Noble - Oscillator Design and Computer Simulation

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Oscillator Design and Computer SimulationRandall W. Rhea

1995, hardcover, 320 pages, ISBN l-8849-32-30-4

This book covers the design of L-C, transmission line, quartz crystaland SAW oscillators. The unified approach presented can be used with awide range of active devices and resonator types. Valuable to experi-enced engineers and those new to oscillator design. Topics include: limit-ing and starting, biasing, noise, analysis and oscillator fundamentals.

The electronic text that follows was scanned from the Noble publish-ing edition of Oscillator Designand Computer Simulation. The book isavailable from the publisher for $49.00 (list price $64.00). Please mentionEagleware offer to receive this discount. To order, contact:

Noble Publishing Corporation

630 Pinnacle CourtNorcross, GA 30071 USA

Phone: 770-449-6774Fax: 770-448-2839E-mail: [email protected]

Dealer discounts and bulk quantity discounts available.


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Contents

Preface

1 Analysis Fundamentals

1.1 Voltage Transfer Functions1.2 Power Transfer Functions1.3 Scattering Parameters1.4 The Smith Char t1.5 Radially Scaled Parameters1.6 Matching1.7 Broadband Amplifier Without Feedback 1.8 Stability1.9 Broadband Amplifier With Feedback 1.10 Component Parasitics1.11 Amplifier With Parasitics1.12 References

2 Oscillator Fundamentals

2 . 1 E x a m p l e2 . 2 M i s m a t c h2.3 Relation to Classic Oscillator Theory2 . 4 L o a d e d Q

2.5 L-C Resona tor Configu ra tions2.6 L-C Resona tor P ha se Sh ift2.7 Resonators as Matching Networks2.8 Resonator Voltage2.9 Tra nsm ission Line Resona tors

. . .XII I

1

1348

1011131618212427

29

313234353641414244


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Oscillat or Design a nd Compu ter Simula tion

2.10 Re-entrance 48

2.11 Qua rt z Crysta l Resona tors 492.12 Crysta l Dissipa tion 522.13 Pulling Crystal Oscillators 532.14 Cera mic Piezoelectr ic Resona tors 562.15 SAW Resona tors 582.16 Multiple Resona tors 592.17 Phase 622.18 Negat ive Resistance Ana lysis 65

2.19 Em itter Capa cita nce in Negat ive-R Oscillat ors 702.20 Looking Th rough t he Resona tor 732.21 Negat ive Resista nce Oscillator Noise 742.22 Negative Conductance Oscillators 752.23 Stability Factor and Oscillator Design 802.24 Output Coupling 812.25 Pulling 842.26 Pushing 862.27 References 86

3 Limiting and Starting 89

3.1 Limiting 893.2 Amplitude and Frequency Stability 903.3 Class-A Opera tion 913.4 Nea r-Cla ss-A Exam ple 92

3.5 Predicting Output Level 963.6 Outpu t H ar monic Cont ent 1023.7 Class-C Power Oscillators 1043.8 Star t ing 1053.9 Sta rt ing Time 1063.10 Bias Time Const an t 1083.11 Fr equen cy Effects of Limitin g 1093.12 References 110


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Contents vii

4 N o i s e 111

4.1 Single-Sideba nd Ph ase Noise 1114.2 Amplifier Noise 1124.3 Amplifier Flicker Noise 1134.4 Oscillator Noise 1144.5 Oscillator Noise Nomograph 1164.6 Residual Phase and Frequency Modulation 1184.7 Varactor Modulation Phase Noise 1204.8 Buffer Amplifiers 1214.9 Frequency Multiplication 1244.10 Discrete Sidebands 1244.11 Power Supply Noise 1254.12 Low-Noise Design Suggestions 1264.13 Typical Oscillator Noise Performance 1284.14 References 131

5 Biasing 1335.1 Bipolar Tra nsist or Biasing 1335.2 Simple Feedback Biasing 1345.3 One-Battery Biasing 1365.4 CC Negative Supply Biasing 1375.5 Dua l Supply Biasing 1385.6 JFET Biasing 1395.7 Ground ed Sour ce 1395.8 Self-Bias 1405.9 Dual-gate F ET 1415.10 Active Bipolar Biasing 1435.11 Hybrid Biasing 1445.12 References 145


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. . .Vlll Oscillator Design an d Compu ter Simula tion

6 Computer Techniques 147

6.16.26.3

6.4

6.56.66.7

6.8

6.9

6.10

7 Circui ts 183

7.1 Frequency Range 1847.2 Stability 1857.3 Tuning Bandwidth 1857.4 Phase Noise 1877.5 Simplicity 1877.6 General Comments 1877.7 Output Coupling 1897.8 References 189

Oscillat or Simu lat ion 148Simple Resona tor E xam ple 149Oscillator Synthesis 1536.3.1 Synthesis Example 1536.3.2 Analysis of the Example 1556.3.3 Optimizat ion of th e Example 1556.3.4 Noise Per form ance of th e Example 157SPICE Analysis of Oscillators 159

Loaded Q Limitation 1611OOMHz Loop Oscillat or Measu r ed Da ta 161Negative-Resistance Oscillator Computer 1646.7.1 Analysis Fundamentals 1656.7.2 Device Selection 1666.7.3 Circuit Enhancements 168Broad Tun ing UHF VCO Example 1706.8.1 Frequency Tuning Linearity 174Spice Analysis of the UHF VCO 1776.9.1 Oscillator Sta rt ing Time 1786.9.2 The Oscillator Spectrum 181References 182


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Contents

8 L-C Oscillators

8.18.28.3

8.4

8.5

8.68.7

Capacitors 191Inductors 193L-C Colpit t s Oscilla t or 1968.3.1 Output Coupling 199L-C Clapp Oscillator 1998 .4 .1 Tu n in g 2008.4.2 Output Coupling 2028.4.3 Circuit Vagaries 202

8.4.4 Operating Frequency 202L-C Bipolar Tran sistor Oscillat or 2028 .5 .1 Tun ing 2058.5.2 Coupling Capacitor Inductance 2068.5.3 Controlling the Phase 2068.5.4 The Bipolar Amplifier 207L-C Hybrid Oscillator 207References 211

9 Distributed Oscillators

9.1 Negat ive Resista nce UH F Oscillat or9.1.1 Circuit Vagaries9.1.2 L-C Resonator Form9.1.3 Output Coupling9.1.4 Advantages9.1.5 Circuit Variations

9.2 Negat ive-R Oscillat or with Tra ns form er9.3 Bipolar Cavity Oscillat or

9 .3 .1 Tun ing9.3.2 Example

9.4 Hybr id Cavity Oscillat or9 . 5 R e f e r e n c e s

ix

191

213

215219220220221222224225227229231234


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X Oscillat or Design a nd Compu ter Simulat ion

IO SAW Oscillators 235

10.1 SAW Bipolar Oscillator 23610.1.1 Output Coupling 238

10.2 SAW Hybr id Oscillator 23810.2.1 Tuning 23910.2.2 E lemen t Valu es 24110.2.3 Output Coupling 242

10.3 SAW Dual-gate FET Oscillator 24210.3.1 E lement Va lues 245

10.3.2 Output Coupling 24610.4 References 246

11 Quartz Crystal Oscillators 2 4 7

11.1 Pier ce Cryst a l Oscillat or 24811.1.1 Loaded Q 24911.1.2 E lemen t Valu es 250

11.1.3 Dissipation 25111.2 Colpitt s Crysta l Oscillat or 25111.2.1 Limitations 25111.2.2 E lement Va lues 25311.2.3 Comments 254

11.3 High-Performance Crystal Oscillator 25411.3.1 Performance 25511.3.2 Low-Frequency Overtone Crystals 25611.3.3 Example 256

11.3.4 E lemen t Valu es 25811.3.5 Frequency Pulling 25911.3.6 Phase Noise 25911.3.7 AM-to-PM Conversion 261

11.4 But ler Overt one Crysta l Oscillat or 26111.4.1 Pulling 26311.4.2 Circuit Tips 264

11.5 But ler Oscillat or -Mult iplier 26511.5.1 Example 26611.5.2 Modulation 267

11.6 References 269


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P r e f a c e

The engineer is often confused when faced with his/her firstoscillator design. Other electrical engineering disciplines haveevolved procedures for designing specific networks. A classicexample is electrical filter design where many aspects have beenreduced to cook-book simplicity With experience, the engineerdevelops a feel for the practical problems involved in filter design,and applies creative solutions to these problems. But the appren-tice has many references with well-outlined approaches to theproblems. For the RF and microwave oscillator apprentice, theapproach is often less effective. Typically, the literature issearched for an oscillator type similar to that needed for thepresent requiremen t. Component values ar e modified an d aprototype is constructed to complete the design. This approach isfraught with difficulty Lacking is an understanding of the fun-da m en ta l principles involved. A lar ge nu m ber of var iables a ffectoscillator operation, and if the performance is inadequate, theappr entice is un certa in a bout a solution. Although mu ch litera -ture exists concerning oscillators, each reference typically ad-dresses a specific oscillator type. A fundamental understandingof the concepts is all too often buried in pages of equations.

The purpose of this book is to demystify oscillator design andprovide a pra ctica l referen ce on th e design of RF an d m icrowaveoscillators. The thrust of the book is on concepts, a unified designapproach to a variety of oscillators, and verification of the designvia compu ter s imu lat ion. This is not a book of ma th ema t ics.Equa tions ar e included only when t hey cont ribute t o fun dam en-tal understanding, determine component values, or predictoscillator performance.


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xiv Oscillator Design an d Comp ut er Simu lation

Design begins with a linear approach. An active amplifier iscascaded with a passive frequency-selective resonator. The cir-cu it s ma ll signa l open -loop ga in/pha se (Bode) plot is con sidered.To form the oscillat or , th e loop is closed. Oscilla t ion bu ilds un tillimiting occurs which reduces the loop gain to unity The linearBode plot describes many aspects of oscillator performance. Thenon linear characteristics of the loop amplifier are consideredindependently Together, these considerations predict nearly allaspects of oscillator behavior including the gain/phase oscillationma rgin, oscillation frequ ency, noise per form an ce, sta rt -u p t ime,output level, harmonic level, and conditions conducive to spuriousoscillation. This design approach is applied to a variety of oscil-lators using bipolar, JFET, MOSFET, and hybrid/MMIC activedevices with L-C (inductor-capacitor), transmission line, SAW,and piezoelectric resonators.

