# A Guide to Noise in Microwave Circuits

## Devices, Circuits and Measurement

1. Edition January 2022

512 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-85936-9**

A GUIDE TO NOISE IN MICROWAVE CIRCUITS

A fulsome exploration of critical considerations in microwave circuit noise

In A Guide to Noise in Microwave Circuits: Devices, Circuits, and Measurement, a team of distinguished researchers deliver a comprehensive introduction to noise in microwave circuits, with a strong focus on noise characterization of devices and circuits. The book describes fluctuations beginning with their physical origin and touches on the general description of noise in linear and non-linear circuits.

Several chapters are devoted to the description of noise measurement techniques and the interpretation of measured data. A full chapter is dedicated to noise sources as well, including thermal, shot, plasma, and current.

A Guide to Noise in Microwave Circuits offers examples of measurement problems--like low noise block (LNB) of satellite television - and explores equipment and measurement methods, like the Y, cold source, and 7-state method. This book also includes:

* A thorough introduction to foundational terms in microwave circuit noise, including average values, amplitude distribution, autocorrelation, cross-correlation, and noise spectra

* Comprehensive explorations of common noise sources, including thermal noise, the Nyquist formula and thermal radiation, shot noise, plasma noise, and more

* Practical discussions of noise and linear networks, including narrowband noise

* In-depth examinations of calculation methods for noise quantities, including noise voltages, currents, and spectra, the noise correlation matrix, and the noise of simple passive networks

Perfect for graduate students specializing in microwave and wireless electronics, A Guide to Noise in Microwave Circuits: Devices, Circuits, and Measurement will also earn a place in the libraries of professional engineers working in microwave or wireless circuits and system design.

