnetic flux.

4.2 FASTHENRY:-

FASTHENRY was developed for the solution of Maxwell equations and ex-

traction of inductances and resistances. FASTHENRY is a software for computing

the frequency-dependent self and mutual inductances and resistances of a generic

tridimensional conductive structure, in the magnetoquasistatic approximation[10]

as explained later.

FASTHENRY specifies every conductor as a sequence of rectilinear segmentsconnected between nodes. Every segment has a finite conductivity and the shape of

a parallelepiped, whose height and width can be assigned. A node is a point in the

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3D space. The section of a segment can be divided, if required, into an arbitrary

number of parallel filaments, the whole of which constitutes the segment itself; it is

then assumed that every filament carries a uniform current.

In this way is possible to model the high-frequency effects on the segments. In

fact, when the frequency increases, the current is no longer uniformly distributed

along the cross section of a conductor. However, in limited regions of the section,

the current can be reasonably approximated as uniform. Therefore, being able to

specify an arbitrary discretization of the volume of the conductors, the accuracy of

the results is affected accordingly and in general is better as the discretization is re-

fined.The results are provided in form of a Maxwell impedance matrix Z = R + jL,

where R is the resistance and L is the inductance[19].

4.3 VERIFICATION OF PARTIAL INDUCTANCE

APPROACH ON MEANDER:-

Meander structure is composed of horizontal and vertical conductive segments.

We decomposed the meander structure and consider only the vertical conductive

segments surrounded by ground on both the sides and then compute the self and

mutual inductance. Previous work done by people consider the effect of one side

ground only but in our case we are considering the effect of both sides ground.

Initially we are considering that the length of the horizontal conductive segment is

negligibly small compared to that of ground. The total self inductance is equal to

the sum of self inductance of all horizontal and vertical conductive segments. The

simple layout of meander structure is shown in Figure 4.1.

Figure 4.1: Layout of Meander structure surrounded with Ground

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Ground Signal and Ground Signal Ground structure on FASTHENRY:-

Figure 4.2: Ground Signal Ground structure

In this section we analyze the performance of ground signal and ground signal

ground structure. By varying the spacing between the signal and ground variation

in the value of inductance is observed. As we increase the spacing between signal

and ground the inductance is going to increase. The parameters of a Ground signal

Ground structure are shown in Table 4.1. In case of ground signal ground struc-

ture(GSG)(Figure 4.2), two cases are there:

Table 4.1: Parameters of Ground Signal Ground structure

Sr.No.

Parameter Description

1 wg Width of the ground segment2 ws Width of the signal conductor3 dg Distance from the center of the ground

segment to edge of the signal segment4 lw Length of the signal conductor5 ds Spacing between the two signal conduc-

tors

• In first case the spacing between signal and ground is same on both the sidesi.e. Symmetric structure.

• On the other hand in second case the spacing between signal and ground isnot same on both the sides i.e. Asymmetric structure.

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Figure 4.3: Ground Signal Structure

Figure 4.4: Plot of self inductance by varying the dg

Figure 4.5: Plot of self inductance by Varying the length lw of the conductor

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Figure 4.6: Ground Signal Ground- Symmetric Structure

Figure 4.7: Ground Signal Ground- Asymmetric Structure

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4.4 CIRCUIT DIAGRAM OF MEANDER ON

SILICON SUBSTRATE:-

• Rsm is the series resistance, which models skin effect and ohmic losses.

Rsm = ρ.lw.δ.(1 − e−tδ )

(4.4.1)

where parameter t represent conductor thickness and δ is the skin depth.

• Lsm is the series self Inductance, consists of positive mutual inductance andnegative mutual inductance.

• C ox represents the capacitance between the meander arm and the substrate.

Figure 4.8: Circuit diagram of meander

Figure 4.9: Meander structure on Silicon substrate

C ox

= ltotal.w.ox

2.tox(4.4.2)

where tox is the oxide thickness, ltotal is the total inductor length and ox is the

permittivity of the oxide.

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• R p and C p are the coupling resistance and capacitance associated with Sisubstrate.

C p = ltotal.w.si

2.tsi(4.4.3)

R p = 2.ρsi.tsi

ltotal.w

(4.4.4)

where si is the permittivity of the silicon, ρsi and ρ resistance of the conductor

line and silicon substrate.

4.5 MODELING OF INDUCTANCE:-

Ampere’s law:

∇× B = µ J + ∂ E

∂t (4.5.1)

where µ is the magnetic permeability and is the electric permittivity of the

material. The first term on the right-hand side represents a magnetic field gen-

erated from a wire carrying an electric current. The second term corresponds to

the magnetic field generated from the displacement current, which represents the ac

current flowing between two conductors due to their capacitive coupling as shown

in Figure 4.9. In integrated circuits, the second term is usually neglected because

the current flowing in the conductor is much larger than the displacement current.