If the amplifier and resonator were ideally simple, the mathemat-ics involved for a complete linear solution would be simple.Accurate active-device models at RF and microwave frequenciesa r e complex. Th erefore, solvin g th e equa t ion s for the loop, whilenot conceptually difficult, is typically tedious. The exact equa-tion s a n d t echn iques a re differen t for each oscillator type, whichdiscourages a unified design approach. Instead, why not leavethe burden of computing the network responses to a general-pur-pose circuit simulation computer program? The accuracy andconvenience of these programs is now mature. Dealing with apleth ora of pra ctical pr oblems , such a s comp on ent parasitics, issimple for a simulation progra m. The designer ma y ponder th econcepts and solutions, while the computer handles the tediumof analysis. A unified approach to oscillator design is encouragedin t his environment .

Most of the specific oscillator designs covered in this book are oldfriends of mine. Over 1 million units of a 300-MHz crystalcontrolled transmitter based on the Butler overtone oscillator

with built-in frequency multiplier were constructed by Scientific-Atlanta, my former employer. Many other designs have beenconst ru cted by the th ousa nds.


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Preface xv

This is the second edition of a book originally published byPrentice Hall. This second edition includes significant updatesand over 100 pages of new material. The new material includesremarks relating open loop oscillator theory to classical terminol-ogy It expan ds r eson at or th eory t o include a ddition al L-C form san d popular cera mic load ed coaxial resona tors. The m at erial onnegat ive r esista nce oscillators is subst an tially upda ted a nd ex-pan ded. Cha pter 6 on compu ter a ided techn iques is rewritten toinclude recent advances and Spice-based oscillator analysis. Anew Chapter 12 includes case studies of typical oscillator specifi-cat ions a nd descript ions of th e design pr ocedur es us ed to sat isfythose requirements.

I would like to thank Larry McKinney of Scientific-Atlanta forthought-provoking discussions and sharing design experiencesand data. I would also like to thank Crawford Patterson forlayout and edit work on this second edition.

Randall W. RheaS tone M oun tain , Georgia

January 2,1995


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A n a l y s i s F u n d a m e n t a l s

For this section, we assume that networks are linear and timeinvariant. Time invariant signifies that the network is constant with time. Linear signifies the output is a linear function of theinput. Doubling the input driving function doubles the resultant output. The network may be uniquely defined by a set of linear equations r e l a t i n g port voltages and currents.

1 .I Voltage Transfer FunctionsConsider the network in Figure 1-1A terminated at the generatorwith Rg, terminated at the load with R I, and driven from a voltagesource I& [l]. Et is th e volta ge acr oss t he load .

The qua nti ty E avail is the voltage across the load when all of theavailable power from t he genera tor is tr an sferr ed to th e load.

E 4 - I E Aavait = R &? 2

1.1

For the case of a null network with R I = R g,

1.2

since one-half of E g is dropped across R g and one-half is droppedacross R t. For the case of a non-null network, dividing both sidesof t he equ a t ion 1.1 by Et gives

Eavai tP &-=

- - E l R g 2E 1

1.3


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Oscillator Design and Computer Simulation

Rg I I

A

-_ ,a l

f-a2

c - -b l B b 2

Figu re l -l A l inea r, t ime- invar ian t ne twork def ined in t e rms o f t er m i na l vol t ages (A) and i n t e r m s o f p o r t inc iden t and r e f l ec ted w a v e s (B).

We can then define the voltage transmission coefficient as thevoltage across the load, E l, divided by the maximum availablevoltage across the load Eaaa iz ,or

Elt=== F m

-RI Eg

1 .4

This voltage transmission coefficient is the voltage gain ratio.For th e case with R g = R z , since E g = 2Ez, th e t ra nsmissioncoefficient is 1.


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Analysis Fundamentals 3

1.2 Power Transfer Functions

The power inser t ion loss is defined a s

1.5

where the voltages and resistances are defined as before, P n u z z isthe power delivered to the load with a null network and Pl is thepower delivered to the load with a network present. Figure l-2

depicts Pd as a function of RI with a null network, I&T= 1.414 voltsand R g = 1 ohm. Notice the maximum power delivered to the loadoccurs with R I = 1 ohm = R g.

When R I is not equal to R g, a network such as an ideal transformeror a reactive matching network may re-establish maximum

Figu re 1-2 Power delivered to the load versus the terminationresistance ratio.


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4 Oscillat or Design a nd Compu ter Simula tion

power tr an sfer. When inserted, th is passive net work ma y there-fore r esult in more power being delivered t o th e load th a n wh enabsent. The embarrassment of power gain from a passive deviceis avoided by an alternative definition, the power transfer func-tion

P a v a i l RI Eg 2 1-_= _ - _I II 4R g E l t2 1.6where

1.7

When R I = R g, th ese definitions ar e ident ical.

1.3 Scattering Parameters

The networks depicted in Figure l-l may be uniquely describedby a number of two-port parameter sets including H, E : 2, A B C D ,S an d oth ers which ha ve been used for th is purpose. Ea ch h aveadvantages and disadvantages for a given application. Carson[2] an d Altman [3] consider network parameter sets in detail .

S-parameters have earned a prominent position in RF circuitdesign, an alysis an d measu rem ent [4,5]. Other pa ra meters , such

as E : 2 and H parameters, require open or short circuits on portsdur ing mea su rem ent . This poses serious pr actical d ifficulties forbroadba nd high frequency measu remen t. Scat ter ing par am eters(S-parameters) are defined and measured with ports terminatedin a reference impedance. Modern network analyzers are wellsuited for a ccur at e measur ement ofS-parameters. S-parametershave the additional advantage that they relate directly to impor-ta nt system specificat ions such a s gain a nd ret ur n loss.

As depicted in Figure 1-lB, two-port S-parameters are defined byconsidering a set of voltage waves. When a voltage wave from asour ce is incident on a net wor k, a port ion of th e volta ge wave istransmitted through the network, and a portion is reflected back


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toward the source. Incident and reflected voltage waves may also

be present at th e out put of th e network . New variables ar edefined by dividing the voltage waves by the square root of thereference impedance. The square of the magnitude of these newvariables may be viewed as traveling power waves.

I al I 2 = incident pow er wave at th e netw ork in pu t 1.8

I bl I 2 = reflected power wave at the network input 1.9

I a2 I 2 = incident power wave at th e network outpu t 1.10

I b2 I 2 = reflected power wave at the netw ork ou tpu t 1.11

These new variables andby the expressions

bl = a & l + m & 2

b2 = a A l + a & 2

blSi2 = --,a1 = 0a2

b2S zl=za 2=0

bz&2=gal=O

the network S-parameters are related

1.12

1.13

1.14

1.15

1.16

1.17

Terminating the network with a load equal to the referenceimpedance forces ag = 0. Under these conditions.

bis11=--& 1.18

bzs21=--& 1.19

S 11 is then the network input reflection coefficient and S21 is theforward voltage transmission coefficient t of the network. When


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6 Oscillator Design and Computer Simulation

the generator and load resistance are equal, the voltage trans-

mission coefficient defined t earlier is equal to S21. Terminatingth e network at th e inpu t with a load equa l to th e referenceimpedance and driving the network from the output port forcesal = 0. Under these conditions.

1.20

bls12=-& 1.21

S 22 is then the output reflection coefficient and S 12 is the reversetransmission coefficient of the network.

The S-parameter coefficients defined above are linear ratios. TheS-parameters also may be expressed as a decibel ratio.

Because S-parameters are voltage ratios, the two forms arerelated by the simple expressions

I SII I = input reflection gain (dB) = 20 log I SII I 1.22

I S22 I = output reflection gain (dB) = 20 log I S22 I 1.23

I SZI I = forward gain (dl3) = 201ogI SZI I 1.24

I S12 I = reverse gain ( d B ) = 201og I SE I 1.25

To avoid confusion, in this book, the linear form of the scat ter ing

coefficients are referred to as CII, C21, Cl2 and C.22. The decibelform of S 21 and S 12 are often simply referred to as the forwardand reverse gain. With equal generator and load resistance, S 21and S 12 ar e equal to th e power insert ion gain defined ear lier.

The reflection coefficients magnitudes, I S 11 I and I S 22 I are lessthan 1 for passive networks with positive resistance. Therefore,the decibel input and output reflection gains, I S 11 I and I S 22 I ,are negative numbers. Throughout this book, S 11 and S22 arereferred to as return losses, in agreement with standard industryconvention. Therefore, the expressions above relating coefficientsand the decibel forms should be negated for S 11 and S22.


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Input VSWR and S 11 are related by

VSWR = 1 +I is11 I

l- ISill1.26

The output VSWR is related to S22 by an analogous equation.Table l-l relates various values of reflection coefficient, returnloss, and VSWR .

The complex input impedance is related to the input reflectioncoefficients by the expression

1.27

The output impedance is defined by an analogous equation usings22.

