Practical rf system design wiley ieee ebook pdf version

Practical RF System Design (Wiley – IEEE) – Ebook PDF Version Visit to download the full and correct content document: https://ebookmass.com/product/practical-rf-system-design-wiley-ieee-ebook-pdf-versi on/

PRACTICAL RF SYSTEM DESIGN

x

5

CONTENTS

NOISE AND NONLINEARITY

123

5.1 Intermodulation of Noise / 123 5.1.1 Preview / 124 5.1.2 Flat Bandpass Noise / 125 5.1.3 Second-Order Products / 125 5.1.4 Third-Order Products / 130 5.2 Composite Distortion / 133 5.2.1 Second-Order IMs (CSO) / 134 5.2.2 Third-Order IMs (CTB) / 136 5.2.3 CSO and CTB Example / 136 5.3 Dynamic Range / 137 5.3.1 Spurious-Free Dynamic Range / 137 5.3.2 Other Range Limitations / 139 5.4 Optimizing Cascades / 139 5.4.1 Combining Parameters on One Spreadsheet / 139 5.4.2 Optimization Example / 143 5.5 Spreadsheet Enhancements / 146 5.5.1 Lookup Tables / 146 5.5.2 Using Controls / 147 5.6 Summary / 147 Endnotes / 147 6

ARCHITECTURES THAT IMPROVE LINEARITY

6.1 Parallel Combining / 149 6.1.1 90◦ Hybrid / 150 6.1.2 180◦ Hybrid / 152 6.1.3 Simple Push–Pull / 154 6.1.4 Gain / 155 6.1.5 Noise Figure / 156 6.1.6 Combiner Trees / 156 6.1.7 Cascade Analysis of a Combiner Tree / 157 6.2 Feedback / 158 6.3 Feedforward / 159 6.3.1 Intermods and Harmonics / 160 6.3.2 Bandwidth / 161 6.3.3 Noise Figure / 161 6.4 Nonideal Performance / 162 6.5 Summary / 163 Endnotes / 163

149

CONTENTS

7 FREQUENCY CONVERSION

xi

165

7.1 Basics / 165 7.1.1 The Mixer / 165 7.1.2 Conversion in Receivers / 167 7.1.3 Spurs / 168 7.1.4 Conversion in Synthesizers and Exciters / 170 7.1.5 Calculators / 170 7.1.6 Design Methods / 170 7.1.7 Example / 171 7.2 Spurious Levels / 171 7.2.1 Dependence on Signal Strength / 171 7.2.2 Estimating Levels / 173 7.2.3 Strategy for Using Levels / 175 7.3 Two-Signal IMs / 176 7.4 Power Range for Predictable Levels / 177 7.5 Spur Plot, LO Reference / 180 7.5.1 Spreadsheet Plot Description / 180 7.5.2 Example of a Band Conversion / 182 7.5.3 Other Information on the Plot / 184 7.6 Spur Plot, IF Reference / 186 7.7 Shape Factors / 196 7.7.1 Definitions / 197 7.7.2 RF Filter Requirements / 197 7.7.3 IF Filter Requirements / 200 7.8 Double Conversion / 202 7.9 Operating Regions / 203 7.9.1 Advantageous Regions / 203 7.9.2 Limitation on Downconversion, Two-by-Twos / 206 7.9.3 Higher Values of m / 209 7.10 Examples / 211 7.11 Note on Spur Plots Used in This Chapter / 216 7.12 Summary / 216 Endnotes / 217 8 CONTAMINATING SIGNALS IN SEVERE NONLINEARITIES

8.1 Decomposition / 220 8.2 Hard Limiting / 223 8.3 Soft Limiting / 223

219

xii

CONTENTS

8.4 Mixers, Through the LO Port / 225 8.4.1 AM Suppression / 225 8.4.2 FM Transfer / 226 8.4.3 Single-Sideband Transfer / 226 8.4.4 Mixing Between LO Components / 228 8.4.5 Troublesome Frequency Ranges in the LO / 228 8.4.6 Summary of Ranges / 235 8.4.7 Effect on Noise Figure / 236 8.5 Frequency Dividers / 240 8.5.1 Sideband Reduction / 240 8.5.2 Sampling / 241 8.5.3 Internal Noise / 242 8.6 Frequency Multipliers / 242 8.7 Summary / 243 Endnotes / 244 9

PHASE NOISE

9.1 Describing Phase Noise / 245 9.2 Adverse Effects of Phase Noise / 247 9.2.1 Data Errors / 247 9.2.2 Jitter / 248 9.2.3 Receiver Desensitization / 249 9.3 Sources of Phase Noise / 250 9.3.1 Oscillator Phase Noise Spectrums / 250 9.3.2 Integration Limits / 252 9.3.3 Relationship Between Oscillator Sϕ and Lϕ / 252 9.4 Processing Phase Noise in a Cascade / 252 9.4.1 Filtering by Phase-Locked Loops / 253 9.4.2 Filtering by Ordinary Filters / 254 9.4.3 Implication of Noise Figure / 255 9.4.4 Transfer from Local Oscillators / 255 9.4.5 Transfer from Data Clocks / 256 9.4.6 Integration of Phase Noise / 258 9.5 Determining the Effect on Data / 258 9.5.1 Error Probability / 258 9.5.2 Computing Phase Variance, Limits of Integration / 259 9.5.3 Effect of the Carrier-Recovery Loop on Phase Noise / 260

245

CONTENTS

xiii

9.5.4 Effect of the Loop on Additive Noise / 262 9.5.5 Contribution of Phase Noise to Data Errors / 263 9.5.6 Effects of the Low-Frequency Phase Noise / 268 9.6 Other Measures of Phase Noise / 269 9.6.1 Jitter / 269 9.6.2 Allan Variance / 271 9.7 Summary / 271 Endnote / 272 APPENDIX A

OP AMP NOISE FACTOR CALCULATIONS

273

A.1 Invariance When Input Resistor Is Redistributed / 273 A.2 Effect of Change in Source Resistances / 274 A.3 Model / 276 APPENDIX B REPRESENTATIONS OF FREQUENCY BANDS, IF NORMALIZATION

279

B.1 Passbands / 279 B.2 Acceptance Bands / 279 B.3 Filter Asymmetry / 286 APPENDIX C

CONVERSION ARITHMETIC

289

C.1 Receiver Calculator / 289 C.2 Synthesis Calculator / 291 APPENDIX E

EXAMPLE OF FREQUENCY CONVERSION

293

APPENDIX F

SOME RELEVANT FORMULAS

303

F.1 Decibels / 303 F.2 Reflection Coefficient and SWR / 304 F.3 Combining SWRs / 306 F.3.1 Summary of Results / 306 F.3.2 Development / 307 F.3.3 Maximum SWR / 308 F.3.4 Minimum SWR / 309 F.3.5 Relaxing Restrictions / 309 F.4 Impedance Transformations in Cables / 310 F.5 Smith Chart / 310

xiv

CONTENTS

APPENDIX G TYPES OF POWER GAIN

G.1 G.2 G.3 G.4 G.5

Available Gain / 313 Maximum Available Gain / 313 Transducer Gain / 314 Insertion Gain / 315 Actual Gain / 315

APPENDIX H FORMULAS RELATING TO IMs AND HARMONICS

H.1 H.2 H.3 H.4 H.5

313

317

Second Harmonics / 317 Second-Order IMs / 318 Third Harmonics / 318 Third-Order IMs / 319 Definitions of Terms / 320