This assumption is called the magnetoquasistatic[10] approximation.

Figure 4.10: Return current through Capacitors

now by using this assumption the integral’s form of Ampere’s law is:

B. dl = µ0I (4.5.2)Faraday’s law: According to faraday, a steady magnetic field produces no cur-

rent flow, but a time varying field produces an induced voltage in a closed circuit,

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which causes a flow of current.

∇× E = −∂ B

∂t (4.5.3)

The integral form of Faraday’s law is:

E. dl = −∂

s

B. ds

∂t (4.5.4)

With the help of both the two Equations 4.5.2 and 4.5.4, We can characterize the

interaction between electric field and magnetic field and derive the expression for

inductance. Inductance is a property of the physical layout of a conductor and is a

measure of the ability of a conductor configuration to link magnetic flux. The flux

linkage is the total magnetic field enclosed by a closed circuit. The inductance of acurrent-carrying loop is defined as the ratio of the total magnetic flux penetrating

the surface of the loop and the current of the loop that produced it:

Lij = ψij

I =

s

B. ds

I (4.5.5)

Figure 4.11: Two magnetically coupled loops

For two current loops in Figure 4.10, the magnetic flux produced by current I 1

is linked inside the area S 2 which is enclosed by C 2. The flux linkage enclosed in C 2

is the following:

ψ12 =

s

B1.ds (4.5.6)

If the C 1 consists of multiple turns N 1, the total flux produced is N 1 times larger,

Λ12 = N 1ψ12. When two current carrying conductors are in proximity, their mag-

netic flux lines interact with each other. If the currents flow in same directions,

inductance of each conductor is increased. Currents flowing in the opposite direc-

tion decrease each conductors inductance. The change in an isolated conductors

inductance when in proximity to another conductor is known as their mutual induc-

tance. The mutual inductance M 12, between two loops is defined as the following:

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M 12 = N 1N 2

I 1(4.5.7)

The self inductance, L11, is defined from the magnetic flux produced by I 1 en-

closed by the contour C 1 as the following:

L11 = Λ11

I 1(4.5.8)

Figure 4.12: Circular conductor with radius b

Bin = âφµ0r1I i2πb2

, r1 ≤ b (4.5.9)

Bout = âφµ0I i2πr2

, r2 ≥ b (4.5.10)

where Bin is the magnetic field density vector inside the conductor and Bout is the

vector outside the conductor. The current in a unit length of this annular ring[Figure

4.11] is linked by the flux that can be obtained by integrating Equation 4.5.5, through

this we are able to derive the formula of the internal inductance per unit length.

dφin =

br

Bindr = µ0I i4πb2

(b2 − r2) (4.5.11)

d∧in = 2rdr

b2 dφ (4.5.12)

∧in = b0

d∧in (4.5.13)

∧

in = b

0

2rdr

b2

µ0I i

4πb2

(b2

−r2)dr (4.5.14)

∧in = 2

b2

µ0I i4πb2

b0

[rb2 − r3]dr (4.5.15)

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= µ0I i2πb4

b2r2

2 − r

4

4

b0

(4.5.16)

= µ0I i2πb4

b4

4

(4.5.17)

Lin = ∧in

I i=

µ08π

(4.5.18)

Total Inductance per unit length of two wire system:

Lin = ∧in

I i=

µ04π

(4.5.19)

4.6 SELF INDUCTANCE:-

The total magnetic flux generated by a current can be partitioned into the por-

tion lying outside the conductor plus the flux that lies inside the conductor. The

storage of energy associated with the internal flux leads to internal inductance and

that associated with the external flux is represented by external inductance[8].

Now compute the self inductance when there is only one signal conductor sur-

rounded by ground on both the sides as shown in Figure 4.12. Considering that the

distance between the signal and ground is equal on both the sides.