T a b le l -l Radially Scaled Reflection Coefficient Parameters

V S W R S r l ( d B ) C l 7 V S W R

4 0 . 0 0 . 0 1 0 1 1 . 0 2 03 0 . 0 0 . 0 3 2 1 . 0 6 52 5 . 0 0 . 0 5 6 1 . 1 1 92 0 . 0 0 . 1 0 0 1 . 2 2 21 8 . 0 0 . 1 2 6 1 . 2 8 81 6 . 0 0 . 1 5 8 1 . 3 7 71 5 . 0 0 . 1 7 8 1 . 4 3 31 4 . 0 0 . 2 0 0 1 . 4 9 91 3 . 0 0 . 2 2 4 1 . 5 7 71 2 . 0 0 . 2 5 1 1 . 6 7 11 0 . 5 0 . 2 9 9 1 . 8 5 11 0 . 0 0 . 3 1 6 1 . 9 2 59 . 5 4 0 . 3 3 3 2 . 0 0 09 . 0 0 0 . 3 5 5 2 . 1 0 08 . 0 0 0 . 3 9 8 2 . 3 2 37 . 0 0 0 . 4 4 7 2 . 6 1 5

6 . 0 2 0 . 5 0 0 3 . 0 0 05 . 0 0 0 . 5 6 2 3 . 5 7 04 . 4 4 0 . 6 0 0 3 . 9 9 74 . 0 0 0 . 6 3 1 4 . 4 1 93 . 0 1 0 . 7 0 7 5 . 8 2 92 . 9 2 0 . 7 1 4 6 . 0 0 52 . 0 0 0 . 7 9 4 8 . 7 2 41 . 9 4 0 . 8 0 0 8 . 9 9 21 . 7 4 0 . 8 1 8 1 0 . 0 21 . 0 0 0 . 8 9 1 1 7 . 3 90 . 9 1 5 0 . 9 0 0 1 9 . 0 00 . 8 6 9 0 . 9 0 5 2 0 . 0 00 . 4 4 6 0 . 9 5 0 3 9 . 0 00 . 1 7 5 0 . 9 8 0 9 9 . 0 00 . 0 8 7 3 0 . 9 9 0 1 9 9 . 0


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8 Oscillator Design an d Compu ter Simulat ion

1.4 The Smith Chart

In 1939, Phill ip H. Smith published an article describing a circu-lar chart useful for graphing and solving problems associatedwith transmission systems [6]. Although the characteristics of transmission systems are defined by simple equations, prior tothe advent of scientific calculators and computers, evaluation of these equations was best accomplished using graphical tech-niques. The Smith chart gained wide acceptance during animportant developmental period of the microwave industry Thechart has been applied to solve a wide variety of transmissionsystem pr oblems , man y which ar e described in a book by PhillipSmith 171.

The design of broadban d t ra nsm ission systems using th e Smithchart involves graphic constructions on the chart repeated forselected frequencies th roughout t h e ra nge on int erest . Alth ougha vast improvement over the use of a slide rule, the process is

tedious except for single frequencies and useful primarily fortraining purposes. Modern interactive computer circuit simula-tion programs with high-speed tuning and optimization proce-dures are much more efficient. However, the Smith chart remainsan import an t tool as a n insight ful display overlay for computer-generated data. An impedance Smith Chart with unity reflectioncoefficient r ad ius is sh own in F igu re l-3.

The impedance Smith chart is a mapping of the impedance plane

and the reflection coefficient. Therefore, the polar form of areflection coefficient plotted on a Smith chart provides the corre-sponding impedance. All values on the chart are normalized toth e referen ce imp edan ce such as 50 oh ms. The ma gnitu de of th ereflection coefficient is plott ed a s t h e dist ance from t he cen ter of th e Smith char t. A perfect m at ch plott ed on a Smith cha rt is avector of zero length (the reflection coefficient is zero) and istherefore located at the center of the chart which is l+ j 0 , or 50

ohms. The radius of the standard Smith chart is unity Admit-tance Smith charts and compressed or expanded charts withoth er t ha n u nity ra dius at th e circum ference ar e available.


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Analysis Fundamentals

1 8 0

9

Figu re 1-3 Impedance Smith chart with unity reflectioncoefficient radius.

Purely resistive impedances map to the only straight line of thechart with zero ohms on the left and infinite resistance on the

right. Pure reactance is on the circumference. The completecircles with centers on the real axis are constant normalizedresistance circles. Arcs rising upwards are constant normalizedinductive reactance and descending arcs are constant normalizedcapacitive reactance.


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High impedances are located on the right portion of the chart, low

impedan ces on t he left port ion, indu ctive reacta nce in t he u pperha lf, an d capacitive rea cta nce in th e lower h alf. The a ngle of th ereflection coefficient is measured with respect to the real axis,with zer o-degrees to th e right of th e cent er, 90 stra ight u p, an d-90 st r a igh t d own. A vector of length 0.447 a t 63.4 exten ds t othe intersection of the unity real circle and unity inductiverea cta nce ar e 1 +jl , or 50 +j50 when d emoralized.

The impedance of a load as viewed through a length of lossless

transmission line as depicted on a Smith chart rotates in aclockwise direction with const a nt ra dius a s length of line or th efrequency is increased. Transmission line loss causes the reflec-tion coefficient to spiral inward.

1.5 Radially Scaled Parameters

The reflection coefficient, return loss VSWR, and impedance of anetwork port are dependent parameters. A given impedance,whether specified as a reflection coefficient or return loss, plotsat the same point on the Smith chart. The magnitude of theparameter is a function of the length of a vector from the chartcenter to the plot point. Therefore, these parameters are referredto as radially scaled parameters. For a lossless network, thetransmission characteristics are also dependent on these radiallyscaled parameters. The length of this vector is the voltage reflec-tion coefficient, p, and is essentially the reflection scatteringparameter of that port. The complex reflection coefficient at agiven por t is r elat ed to the impeda nce by

z-z,p=z+zo 1.28

where Z is the port impedance and Z, is the reference impedance.

ThenRL&?=-2010g IpI 1.29


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1.30

LA=-lolOg(l- Ip12) 1.31Table l-l includes representative values relating these radiallyscaled parameters.

1.6 MatchingGain (or loss) is clearly an important parameter of a network. Thedefinition of gain t ha t will be used is th e tr an sdu cer power ga in.The transducer power gain is defined as the power delivered tothe load divided by the power available from the source.

G t = p p,,a v a i a e

1.32

The S-parameter data for the network is measured with a sourceand load equal to the reference impedance. The transducer powergain with the network inserted in a system with arbitrary sourceand load reflection coefficients is 151

Gt =I c21 I 2(1 - I l-s I 2)u - I EC I 2 ,

~(i-cllr~)(i-c22r~)-c21c12r~rsi21.33

where

I s = reflection coefficient of the source

IL = reflection coefficient of the load

If I s an d IL ar e both zero, th en

Gt=C212

or

Gt(dB) = 20 log I C21 I = IS21

1.34

1.35

1.36

1.37


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When both ports of the network are conjugately matched, and Clz= 0,

G1

Ic21121

u r n a x =1- IC~~12 I - IC2212

1.40

The first and third terms are indicative of the gain increaseachievable by matching the input and output, respectively If Cl1or C22 ar e much larger th an zero, substa nt ial gain impr ovementis achieved by matching. Matching not only increases the net-

work gain, but reduces reflections from the network.It is more desirable for network gain to flatten across a frequencyband than minimum reflections. The lossless matching networksar e designed to provide a better ma tch at frequen cies where t hetwo-port gain is lower. By careful design of amplifier matchingnetworks, it is frequently possible to achieve a gain response flatwithin fractions of a decibel over a bandwidth of more than anoctave.

1.7 Broadband Amplifier Without Feedback

An example of 2 to 4 GH z amplifier design using the foregoingprinciples is considered next. An Avantek AT60585 bipolar tran-sistor with the S-parameter data given in Table l-2 for thecommon-emitter configuration is used. This data is graphed in

Figur e 1-5. The t r an sistor gain in d ecibels, S21, is plot ted on th eleft. The gain is 11.4 dB a t 2 GH z and 5.8 dB a t 4 GHz. Th etr an sistor input a nd out put retu rn loss plot t ed on a Smith cha rtare shown on the right in Figure l-5. The input impedance is lessthan 50 ohms and slightly inductive. The output impedance isgreat er th an 50 ohm s an d capa citive. The ma rk ers ma y be usedto discern th e tr aces an d r ead specific values (along t he bott omof the screen).

As is typical, the transistor gain decreases with increasing fre-quen cy Using equa tion 1.29 an d S11 an d S.22 from Ta ble 1-2, th ead ditiona l gain a chievable at 4 GH z by mat ching both th e inputand output is 1.0 + 1.4 = 2.4 dB. The conjugately matched gain


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frequencies to increa se th e gain, an d to worsen th e ma tch at th elower frequencies to decrease the gain. First, placing a shuntcapa citor at th e input of th e t r an sistor rota tes th e 4 GH z end of the SII trace toward the center of the Smith chart, improving theinput match at 4 GH z and increasing the gain. By trying differentvalues for the shunt capacitor, it was discovered that a value of 1pF results in the maximum increase in gain. The results areshown in Figure 1-6. The gain at 4 GH z has been increased to 6.7dB, up from 5.8 dB. The gain a t 2 GH z was unaffected.

A shorted transmission line stub followed by a series transmis-sion line are used to match the output. The values for thisnetwork were determined using optimization with the =Super-Sta r= compu ter pr ogram .

The results are shown in Figure 1-7. The flatness is within a fewtenths of a decibel. Actual results should agree closely with thesecalculat ed results becau se the assu mpt ion t ha t CD = 0 is unnec-

2 mo 3 c m 4 0 0 0

s 2 1 - s 2 1 - S l l - s 2 2 -2 o o a 2 o o o1 1 . 3 3 E %S l i E 5 5 E & 3 - 7 . 5 7 9 6 2 F E 3 3 4 E 8 1 8 z &3 2

1 1 . 3 3 9 . 8 4 2 6 1 8 . 5 6 1 5 5 6 . 6 9 8 4 3 - 4 . 8 4 9 1 3 - 4 . 8 6 4 9 7 - 4 . 8 0 8 2 2 - 3 . 6 6 8 3 4C.

11

Figu re 1 -6 Au an tek A T 60585 transistor responses with 1 p Fshu n t capacitance at th e input .


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2 G 0 0 3 i mO 4 O Gt l

s 2 1 - s 2 1 - S l l - s 2 2 -k a c l 2 6 0 0 3 4 0 0 4 0 0 0 2 o c a 3 4 0 07 . ' j 8 6 8 9 7 . 9 4 1 8 8 8 . 0 3 4 9 8 7 . 8 7 7 0 8 - 6 . 0 9 5 2 z T i 9 9 6 - 8 . 0 5 3 3 5 %% m7 . 9 8 6 8 9 7 . 9 4 1 8 8 8 . 0 3 4 9 8 7 . 8 7 7 0 8 - 1 . 6 2 0 9 - 2 . 6 6 3 1 9 - 4 . 8 9 38 1 - 5 . 9 8 7 9 9

Figu re 1-7 Avantek AT60585 transistor with matching at theinput and output.

essary when computer simulation techniques are utilized. Thesch ema t ic of t h e comp leted design is given in Figur e 1-8.