APPENDIX I CHANGING THE STANDARD IMPEDANCE

321

I.1 General Case / 321 I.2 Unilateral Module / 323 APPENDIX L

POWER DELIVERED TO THE LOAD

325

APPENDIX M

MATRIX MULTIPLICATION

327

APPENDIX N NOISE FACTORS — STANDARD AND THEORETICAL

N.1 N.2 N.3 N.4 N.5 N.6

Theoretical Noise Factor / 329 Standard Noise Factor / 331 Standard Modules and Standard Noise Factor / 332 Module Noise Factor in a Standard Cascade / 333 How Can This Be? / 334 Noise Factor of an Interconnect / 334 N.6.1 Noise Factor with Mismatch / 335 N.6.2 In More Usable Terms / 336 N.6.3 Verification / 338 N.6.4 Comparison with Theoretical Value / 340 N.7 Effect of Source Impedance / 341 N.8 Ratio of Power Gains / 342 Endnote / 343

329

CONTENTS

xv

APPENDIX P

IM PRODUCTS IN MIXERS

345

APPENDIX S

COMPOSITE S PARAMETERS

349

APPENDIX T

THIRD-ORDER TERMS AT INPUT FREQUENCY

353

APPENDIX V FIGURE

SENSITIVITIES AND VARIANCE OF NOISE

APPENDIX X

CROSSOVER SPURS

359

APPENDIX Z

NONSTANDARD MODULES

363

Z.1 Z.2 Z.3 Z.4 Z.5

355

Gain of Cascade of Modules Relative to Tested Gain / 363 Finding Maximum Available Gain of a Module / 366 Interconnects / 367 Equivalent S Parameters / 367 S Parameters for Cascade of Nonstandard Modules / 368 Endnote / 369

REFERENCES Endnote / 377

371

INDEX

379

PREFACE

This book is about RF system analysis and design at the level that requires an understanding of the interaction between the modules of a system so the ultimate performance can be predicted. It describes concepts that are advanced, that is, beyond those that are more commonly taught, because these are necessary to the understanding of effects encountered in practice. It is about answering questions such as: • How will the gain of a cascade (a group of modules in series) be affected by the standing-wave ratio (SWR) specifications of its modules?

• How will noise on a local oscillator affect receiver noise figure and desensitization?

• How does the effective noise figure of a mixer depend on the filtering that precedes it?

• How can we determine the linearity of a cascade from specifications on its modules?

• How do we expect intermodulation products (IMs) to change with signal amplitude and why do they sometimes change differently?

• How can modules be combined to reduce certain intermodulation products or to turn bad impedance matches into good matches?

• How can the spurious responses in a conversion scheme be visualized and how can the magnitudes of the spurs be determined? How can this picture be used to ascertain filter requirements? xvii

xviii

PREFACE

• How does phase noise affect system performance; what are its sources and how can the effects be predicted?

I will explain methods learned over many years of RF module and system design, with emphasis on those that do not seem to be well understood. Some are available in the literature, some were published in reviewed journals, some have developed with little exposure to peer review, but all have been found to be important in some aspect of RF system engineering.

I would like to thank Eric Unruh and Bill Bearden for reviewing parts of the manuscript. I have also benefited greatly from the opportunity to work with many knowledgeable colleagues during my years at Sylvania-GTE Government Systems and at ESL-TRW in the Santa Clara (Silicon) Valley and would like to thank them, and those excellent companies for which we worked, for that opportunity. I am also grateful for the education that I received at Santa Clara and Stanford Universities, often with the help of those same companies. However, only I bear the blame for errors and imperfections in this work.

WILLIAM F. EGAN Cupertino, California February, 2003

GETTING FILES FROM THE WILEY ftp AND INTERNET SITES

To download spreadsheets that are the bases for figures in this book, use an ftp program or a Web browser.

FTP ACCESS

If you are using an ftp program, type the following at your ftp prompt: ftp://ftp.wiley.com

Some programs may provide the first “ftp” for you, in which case type ftp.wiley.com

Log in as anonymous (e.g., User ID: anonymous). Leave password blank. After you have connected to the Wiley ftp site, navigate through the directory path of: /public/sci_tech_med/rf_system

WEB ACCESS

If you are using a standard Web browser, type URL address of: xix

xx

GETTING FILES FROM THE WILEY ftp AND INTERNET SITES

ftp://ftp.wiley.com

Navigate through the directory path of: /public/sci_tech_med/rf_system

If you need further information about downloading the files, you can call Wiley’s technical support at 201-748-6753.

SYMBOLS LIST AND GLOSSARY

The following is a list of terms and symbols used throughout the book. Special meanings that have been assigned to the symbols are given, although the same symbols sometimes have other meanings, which should be apparent from the context of their usage. (For example, A and B can be used for amplitudes of sine waves, in addition to the special meanings given below.)

≡

= ∼

X|y y2 X|y1 x

∼

∼

acceptance band

contaminant passband

is identically equal to, rather than being equal only under some particular condition is defined as (superscript) indicates rms variable X with the condition y or referring to y variable X with y between yl and y2 angle or phase of x low-pass filter band-pass filter band of frequencies beyond the passband where rejection is not required; used to indicate the region between the passband and a rejection band undesired RF power band of frequencies that pass through a filter with minimal attenuation or with less than a specified attenuation xxi

xxii

SYMBOLS LIST AND GLOSSARY

rejection band

band of frequencies that are rejected or receive a specified attenuation (rejection) signal in relation to a larger signal

sideband

Generic Symbols (applied to other symbols)

* |x| x̆

complex conjugate magnitude or absolute value of x x is an equivalent noise factor or gain that can be used in standard equations to represent cascades with extreme mismatches (see Section 3.10.4)

Particular Symbols

A a |a| AM an aRT B Br Bv BW c(n, j ) cas CATV cbl CSO CTB dB DBM dBm dBc dBV dBW e F f fˆ

voltage gain in dB. Note that G can as well be used if impedances are the same or the voltage is normalized to R0 . voltage transfer ratio. voltage gain (not in dB) amplitude modulation nth-order transfer coefficient [see Eq. (4.1)] round-trip voltage transfer ratio noise bandwidth RF bandwidth video, or postdetection, bandwidth bandwidth j th binomial coefficient for (a + b)n (Abromowitz and Stegun, 1964, p. 10) subscript referring to cascade cable television subscript referring to cable composite second-order distortion (Section 5.2) composite triple-beat distortion (Section 5.2) decibels doubly balanced mixer decibels referenced to 1 mW decibels referenced to carrier decibels referenced to 1 V decibels referenced to 1 W voltage from an internal generator noise figure, F = 10 dB log10 f or fundamental (as opposed to harmonic or IM). noise factor (not in dB) or standard noise factor (measured with standard impedances) or frequency theoretical noise factor (measured with specified driving impedance) (see Sections 3.1, N.1)

SYMBOLS LIST AND GLOSSARY

FDM fc fosc fI or fIF fL or fLO FM fm fR or fRF G gk gpk H I , IF i IF IIP IM IMn in int(x) IP IPn ISFDR k kT0 L L, LO

Lϕ M m m̃ ma MAX{a, b} m×n N0 NT o

xxiii

frequency division multiplex center frequency oscillator center frequency intermediate frequency, frequency at a mixer’s output local oscillator frequency frequency modulation modulation frequency radio frequency, the frequency at a mixer’s input power gain, sometimes gain in general, in dB. power gain of module k, sometimes gain in general, not in dB. power gain preceding module k subscript referring to harmonic intermediate frequency, the result of converting RF using a local oscillator subscript indicating a signal traveling in the direction of the system input intermediate frequency, frequency at a mixer’s output input intercept point (IP referred to input levels) intermodulation product (intermod) nth-order intermod or IM for module n subscript indicating a signal entering a module (1) at the port of concern or (2) at the input port integer part of x intercept point intercept point for nth-order nonlinearity or for module n instantaneous spur-free dynamic range (see Section 5.3) Boltzmann’s constant approximately 4 × 10−21 W/Hz single-sideband relative power density local oscillator, the generally relatively high-powered, controllable, frequency in a frequency conversion or the oscillator that provides it single-sideband relative power density due to phase noise a matrix (bold format indicates a vector or matrix) modulation index (see Section 8.1) rms phase deviation in radians subscript for “maximum available” the larger of a or b m refers to the exponent of the LO voltage and n refers to the exponent of the RF voltage in the expression for a spurious product; if written, for example, 3 × 4, m is 3 and n is 4 noise power spectral density available thermal noise power spectral density at 290 K, kT0 subscript indicating a signal traveling in the direction of the system output.