Total self inductance(Lself ) = lw (Lint + Lext)

Lint =µ08π

(1 + 0.5) = 1.5µ08π

Due to ground:

Bg = 0.5µ0I i2

πx

Due to signal:

Bsi = µ0I i

2π[rg + dg + rs − x]

Lext = ψx

I i

ψx =

rg+dg

rg(BG + Bsi)dx

ψxI i

=

rg+dg

rg

0.5µ02πx

+ µ0

2π[rg + dg + rs − x]

dx

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Figure 4.13: GSG: Symmetric

= µ02π

rg+dg

rg

0.5

x +

1

[rg + dg + rs − x]

dx

= µ02π

ln x0.5

rg + dg + rs − x

rg+dgrg

=

µ0

2π ln (rg + dg)0.5

rs − ln (rg)

0.5

rs + dg

= µ02π

ln (rg + dg)

0.5.(rs + dg)

rs.r0.5g

Effect of Ground G2:-

Bsi = µ0I i

2π[0.5wg + dg + ws + dg1 − x]

ψxI i

=

dg1+0.5ws

dg1

0.5µ02πx

+ µ0

2π[0.5wg + dg + ws + dg1 − x]

dx

= µ02π

ln x0.5

0.5wg + dg + ws + dg1 − x

dg1+0.5wsdg1

= µ0

2π

ln (dg1 + 0.5ws)0.5.(0.5wg + dg + 0.5ws)

(0.5wg + dg + ws)d0.5g1

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Lself 1 = lwµ02π

15

40

+ln (rg + dg)

0.5.(rs + dg)

rs.r0.5g

− ln (dg1 + 0.5ws)0.5.(0.5wg + dg + 0.5ws)

(0.5wg + dg + ws)d0.5g1

(4.6.1)

Similarly the effect of ground G1 can be calculated but as the distance is equal on

both the sides, the expression will be identical.

Self InductanceLself 1 = G1S 1 - Effect of ground(G2)

Total Self Inductance = 2.Lself 1

Compute the self inductance for Asymmetric structure, When the distance between

the signal and the ground is not equal on both the sides as shown in Figure 4.13.

Lint = µ08π

(1 + 0.5) = 1.5µ08π

Self Inductance due to GroundG1:-

Lext == µ02π

ln (rg + dg)0.5

.(rs + dg)rs.r0.5g

Self inductance due to GroundG2:-

Bsi = µ0I i

2π[rg + dg1 + rs − x]

ψxI i = rg+dg1rg

0.5µ02πx +

µ02π[rg + dg1 + rs − x]

dx

= µ02π

ln (rg + dg1)

0.5.(rs + dg1)

rs.r0.5g

Self Inductance due to the effect of Ground G2 on Ground G1:-

Bsi =

µ0I i

2π[0.5wg + dg1 + ws + dg2 − x]

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Figure 4.14: GSG: Asymmetric

ψxI i

=

dg2+0.5ws

dg2

0.5µ02πx

+ µ0

2π[0.5wg + dg1 + ws + dg2 − x]

dx

= µ02π

ln x0.5

0.5wg + dg1 + ws + dg2 − x

dg2+0.5wsdg2

= µ02π

ln (dg2 + 0.5ws)

0.5.(0.5wg + dg1 + ws)

(0.5wg + dg1 + 0.5ws)d0.5g2

Self Inductance due to the effect of Ground G1 on Ground G2:-

Bsi = µ0I i

2π[0.5wg + dg + ws + dg3 − x]

ψxI i

=

dg3+0.5ws

dg3

0.5µ02πx

+ µ0

2π[0.5wg + dg + ws + dg3 − x]

dx

= µ02π

ln x0.5

0.5wg + dg + ws + dg3 − xdg3+0.5ws

dg3

= µ02π

ln (dg3 + 0.5ws)

0.5.(0.5wg + dg + ws)

(0.5wg + dg + 0.5ws)d0.5g3

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Figure 4.15: Coupling inductance

Lself 1 = lwµ02π

15

40

+

ln (rg + dg)0.5.(rs + dg)

rs.r0.5g

+ln (rg + dg1)

0.5.(rs + dg1)

rs.r0.5g

− ln (dg2 + 0.5ws)0.5.(0.5wg + dg1 + ws)

(0.5wg + dg1 + 0.5ws)d0.5g2

− ln (dg3 + 0.5ws)0.5.(0.5wg + dg + ws)

(0.5wg + dg + 0.5ws)d0.5g3

(4.6.2)

4.7 MUTUAL INDUCTANCE:-The coupling inductance Lij is proportional to the overlapping area of S i and S j .

If there are two signal conductors as shown in Figure 4.14 then:

Due to ground:

Bg = µ0I i2πx

Due to signal:

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Bsi = µ0I i

2π[0.5wg + dg + ds + 1.5ws − x]

ψx = 1

l SlapBdsoverlap

ψ

x = 0.5wg+dg+ws0.5wg (BG + Bsi)dx

ψxI i

=

0.5wg+dg+ws

0.5wg

µ02πx

+ µ0

2π[0.5wg + dg + ds + 1.5ws − x]

dx

= µ02π

0.5wg+dg+ws

0.5wg

1

x +

1

[0.5wg + dg + ds + 1.5ws − x]

dx

= µ02π

ln x

0.5wg + dg + ds + 1.5ws−

x

0.5wg+dg+ws0.5wg

= µ02π

ln(0.5wg + dg + ws)