1.8 Stability

The type of networks used at the input and output of an amplifiermust be selected based on an additional criterion, stability Thefact that Cl2 is not equal to zero represents a signal path fromthe transistor output to the input. This feedback path is anopportunity for oscillation to occur. The reflection coefficientspresent ed to th e tr an sistor by the m at ching network s affect t hestability of the amplifier. A stability factor, K, is


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TI 3

AT605>

T1PF T 75__L -I-- -

Figure 1-8 Bipolar 2 - 4 GHz transistor amplifier withmatching at the input and output optimized to flatten the gain.

K = w 111 2 - I C 2 2 1 2 + lOI2

2 IC12I IC2111.41

whereD= CllC22 - Cl2 c21 1.42

When K>l, C11


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18 Oscillator Design a nd Compu ter Simula tion

This circle is the locus of loads for which Cl1 = 1. The region insideor outside the circle may be the stable region.

The input plane stabili ty circle equations are the same as theoutput plane equations, with 1 and 2 in the subscripts inter-changed. Reference [4 ] includes a more detailed tutorial onstability

1.9 Broadband Amplifier With Feedback

Achieving a flat frequ ency response by sha ping th e ma tch of th einput and output networks has the advantage that the gain athigher frequencies can exceed S21. This advantage is especiallyuseful at higher frequencies where gain is more expensive toachieve, so this technique is a common practice in microwaveamplifier design. Unfortunately, this technique has several dis-advantages:

(a) The match is necessarily poor at lower frequencies.(b ) The ba nd widt h of flat gain r esponse is limited.

(c) Sta bility consider at ions a re crit ica l.

Anoth er m eth od of flat ten ing th e frequ ency response is to applyresistive negative feedback. This method overcomes the abovedisadvantages but the gain is less than the gain of the transistorat the highest frequency However, at UHF and lower frequencies,where transistor gain is naturally higher and less expensive, thisdisadvan ta ge is less significa nt .

Amplifiers designed using negative feedback can possess widebandwidth, excellent match, excellent stability, and excellentflatness. Consider the simple amplifier shown in Figure l-9.Shunt (collector to base) feedback and series (emitter) feedback ar e applied to an MRF901 t ransis tor. S-parameter data for the

tr an sistor is given in Ta ble l-3.The results shown in Figure l-10 were computed using the=SuperStar= program . These results illus t ra te excellent gainflatness and match from low frequencies to 450 MHz. In practice,


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BROADAMP

t

Re11 ohm

-

Figu re 1-9 Simple broadband amplifier using resistive series feedback in the emitter, R et an d shu nt feed back from collector tobase, L p and R F

the low-frequency response is limited by the values of the inputand output coupling capacitors.

The inductor, L p, in series with the shunt feedback resistor iscalled a peaking inductor. It is used to extend the bandwidth of the amplifier. At higher frequencies, where the amplifier gainbegins to fall because the open-loop transistor gain is falling, the

reactance of the peaking inductor effectively reduces the shunt

T a ble 1-3 S -Param eter Data for a M otorola MRF901T ransistor Biased at 10 V an d 15 m A

Freq CII Angle CZI Angle(MHz) (ratio) (deg) (ratio) (deg)

50 .5 -23 24.0 160100 .51 -66 20.4 141200 .47 -112 14.5 119500 .50 -166 6.81 92

Cl2 Angle(ratio) (deg)

.Ol 69.02 63

.03 54

.05 57

C 2 2 A n g l e(ratio) (deg)

.90 -12.83 -22

.63 -31

.41 -35


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2 0 Oscillator Design an d Comp ut er Simu lation

1 e-06 2 2 5 4 5 0s 2 1 -

m l 3 0s 2 1 - S l l - s 2 2 -

8 . Ol Y5 5 7 . 8 5 8 7 3 7 . 9 5 0 9 7 3 1 04 5 0 7 . 9 7 5 2 - 2 8 . 0 8 9 9le-06 - 2 9 . 4 5 8 3 1 3 0- 2 3 . 0 8 3 1 0 - %5 9 4 98 . 0 1 4 5 5 7 . 8 5 8 7 3 7 . 9 5 0 9 7 7 . 9 7 5 2 - 3 6 . 2 5 8 9 - 2 4 . 7 1 6 3 - 1 8 . 9 7 1 2 - 1 5 . 9 8 9

F i g u r e l -1 0 Gain a nd m atch responses of th e broad band amplifier with resistive feedback.

feedback and extends the frequency response. A similar tech-nique m ay be employed in th e emitt er by placing a capa citor inpar allel with th e emitter series feedback resistor. The ma tch atthese extended frequencies is not as good as the match at lowerfrequencies.

When the transistor open-loop gain is much greater than the gainwith feedback, t he gain with feedback is given by

Gf (dB) = 20 log 1.45

w h e r e

R f R f i =z0

1.46


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R,,GRf 1.47The shunt feedback resistor, R f, reduces both the input and outputimpedance. The series feedback resistor, R e, increa ses both th einpu t a nd outpu t impeda nce. As gr eat er feedback is applied, th einput an d out put impedan ces asymptotically approach th e rela-tion

z. = (R f IL? 1.48This expression, which indicates the proper relationship of Rf andR e to achieve a desired &, is most valid when the device inputand output impedances are already near the desired ZO. Whenth e inpu t a nd outpu t impeda nces differ from ZO , other valu es forRf and R e may yield a better match. For example, if both the inputand output impedances are higher than ZO , more shunt feedback (lower R f) and less series feedback (lower R e) will yield a bett ermatch.

In Figure l-11, the frequency response of MRF901 t ransis toramplifiers with a 50 ohm source and load is compared for differingvalues of shu nt a nd series feedback. A peak ing indu ctor is used,but not an emitter peaking capacitor. The peaking inductorvalues have been optimized to achieve the greatest possiblebandwidth.

1 .I 0 Component Parasitics

Component s u sed in t he const ru ction of electr onic network s a reseldom as ideal as we would wish. An example is Cl2 not equa lto zero for active devices. Even relatively simple componentssuch as resistors, capacitors, and inductors have significantparasitics. Through UHF frequencies, some of the more impor-ta nt par as itics of pa ssive component s ar e

(a ) Inductance of capacitor leads

(b) Self-capacitance of inductors


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22 Oscillat or Design a nd Compu ter Simulat ion

0 1 I I I111111 I I , ,,,111 I10 3 0 50 100 3 0 0 5 0 0 1 0 0 0 2 0 0 0

F r e q u e n c y (MHz)

Figure l-11 Closed loop frequency response of a transistor amplifier with varying degrees of feedback applied.

(c) Fin ite Q of indu ctors

(d ) Coupling between inductors

Other parasitics may be significant as well, but an experienced

high-frequency designer will consider the effects of these fourparasitic types on every passive component used in the design.The importa nce of th is ca nn ot be over st ressed. Coun tless hour sof breadboard trouble shooting can be saved by considering theseeffects dur ing th e design.

The vast majority of design equations published in engineeringliterature do not include the effects of these parasitics becauseth e r esultin g complexity would h opelessly redu ce t he usefulnessof the expressions. Th is places RF an d m icr owave design in t hecategory of black magic, to be delved in only by those initiated inthe art. Often, successful practitioners are simply those who havethe experience of knowing which parasitics to worry about, what


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to do about th ose, an d which ar e insignifican t in a given a pplica -

tion.Compu ter simu lat ion pr ogra ms offer a powerful tool for dealingwith these effects. All parasitics are not included directly incompu ter pr ogra m comp onen t models simply becau se t he p ossi-bilities ar e endless. However, th e designer can easily a dd t o th enetwork description parasitics appropriate for the componentsbeing used. In addition to simulating and identifying theseeffects, tuning and optimization in the computer program canassist in determining a remedy. Listed in Table l-4 are typicalpassive component parasitics for high frequencies. Reference [8]includes an entire chapter devoted to components and parasitics.

Ta b le 1 -4 T ypical Com ponen t Para sitic E ffects at HighFrequencies and Possible Remedies

Parasitic EffectsTypical Values Remedies

Capacitorleadinductance

Lead spacing L0.25 in. 9 nH0.20 in. 8 nH

0.10 in. 4 nHLeadless 1 nH

Inductorselfcapacitance

Inductor Q

Refer to Chapter 8

Refer to Chapter 8

inductor coupling Varies significantly

Use capacitors in parallel

Use smaller diameter coilUse toroidReduce required inductance

Increase inductor volumeAt lower frequencies use

pot coresIncrease inductor spacingReorient inductorsUse toroidsUse magnetic shielding


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1 .I 1 Amplifier With Parasitics

An approximate model for a l/4-watt leaded ca rbon composit ionor film resistor is shown in Figure 1-12. For higher resistancevalues, the reactance of the lead inductance is less significantthan the resistance. In this case, the parallel capacitance isimportant at higher frequencies. For lower resistance values andhigh frequencies, the reactance of the lead inductance is moresignificant.

Figure 1-13 shows the schematic of a simple 108 to 300 MHzamplifier similar to the amplifier in Figure 1-9, but using a2N5179 tr an sistor a nd feedback r esistors with par asitics.

The results are shown in Figure 1-14. The solid traces in eachcase are with ideal resistors with no parasitics. On the upper left(LRF), th e dash ed response is with 9 nH of indu cta nce added t othe resistor Q . Notice the gain is increased and the flatness isimproved. The resistor parasitic inductance adds to the requiredpeaking inductance and aids amplifier performance. At the upperright (CRF), the resistor parallel capacitance is added. The gainis reduced and the flatness is degraded. Therefore, inductance inthe shunt feedback resistor is not a problem but capacitancedegrades performance somewhat.