xxiv

SYMBOLS LIST AND GLOSSARY

OIP out P p pavail,j PM pout,j PPSD PSD R, RF R0 RT S Ŝ Sij k SF SFDR S/N SSB SWR T T0 Tij k Tk UUT V v V v̂ ṽ vi , vin , vo , vout ± f ρ σ

output intercept point (IP referred to output levels) subscript indicating a signal exiting a module (1) at the port of concern or (2) at the output port power in dB. power (not in dB). available power at interface j (preceding module j ) phase modulation output power at interface j (preceding module j ) phase power spectral density power spectral density radio frequency, the frequency at a mixer’s input agreed-upon interface impedance, a standard impedance (e.g., 50 ); characteristic impedance of a transmission line subscript for “round trip” power spectral density or S parameter (see Section 2.2.1) sensitivity (see Section 2.5) S parameter of row i and column j in the parameter matrix for module (or element) number k shape factor, ratio of bandwidth where an attenuation is specified to passband width spur-free dynamic range (see Section 5.3.1) signal-to-noise power ratio single-sideband; refers to a single signal in relation to a larger signal standing wave ratio (see Section F.2) absolute temperature or subscript referring to conditions during test temperature of 290 K (16.85◦ C) T parameter (see Section 2.2.3) of row i and column j in the parameter matrix for module (or element) number k noise temperature of module k (see Section 3.2) unit under test a vector (bold format indicates a vector or matrix) normalized wave voltage (see Section 2.2.2) or voltage (not in dB.) voltage in dB phasor representing the wave voltage (see Section 2.2.2) phasor whose magnitude is the rms value of the voltage √ ṽ = v̂/ 2 (see Section 2.2.2) see Fig. 2.2 and Section 2.2.1 maximum ± deviation in dB of cable gain Acbl , from the mean peak frequency deviation or frequency offset from spectral center reflection coefficient (see Section F.2) standard deviation

SYMBOLS LIST AND GLOSSARY

σ2 τ ϕ(t)

xxv

variance voltage transfer ratio of a matched cable (i.e., no reflections at the ends) ωt + θ

Practical RF System Design. William F. Egan Copyright  2003 John Wiley & Sons, Inc. ISBN: 0-471-20023-9

CHAPTER 1

INTRODUCTION

This book is about systems that operate at radio frequencies (RF) (including microwaves) where high-frequency techniques, such as impedance matching, are important. It covers the interactions of the RF modules between the antenna output and the signal processors. Its goal is to provide an understanding of how their characteristics combine to determine system performance. This chapter is a general discussion of topics in the book and of the system design process.

1.1 SYSTEM DESIGN PROCESS

We do system design by conceptualizing a set of functional blocks, and their specifications, that will interact in a manner that produces the required system performance. To do this successfully, we require imagination and an understanding of the costs of achieving the various specifications. Of course, we also must understand how the characteristics of the individual blocks affect the performance of the system. This is essentially analysis, analysis at the block level. By this process, we can combine existing blocks with new blocks, using the specifications of the former and creating specifications for the latter in a manner that will achieve the system requirements.

The specifications for a block generally consist of the parameter values we would like it to have plus allowed variations, that is, tolerances. We would like the tolerances to be zero, but that is not feasible so we accept values that are compromises between costs and resulting degradations in system performance.

Not until modules have been developed and measured do we know their parameters to a high degree of accuracy (at least for one copy). At that point we might insert the module parameters into a sophisticated simulation program to compute 1

2

CHAPTER 1

INTRODUCTION

the expected cascade performance (or perhaps just hook them together to see how the cascade works). But it is important in the design process to ascertain the range of performance to be expected from the cascade, given its module specifications. We need this ability so we can write the specifications.

Spreadsheets are used extensively in this book because they can be helpful in improving our understanding, which is our main objective, while also providing tools to aid in the application of that understanding.

1.2

ORGANIZATION OF THE BOOK

It is common practice to list the modules of an RF system on a spreadsheet, along with their gains, noise figures, and intercept points, and to design into that spreadsheet the capability of computing parameters of the cascade from these module parameters. The spreadsheet then serves as a plan for the system.

The next three chapters are devoted to that process, one chapter for each of these parameter.

At first it may seem that overall gain can be easily computed from individual gains, but the usual imperfect impedance matches complicate the process. In Chapter 2, we discover how to account for these imperfections, either exactly or, in most cases, by finding the range of system gains that will result from the range of module parameters permitted by their specifications.

The method for computing system noise figure from module noise figures is well known to many RF engineers but some subtleties are not. Ideally, we use noise figure values that were obtained under the same interface conditions as seen in the system. Practically, that information is not generally available, especially at the design concept phase. In Chapter 3, we consider how to use the information that is available to determine system noise figure and what variations are to be expected. We also consider how the effective noise figures of mixers are increased by image noise. Later we will study how the local oscillator (LO) can contribute to the mixer’s noise figure.

The concept of intercept points, how to use intercept points to compute intermodulation products, and how to obtain cascade intercept points from those of the modules will be studied in Chapter 4. Anomalous intermods that do not follow the usual rules are also described.

The combined effects of noise and intermodulation products are considered in Chapter 5. One result is the concept of spur-free dynamic range. Another is the portrayal of noise distributions resulting from the intermodulation of bands of noise. The similarity between noise bands and bands of signals both aids the analysis and provides practical applications for it.

Having established the means for computing parameters for cascades of modules connected in series, in Chapter 6 we take a brief journey through various means of connecting modules or components in parallel. We discover the advantages that these various methods provide in suppressing spurious outputs and how their overall parameters are related to the parameters of the individual components.

TEST AND SIMULATION

3

Then, in Chapter 7, we consider the method for design of frequency converters that uses graphs to give an immediate picture of the spurs and their relationships to the desired signal bands, allowing us to visualize problems and solutions. We also learn how to predict spurious levels and those, along with the relationships between the spurs and the passbands, permit us to ascertain filter requirements.

The processes described in the initial chapters are linear, or almost so, except for the frequency translation inherent in frequency conversion. Some processes, however, are severely nonlinear and, while performance is typically characterized for the one signal that is supposed to be present, we need a method to determine what happens when small, contaminating, signals accompany that desired signal. This is considered in Chapter 8. The most important nonlinearity in many applications is that associated with the mixer’s LO; so we emphasize the system effects of contaminants on the LO. Lastly, in Chapter 9, we will study phase noise: where it comes from, how it passes through a system, and what are its important effects in the RF system.

1.3 APPENDIXES

Material that is not essential to the flow of the main text, but that is nevertheless important, has been organized in 17 appendixes. These are designated by letters, and an attempt has been made to choose a letter that could be associated with the content (e.g., G for gain, M for matrix) as an aid to recalling the location of the material. Some appendixes are tutorial, providing a reference for those who are unfamiliar with certain background material, or who may need their memory refreshed, without holding up other readers. Some appendixes expand upon the material in the chapters, sometimes providing more detailed explanations or backup. Others extend the material.

1.4 SPREADSHEETS 

The spreadsheets were created in Microsoft Excel and can be downloaded as Microsoft Excel 97/98 workbook files (see page xix). This makes them available for the readers’ own use and also presents an opportunity for better understanding.

One can study the equations being used and view the charts, which appear in black and white in the text, in color on the computer screen. One can also make use of Excel’s Trace Precedents feature (see, e.g., Fig. 3.5) to illustrate the composition of various equations.