(ds + 0.5ws) − ln(0.5wg)

(dg + ds + 1.5ws)

= µ02π

ln(0.5wg + dg + ws)(dg + ds + 1.5ws)

(ds + 0.5ws)0.5wg

Total Mutual Inductance (M 12)=l (L

int + L

ext)

(M 12) =lµ02π

1

4 +

ln(0.5wg + dg + ws)(dg + ds + 1.5ws)

(ds + 0.5ws)0.5wg

If there are three signal conductors then:

Total Mutual Inductance = (M 12) + (M 23) + (M 13)

Due to signal:

Bsi = µ0I i

2π[0.5wg + dg + 2ds + 2.5ws − x]

(M 13) = 0.5wg+dg+ws

0.5wg

µ02πx

+ µ0

2π[0.5wg + dg + 2ds + 2.5ws − x]

dx

= µ0

2π

0.5wg+dg+ws

0.5wg

1x

+ 1[0.5wg + dg + 2ds + 2.5ws − x]

dx

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= µ02π

ln x

0.5wg + dg + 2ds + 2.5ws − x

0.5wg+dg+ws0.5wg

= µ0

2π ln(0.5wg + dg + ws)

(2ds + 1.5ws) − ln(0.5wg)

(dg + 2ds + 2.5ws)

= µ02π

ln(0.5wg + dg + ws)(dg + 2ds + 2.5ws)

(2ds + 1.5ws)0.5wg

Due to signal:

Bsi = µ0I i2π[0.5wg + dg + 2ds + 2.5ws − x]

(M 23) =

0.5wg+dg+2ws+ds

0.5wg

µ02πx

+ µ0

2π[0.5wg + dg + 2ds + 2.5ws − x]

dx

= µ02π

0.5wg+dg+2ws+ds

0.5wg

1

x +

1

[0.5wg + dg + 2ds + 2.5ws − x]

dx

= µ0

2π

ln x

0.5wg + dg + 2ds + 2.5ws − x0.5wg+dg+2ws+ds

0.5wg

= µ02π

ln(0.5wg + dg + 2ws + ds)

(ds + 0.5ws) − ln(0.5wg)

(dg + 2ds + 2.5ws)

= µ02π

ln(0.5wg + dg + 2ws + ds)(dg + 2ds + 2.5ws)

(ds + 0.5ws)0.5wg

Total Mutual Inductance M =l (Lint + L

ext)

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M = lµ0

2π

1

4

+ln(0.5wg + dg + ws)(dg + ds + 1.5ws)

(ds + 0.5ws)0.5wg

+ln(0.5wg + dg + 2ws + ds)(dg + 2ds + 2.5ws)

(ds + 0.5ws)0.5wg

+ln(0.5wg + dg + ws)(dg + 2ds + 2.5ws)

(2ds + 1.5ws)0.5wg

(4.7.1)

4.8 EXPERIMENTAL RESULTS:-

When line spacing decreases, it is observed that the inductance of the mean-

der coil decreases. This is because meander coil has negative mutual inductance as

shown in Figure 4.3. The plot for symmetric structure and asymmetric structure

is shown in Figure 4.6 and Figure 4.7. A comparison is made between derived and

FASTHENRY for self inductance variation with increasing distance of ground plane

dg, as shown in Figure 4.4. Initially the error is very small but as the distance

increases error increases linearly.

The plot for self inductance calculated from FASTHENRY as well as analytically

by varying the length of the conductor and keeping the distance between ground

and signal fixed is shown in Figure 4.5.

