INTRODUCTION

The sensitivity of communications systems is limited by noise. The broadest definition of noise as “everything except the desired signal” is most emphatically not what we will use here, however, because it does not separate, say, artificial noise sources (e.g., 60-Hz power-line hum) from more fundamental (and therefore irreducible) sources of noise that we discuss in this chapter.

That these fundamental noise sources exist was widely appreciated only after the invention of the vacuum tube amplifier, when engineers finally had access to enough gain to make these noise sources noticeable. It became obvious that simply cascading more amplifiers eventually produces no further improvement in sensitivity because a mysterious noise exists that is amplified along with the signal. In audio systems, this noise is recognizable as a continuous hiss while, in video, the noise manifests itself as the characteristic “snow” of analog TV systems.

The noise sources remained mysterious until H. Nyquist, J. B. Johnson and W. Schottky published a series of papers that explained where the noise comes from and how much of it to expect. We now turn to an examination of the noise sources they identified.

THERMAL NOISE

Johnson was the first to report careful measurements of noise in resistors, and his colleague Nyquist explained them as a consequence of Brownian motion: thermally agitated charge carriers in a conductor constitute a randomly varying current that gives rise to a random voltage (via Ohm's law).

INTRODUCTION

This chapter focuses attention on those aspects of transistor behavior that are of immediate relevance to the RF circuit designer. Separation of first-order from higher-order phenomena is emphasized, so there are many instances when crude approximations are presented in the interest of developing insight. As a consequence, this review is intended as a supplement to – rather than a replacement for – traditional rigorous treatments of the subject. In particular, we must acknowledge that today's deepsubmicron MOSFET is so complex a device that simple equations cannot possibly provide anything other than first-order (maybe even zeroth-order) approximations to the truth. The philosophy underlying this chapter is to convey a simple story that will enable first-pass designs, which are then verified by simulators using much more sophisticated models. Qualitative insights developed with the aid of the zeroth-order models enable the designer to react appropriately to bad news from the simulator. We design with a simpler set of models than those used for verification.

With that declaration out of the way, we now turn to some history before launching into a series of derivations.

A LITTLE HISTORY

Attempts to create field-effect transistors (FETs) actually predate the development of bipolar devices by over twenty years. In fact, the first patent application for a FET-like transistor was filed in 1926 by Julius Lilienfeld, but he never constructed a working device. Before co-inventing the bipolar transistor, William Shockley also tried to modulate the conductivity of a semiconductor to create a field-effect transistor.

Since publication of the first edition of this book in 1998, RF CMOS has made a rapid transition to commercialization. Back then, the only notable examples of RF CMOS circuits were academic and industrial prototypes. No companies were then shipping RF products using this technology, and conference panel sessions openly questioned the suitability of CMOS for such applications – often concluding in the negative. Few universities offered an RF integrated circuit design class of any kind, and only one taught a course solely dedicated to CMOS RF circuit design. Hampering development was the lack of device models that properly accounted for noise and impedance at gigahertz frequencies. Measurements and models so conflicted with one another that controversies raged about whether deep submicron CMOS suffered from fundamental scaling problems that would forever prevent the attainment of good noise figures.

Today, the situation is quite different, with many companies now manufacturing RF circuits using CMOS technology and with universities around the world teaching at least something about CMOS as an RF technology. Noise figures below 1 dB at gigahertz frequencies have been demonstrated in practical circuits, and excellent RF device models are now available. That pace of growth has created a demand for an updated edition of this textbook.

I have written this book mainly for students who will need to apply maths in science or engineering courses. It is particularly designed to help the foundation or first year of such a course to run smoothly but it could also be useful to specialist maths students whose particular choice of A-level or pre-university course has meant that there are some gaps in the knowledge required as a basis for their University course. Because it starts by laying the basic groundwork of algebra it will also provide a bridge for students who have not studied maths for some time.

The book is written in such a way that students can use it to sort out any individual difficulties for themselves without needing help from their lecturers.

A message to students

I have made this book as much as possible as though I were talking directly to you about the topics which are in it, sorting out possible difficulties and encouraging your thoughts in return. I want to build up your knowledge and your courage at the same time so that you are able to go forward with confidence in your own ability to handle the techniques which you will need. For this reason, I don't just tell you things, but ask you questions as we go along to give you a chance to think for yourself how the next stage should go.

I have thoroughly revised all the ten chapters in the original edition, both making some changes due to comments from my readers and also checking for errors. I've also added a chapter on vectors which continues naturally from the present chapter on complex numbers.

I wrote the first version of this new chapter as an extension to the book's website (which is now at http://www.mathssurvivalguide.com) building up the pages there gradually. Their content was influenced by emails from visitors, often with particular problems with which they hoped for help. I've now extensively rewritten and rearranged this material. Writing in book form, it was possible to structure the content much more closely than on the Web so that it's easy to see the connections between the different areas and how results can be applied to later problems. The new chapter also has, of course, many practice exercises with complete solutions just as the earlier chapters have.

I'm once again very grateful to Rodie and Tony Sudbery and to David Olive for their helpful suggestions and comments. I must also thank all the people who emailed me, both with comments on the original ten chapters, and also with particular needs in using vectors which I've tried to fulfil here.

I hope that this two-way communication will continue. You can email me from the book's website if you would like to.