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CIRCUIT MODELLING OF VARACTOR DIODES

IN ELECTRONICALLY TUNED MICROWAVE

OSCILLATORS

GORDON R. DYER.

Submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy.

Department of Electrical and Electronic Engineering

The University of Leeds

December 1981

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SUMMARY

The Schottky barrier Varactor is still unique in its ability to tune a microwave oscillator at very high modulation rates, so it is widely used despite having some very restrictive fundamental performance limitations such as high r.f. loss and low capacitance ratio. In this thesis both the theoretical performance limitations and the practical limitations of the Varactor are considered in terms of circuit models and the device structure.

Both the Varactor diode and the transferred electron device are non linear and two methods of modelling non linear devices under large signal conditions are reviewed in Chapter 2. The simplest possible non linear oscillator has been simulated on an analogue computer to verify the modelling techniques.

The maximum possible theoretical cut-off frequency has been calculated for a Varactor with constant doping and it is compared with commercial devices in Chapter 3. An outline of the procedure required to design a Varactor is given and optimum device parameters are calculated for constant doping. The diode impedance has been calculated under large signal conditions over the full bias range.

The transferred electron device is more difficult to model than the Varactor and a time-domain computer simulation which solves quasi-static current flow equations has been used. The results from this model have been used to predict the power output from the oscillator circuits which are analysed in Chapter 5.

A fundamental trade-off in performance between tuning range, Q-factor and power output is analysed in Chapter 5 with no narrowband approximations. The r.f. voltage swing across the Varactor is found to cause a severe tuning limitation because of the high noise level and low Q-factor which result from any excursion into forward bias.

Measurements made on Varactor diodes and oscillators are presented in Chapter 6. A novel method of de-embedding the package from the Varactor at microwave frequencies is developed and a detailed comparison made with previously published results. Two waveguide oscillator; have been constructed and measurements made on them are compared with the calculations of Chapter 5.

To study the f.m. noise of the oscillators a waveguide frequency discriminator has been developed which allows automated measurements to be made with great ease when compared with other methods such as cavity discriminators. This has allowed the noise properties of the oscillator and devices to be compared quantitatively with theory.

Finally, because of the unacceptable oscillator performance when the Varactor is swung into forward bias, a current limited Varactor is proposed in Chapter 8 which should allow the full capacitance range to be used without a degradation of the noise performance at zero bias, even in the presence of moderately large r.f. voltage swings.

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CONTENTS

Page no.