1.5 TEST AND SIMULATION

Ultimately, we know how a system performs by observing it in operation. We could also observe the results of an accurate simulation, that being one that

4

CHAPTER 1

INTRODUCTION

produces the same results as the system. Under some conditions, it may be easier, quicker, or more economical to simulate a system than to build and test it. Even though the proof of the simulation model is its correspondence to the system, it can be valuable as an initial estimate of the system to be improved as test data becomes available. Once confidence is established, there may be advantages in using the model to estimate system performance under various conditions or to predict the effect of modifications. But modeling and simulating is basically the same as building and testing. They are the means by which system performance is verified. First there must be a system and, before that, a system design.

In the early stages of system design we use a general knowledge of the performance available from various system components. As the design progresses, we get more specific and begin to use the characteristics of particular realizations of the component blocks. We may initially have to estimate certain performance characteristics, possibly based on an understanding of theoretical or typical connections between certain specifications. As the design progresses we will want assurance of important parameter values, and we might ultimately test a number of components of a given type to ascertain the repeatability of characteristics.

Finally we will specify the performance required from the system’s component blocks to ensure the system meets its performance requirements.

Based on information concerning the likelihood of deviations from desired performance provided by our system design analysis, we may be led to accept a small but nonzero probability of performance outside of the desired bounds.

Once the system has been built and tested, it may be possible to use an accurate simulation to show that the results achieved, even with expected component variations, are better than the worst case implied by the combination of the individual block specifications. To base expected performance on simulated or measured results, rather than on functional block specifications, however, requires that we have continuing control over the construction details of the components of various copies of the system, rather than merely ensuring that the blocks meet their specifications. For example, a particular amplifier design may produce a stable phase shift that has a fortuitous effect on system performance, but we would have to control changes in its design and in that of interacting components.

Another important aspect of test is general experimentation, not confined to a particular design, for the purpose of verifying the degree of applicability of theory to various practical components. Examples of reports giving such supporting experimental data can be seen in Egan (2000), relative to the theory in Chapter 8, and in Henderson (1993a), relative to Chapter 7. We can hope that these, and the other, chapters will suggest opportunities for additional worthwhile papers.

1.6

PRACTICAL SKEPTICISM

There is a tendency for engineering students to assume that anything written in a book is accurate. This comes naturally from our struggle just to approach the knowledge of the authors whose books we study (and to be able to show this on

REFERENCES

5

exams). With enough experience in using published information, however, we are likely to develop some skepticism, especially if we should spend many hours pursuing a development based on an erroneous parameter value or perhaps on a concept that applies almost universally — but not in our case. Even reviewed journals, which we might expect to be most nearly free of errors, and classic works contain sources of such problems. But the technical literature also contains valuable, even essential, information; so a healthy skepticism is one that leads us to consult it freely and extensively but to continually check what we learn.

We check for accuracy in our reference sources, for accuracy in our use of the information, and to ensure that it truly applies to our development. We check by considering how concepts correlate with each other (e.g., does this make sense in terms of what I already know), by verifying agreement between answers obtained by different methods, and by testing as we proceed in our developments. The greater the cost of failure, the more important is verification. Unexpected results can be opportunities to increase our knowledge, but we do not want to pay too high a price for the educational experience.

1.7 REFERENCES

References are included for several reasons: to recognize the sources, to offer alternate presentations of the material, or to provide sources for associated topics that are beyond the scope of this work. The author–date style of referencing is used throughout the book. From these, one can easily find the complete reference descriptions in the References at the end of the text. Some notes are placed at the end of the chapter in which they are referenced.

Practical RF System Design. William F. Egan Copyright  2003 John Wiley & Sons, Inc. ISBN: 0-471-20023-9

CHAPTER 2

GAIN

In this chapter, we determine the effect of impedance mismatches (reflections) on system gain. For a simple cascade of linear modules (Fig. 2.1), we could write the overall transfer function or ratio as g = g1 g2 · · · gN , where

gj =

uj +1 uj

(2.1)

(2.2)

and u is voltage or current or power. The gain is |g|, which is the same as g if u is power. This would require that we measure the values of u in the cascade.

If we measure them in some other environment, we could get different gains because of differing impedances at the interfaces. However, it may be difficult to measure u in the cascade, and a gain that must be measured in the final cascade has limited value in predicting or specifying performance. For example, a variation of about ±1 dB in overall gain can occur for each interface where the standing-wave ratios (SWRs) are 2 and a change as high as 2.5 dB can occur when they are 3. (See Appendix F.1 for a discussion of decibels (dB).)

Here we consider how the expected gain of a cascade of linear modules can be determined, as well as variations in its gain, based on measured or specified parameters of the individual modules. Throughout this book, gains and other parameters are so generally functions of frequency that the functionality is not shown explicitly. Equations whose frequency dependence is not indicated will apply at any given frequency.

We begin with a description, for modules and their cascades, that applies without limitations but which requires detailed knowledge of impedances and 7

8

CHAPTER 2

GAIN

Cascade u2 u1

g1

u3 g2

gn

un + 1

Modules

Fig. 2.1 Transfer functions in a simple cascade.

which can be complicated to use. Then we will discover a way to simplify the description of the overall cascade by taking into account special characteristics of some of its parts. This will lead us to a standard cascade, composed of unilateral modules separated by interconnects (e.g., cables) that have well-controlled impedances. The unilateral modules, usually active, have negligible reverse transmission. The passive cables are well matched at the standard impedance (e.g., 50 ) of the cascade interfaces; these are the impedances used in characterizing the modules.

It is common to specify the desired performance of each module plus allowed variations from that ideal. The desired performance includes a gain and standard interface impedances. The allowed variations are given by a gain tolerance and the required degree of input and output impedance matches, expressed as maximum SWRs or, equivalently, return losses or reflection coefficient magnitudes (see Appendix F.2). These are the parameters required for determination of the performance of the standard cascade. We will also find ways to fit bilateral modules into this scheme.

We will also consider the case where the modules are specified in terms of their performance with various nonstandard interface impedances (e.g., 2000 –j 500 ), and we will discover how to characterize cascades of these modules. For cases where it may be desirable to include these nonstandard cascades as parts of a standard cascade, we will determine how to describe them in those terms.

Finally, we will study the use of sensitivities in analyzing cascade performance.

Many varieties of power gains are described in Appendix G. If all interfaces were at standard impedance levels (e.g., 50 everywhere), these gains would all be the same, but the usually unintended mismatches lead to differing values for gain, depending on the definitions employed.

2.1

SIMPLE CASES

In some cases these complexities are unimportant. For example, where operational amplifiers (op amps) are used at lower frequencies, measurements of voltages at interfaces can be practical and their low output impedances and high input impedances allow performance in the voltage-amplifier cascade to duplicate what was measured during test. However, this luxury is rare at radio frequencies.

GENERAL CASE

9

In other cases, complexities may be ignored in an effort to get an answer with minimum effort or with the available information. That answer may be adequate for the task at hand; at least it is better than no estimate. Commonly, we simply assume that gains will be the same as when a module or interconnect was tested in a standard-impedance environment. We try to make this so by keeping input and output impedances close to that standard impedance when designing or selecting modules.

While this simplified approach can be useful, we will consider here how to make use of additional information about modules to get a better estimate of cascade performance, one that includes the range of gain values to be expected.

2.2 GENERAL CASE

To characterize the modules so their performance in the system can be predicted, we need more parameters, a set of four (generally called two-port parameters; we are characterizing our modules as having two ports, an input port and an output port) for each module (Gonzalez, 1984, pp. 1–31; Pozar, 2001, pp. 47–55). We begin by considering the parameters that we can use to describe the modules.

2.2.1

S Parameters

Individual RF modules are usually defined by their S (scattering) parameters (Pozar, 2001, pp. 50–53; Gonzalez, 1984, pp. 9–10). This can be done with the help of the matrix (see Appendix M for help in using matrices),

vout,1 vout,2

S11 = S21

S12 S22

1

vin,1 . vin,2

(2.3)

The subscripts in and out refer to waves propagating1 into and out of the module at either port (1 or 2). The other subscripts on the vector components indicate the input port 1 or output port 2, whereas the subscript on each matrix element is its row and column, respectively. Subscript 1 on the matrix indicates module 1. We use the same index for the module and for its input port (port 1 here).