Preface xv

1 Introduction 1

Preliminary Remarks 1

History 6

References 7

2 Basic Terms 9

Average Values 9

Amplitude Distribution 10

Autocorrelation 12

Cross-Correlation 15

Noise Spectra 18

Autocorrelation Function and Spectral Power Density 19

Band-Limited Noise on the Spectrum Analyzer 20

References 22

3 Noise Sources 23

Thermal Noise 23

Nyquist Formula and Thermal Radiation 24

Validity and Experimental Confirmation of the Nyquist Formula 27

Thermal Noise Under Extreme Conditions 28

Shot Noise 29

Plasma Noise 33

Current Noise of Resistors and Contacts 34

Technical Resistors 34

Resistors Consisting of Semiconductor Material 36

Contact Noise 37

Generation-Recombination Noise 38

LF Noise from Transistors 40

References 42

4 Noise and Linear Networks 45

Narrowband Noise 45

Calculating with Phasors 45

Noise Source with Complex Internal Resistance 51

The Equivalent Noise Bandwidth 52

Network Components at Different Temperatures 54

Noise Generator and Attenuator 58

References 58

5 Nonlinear Networks 59

Mixing 59

Band-Limited RF Noise at Input 59

Amplitude Clipping 62

The Detector as a Nonlinear Network 63

The Noise Spectrum Behind a Quadratic Detector 65

The Noise Spectrum Behind a Linear Detector 69

The Sensitivity Limit 70

Noise with Signal 73

The Phase Sensitive Rectifier 74

Trace Averaging 76

References 78

6 The Noise Factor 79

Amplifier and Noise Power 79

The Noise FactorF 80

Cascaded Amplifiers 83

The Noise MeasureM 85

Definitions of Gain 85

Source and Load 89

Broadband and Spot Noise Factor 91

Noise Factor of a Passive Network 92

Antenna Temperature 93

The Reference Temperature T0 = 290 K 98

Noise Factor and Detection Limit 99

References 100

7 Noise of Linear Two-Ports 101

Representation of Two-Ports 101

Noise Modeling Using the Chain Matrix 102

References 108

8 Calculation Methods for Noise Quantities 109

Noise Voltages, Currents, and Spectra 109

Calculating with Current, Voltage, and Noise Waves 112

The Noise Correlation Matrix 115

The Correlation Matrix of Passive Components 117

The Noise of Simple Passive Networks 119

Transformation of Noise Sources in Different Network

Representations 128

Correlation Matrix and IEEE Elements 131

FET-Like Network with the Y-Correlation Matrix 134

Noise Sources at Input with ABCD Correlation Matrix 138

References 142

9 Diodes and Bipolar Transistors 143

Semiconductor Diode 143

Bipolar Transistor 145

Small-Signal Equivalent Circuit 147

Hawkins BJT Noise Model 148

Two Approaches for the Collector Noise Current Source 155

BJT Noise Model with Correlation Matrices 157

The Pi-Model 157

The T-Model with Correlation Matrices 161

Transformation of the Y-Sources to the Input 165

Modeling of a Microwave Transistor with Correlation Matrices 168

Simplest Pi-Model 174

Contour Diagram 177

Transistor in the Circuit 179

Using the Contour Diagram 183

References 185

10 Operational Amplifier 187

Operational Amplifier as Circuit Element 187

Noise Sources of the Operational Amplifier 188

Consideration of 1/f Noise 193

Operational Amplifier as an Active Low-Pass Filter 195

References 198

11 Field Effect Transistors 201

JFET 201

Mode of Operation of the FET 201

The Channel Noise 204

Noise Sources at the Gate 205

The Correlation 206

Transformation to the Input 206

Simple Approximations 211

Field Effect Transistors for the Microwave Range (MESFET, HFET) 214

The Pucel Model 215

The Pospieszalski model 218

Discussion of the Results 225

Criteria for Noise Data 225

References 229

12 Theory of Noise Measurement 231

Measurements of Two-Ports 231

The Equivalent Noise Resistance 234

Voltage and Current Source 235

Voltage and Current Source with Correlation 237

3 dB and Y-Method 241

References 243

13 Basics of Measuring Technique 245

Principles of the RF-Receiver 245

The Detection Limit 245

Diode as RF Receiver (Video Detector) 249

RF and Microwave Range Receiver 254

Dicke Radiometer 258

Correlation Radiometer in the Microwave Range 261

Network Analyzer as a Noise Measurement Device 263

References 265

14 Equipment and Measurement Methods 267

Noise Measurement Receiver 267

Spectrum Analyzer 269

The Y-Method 273

Measurements in the Microwave Range 275

Selection Criteria of the Mixer 278

Image Rejection 279

Complete Noise Characterization 282

Analysis of Multi-impedance Measurements 283

Cold Source Method 285

The 7-State Method 287

On-Wafer Measurement of Cold Source 288

On-Wafer with Noise Generator According to the Y-method 293

References 296

15 Noise Generators 299

Vacuum Diode 299

Gas Discharge 300

Semiconductor Diodes 302

Excess Noise Ratio (ENR) 303

Hot-Cold Sources 305

References 307

16 Impedance Tuners 309

Impedance Transformation with Simple Methods 309

Mechanical Components for the Microwave Range 311

Electronic Components 313

Precision Automatic Tuner 315

Attenuation of the Tuner 317

References 318

17 Examples of Measurement Problems 319

Transistor in a Test Fixture 319

The Low Noise Block (LNB) of Satellite Television 322

Verification of a Noise Measurement 325

References 327

18 Measurement and Modeling of Low-Frequency Noise 329

Correlation Radiometer for Low Frequencies (f 10 MHz) 329

The Low-Frequency Noise of Transistors 333

Measurement Setup for LF Noise 334

Examples of LF Noise Measurements on GaAs-HBT 336

Modeling of LF Noise 337

The Noise of the Microphone 337

References 342

19 Measurement Accuracy and Sources of Error 345

Accuracy of Measured Data 345

Error of Measurements 345

Inaccuracies of the Noise Measurement 346

Uncertainty of the ENR Calibration 349

Noise Source Mismatch 350

T0 = 290 K Is not TOFF 352

Mismatch in the System 353

Linearity of the Receiver 356

References 357

20 Phase Noise 359

Basics 359

Reciprocal Mixing 361

Description of Phase Noise 363

Spectral Power Density of Phase Fluctuations S Phi (f ) 364

The Single Sideband Phase Noise L(f ) 365

Spectral Power Density of Frequency Fluctuations S Delta f (f ) 365

Excursus on Frequency and Phase Modulation 366

The Allan Variance 368

Residual FM 370

Multiplication and Division 371

Amplitude Noise 371

Phase Noise and Jitter 372

References 374

21 Physics of the Oscillator 377

Oscillation Condition [1] 377

Simple Model of the Phase Disturbance [2] 378

Phase Slope, Resonator Quality, and Frequency Stability [3] 379

The Formula of Leeson [4] 382

Components of Oscillators 384

Influence of the Varactor Diode 386

Upward Mixing of LF Noise 390

The Influence of Microwave Noise on Phase Noise 393

References 396

22 Phase Noise Measurement 399

Basic Parameters 399

Spectrum Analyzer 399

Phase Detector Method 406

The Sensitivity of the Phase Detector 407

Example Calibration and Measurement 409

Keeping the Quadrature by a PLL 410

Delay Line as Frequency Discriminator 412

The Sensitivity of the Delay-Line Method 414

Configuration and Calibration 418

Resonator as Frequency Discriminator 420

Detection Limit 421

Comparison of Measurement Systems 422

Cross-Correlation Technique 423

Amplitude Noise 425

Problems with On-Wafer Measurement 429

Residual Phase Noise 430

References 432

Appendix 435

Noise Signals and Deterministic Signals 435

Random Signals 436

Characteristic Values 437

The Probability Density Function 438

Example Sine Function 439

Example Sawtooth Voltage 440

Example White Noise 440

Example Sinusoidal Signal with Noise 441

Example Narrowband Noise 441

The Autocorrelation Function 444

Example Sine 444

Example Sawtooth 444

Example Noisy Sine 445

Example White Noise 446

Example Low-Pass Noise 447

Example Bandpass Noise 449

Fourier Series 451

Sine-Cosine Spectrum 452

Amplitude-Phase Spectrum 452

Complex Fourier Series 452

The Fourier Integral 453

Energy and Power Signals 456

Example Transient Time Function 457

The Parseval Equation 459

Example Voltage Pulse 460

Fourier Transform and Power Spectral Density 462

Example Rectangular Pulse 463

Time-Limited Noise Signal 465

Example of a Time-LimitedWave Train 466

TheWiener-Khinchin Theorem 468

Cross Correlation 470

Example of Two Sine Functions 471

Example of Two White Noise Signals 472

Example of Two Bandpass Noise Signals 472

Example White Noise and Bandpass Noise 474

Cross-Correlation After Splitting into Two Branches 474

Power Spectral Density Real and Complex 477

The Cross-Spectral Density 478

Complex Representation of the Cross-Spectral Density 479

Transmission of Noise by Networks 479

References 485

Glossary of Symbols 487

Index 491

Dr. Matthias Rudolph, is Ulrich L. Rohde Professor for RF and Microwave Techniques at Brandenburg University of Technology in Cottbus, Germany. He heads the Low-Noise components laboratory at the FBH.