Next consider the effects of the same parasitics in the seriesfeedback resistor, Re. On the lower left (LRE) adding the resistor

0 . 6 pF

Figure 1-12 Model of a l/4-watt composition or film resistor with first-order parasitics.


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Rf Lr

I, I I

0.5 pF

50 nHI

AMP> q 2N5179

Re4.7 ohm

Le9 nH

Ce

0.5 pF

Figure I-13 Schematic of a broadband feedback amplifier with resistorparasitics included.

inductance causes significant performance degradation. How-ever, parasitic capacitance has no discernible effect in Re (CRE).

Parasitic sensitivities are highest for capacitance in Rf becausethe resistor value is higher A small series reactance has littleeffect wh ile par allel capa cita nce shu nt s t he h igh r esista nce. Onthe other hand, the low resistance of Re makes it extremelysusceptible to small values of series inductive reactance butinsens itive to par a llel ca pa citive rea cta nce.

The effect of emitter resistor inductance is reduced by smallerresistor length (such as l/8 watt or chip resistors) or by using twoor more resistors in parallel. Two resistors effectively reduce the


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,

i : / / / /. , : .:,:

Ena: g rl0ts-i 0 Tut Da: m 121S5S ,994 OScSFIE.ScHF,H~ Fz-save F3-0pl FI-Tua FPNsld FBEdiu F7 1~: 52 FS

Figure 1-14 Gain responses of the 2N5179 tran sistor am plifier w ith ideal elem en ts (all solid traces). Dash ed responses arewith 9 n H ind uctance in R f (LRF), 0.5pF in Rf (CRF), 9 nHindu ctance in R e ( L R E ) and 0.5pF in Re (CRE) .

lead inductance by a factor of 2. The increased parasitic capaci-tance is unimportant because it has little impact on the response.

Anoth er potent ial problem is th e lead indu cta nce of th e 2N5179transistor. The emitter lead must be very short. A better solutionwould be to use a leadless form of this transistor.

These are but a few of the parasitic considerations with which thehigh-frequ ency designer mu st deal. Remember, it is very impor-tant to become habitual about considering these effects for everycomponent . It h as been t he a ut hor s experience th at designer sreadily find solutions to these problems once the problems arerecognized.


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1 .I 2 References

[l] G. Mat th aei, L. Youn g an d E.M.T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, ArtechHouse Books, Norwood, Massachusetts, 1980, p. 36.

[21 Ralph S. Carson, High-Frequency Amplifiers, John Wiley &Sons, New York, 1982.

[3] Jerome L. Altman, Microwave Circuits, D. Van Nostrand,Princeton, NJ, 1964.

[4] Applicat ion Note 95, S-Pa ra met ers-Circuit Ana lysis a nd De-sign, Hewlett-Packard, Palo Alto, CA, September 1968.

[5] Application Note 154, S-Parameter Design, Hewlett-Packard,Palo Alto, CA, April 1972.

[6] Philip H. Smith, Transmission Line Calculator, Electronics,Vol. 12, January 1944, p. 29.

[7] Ph ilip H. Smith, Electronic Applications of the Smith Chart,McGraw-Hill, New York, 1969.

[8] Randall W. Rhea, HF Filter Design and Computer Simulation,Noble Publishing, Atlanta, 1994.


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O s c i l l a t o r F u n d a m e n t a l s

?too methods of oscillator analysis and design are considered inthis book. One method involves the open-loop gain and phaseresponse versus frequency. T his B ode response [1] and nonlinear effects d iscussed later predict m an y aspects of oscillator perform -an ce. A second m eth od considers th e oscillator as a one-port witha n egative real im pedan ce to wh ich a resona tor is atta ched. T heloop methodprovides a more complete and intuitive analysis whileth e negative resistance m eth od is m ore suitable for broad tu n ing

oscillators operating above several hundred megahertz.The loop m eth od is st ud ied first . Consider th e a mplifier-resona -tor cascad e in Figur e 2-l. The cascade is dr iven by a sour ce witha resistance of 2 , and is terminated in a load resistance of ZO .The ga in (forward) a t a given fr equen cy is

Gf = 20 log 1 C21I 2.1

where Cz1 is the m agnitu de of th e forwa rd-scat ter ing para meterfor th e cascade at a given frequency The t ra nsm ission pha se ata given frequency is the angle of C21. If the cascade is matchedat the input and output to 2 , the magnitudes of Cl1 and C22, theinput and output scattering parameters, are zero.

The ga in-pha se r esponse for a typical cascade is given in F igur e2-2. The normal convention of the Bode response is to plotfrequency on a logarithmic scale. Because oscillators typically

operate over less than a decade of bandwidth, we will use a linearfrequency scale. The curve on the left with a peak just above 100MHz is t h e gain plot t ed on a scale of -20 to 20 dB. The S-shapedtrace is the transmission phase plotted on a scale of -225 to 225.Plotted on the right Smith chart are the cascade input return loss,


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Oscillator Fundamentals 31

L c& YZ- II025S l l - s 2 2 -

1 0 . 0 0 3 7 1 2 . 7 2 9 1 1 0 . 6 9 - T9 0 8 9 6 _ ' I %4 2 ' I : 6 4 21 1 0- 4 . 2 7 2 7 7

3 3 . 9 4 9 9 - 2 . 4 8 80 8 - 2 . 4 8 80 8 - 5 7 . 0 0 88- 7 . 3 5 5 5 6 - 1 2 . 8 8 6 6- 1 2 . 8 8 6 6 3 . 6 3 5 1 8E R a I : Rmml t 0 We dLI c t 1 916: 1 *351334 TEYP. SCH~ , ~ ~ F * - S a r s F 3 0 p ( F , - T ms F Ws r d F BI d t F 7 Tuns:5ZFS

Figu re 2-2 Open-loop transmission gain and phase (left) and inpu t an d out pu t m atch (right) of a reson ator-am plifier cascad e.

2.1 An Example

Figur e 2-3 sh ows th e schem at ic of th e ca sca de u sed t o compu tethe open-loop response given in Figure 2-2. A pi-network resona-tor is cascaded with a common-emitter 2N5179 bipolar NPNtransistor amplifier. Rc is the collector DC load resistance andRb provides base bias. Rf is an RF feedback resistor which isdecoupled for biasing through a 1000 pF capacitor. The output1000 pF ca pa citor is us ed for DC decou pling of t h e collector an dbase when the oscillator is finally formed by connecting the

output to the input.A characteristic of well-designed oscillators is a gain peak nearthe phase zero crossing frequency A second desirable charac-ter istic is a ph ase-zero-crossing nea r t he m aximu m ph as e slope.


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1 0 0 0 pF - 4 7 0 0 0 o h mI

Rb II >

o s c > 2N.5179

c c _

Figure 2-3 S c h em a t i c of a 100 MHz example osc i l l a to r.

These criteria are approximately satisfied in this example withthe gain peak and maximum phase slope occurring near 102 MHz.The gain margin is large for this example, ensuring that vari-ations in production transistor parameters, passive componenttolerances, and temperature effects are unlikely to prevent oscil-lat ion.

2.2 Mismatch

The input and output scattering parameters for the cascade, Cl1

and C22, are plotted on the Smith chart in Figure 2-2. Marker 7is at the phase zero crossing frequency of 100 MHz. Cl1 is 0.24at 133 and C22 is 0.23 at 40. When the cascade input and outputimpedances ar e not equa l to Zo, th e misma tch results in ananalyzed gain that differs from the maximum available gain. If


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the input and output impedances are equal to each other and real,but not equal to ZO, then in the analysis, Z0 may be readjustedto obtain a correct analysis. The gain and phase are then accu-rately modeled. To simplify measurement of the Bode response,it is generally desirable to design the oscillator network so thatthe input and output impedances are equal to the impedance of available measurement equipment, typically 50 or 75 ohms.

For this first example, C22 is not exactly equal to Z,, and thecalculated and displayed loop gain is less than it would be if the

output were matched [2]. When the output of this cascade isconnected to the input to form the oscillator, the mismatch willreduce the loop gain below the maximum available value.

If the amplifier reverse isolation is adequate,Cl2 may be assumedzero. The loop gain, with th e output driving the inpu t, may thenbe derived from equation 1.38.

Gopen loop =

l-IC2212C212

1-IC1112

I l-CllC22 I 2 I l-CllC22 I 2 2.2

where

Cl1 = cascade input reflect ion coefficient

Czz = cascade outpu t reflection coefficient

l-s = c22

I-L = Cl1

For this example,

2.3

2.4

2.5

2.6

G = 0.851 x 18.66 x 0.847 = 13.45 = 11.3dB 2.7

In this case the mismatch reduces the open-loop gain by 1.4 to11.3 dB. Because feedback is often employed in the amplifier, theassumption that C12=0 may not be valid. In this case, equation

2.2 only approximately represents the open-loop gain with thecascade terminating itself The best policy is to design the cas-cade for at least a reasonable match at both the input and output.The cascade may include matching networks at the input and


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output but this level of complexity is typically not required or

justified.

2.3 Relation to Classic Oscillator Theory

Th e open -loop con cept of oscillat or design is oRen met withconsiderable skepticism by engineers familiar with classicoscillator terminology For comfort consider Figure 2-4A wherethe oscillator cascade is drawn with only the RF components.Next, th e circuit is redra wn in F igur e 2-4B with th e out putconnected to the input and the ground floated. In Figure 2-4Cthe emitter is selected as the ground reference point. Notice theconfiguration is the familiar common-emitter Pierce oscillator. InFigure 2-4D the circuit is again redrawn, this time with the base

I+ 7-T-l

oscH kfQ-LTI 1 I A

yI I- -C

I

0

a -- -I

D F

Figu re 2-4 Va rious d efin itions of th e loop oscillat or based onthe selected ground reference point.


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Oscillator Fu nda ment als 35

selected as ground reference. The result is the familiar common-

base Colpitts. These open-loop, Pierce and Colpitts oscillators arein fact the same oscillator!