We can also write the subscripts in terms of the system with i or o, referring to waves traveling toward the input or toward the output of the system, respectively.

Refer to Fig. 2.2. With this notation, Eq. (2.3) becomes

vi1 vo2

S11 = S21

S12 S22

1

vo1 . vi2

(2.4)

vo,j . vi,j +1

(2.5)

More generally, for the j th module,

vi,j vo,j +1

S11 = S21

S12 S22

j

10

CHAPTER 2

GAIN

vin,3 = vo,3

vout,4 = vo,4 Module 3

Cable 2 vout,3 = vi,3

System input

vin, j = vo, j

Cable 4 vin,4 = vi,4

vout, j +1 = vo, j +1 Module j

Cable j − 1 vout, j = vi, j

vin, j +1 = vi, j +1

Module

Module

THESE ARE in

THESE ARE out

Module

Module

THESE ARE i

THESE ARE o

Fig. 2.2

Cable j + 1

System output

Definitions of wave subscripts.

By normal matrix multiplication then, vi,j = S11j vo,j + S12j vi,j +1

(2.6)

vo,j +1 = S21j vo,j + S22j vi,j +1 .

(2.7)

and

This is a convenient form for measurements. It relates signals coming “out” of the module, at either port, to those going “in” at either port. We can control the inputs, ensuring that there is only one by terminating the port to which we do not apply a signal, and measuring the two resulting outputs, one at each port (Fig. 2.3). These give us two of the four parameters and a second measurement, with input to the other port, gives the other two.

R0 Calibrated coupler Calibrated generator

R0 vin,1

vout,1 Sample

Module under test

vout,2 vin,2 = 0

Measure reflection

Fig. 2.3 Measurement setup.

Measure output

GENERAL CASE

11

Thus, for module 1, with port 2 terminated (vin,2 ≡ vi2 = 0), we measure the reflected signal at port 1 to give the reflection coefficient for that port, S11 =

vout,1 vi1 ≡ vin,1 vo1

(2.8)

and the transmission coefficient from port 1 to port 2, S21 =

vout,2 vo2 ≡ . vin,1 vo1

(2.9)

Then we turn the module around and input to port 2 while terminating port 1, giving the reverse transmission coefficient and port 2 reflection coefficient, respectively: vout,1 vi1 ≡ , vin,2 vi2 vout,2 vo2 ≡ .

S22 = vin,2 vi2

S12 =

(2.10) (2.11)

(We are using both subscript forms here as an aid in understanding their equivalency.) In each case the S parameter subscripts represent the ports of effect and cause, respectively, Seffect cause , where “effect” is the port where “out” occurs and “cause” is the port where “in” occurs.

2.2.2

Normalized Waves

We have called vx (i.e., vo , vi , vout , or vin ) a “wave,” but the symbol implies a voltage. It is customary to use normalized voltages with S parameters, and the usual way √ to normalize them is by division of the root-mean-square (rms) voltage by R0 , where R0 is the real part of the characteristic impedance Z0 of the transmission line in which the waves reside. We will assume that Z0 is real.2 An RF voltage corresponding to vx can be represented by Vmx cos(ωt + θ ) = Re Vmx ej (ωt+θ ) .

(2.12)

This can be abbreviated v̂x (t) = v̂x ej ωt ,

(2.13)

v̂x = Vmx ej θ .

(2.14)

where

Sometimes a phasor is employed whose magnitude is the effective (rms) value (Hewlett-Packard, 1996; Yola, 1961; Kurokawa, 1965): √ ṽx = (Vmx / 2)ej θ .

(2.15)

12

CHAPTER 2

GAIN

Our normalized voltage,

vx = ṽx / R0 ,

(2.16)

uses this form, which has the advantage that the available power in the traveling wave can be expressed simply as px = |vx |2 .

(2.17)

Traditionally, the symbol an is used for vin,n and bn is used for vout,n .

If, on the other hand, the phasor employed in Eq. (2.16) is v̂x rather than ṽx (Pozar, 1990, p. 229, 1998, p. 204), the power will be |vx |2 /2. In most cases the module parameters are ratios of two waves at the same impedance; so it makes no difference whether they are ratios of vx or of v̂x or of ṽx .

2.2.3

T Parameters

Unfortunately, we cannot use S matrices conveniently for determining overall response because we cannot multiply them together to produce anything useful.

We require a matrix equation for overall transfer function of the form V1 = MVn+1 = M1 M2 M3 · · · Mn Vn+1 .

(2.18)

Here the vector Vj , representing a module input, has the same identifying number (subscript) as the matrix Mj , representing the module. Note that we are operating on outputs to give inputs. This is nice in that the matrices are then written in the same order in which the modules are traditionally arrayed in a drawing (left to right from input to output, as in Fig. 2.1). There is also an even better reason.

The vector on which the matrix operates (multiplies) must contain the information needed to produce the resulting product. Unilateral modules that have little or no reverse transmission do not provide significant information about the output to the input; thus a mathematical representation in which the matrix operated on that input would not work well. On the other hand, all modules of interest produce outputs that are functions of their inputs; so there is sufficient information in the vector representing the output to form the input.3 Equation (2.18) implies (2.19) V1 = M 1 V2 and in order that

V2 = M 2 V3

(2.20)

V1 = M1 (M2 V3 ) = M1 M2 V3

(2.21)

and so on. All this implies that V2 represents the state between modules 1 and 2 so we define the vector

vo voj = , (2.22) Vj = vi j vij

GENERAL CASE

13

where j represents the port and o and i indicate the voltage wave moving right toward the system output or left toward its input, respectively. Thus the matrix connecting such vectors has the form (Dechamps and Dyson, 1986; Gonzalez, 1984, pp. 11–12)

vo T11 T12 vo = .

(2.23) vi 1 T21 T22 1 vi 2 As before, the module and its input have the same subscript. In many cases it will be more convenient to move the subscript from the vector or matrix to its individual elements, adding the port number as the last subscript:

vo1 vi1

T111 = T211

T121 T221

vo2 . vi2

(2.24)

Each vector, in this representation, describes two waves that occur at a single point in the system whereas, for the S parameters, the vector elements represented waves from different ports.4 However, S-parameter measurements are simpler than T -parameter measurements. Consider that T121 is the ratio between a wave entering the module at port 1, vo1 , and one entering it at port 2, vi2 , while the wave leaving it at port 2, vo2 , is set to zero. To measure this directly, we would require two phase-coherent generators, one driving each port, that would be adjusted so the outputs due to each at port 2 would cancel.

2.2.4

Relationships Between S and T Parameters

It is simpler to measure the S parameters and obtain the T parameters from them.

For example, T22 for module 1 is

vi1 .

T22 = vi2 vo2 =0

(2.25)

Equation (2.7) indicates that the condition vo2 = 0 requires S21 vo1 = −S22 vi2 .

(2.26)

Combining this with Eq. (2.6) we obtain vi1 = −

S11 S22 S11 S22 vi2 + S12 vi2 = S12 − vi2 S21 S21

(2.27)

from which we obtain the T parameter in terms of S parameters, T22 = S12 −

S11 S22 .