List of Symbols 1

CHAPTER 1

VARACTOR TUNED MICROWAVE OSCILLATORS

1.1 Introduction 5

1.2 The Performance of Varactor Tuned TED’s 11

1.3 Physical Structures for Tunable Oscillators 20

1.4 Modelling and Characterisation of Oscillators 23

References 1 27

CHAPTER 2

NONLINEAR CIRCUIT ELEMENTS

2.1 Introduction 30

2.2 Quasi—Linearisation 32

2.3 Describing Functions 33

2.4 Modelling with I—V Characteristics 38

2.5 An Analogue Computer Simulation 43

Appendix 2.1 48

References 2 50

CHAPTER 3

VARACTOR MODELING

3.1 Introduction 52

3.1.1 Historical Development 52

3.1.2 Requirements for Tuning Elements 53

3.1.3 Choice of Junction and Materials 54

3.2 Small Signal Theory

3.2.1 The Potential Barrier 56

3.2.2 Junction Current 57

3.2.3 Junction Capacitance 59

3.2.4 Power Law Doping 60

3.2.5 Frequency Limitations 62

3.3 Small Signal Circuit Modelling 65

3.3.1 The Junction 65

3.3.2 The Epi—Layer 68

3.3.3 The Substrate and Contacts

3.3.4 Leakage Current 71

3.4 Varactor Design 74

3.5 Large Signal Modelling 82

References 3 91

CHAPTER 4

LARGE SIGNAL T.E.D. SIMULATION

4.1 Introduction 94

4.2 Device Modelling

4.3 Field and Current Equations 97

4.4 Electron Velocity and Diffusion 98

4.4.1 The Velocity—Field Characteristic 98

4.1.2 Electron Diffusion 100

4.5 Device Structure and Boundary Conditions 108

4.6 Numerical Algorithm 111

4.6.1 Basic Equations 111

4.6.2 Sign Convention 113

4.6.3 Difference Equations 113

Accuracy and Stability 116

4.7 Stability and Transit Oscillations 119

4.8 Large Signal Characteristics 131

4.8.1 Sinusoidal Voltage 131

4.8.2 LCR Circuit 137

Appendix 4.1 Adams—Bashforth Method 149

References 4 151

CHAPTER 5

OSCILLATOR ANALYSIS

5.1 Introduction 155

5.2 Generalised Oscillator Analysis 156

5.3 Lumped Element Circuits 158

5.3.1 The Parallel Tuned Circuit 160

5.3.2 The Series Tuned Circuit 166

5.3.3 A Comparison of the Parallel and Series

Circuits 168

5.3.4 Parallel Resonance in the Series Circuit 172

5.4 Large Signal Tuning Limitations 178

5.5 A Reduced Height Waveguide Oscillator 183

5.6 A Full Height Waveguide Oscillator 189

References 5 203

CHAPTER 6

EXPERIMENTAL RESULTS ON VARACTCRS AND VARACTOR

TUNED OSCILLATORS

6.1 Introduction 205

6.2 Varactor Measurements 205

6.2.1 Low Frequency Measurements 208

(a) I—V Characteristics 208

(b) C—V Characteristics at 1MHz 212

6.2.2 U.H.F. Measurements 220

6.2.3 Microwave Measurements 223

(a) Varactor Impedance Measurements 234

(b) Varactor Package Extraction 246

6.3 Oscillator Measurements 250

6.3.1 A Reduced Height Waveguide Oscillator 252

6.3.2 A Double Height Waveguide Oscillator 256

Appendix 6.1 De—Embedding the Package of the Varactor 263

References 6 265

CHAPIER 7

F.M. NOISE

7.1 Introduction 267

7.2 Mathematical Representation of Noise 268

7.3 Noise Mechanisms

7.4 The Varactor Diode 274

7.5 Measurement of F.M. Noise 278

7.6 A Delay Line Discriminator 281

The Complete System 290

Discriminator Bandwidth 296

7.7 Circuit Effects 298

7.8 Noise Measurements 303

7.8.1 The Transferred Electron Device 303

7.8.2 A Varactor Tuned Oscillator 309

7.9 Conclusions 316

Appendix 7.1 Narrowband F.M. Validity 318

Appendix 7.2 F.M. Discriminator Validity 320

Appendix 7.3 The Ideal Unmatched Hybrid 332

Appendix 7.1. Discriminator Linearity 333

Appendix 7.5 Noise Analysis with Amplitude Dependent

Circuit 335

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References 7 338

Page no.

CHAPTER 8

DISCUSSION AND CONCLUSIONS 341

Appendix 8.1 The Current Limited Varactor 348

PUBLICATIONS 350

LIST OF SYMBOLS

A signal amplitude

area

A** Richardson's constant

B susceptance

magnetic flux density

bandwidth

C capacitance

Cj Varactor junction capacitance

C0 Varactor capacitance at zero bias

CB Varactor capacitance at breakdown

c speed of light

D electric displacement

diffusion coefficient

describing function

A,B,C,D two-port transmission parameters

E electric field strength

E Electronic charge

F force

f frequency

fco Varactor cut-off frequency

G conductance

Gj Varactor junction capacitance

H magnetic field strength

I direct current

Is saturation current

i instantaneous current

J current density

Js saturation current density,

Jv Bessel function

KT thermal conductivity

k Boltzmann's constant

L inductance

l length

Nd compensated donor density

n electron density

P complex power

p momentum

Q circuit quality factor

quasi-static electric charge

q electric charge

R resistance

Re epilayer resistance

Rc contact resistance

RT thermal resistance

Rii (t) auto correlation function

Rij (t) cross correlation function

Sij scattering parameters

s complex frequency

T absolute temperature

t time

V d.c. voltage

V a.c. voltage amplitude

VB breakdown voltage

V instantaneous voltage

Vj varactor junction voltage

W real power

epilayer width

w(t) weighting function

X reactance

Y admittance

Z impedance

Zo impedance of free space

a attenuation coefficient 3 propagation coefficient

B complex propagation coefficient

detector diode sensitivity

F reflection coefficient

d skin depth

the delta function

e dielectric permittivity

eo dielectric permittivity of free space

Ao wavelength in free space

Ag wavelength in waveguide

Uo magnetic permeability of free space

u electron mobility

P resistivity

t time, relaxation time

0 phase

effective barrier height magnetic flux

0m work function of metal

0n potential difference between fermi level

and conduction band

phase

spectral density

Xs electron affinity

W angular frequency

O instantaneous angular frequency