2.4 Loaded Q

The oscillator loaded Q is a critical parameter. The loaded Q is adirect indication of many oscillator performance parameters. Ahigh loaded Q

(a ) Reduces ph ase noise

(b) Reduces fr equen cy dr ift

(c) Isola t es per forma n ce from a ctive-device var iat ion

Ph a se noise is invers ely pr oport iona l to the squ a re of th e load edQ [4]. D ft s d~ 1 r e uced because the resonator solely determinesthe oscillation frequency in high-Q designs. Isolating the resona-t or fr om act ive device reactances r educes th e effect of t emper a -ture. Many oscillator designs have low loaded Qs. The phasenoise a nd long-ter m sta bility of th ese designs ar e far fr om opti-mum. An oscillator with a low loaded Q is often the root problemeven though designers offer imaginative and esoteric descriptionsof th e problem. Noise is discuss ed fur th er in Cha pter 4.

Th e open-loop loaded Q of a cascad e is

For the 100 MHz example the loaded Q is approximately 5.2. Theloaded Q in terms of the phase slope is

d9&I = o.5fo d f

where cpis in r adia ns orn fo dv-

= 360 d f 2.10


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LCl

L3A L3B

LC3

9.2 pF

I-

C5A C5B

LC+pjskJI l s p F

33pFI-

220 pF

T T

_!- C9A

T33 pFi

L9A77 nH

-L-

-I_-

-L -L- -

C6

LC6

- -

C8A C8B

75 nH 16pF

Figure 2-5 L-C reson a tor s t ru ctu res wi t h a r esona n t f requ encyof 100 MHz a n d a load ed Q of 6.9 when te rm ina ted in 50 oh m s .


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The difficulty with the simple series and parallel resonator is

extr eme element values with 50 ohm ter mina tions a s th e loadedQ is increa sed. Notice in Figur e 2-5 th at th e series indu ctor, Ll,is 1100 nH and the shunt inductor, L2, is 5.6 n H . If a higherloaded Q is desired the values become even more extreme.

LC3 through LC6 are three element resonators. LCl and LC2are bandpass structures. LC3 and LC4 are lowpass and LC5 andLC6 ar e highpass str uctu res. At high load ed Q (6.9 in th is ca se),t h e lowpass an d highpass structures have responses which aresimilar to bandpass, at least near the resonant frequency Apotential hazard of lowpass and h ighpass structures is that signaltransmission with only small attenuation may occur over a broadband of frequencies. Unless care is exercised, additional reac-tances in the oscillator circuit for biasing and decoupling maycause an additional transmission phase zero and result in am-biguous oscillation frequencies. The three element forms do offermore reasonable element values. LC3 and LC5 have large but

moderat ed indu cta nce values a nd LC4 an d LC6 have sma ll butmoderated element values.

Resonator LC3 is analyzed by converting each series inductor andtermination resistance combination to a parallel equivalent. Theresulting two shunt inductors and two shunt resistors for apa ra llel resona nt circuit. Th e load ed Q for LC3 is t hen

Q1=$ 02.13

where Xl is the reactance of L3A or B. The reactance of theresonating capacitor, Cs, is then

R, + Xl2x c 3 = u r i 2.14

Element values for t he simple an d th ree element resona tors ar e

unique. Only one set of values satisfy a given loaded Q andtermination resistance.

Although values for th e th ree elemen t resona tors ar e more m od-erate than the simple resonators, as the loaded Q is increased


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Oscillat or Fu nda ment als 39

further, even those values become impractical. Lower termina-

tion resistance moderates values in LCl, 3 and 5 while highertermination resistance moderates values in LC2,4, and 6. Thisis th e basis for rem ar ks often foun d in oscillat or liter at ur e suchas a FE T tr an sistor is more su itable becau se th e higher imped-ances load the parallel resonator lightly and provide higher Q.In t he a ut hor s view this repr esent s a n ar row perspective onoscillator design. We should learn an important lesson from filterdesign t heory. How ar e na r rowban d filter s (high load ed Q) con-structed with reasonable element values and 50 ohm termina-tions? The answer is found in the use of coupling elements.

C7A and B in the four element resonator LC7 are examples of coupling elements. At 100 MHz the shunt 33 pF capa citors ar eapproximately 50 ohms of reactance which are in parallel withthe terminations. The resulting series equivalent R-C networksand the input and output are 25 ohms resistance and 25 ohmsreactance (the reactance of a 66 pF capacitor). The effective

termination resistance is halved and the required resonatorser ies in du ctor , L7, for a given loaded Q is ha lf th e indu ct an ce of th e simple series r esona tor LCl. Two series capacitors of 66 pFeach increase the resonating capacitor from approximately twicethe simple resonator capacitance of 2.3 pF (4.6 pF ) to 5.5 pF.Increasing the coupling capacitors would further reduce therequired inductance to achieve a given loaded Q. Thus the fourelement coupled resonators provide a degree of freedom in ele-

ment values.For the LC7 series resonator (shunt-C coupled series resonator),the effective capacitance which resonates with the series inductoris

c, = 11 2GA(oo%)2

c , + (oo&c~~)2 + 1

where

CT = series resonator capacitor 2.16

C~A = shunt coupling capacitor 2.17

2.15


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R. = inpu t a nd out put load r esista nce

The required inductan ce to resona te a t f0 is then

2.18

1L 7 = -

00 2ce2.19

The loaded Q, of the LC7 resonator is a function of the shuntcou pling ca pa citors. The r eact an ce requ ired for a given loa ded Qis approximately

where

Qexl l1

2.20

2.21---&l Qu

and Qu is i n d u c t o r u n l o a d e d Q .

F or LC8 (t op-C-cou pled pa r a llel r eson a tor ) t h e effect ive r esona t -ing ca pa citor is

Ce=Cs+2CsA

WoRG3A~2 + 12.22

The top-C coupled resonator in Figure 2-4 requires series cou-pling reactances of approximately

2.23

where

BL~ = admitt an ce of th e shu nt inductor 2.24

The coup ling element s m a y be indu ctors or m ixed, as d iscuss edin the series resonator case above.


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2.6 L-C Resonator Phase Shift

The tra nsm ission ph ase shift a t r esona nce (ma ximu m t ra nsm is-sion a nd m aximu m ph a se slope) of th e simple resona tors is zerodegrees. The tr an smission pha se shift of th e th ree elemen t reso-na tors at r esona nce is 180.

The four element coupled resonators also provide a degree of freedom in t ra nsm ission pha se shift at resona nce. For exam ple,with LC7, for a given Q, smaller values of shunt capacitance lead

to larger series inductance up the the value of inductance for thesimple resonator. At this extreme, the transmission phase ap-proaches zero-degrees. Large values of shun t capacita nce de-crease the ser ies inductance and the t r ansmiss ion phaseapproaches -180 at resona nce. The LC8 resonat or has a t ra ns-mission phase shift of zero-degrees for large CSA and B and +180for small C8A and B. The designer therefor has available reso-na tors of ar bitr ar y tr an smission ph ase at resona nce!

2.7 Resonators as Matching Networks

The element values of the resonators in Figure 2-5 are symmetricwith respect to the input and output. If the elements are lossless(high unloaded Q), at resonance the input impedance is purelyresistive and equal to the termination resistance. If the termina-

t ion resistan ce is 50 ohm s t he inpu t resistan ce is 50 ohm s a nd if th e term inat ion r esista nce is 1000 ohm s th e inpu t r esista nce is1000 ohms. Although the resonant frequency shifts with termi-nation resistance for the three and four element resonators, atresonance the input impedance equals the termination resis-tance.

Earlier it was stated that one oscillator design goal was a matchedcascade input and output impedance. The resonator behavior

described above naturally maintains this criteria provided thecascade amplifier is matched at the input and output. If theamplifier input and output impedance are not matched, it is oftenpossible to use t he r esona tor a s a m a tching device by pert ur bingthe symmetry of a three or four element resonator. This is


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preferred to adding matching networks because the number of

elements and the possibility of introducing additional resonancesare minimized.

For example, consider resonator LC7 cascaded with an amplifierwith an input resistance of 200 ohms and an output resistance of 50 ohms. The resonator is terminated in 200 ohms. The inputresistance looking into the resonator would be 200 ohms if LC7were symmetric. When C~A is reduced to approximately 20 pF,LC7 acts as a ma tching network with an input resistan ce of 50ohm s, therefore m at ching th e cascade inpu t a nd outpu t imped-ance. The resonators are also capable of absorbing terminationrea cta nce by ad just men t of resona tor reac tances .

2.8 Resonator Voltage

In an earlier section we listed desirable attributes of high loaded

Q. However, th ere a re funda ment al l imita tions t o th e ma ximu mloa ded Q. As t he ca scade loa ded Q a ppr oaches t he u nload ed Q of components in the resonator the resonator insertion loss ap-proaches infinity The insert ion loss for th e r esona tor is

IL = -20 log 2.25

where I L is a positive decibel number. I L is therefore equal to

-Szl dB. For example, if Qu = 100 and &I = 21.5, IL = 2.1 dB. If the cascade amplifier has adequate gain then significant loss canbe tolerated in the resonator. Nevertheless, the loaded Q can notexceed t he component un load ed Q.

A second factor which may limit the maximum loaded Q isresonator voltage. This is particularly a problem with high-poweroscillators and oscillators with varactor tuning elements. Thevoltage at resonance across the shunt inductor and capacitor inLC8, th e top-C cou pled pa r a llel resona tor , is given by


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VS1

vr= RO - jxcu 2.26

where V. is source voltage intoR . ohms and BL~ is the admittanceof the shunt inductor.

The insertion loss, &I and V r versus XS ar e given in F igur e 2-6 forQU = 200, R . =50 ohms, VS = 0.707 Vm S ( +lO dBm,) and BL~ = .Olmhos. Notice with only 0.707 volts drive the resonator voltagereaches 5 Vrms or 14.1 VP, a t Qz/QU= 0.5! The insertion loss atQ$QU = 0.5 is 6.02 dB. A varactor used for C8 would be driven

Ql

5 --

ts210 dB --

6000 4 0 0 0 2000 0

- xs

5 v

4 v

3V

2 vRes onat orVol t age(rms>

- 1v

f igure 2-6 In sertion loss, load ed Q an d resonator voltage as a function of the coupling reactance in top-C coupled parallelresonators.