S21

(2.28)

14

CHAPTER 2

GAIN

By a similar process we can obtain the other values of Tij in terms of the Sij :

T11 T21

T12 T22



 S22 1 −  S21  S21  =  S11 S11 S22  S12 − S21 S21

1 1 −S22 = , S21 S11 S12 S21 − S11 S22

(2.29)

(2.30)

and of Sij in terms of Tij ,

2.2.5

S11 S21

S12 S22



 T12 T21 T21  T11 T22 − T11   =  1  T12 − T11 T11

1 T21 T11 T22 − T12 T21 = . −T12 T11 1

(2.31)

(2.32)

Restrictions on T Parameters

We can now show more specifically why the T matrix was designed to give input as a function of output, rather than the converse. For unilateral gain in the forward direction, S12 = 0. This simplifies T22 in Eq. (2.30). On the other hand, unilateral gain in the reverse direction, S21 = 0, causes the elements in Eq. (2.30) to become infinite. As S21 approaches 0, V2 becomes a weak function of V1 , so a large number is required to give V1 in terms of V2 . Moreover, if forward transmission is small, vo2 may become a stronger function of vi2 than of vo1 , in which case V1 becomes dependent on the difference between the two components of V2 and subject to error due to small inaccuracies in M. As a result, M should not represent a process where transmission from V1 to V2 , as defined

by Eq. (2.9), is small or zero. For this reason, Eq. (2.19) is written as it is, since transmission toward the system output S21 is a purpose of a system, and thus is expected to be appreciable, whereas reverse transmission S12 is often minimized.

2.2.6

Cascade Response

Now we can obtain the overall response of a series of modules (a cascade) by multiplying their individual T matrices. The sequence in which the matrices are arrayed must be the same as the sequence, from input to output, of the elements in the cascade and the interface (standard) impedances must be those in which the S or T parameters were measured. If the parameters of adjacent modules are defined for different standard impedances at the same interface, one of them must be recharacterized. This can be done by inserting a T matrix representing the impedance transition, as described in Appendix I.

Another random document with no related content on Scribd:

DANCE ON STILTS AT THE GIRLS’ UNYAGO, NIUCHI

Newala, too, suffers from the distance of its water-supply—at least the Newala of to-day does; there was once another Newala in a lovely valley at the foot of the plateau. I visited it and found scarcely a trace of houses, only a Christian cemetery, with the graves of several missionaries and their converts, remaining as a monument of its former glories. But the surroundings are wonderfully beautiful. A thick grove of splendid mango-trees closes in the weather-worn crosses and headstones; behind them, combining the useful and the agreeable, is a whole plantation of lemon-trees covered with ripe fruit; not the small African kind, but a much larger and also juicier imported variety, which drops into the hands of the passing traveller, without calling for any exertion on his part. Old Newala is now under the jurisdiction of the native pastor, Daudi, at Chingulungulu, who, as I am on very friendly terms with him, allows me, as a matter of course, the use of this lemon-grove during my stay at Newala.

FEET MUTILATED BY THE RAVAGES OF THE “JIGGER” (Sarcopsylla penetrans)

The water-supply of New Newala is in the bottom of the valley, some 1,600 feet lower down. The way is not only long and fatiguing, but the water, when we get it, is thoroughly bad. We are suffering not only from this, but from the fact that the arrangements at Newala are nothing short of luxurious. We have a separate kitchen—a hut built against the boma palisade on the right of the baraza, the interior of which is not visible from our usual position. Our two cooks were not long in finding this out, and they consequently do—or rather neglect to do—what they please. In any case they do not seem to be very particular about the boiling of our drinking-water—at least I can attribute to no other cause certain attacks of a dysenteric nature, from which both Knudsen and I have suffered for some time. If a man like Omari has to be left unwatched for a moment, he is capable of anything. Besides this complaint, we are inconvenienced by the state of our nails, which have become as hard as glass, and crack on the slightest provocation, and I have the additional infliction of pimples all over me. As if all this were not enough, we have also, for the last week been waging war against the jigger, who has found his Eldorado in the hot sand of the Makonde plateau. Our men are seen all day long—whenever their chronic colds and the dysentery likewise raging among them permit—occupied in removing this scourge of Africa from their feet and trying to prevent the disastrous consequences of its presence. It is quite common to see natives of this place with one or two toes missing; many have lost all their toes,

or even the whole front part of the foot, so that a well-formed leg ends in a shapeless stump. These ravages are caused by the female of Sarcopsylla penetrans, which bores its way under the skin and there develops an egg-sac the size of a pea. In all books on the subject, it is stated that one’s attention is called to the presence of this parasite by an intolerable itching. This agrees very well with my experience, so far as the softer parts of the sole, the spaces between and under the toes, and the side of the foot are concerned, but if the creature penetrates through the harder parts of the heel or ball of the foot, it may escape even the most careful search till it has reached maturity.

Then there is no time to be lost, if the horrible ulceration, of which we see cases by the dozen every day, is to be prevented. It is much easier, by the way, to discover the insect on the white skin of a European than on that of a native, on which the dark speck scarcely shows. The four or five jiggers which, in spite of the fact that I constantly wore high laced boots, chose my feet to settle in, were taken out for me by the all-accomplished Knudsen, after which I thought it advisable to wash out the cavities with corrosive sublimate. The natives have a different sort of disinfectant—they fill the hole with scraped roots. In a tiny Makua village on the slope of the plateau south of Newala, we saw an old woman who had filled all the spaces under her toe-nails with powdered roots by way of prophylactic treatment. What will be the result, if any, who can say?

The rest of the many trifling ills which trouble our existence are really more comic than serious. In the absence of anything else to smoke, Knudsen and I at last opened a box of cigars procured from the Indian store-keeper at Lindi, and tried them, with the most distressing results. Whether they contain opium or some other narcotic, neither of us can say, but after the tenth puff we were both “off,” three-quarters stupefied and unspeakably wretched. Slowly we recovered—and what happened next? Half-an-hour later we were once more smoking these poisonous concoctions—so insatiable is the craving for tobacco in the tropics.

Even my present attacks of fever scarcely deserve to be taken seriously. I have had no less than three here at Newala, all of which have run their course in an incredibly short time. In the early afternoon, I am busy with my old natives, asking questions and making notes. The strong midday coffee has stimulated my spirits to

an extraordinary degree, the brain is active and vigorous, and work progresses rapidly, while a pleasant warmth pervades the whole body. Suddenly this gives place to a violent chill, forcing me to put on my overcoat, though it is only half-past three and the afternoon sun is at its hottest. Now the brain no longer works with such acuteness and logical precision; more especially does it fail me in trying to establish the syntax of the difficult Makua language on which I have ventured, as if I had not enough to do without it. Under the circumstances it seems advisable to take my temperature, and I do so, to save trouble, without leaving my seat, and while going on with my work. On examination, I find it to be 101·48°. My tutors are abruptly dismissed and my bed set up in the baraza; a few minutes later I am in it and treating myself internally with hot water and lemon-juice.

Three hours later, the thermometer marks nearly 104°, and I make them carry me back into the tent, bed and all, as I am now perspiring heavily, and exposure to the cold wind just beginning to blow might mean a fatal chill. I lie still for a little while, and then find, to my great relief, that the temperature is not rising, but rather falling. This is about 7.30 p.m. At 8 p.m. I find, to my unbounded astonishment, that it has fallen below 98·6°, and I feel perfectly well. I read for an hour or two, and could very well enjoy a smoke, if I had the wherewithal—Indian cigars being out of the question.

Having no medical training, I am at a loss to account for this state of things. It is impossible that these transitory attacks of high fever should be malarial; it seems more probable that they are due to a kind of sunstroke. On consulting my note-book, I become more and more inclined to think this is the case, for these attacks regularly follow extreme fatigue and long exposure to strong sunshine. They at least have the advantage of being only short interruptions to my work, as on the following morning I am always quite fresh and fit.