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n7 x=-2 y=2

n8 x=-2 y=-2

n9 x=2 y=-2

n10 x=2 y=3

n11 x=-3 y=3

n12 x=-3 y=-3

n13 x=3 y=-3

n14 x=3 y=4

n15 x=-4 y=4

n16 x=-4 y=-4

n17 x=4 y=-4

*Segment Declaration

e1 n1 n2 w=0.1 h=0.1

e2 n2 n3 w=0.1 h=0.1

e3 n3 n4 w=0.1 h=0.1

e4 n4 n5 w=0.1 h=0.1

e5 n5 n6 w=0.1 h=0.1

e6 n6 n7 w=0.1 h=0.1

e7 n7 n8 w=0.1 h=0.1

e8 n8 n9 w=0.1 h=0.1e9 n9 n10 w=0.1 h=0.1

e10 n10 n11 w=0.1 h=0.1

e11 n11 n12 w=0.1 h=0.1

e12 n12 n13 w=0.1 h=0.1

e13 n13 n14 w=0.1 h=0.1

e14 n14 n15 w=0.1 h=0.1

e15 n15 n16 w=0.1 h=0.1

e16 n16 n17 w=0.1 h=0.1

.external n1 n17

*Frequency Range

.freq fmin=1e9 fmax=1e10

*Programme end

.end

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Appendix B

FASTHENRY CODING OF

TRANSFORMER:-

* Transformer

Figure B.1: Screen shot of TRANSFORMER

.units um

.default z=0 sigma=5.8e-7

*Nodes Declarationn1 x=40 y=0

n2 x=40 y=40

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n3 x=0 y=40

n4 x=0 y=-40

n5 x=80 y=-40

n6 x=80 y=80

n7 x=-40 y=80

n8 x=-40 y=-80

n9 x=120 y=-80

n10 x=120 y=120

n11 x=-80 y=120

n12 x=-80 y=-120

n13 x=160 y=-120

n14 x=160 y=160

n15 x=-120 y=160

n16 x=-120 y=-160n17 x=200 y=-160

n18 x=20 y=20

n19 x=20 y=-20

n20 x=60 y=-20

n21 x=60 y=60

n22 x=-20 y=60

n23 x=-20 y=-60n24 x=100 y=-60

n25 x=100 y=100

n26 x=-60 y=100

n27 x=-60 y=-100

n28 x=140 y=-100

n29 x=140 y=140

n30 x=-100 y=140

n31 x=-100 y=-140

n32 x=180 y=-140

n33 x=180 y=180

n34 x=-140 y=180

*Segment Declaration

e1 n1 n2 w=10 h=1

e2 n2 n3 w=10 h=1

e3 n3 n4 w=10 h=1

e4 n4 n5 w=10 h=1

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e5 n5 n6 w=10 h=1

e6 n6 n7 w=10 h=1

e7 n7 n8 w=10 h=1

e8 n8 n9 w=10 h=1

e9 n9 n10 w=10 h=1

e10 n10 n11 w=10 h=1

e11 n11 n12 w=10 h=1

e12 n12 n13 w=10 h=1

e13 n13 n14 w=10 h=1

e14 n14 n15 w=10 h=1

e15 n15 n16 w=10 h=1

e16 n16 n17 w=10 h=1

e17 n18 n19 w=10 h=1e18 n19 n20 w=10 h=1

e19 n20 n21 w=10 h=1

e20 n21 n22 w=10 h=1

e21 n22 n23 w=10 h=1

e22 n23 n24 w=10 h=1

e23 n24 n25 w=10 h=1

e24 n25 n26 w=10 h=1

e25 n26 n27 w=10 h=1e26 n27 n28 w=10 h=1

e27 n28 n29 w=10 h=1

e28 n29 n30 w=10 h=1

e29 n30 n31 w=10 h=1

e30 n31 n32 w=10 h=1

e31 n32 n33 w=10 h=1

e32 n33 n34 w=10 h=1

g1 x1=220 y1=180 z1=0

+ x2=220 y2=-170 z2=0

+ x3=205 y3=-170 z3=0

+thick=2

+seg1=20 seg2=20

+nin1 (213,-170,0)

+nout1 (213,180,0)

g2 x1=-150 y1=205 z1=0

+ x2=-150 y2=190 z2=0

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+ x3=210 y3=190 z3=0

+thick=2

+seg1=20 seg2=20

+nin2 (30,205,0)

+nout2 (30,190,0)

g3 x1=-145 y1=199 z1=0

+ x2=-160 y2=199 z2=0

+ x3=-160 y3=-170 z3=0

+thick=2

+seg1=20 seg2=20

+nin3 (-152.5,199,0)

+nout3 (-152.5,-170,0)

g4 x1=-155 y1=-170 z1=0

+ x2=-155 y2=-185 z2=0

+ x3=213 y3=-185 z3=0

+thick=2

+seg1=20 seg2=20

+nin4 (29,-170,0)

+nout4 (29,-185,0)

.equiv n1 nin1 nin2

.equiv n2 nin2 nin1

.external n1 n17

*Frequency range

.freq fmin=1e9 fmax=1e10

.end

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Appendix C

DIMENSIONS OF INDUCTOR

LAYOUT:-

30µm

30µm

30µm

50µm

50µm

50µm

50µm

50µm

50µm

30µm30µm

30µm30µm

30µm30µm

40µm

335µm

30µm

30µm

30µm

20µm

970µm

100µm

100µm

1170µm

100µm 100µm

1170µm

50µm

970µm

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