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into heavy forward conduction and perhaps even reverse break-

down by the RF voltage. Since this significantly degrades reso-nator unloaded Q and increases loss, limiting in the cascadeoccurs in the resonator instead of the amplifier, an intolerablesituation leading to erratic tuning and poor stability

The var a ctor m ay be decoupled from t he resona tor by placing avery small capacitor in series with the varactor, therefore drop-ping most of the voltage across the series capacitor. However, thevaractor now has much less ability to shift the oscillation fre-quency Thus, we face a fundamental tradeoff; high loaded Qresults in h igh resona tor voltage an d impedes broadban d varac-tor tuning. Keep in mind that broad tuning and high Q are notinherently impossible. The problem is resonator voltage. Whentuning elements are not effected by high voltage, such as withmechanically tuned capacitors and cavities, broad tuning andhigh Q are possible. A wonderful example is the venerableHewlett-Packard model HP608 signal source.

2.9 Transmission Line Resonators

Over limited bandwidth there are important lumped (L-C) anddistributed (transmission-line) equivalences. For example, ashunt inductor may be replaced with a shorted transmission linestub. The equivalent inductive reactance of a shorted stub less

th an 90 long isXl = 2 , t an Oe 2.27

where

9, = electr ical length of the stu b 2.28

Z. = cha ra cter istic impedan ce of the line 2.29

Similarly, an open stub less than 90 long may replace a capacitor.The equ ivalent ca pacitive rea cta nce is

X, = Z. t an ee 2.30


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The reactance of inductors and capacitors vary linearly withfrequency over the entire frequency range for which componentparasitics are not a problem. From the above expressions we seethe reactance of transmission line stubs are trigonometric func-tions of frequ ency which ar e linea r an d t her efore simu latelumped reactance when the electrical length is short. The erroris about 1% at 10 and 10% at 30. The reactance is predictedaccurately by the above equations for any length less than 90.It is not absolutely necessary that the reactance varies linearlywith frequency unless the oscillator is to be tuned over a widefrequency range. Electrical lengths of 45 or even 60 are some-times used. However, as the length approaches90, the reactanceapproaches infinity Unlike lumped elements, transmission lineelements do not have unique solutions forZ0 and &. For example,50 ohm s of indu ctive reacta nce is simu lated with a 50 ohmshorted stub 45 long or a 100 ohm shorted stub 26.56 long.

The equations above describe the equivalence between a single

lumped and a distributed element. A distributed element alsomay serve as an equivalent to an L-C pair. A high-impedancetransmission line which is 180 long at f. behaves like a seriesL-C resonator at f. with an inductive reactance given by

d 0

Xl = 2 2.31

Likewise, a transmission line shorted stub which is 90 long at f.

behaves like a parallel L-C resonator at f. with an inductivereactance given by

2.32

Shown in Figure 2-7 are various transmission line resonatorswith a resonant frequency of 100 MHz and a loaded Q of 5.2 whenterminated in 50 ohms. TLl and TL2 are analogous to LCl andLC2 and are a direct implementation of the above equations.Notice the extreme values of line impedance. This is a directcarry-over of the extreme L-C values for these simple resonatorforms. As with the L-C resonators, the transmission line imped-


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TLITLI w-/----+

440 ohm180

TL2

TL2

T

2.8 ohm90

140

C3A 50 ohm C3B

TL3)--1 m I+12 pF TL3 12 pFTL5A

C7A C7B

-

TL4

TLGA TLGB TLGC

~~6~---~-t+-----l\

34 ohm90

155 ohm180

34 ohm90

TL8

L8A LaB

Figure 2-7 Transmission line structures with a resonant frequency of 100 MHz an d a loaded Q of 5.2 w hen t erm inat ed in 50 ohms.


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an te values are moderated by lower termination resistance forthe series resonator and higher termination resistance for theshunt resonator.

Again, as with the L-C resonators, the solution is to use couplingtechniques. Examples are given as TL3 through TL8 in Figure2-7. Most of these examples use transmission lines with a char-acteristic impedance of 50 ohms. However, transmission lineresonator solutions are typically not unique and alternative reso-na tors with eith er h igher or lower lin e imp edan ce ar e possible.

End coupling capacitances are used in TL3. 12 pF is far too muchcapacitance to realize as a gap in microstrip and lumped elementswou ld be u sed. At higher microwave frequ encies a gap becomesfeasible. The capacitive loading shortens the required transmis-sion line electrical length at the resonant frequency In this casethe line length is approximately 140. For higher loaded Q theend capacitors must be smaller and transmission line shorteningis reduced.

The end -coupling capa citors a s a fu nction of Q ar e

2.33

The r equired length of th e tr an sm ission line for r esona n ce is

$ = 180 - t a r-5 26&&G 2.34

assuming the Z. of the resonator transmission line equals theinput an d out put load r esista nce. In pr actice, th e resona tor ma ybe higher or lower in impedance if the coupling capacitors andresonator length are adjusted. This technique may be used toshift slightly the location of the phase zero crossing on the phaseslope, particularly for lower Qs.

In TL4 a shu nt resona tor is tapped t o increa se th e loaded Q for

th e moderat e 50 oh m line impeda n ce. The t ota l electr ical lengthof the two sections is somewhat greater than 90 because of term inat ion loading.


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2.11 Quartz Crystal Resonators

A quartz crystal resonator is a thin slice of quartz with conductingelectrodes on opposing sides [6]. Applying a voltage across thecrystal displaces the surfaces, and vice versa. The quartz is stiff,and the crystal has natural mechanical resonant frequencieswhich depend on the orientation of the slice in relation to thecrystal lattice (cut). Although there are many crystal cuts, themost common cut for high-frequency application is AT FT-243cryst als, in comm on use dur ing World War II, ha d spr ing-load edth ick m eta l plat es pressing aga inst ea ch side of th e qua rt z slice.These crystals could be disassembled and the quartz etched toreduce the r esona nt frequency Drawing a gra phite pencil ma rk on the quartz lowered the resonant frequency Modern quartzcrystals use electrodes plated directly onto the quartz disk.

Quartz crystals have very desirable characteristics as oscillatorresonators. The natural oscillation frequency is very stable. In

addition, the resonance has a very high Q. Qs from 10,000 toseveral hundred thousand are readily obtained. Qs of 2 millionare achievable. Crystals of high performance can be mass pro-duced for a few dollars. The crystal merits of high Q and stabilityare also its principal limitations. It is difficult to tune (pull) acrystal oscillator.

Qua r tz cryst a l resona tors a re a vailable for frequen cies a s low as1 k H z . The practical frequency range for fundamental-mode

AT-cut crystals is 0.6 to 20 MHz. Crystals for fundamentalfrequencies higher than 20 to 30 MHz are very thin and thereforefragile. Crystals are used at higher frequencies by operation atodd h ar monics (overtones) of the fun dam ent al frequen cy Ninth-overtone crystals are used up to about 200 MHz, the practicalupper limit of crystal oscillators.

It is possible to extend the maximum operating frequency andstil l maintain adequate mechanical strength by surrounding avery thin quartz disk with a thicker concentric outer support ringintegral to the qua rt z. This stru ctu re is ma nu factur ed by chemi-cal etching techniques [7]. Due to the resulting shape, the result-ing structure is referred to as inverted mesa.


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50 Oscillator Design a nd Compu ter Simula tion

A simplified elect r ica l equ iva lent circuit for t h e qua r t z cr ysta l isgiven in Figure 2-8. Co is the capacitance formed by the electrodessepar at ed by th e quar tz dielectr ic. It is stat ic an d ma y be meas-ur ed a t a ny frequen cy well below r esona nce. R m , L m and Cm a r eth e motional parameters of the crystal. L m an d Cm resonate atthe series resonant frequency of the crystal. R m is associated withthe loss of the resonator. The model in Figure 2-8 represents oneoscillation mode. A more complex model can represent a crystalthrough as many overtones modes as desired. For the sake of simplicity, this simple model is usually employed and differentvalues are used to model fundamental or overtone modes. Spu-rious resonances occur at frequencies near the desired resonance.In a high-quality crystal, the motional resistance of spuriousmodes are at least two or three times the primary resonanceresistan ce an d t he spu rious modes ma y be ignored.

Crystal manufactures can provide specific data on model parame-ters. Nominal values are a function of the fundamental frequency

and the overtone being used. The manufacturer has some controlover par am eter values. Typical funda ment al-mode values a re

Co =3 to 8 p F 2.35

Cm = 0.004c0 2.36

L m = 1Gw3 2 C m

2.37

R m = 400 ohm s at 1 MHz to 20 ohm s a t 20 MHz 2.38

where

F i g u r e 2-8 L u m p e d R -L -C m o d el f or a q u a r t z c r ys t a l .


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Oscillator Fundamentals

fs = series resona nt frequ ency

For overtone crystals

C o = 3 t o 8 p F

C ,=Cm fund

overtone 2

L m = l(2~fs)2Cm

2.42

Rm = 35 ohm s for t hir d overt one 2.43

= 55 ohms for fifih overtone 2.44

= 90 ohms for seventh overtone 2.45

= 150 ohm s for nin th overtone 2.46

51

2.39

2.40

2.41

The parameters may be determined by measuring S21 versusfrequ ency with a h igh-quality scala r or vector n etwork a na lyzer.The crystal is inserted in series in the transmission path. Aresponse similar to Figure 2-9 should be observed. The crystal Qis very high. Careful tuning of the analyzer center frequency anda narrow scan width ar e required. The peak t ra nsm ission pointoccurs a t th e series resona nt frequency, fS . T he insertion loss at

0 dBt

-2 dB

Figu re 2-9 Insertion loss response of a quartz crystal resonator.