My treasure of a cook is suffering from an enormous hydrocele which makes it difficult for him to get up, and Moritz is obliged to keep in the dark on account of his inflamed eyes. Knudsen’s cook, a raw boy from somewhere in the bush, knows still less of cooking than Omari; consequently Nils Knudsen himself has been promoted to the vacant post. Finding that we had come to the end of our supplies, he began by sending to Chingulungulu for the four sucking-pigs which we had

bought from Matola and temporarily left in his charge; and when they came up, neatly packed in a large crate, he callously slaughtered the biggest of them. The first joint we were thoughtless enough to entrust for roasting to Knudsen’s mshenzi cook, and it was consequently uneatable; but we made the rest of the animal into a jelly which we ate with great relish after weeks of underfeeding, consuming incredible helpings of it at both midday and evening meals. The only drawback is a certain want of variety in the tinned vegetables. Dr. Jäger, to whom the Geographical Commission entrusted the provisioning of the expeditions—mine as well as his own—because he had more time on his hands than the rest of us, seems to have laid in a huge stock of Teltow turnips,[46] an article of food which is all very well for occasional use, but which quickly palls when set before one every day; and we seem to have no other tins left. There is no help for it—we must put up with the turnips; but I am certain that, once I am home again, I shall not touch them for ten years to come.

Amid all these minor evils, which, after all, go to make up the genuine flavour of Africa, there is at least one cheering touch: Knudsen has, with the dexterity of a skilled mechanic, repaired my 9 × 12 cm. camera, at least so far that I can use it with a little care.

How, in the absence of finger-nails, he was able to accomplish such a ticklish piece of work, having no tool but a clumsy screw-driver for taking to pieces and putting together again the complicated mechanism of the instantaneous shutter, is still a mystery to me; but he did it successfully. The loss of his finger-nails shows him in a light contrasting curiously enough with the intelligence evinced by the above operation; though, after all, it is scarcely surprising after his ten years’ residence in the bush. One day, at Lindi, he had occasion to wash a dog, which must have been in need of very thorough cleansing, for the bottle handed to our friend for the purpose had an extremely strong smell. Having performed his task in the most conscientious manner, he perceived with some surprise that the dog did not appear much the better for it, and was further surprised by finding his own nails ulcerating away in the course of the next few days. “How was I to know that carbolic acid has to be diluted?” he mutters indignantly, from time to time, with a troubled gaze at his mutilated finger-tips.

Since we came to Newala we have been making excursions in all directions through the surrounding country, in accordance with old habit, and also because the akida Sefu did not get together the tribal elders from whom I wanted information so speedily as he had promised. There is, however, no harm done, as, even if seen only from the outside, the country and people are interesting enough.

The Makonde plateau is like a large rectangular table rounded off at the corners. Measured from the Indian Ocean to Newala, it is about seventy-five miles long, and between the Rovuma and the Lukuledi it averages fifty miles in breadth, so that its superficial area is about two-thirds of that of the kingdom of Saxony. The surface, however, is not level, but uniformly inclined from its south-western edge to the ocean. From the upper edge, on which Newala lies, the eye ranges for many miles east and north-east, without encountering any obstacle, over the Makonde bush. It is a green sea, from which here and there thick clouds of smoke rise, to show that it, too, is inhabited by men who carry on their tillage like so many other primitive peoples, by cutting down and burning the bush, and manuring with the ashes. Even in the radiant light of a tropical day such a fire is a grand sight.

Much less effective is the impression produced just now by the great western plain as seen from the edge of the plateau. As often as time permits, I stroll along this edge, sometimes in one direction, sometimes in another, in the hope of finding the air clear enough to let me enjoy the view; but I have always been disappointed.

Wherever one looks, clouds of smoke rise from the burning bush, and the air is full of smoke and vapour. It is a pity, for under more favourable circumstances the panorama of the whole country up to the distant Majeje hills must be truly magnificent. It is of little use taking photographs now, and an outline sketch gives a very poor idea of the scenery. In one of these excursions I went out of my way to make a personal attempt on the Makonde bush. The present edge of the plateau is the result of a far-reaching process of destruction through erosion and denudation. The Makonde strata are everywhere cut into by ravines, which, though short, are hundreds of yards in depth. In consequence of the loose stratification of these beds, not only are the walls of these ravines nearly vertical, but their upper end is closed by an equally steep escarpment, so that the

western edge of the Makonde plateau is hemmed in by a series of deep, basin-like valleys. In order to get from one side of such a ravine to the other, I cut my way through the bush with a dozen of my men.

It was a very open part, with more grass than scrub, but even so the short stretch of less than two hundred yards was very hard work; at the end of it the men’s calicoes were in rags and they themselves bleeding from hundreds of scratches, while even our strong khaki suits had not escaped scatheless.

NATIVE PATH THROUGH THE MAKONDE BUSH, NEAR MAHUTA

I see increasing reason to believe that the view formed some time back as to the origin of the Makonde bush is the correct one. I have no doubt that it is not a natural product, but the result of human occupation. Those parts of the high country where man—as a very slight amount of practice enables the eye to perceive at once—has not yet penetrated with axe and hoe, are still occupied by a splendid timber forest quite able to sustain a comparison with our mixed forests in Germany. But wherever man has once built his hut or tilled his field, this horrible bush springs up. Every phase of this process

may be seen in the course of a couple of hours’ walk along the main road. From the bush to right or left, one hears the sound of the axe— not from one spot only, but from several directions at once. A few steps further on, we can see what is taking place. The brush has been cut down and piled up in heaps to the height of a yard or more, between which the trunks of the large trees stand up like the last pillars of a magnificent ruined building. These, too, present a melancholy spectacle: the destructive Makonde have ringed them— cut a broad strip of bark all round to ensure their dying off—and also piled up pyramids of brush round them. Father and son, mother and son-in-law, are chopping away perseveringly in the background—too busy, almost, to look round at the white stranger, who usually excites so much interest. If you pass by the same place a week later, the piles of brushwood have disappeared and a thick layer of ashes has taken the place of the green forest. The large trees stretch their smouldering trunks and branches in dumb accusation to heaven—if they have not already fallen and been more or less reduced to ashes, perhaps only showing as a white stripe on the dark ground.

This work of destruction is carried out by the Makonde alike on the virgin forest and on the bush which has sprung up on sites already cultivated and deserted. In the second case they are saved the trouble of burning the large trees, these being entirely absent in the secondary bush.

After burning this piece of forest ground and loosening it with the hoe, the native sows his corn and plants his vegetables. All over the country, he goes in for bed-culture, which requires, and, in fact, receives, the most careful attention. Weeds are nowhere tolerated in the south of German East Africa. The crops may fail on the plains, where droughts are frequent, but never on the plateau with its abundant rains and heavy dews. Its fortunate inhabitants even have the satisfaction of seeing the proud Wayao and Wamakua working for them as labourers, driven by hunger to serve where they were accustomed to rule.

But the light, sandy soil is soon exhausted, and would yield no harvest the second year if cultivated twice running. This fact has been familiar to the native for ages; consequently he provides in time, and, while his crop is growing, prepares the next plot with axe and firebrand. Next year he plants this with his various crops and

lets the first piece lie fallow. For a short time it remains waste and desolate; then nature steps in to repair the destruction wrought by man; a thousand new growths spring out of the exhausted soil, and even the old stumps put forth fresh shoots. Next year the new growth is up to one’s knees, and in a few years more it is that terrible, impenetrable bush, which maintains its position till the black occupier of the land has made the round of all the available sites and come back to his starting point.

The Makonde are, body and soul, so to speak, one with this bush.

According to my Yao informants, indeed, their name means nothing else but “bush people.” Their own tradition says that they have been settled up here for a very long time, but to my surprise they laid great stress on an original immigration. Their old homes were in the south-east, near Mikindani and the mouth of the Rovuma, whence their peaceful forefathers were driven by the continual raids of the Sakalavas from Madagascar and the warlike Shirazis[47] of the coast, to take refuge on the almost inaccessible plateau. I have studied African ethnology for twenty years, but the fact that changes of population in this apparently quiet and peaceable corner of the earth could have been occasioned by outside enterprises taking place on the high seas, was completely new to me. It is, no doubt, however, correct.

The charming tribal legend of the Makonde—besides informing us of other interesting matters—explains why they have to live in the thickest of the bush and a long way from the edge of the plateau, instead of making their permanent homes beside the purling brooks and springs of the low country.

“The place where the tribe originated is Mahuta, on the southern side of the plateau towards the Rovuma, where of old time there was nothing but thick bush. Out of this bush came a man who never washed himself or shaved his head, and who ate and drank but little.

He went out and made a human figure from the wood of a tree growing in the open country, which he took home to his abode in the bush and there set it upright. In the night this image came to life and was a woman. The man and woman went down together to the Rovuma to wash themselves. Here the woman gave birth to a stillborn child. They left that place and passed over the high land into the valley of the Mbemkuru, where the woman had another child, which

was also born dead. Then they returned to the high bush country of Mahuta, where the third child was born, which lived and grew up. In course of time, the couple had many more children, and called themselves Wamatanda. These were the ancestral stock of the Makonde, also called Wamakonde,[48] i.e., aborigines. Their forefather, the man from the bush, gave his children the command to bury their dead upright, in memory of the mother of their race who was cut out of wood and awoke to life when standing upright. He also warned them against settling in the valleys and near large streams, for sickness and death dwelt there. They were to make it a rule to have their huts at least an hour’s walk from the nearest wateringplace; then their children would thrive and escape illness.”

The explanation of the name Makonde given by my informants is somewhat different from that contained in the above legend, which I extract from a little book (small, but packed with information), by Pater Adams, entitled Lindi und sein Hinterland. Otherwise, my results agree exactly with the statements of the legend. Washing?

Hapana—there is no such thing. Why should they do so? As it is, the supply of water scarcely suffices for cooking and drinking; other people do not wash, so why should the Makonde distinguish himself by such needless eccentricity? As for shaving the head, the short, woolly crop scarcely needs it,[49] so the second ancestral precept is likewise easy enough to follow. Beyond this, however, there is nothing ridiculous in the ancestor’s advice. I have obtained from various local artists a fairly large number of figures carved in wood, ranging from fifteen to twenty-three inches in height, and representing women belonging to the great group of the Mavia, Makonde, and Matambwe tribes. The carving is remarkably well done and renders the female type with great accuracy, especially the keloid ornamentation, to be described later on. As to the object and meaning of their works the sculptors either could or (more probably) would tell me nothing, and I was forced to content myself with the scanty information vouchsafed by one man, who said that the figures were merely intended to represent the nembo—the artificial deformations of pelele, ear-discs, and keloids. The legend recorded

by Pater Adams places these figures in a new light. They must surely be more than mere dolls; and we may even venture to assume that they are—though the majority of present-day Makonde are probably unaware of the fact—representations of the tribal ancestress.

The references in the legend to the descent from Mahuta to the Rovuma, and to a journey across the highlands into the Mbekuru valley, undoubtedly indicate the previous history of the tribe, the travels of the ancestral pair typifying the migrations of their descendants. The descent to the neighbouring Rovuma valley, with its extraordinary fertility and great abundance of game, is intelligible at a glance—but the crossing of the Lukuledi depression, the ascent to the Rondo Plateau and the descent to the Mbemkuru, also lie within the bounds of probability, for all these districts have exactly the same character as the extreme south. Now, however, comes a point of especial interest for our bacteriological age. The primitive Makonde did not enjoy their lives in the marshy river-valleys.

Disease raged among them, and many died. It was only after they had returned to their original home near Mahuta, that the health conditions of these people improved. We are very apt to think of the African as a stupid person whose ignorance of nature is only equalled by his fear of it, and who looks on all mishaps as caused by evil spirits and malignant natural powers. It is much more correct to assume in this case that the people very early learnt to distinguish districts infested with malaria from those where it is absent.

This knowledge is crystallized in the ancestral warning against settling in the valleys and near the great waters, the dwelling-places of disease and death. At the same time, for security against the hostile Mavia south of the Rovuma, it was enacted that every settlement must be not less than a certain distance from the southern edge of the plateau. Such in fact is their mode of life at the present day. It is not such a bad one, and certainly they are both safer and more comfortable than the Makua, the recent intruders from the south, who have made USUAL METHOD OF good their footing on the western edge of the CLOSING HUT-DOOR plateau, extending over a fairly wide belt of country. Neither Makua nor Makonde show in their dwellings anything of the size and comeliness of the Yao houses in the plain, especially at Masasi, Chingulungulu and Zuza’s. Jumbe Chauro, a Makonde hamlet not far from Newala, on the road to Mahuta, is the

most important settlement of the tribe I have yet seen, and has fairly spacious huts. But how slovenly is their construction compared with the palatial residences of the elephant-hunters living in the plain.

The roofs are still more untidy than in the general run of huts during the dry season, the walls show here and there the scanty beginnings or the lamentable remains of the mud plastering, and the interior is a veritable dog-kennel; dirt, dust and disorder everywhere. A few huts only show any attempt at division into rooms, and this consists merely of very roughly-made bamboo partitions. In one point alone have I noticed any indication of progress—in the method of fastening the door. Houses all over the south are secured in a simple but ingenious manner. The door consists of a set of stout pieces of wood or bamboo, tied with bark-string to two cross-pieces, and moving in two grooves round one of the door-posts, so as to open inwards. If the owner wishes to leave home, he takes two logs as thick as a man’s upper arm and about a yard long. One of these is placed obliquely against the middle of the door from the inside, so as to form an angle of from 60° to 75° with the ground. He then places the second piece horizontally across the first, pressing it downward with all his might.

It is kept in place by two strong posts planted in the ground a few inches inside the door. This fastening is absolutely safe, but of course cannot be applied to both doors at once, otherwise how could the owner leave or enter his house? I have not yet succeeded in finding out how the back door is fastened.

MAKONDE LOCK AND KEY AT JUMBE CHAURO

This is the general way of closing a house. The Makonde at Jumbe Chauro, however, have a much more complicated, solid and original one. Here, too, the door is as already described, except that there is only one post on the inside, standing by itself about six inches from one side of the doorway. Opposite this post is a hole in the wall just large enough to admit a man’s arm. The door is closed inside by a large wooden bolt passing through a hole in this post and pressing with its free end against the door. The other end has three holes into which fit three pegs running in vertical grooves inside the post. The door is opened with a wooden key about a foot long, somewhat curved and sloped off at the butt; the other end has three pegs corresponding to the holes, in the bolt, so that, when it is thrust through the hole in the wall and inserted into the rectangular opening in the post, the pegs can be lifted and the bolt drawn out.[50]

MODE OF INSERTING THE KEY

With no small pride first one householder and then a second showed me on the spot the action of this greatest invention of the Makonde Highlands. To both with an admiring exclamation of “Vizuri sana!” (“Very fine!”). I expressed the wish to take back these marvels with me to Ulaya, to show the Wazungu what clever fellows the Makonde are. Scarcely five minutes after my return to camp at Newala, the two men came up sweating under the weight of two heavy logs which they laid down at my feet, handing over at the same time the keys of the fallen fortress. Arguing, logically enough, that if the key was wanted, the lock would be wanted with it, they had taken their axes and chopped down the posts—as it never occurred to them to dig them out of the ground and so bring them intact. Thus I have

two badly damaged specimens, and the owners, instead of praise, come in for a blowing-up.

The Makua huts in the environs of Newala are especially miserable; their more than slovenly construction reminds one of the temporary erections of the Makua at Hatia’s, though the people here have not been concerned in a war. It must therefore be due to congenital idleness, or else to the absence of a powerful chief. Even the baraza at Mlipa’s, a short hour’s walk south-east of Newala, shares in this general neglect. While public buildings in this country are usually looked after more or less carefully, this is in evident danger of being blown over by the first strong easterly gale. The only attractive object in this whole district is the grave of the late chief Mlipa. I visited it in the morning, while the sun was still trying with partial success to break through the rolling mists, and the circular grove of tall euphorbias, which, with a broken pot, is all that marks the old king’s resting-place, impressed one with a touch of pathos.

Even my very materially-minded carriers seemed to feel something of the sort, for instead of their usual ribald songs, they chanted solemnly, as we marched on through the dense green of the Makonde bush:—