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series resonance is IL. The transmission zero just above fs in

frequency is the parallel resonance, fp. First , Co is obtained bymeasuring the capacitance at a low frequency, such as 1 k H z .Then

where

2

c , = c , [O fp -1 Is

2.47

L m = 1(27rfi)2Cm

2.48

R m = 22, [ldL 2 0 -l] 2.49

ZO= tr an smission m easur ement system impedan ce 2.50

2.12 Crystal Dissipation

Over-excita tion of a qu ar tz crysta l cau ses a long-ter m cha nge of the crystal parameters (aging). Although the change is small,generally a few parts per million, in some applications it issignificant. For best aging, crystal dissipation in the circuitshould be less than 20 pW [S]. Severe over excitation can crack the quartz crystal. For AT cuts, to avoid damaging the crystal,the dissipation should be less than 2 m W, below 1 MHz and above

10 MHz, and less than 5 m W from 1 to 10 MHz.The dissipat ion ma y be compu ted by $/ R m , where E is the rm svoltage across the crystal exactly at series resonance. A moreprecise measurement method is to place a small resistance inseries with t he crysta l and find th e cur rent based on th e voltagedrop across the added resistance. The dissipation is then calcu-lated by 12Rm.


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2.13 Pulling Crystal Oscillators

The long-term stability and close-in phase noise performance of crystal oscillators is superior to L-C and cavity oscillators buttuning more than a fraction of a percent of the resonant frequencyis difficult. L-C oscillators, while less stable and noisier, arereadily tuned a octave or more in frequency For frequencies upto about 100 MHz, there are not many alternatives between thesetwo extremes. The pr imar y limita tion is Co. If it were n ot for Co,crystal oscillators could be pulled much further. The pullingrange for fundamental-mode crystals is approximately

2.51

Since Cm is nominally about 0.004C0, the pull range is about1.002, or 0.2%. The p u llin g ra n ge for overt on e-mode cryst a ls is

Fpu l l overtone = 1 + F pu ll -1overtone 2

2.52

A typical ninth overtone crystal pulling range is 1.000025. Thisis on ly 25 ppm !

Shown in F igur e 2-10 is a crysta l resona tor with a series pullingcapacitor, Ct. Disregarding Co, the pulling capacitor in series witht h e motional ca pa cita nce redu ces t he net effective series capa ci-

tance, increasing the series resonant frequency In Figure 2-11th e tr a nsm ission of such a n etwork with a very lar ge value of Ctis shown as the solid trace. In this case fs is 10 M H z, R m is 30

Lm Cm Rm

Fig u re 2 -1 0 Quartz resonator m odel w ith pu lling capacitor.


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0 225I/

- - _ . _ _ - _ - _ _ _ _ _ _

_, 0

_ _ . ~ _ _ _ _

..i_- ; ::

- 2 09 . 9 8 1 0 1 0 . 0 2

s 2 1 - P 2 1 -9 . 9 9 9 . 9 9 2 1 0 . 0 0 8 1 0 . 0 2- 3 8 . 0 3 3 2 - 3 7 . 2 4 5 3 a . 5 5 7 5 - 7 0 . 0 7 6 28 9 . 2 2 4 6 8 9 . 0 8 6 3 8 7 . 4 6 4 6 7 . 4 1 6 6 6C . C T

Figure 2-11 T ransm ission gain an d phase response of a 10MHz quartz resonator before (solid) and after (dashed) pullingwith a 2.5 p F series capacitor.

ohms, Co is 5 pF and Cm is 0.02 pF. The gain curve peaks and the

transmission phase is zero at series resonance. The insertion lossis 2.5 dB. The transmission phase is again zero at parallelresonance just above 10.016 MHz but the transmission gain isvery low. The dotted curve in Figure 2-11 is the network responsewith Ct at 2 .5 pE Decreasing Ct fr om a lar ge valu e t o 2.5 pF h aspulled the frequency up about 13 k H z . The gain is reduced toabout -12 dB. If Ct is further reduced to pull the frequency higher,th e gain ra pidly fa lls as t he frequen cy appr oaches pa ra llel reso-

nance. This parallel resonance, caused by Co resona ting with anet inductive reactance of the motional arm above series reso-na nce, is t he limitin g factor in crysta l pullability


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th is occur s, th e new r esona tor becomes en tir ely L-C an d crysta lcontrol is lost.

In Figure 2-13 an inductor is placed in parallel with Co. L Oresona tes with Co a t fs . Becau se th e Q of th e LO-CO combinationis much lower than the Q of the crystal, L O effectively cancels thereactance of Co for a broad frequency range around fs. In Figur e2-14 the response of this network is shown with L t equal to 33 ~_LHand Ct from 80 to 2.7 pF. Notice the absence of the parallelresonance and that the gain is flat across the entire frequency

range tuned by Ct. This configuration does have excellent pullingcharacteristics. However, the inductor values in this and theprevious network are large. Inductor parasitic capacitance cancreate additional resonances that cause erratic behavior. Also,the farther a crystal is pulled, the more dependent the operatingfrequency becomes on the L-C pulling elements, eliminating thepurpose of using a crystal.

2.14 Ceramic Piezoelectric Resonators

Piezoelectr ic resona tors cons tr ucted from cera mic ma ter ials ar enow available for the HF frequency range. These devices bridgethe gap in Q U and stabili ty between L-C and quartz crystalresonators. The electrical model is the same as the quartz crystal

Lm CmI I

Rm

H ,Oh bF m30 ohm Lt ?32 pF-

c o

II I I

5 PF

Lo

50700 nH

F ig u re 2 -1 3 Quartz resonator model for pulling above and below fO u sing a series ind uctor an d a com pensat ion ind uctor in

parallel with CO.


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9.38 1 0 l G c l 2s 2 1 -

9 . 9 8- 3 4 . 1 7 8 48 8 . 5 2 3 6

C , C T

9 . 9 9 2- 3 0 . 0 6 3 18 7 . 6 3 3 2

P 2 1 -1 0 . 0 0 8- 1 6 . 1 2 8 27 8 . 1 3 4 9

1 0 . 0 2- 2 2 . 3 3 5 3- 8 4 . 2 1 4 2

Figu re 2-14 T ransm ission gain and phase for a pulled quartzresonator with Co com pen sation . Ct tuned from 80 pF (solid) to4 pF (dashed).

and is shown in Figure 2-8. Typical values for a 4 MHz ceramic

resonator areRm = 6 ohms 2.53

L m = 0.3 m H

Cm = 5.3 pF

C,,=42pF 2.56

The unloaded series resonant Q is about 1200 for this unit. Theratio of C&, is much smaller than an AT-cut quartz crystal, sothe pullability is better. As might be expected, the stability is lessthan the quartz crystal.


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2.15 SAW Resonators

The su rface acous tic wave r esona tor (SAWR) is a h igh-Q resona -tor similar in some aspects to the quartz crystal resonator [9].Interdigitized metal fingers are engraved on the surface of piezo-electr ic subst ra te. Y- cut qua rt z is often us ed. The int erdigitizedtransducer launches and detects surface acoustic waves on thesubst r at e. When t he excited frequ ency is equa l to V,lp, where V,is the propagation velocity and p is the interdigital period, thewaves generated by each finger are in phase. This is the centerfr equ en cy of t he SAWR.

SAWRs used in oscillators have two forms, two-port and two-ter-mina l. The two-port form ha s an input a nd a n output , which a reinterchangeable, and a ground. The two-terminal form is similarto a quartz crystal resonator in that i t has two terminals. Eachform is available in two types. The RP type two-port form hasapproximately 180 degrees of transmission phase shift at reso-

nance and the RS type has approximately zero-degree phase shiftat resona nce. The selection is made dur ing man ufactu re by th eway th e inter digita l fingers a re conn ected t o the port ter mina ls.E lectr ica l models for th e RP SAWR is given in F igu re 2-15A an dth e RS type in Figur e 2-15B. If th e SAWR t wo-por t is con n ect edinternally so that the ports are driven in parallel, the two-termi-nal forms are created. The models are given in Figure 2-15C andD. SAWRs on quartz are practical from 250 to 1200 MHz. De-

signs as low as 50 MHz and as high as 1500 are feasible. Thefrequency range of SAWR satisfies a real need, since quartzcrystals are commonly available only to 200 MHz. SAWR un-loaded Qs are nominally 12,000 at 350 MHz and 6000 at 1000MHz. Typical model parameter values are

R m = 120 ohms 2.57

mL,=Q$

s2.58

Cm= (2rss)2Lm

2.59


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- - - -A l& I DEGREE

C 180 DEGREE D 0 DEGREE

Fig u re 2 -1 5 Two-port SAW resonators ( A & B ) and two-terminal SAW&s (C&D).

CO = 2 .5 pF at 200 MHz

= 1.1 pF at 1200 MHz

Typical manufacturing tolerances

2 . 6 0

2 . 6 1

for SAwRs is f150 ppm. Thefrequ ency shift with a +50 Celsius temperature range is about80 ppm. The aging characteristics of high-quality SAWRs a r eabout 1 to 10 ppm per year. Each of these stability parameters issubstantially better than with L-C resonators but approximatelyan order of ma gnitu de worse t ha n high-qua lity qua rt z crysta ls.Th e SAWR is capable of safe operation at much higher powerlevels than quartz crystals and therefore the ultimate noiseperformance, well removed from the carrier, can be better thancrysta l oscilla t or s. Typica l power diss ipat ion limit s a r e +30 dBmat 250 MHz and +18 dBm at 1000 MHz.

2.16 Multiple Resonators

Thus far we have considered oscillator design using a singleresona tor a s th e frequ ency-selective pha se shift n etwork a nd we


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have identified loaded Q as the critical parameter for high-per-

formance oscillator design. For the single resonator, Q as definedby the loop gain r esponse is r elat ed to th e pha se slope. Fur th er-more, since it is not the loop gain response but the phase slopethat is an indication of oscillator stability and phase noise per-formance, it is best to define the unloaded Q in terms of the phaseslope. The phase slope, d$/do, is the definition of group delaySince the group delay, t d , is

td = dw 2.62

then

2.63

Group delay is a m ore convenient mea sur e of loa ded Q becau semanual computation of the phase slope from phase versus fre-

quency data, or manual computation of the 3-dB bandwidth fromthe amplitude response, is unnecessary.

Recall that a primary limitation of high loaded-Q oscillator designis the insertion loss encountered as the loaded Q approaches theunloa

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Noble - Oscillator Design and Computer Simulation