INTRODUCTION

Many histories of microwave technology begin with James Clerk Maxwell and his equations, and for excellent reasons. In 1873, Maxwell published A Treatise on Electricity and Magnetism, the culmination of his decade-long effort to unify the two phenomena. By arbitrarily adding an extra term (the “displacement current”) to the set of equations that described all previously known electromagnetic behavior, he went beyond the known and predicted the existence of electromagnetic waves that travel at the speed of light. In turn, this prediction inevitably led to the insight that light itself must be an electromagnetic phenomenon. Electrical engineering students, perhaps benumbed by divergence, gradient, and curl, often fail to appreciate just how revolutionary this insight was. Maxwell did not introduce the displacement current to resolve any outstanding conundrums. In particular, he was not motivated by a need to fix a conspicuously incomplete continuity equation for current (contrary to the standard story presented in many textbooks). Instead he was apparently inspired more by an aesthetic sense that nature simply should provide for the existence of electromagnetic waves. In any event the word genius, though much overused today, certainly applies to Maxwell, particularly given that it shares origins with genie. What he accomplished was magical and arguably ranks as the most important intellectual achievement of the 19th century.

Maxwell – genius and genie – died in 1879, much too young at age 48. That year, Hermann von Helmholtz sponsored a prize for the first experimental confirmation of Maxwell's predictions.

INTRODUCTION

Although the focus of this book is the implementation of discrete planar RF circuits, we consider here a number of important components that are fundamentally 3-D in nature: connectors, cables, and wave guides. We'll see that the useful frequency range of these components is bounded in part by the onset of moding, which (in turn) is a function of their physical dimensions. In addition, we'll examine the attenuation characteristics of these various ways to get RF energy from one place to another.

CONNECTORS

MODING AND ATTENUATION

For the flattest response over the largest possible bandwidth, an RF connector should exhibit a constant impedance throughout its length. This requirement is satisfied by maintaining constant dimensions throughout and by filling the intervening volume uniformly with a homogeneous dielectric. As straightforward and obvious as this requirement may seem, we will shortly see that there is at least one extremely popular connector that fails to meet it.

The best and most commonly used RF connectors are coaxial in structure. One important attribute of coaxial geometries is their self-shielding nature; radiation losses are therefore not an issue. One must always take care, however, to maintain transverse electromagnetic (TEM) propagation in which, you might recall from undergraduate electromagnetics courses, neither E nor H has a component in the direction of propagation. At sufficiently high frequencies, non-TEM propagation can occur, and the energy stored or propagated in higher-order modes can cause dramatic impedance changes.

INTRODUCTION

In this chapter, we consider the problems of efficiently and linearly delivering RF power to a load. Simple, scaled-up versions of small-signal amplifiers are fundamentally incapable of high efficiency, so we have to consider other approaches. As usual, there are trade-offs – here, between spectral purity (distortion) and efficiency. In a continuing quest for increased channel capacity, more and more communications systems employ amplitude and phase modulation together. This trend brings with it an increased demand for much higher linearity (possibly in both amplitude and phase domains). At the same time, the trend toward portability has brought with it increased demands for efficiency. The variety of power amplifier topologies reflects the inability of any single circuit to satisfy all requirements.

SMALL- VERSUS LARGE-SIGNAL OPERATING REGIMES

Recall that an important compromise is made in analyzing circuits containing nonlinear devices (such as transistors). In exchange for the ability to represent, say, an inherently exponential device with a linear network, we must accept that the model is valid only for “small” signals. It is instructive to review what is meant by “small” and to define quantitatively a boundary between “small” and “large.”

In what follows, we will decompose signals into their DC and signal components. To keep track of which is which, we will use the following notational convention: DC variables are in upper case (with upper-case subscripts); small-signal components are in lower case with lower-case subscripts.

DEFINITIONS

The title of this chapter should raise a question or two: Precisely what is the definition of RF? Of microwave? We use these terms in the preceding chapter, but purposely without offering a quantitative definition. Some texts use absolute frequency as a discriminator (e.g., “microwave is anything above 1 GHz”). However, the meaning of those words has changed over time, suggesting that distinctions based on absolute frequency lack fundamental weight. Indeed, in terms of engineering practice and design intuition, it is far more valuable to base a classification on a comparison of the physical dimensions of a circuit element with the wavelengths of signals propagating through it.

When the circuit's physical dimensions are very small compared to the wavelengths of interest, we have the realm of ordinary circuit theory, as we will shortly understand. We will call this the quasistatic, lumped, or low-frequency realm, regardless of the actual frequency value. The size inequality simplifies Maxwell's equations considerably, allowing one to invoke the familiar concepts of inductances, capacitances, and Kirchhoff 's “laws” of current and voltage.

If, on the other hand, the physical dimensions are very large compared to the wavelengths of interest, then we say that the system operates in the classical optical regime – whether or not the signals of interest correspond to visible light. Devices used to manipulate the energy are now structures such as mirrors, polarizers, lenses, and diffraction gratings. Just as in the quasistatic realm, the size inequality enables considerable simplifications in Maxwell's equations.

INTRODUCTION

Given the effort expended in avoiding instability in most feedback systems, it would seem trivial to construct oscillators. Murphy, however, is not so kind; the situation is a lot like bringing an umbrella in order to make it rain. An old joke among RF engineers is that every amplifier oscillates, and every oscillator amplifies.

In this chapter, we consider several aspects of oscillator design. First, we show why purely linear oscillators are a practical impossibility. We then present a linearization technique utilizing describing functions that greatly simplify analysis, and help to develop insight into how nonlinearities affect oscillator performance. With describing functions, it is straightforward to predict both the frequency and amplitude of oscillation.

A survey of resonator technologies is included, and we also revisit PLLs, this time in the context of frequency synthesizers. We conclude this chapter with a survey of oscillator architectures. The important issue of phase noise is considered in detail in Chapter 17.

THE PROBLEM WITH PURELY LINEAR OSCILLATORS

In negative feedback systems, we aim for large positive phase margins to avoid instability. To make an oscillator, then, it might seem that all we have to do is shoot for zero or negative phase margins. We may examine this notion more carefully with the root locus for positive feedback sketched in Figure 15.1 This locus recurs frequently in oscillator design because it applies to a two-pole bandpass resonator with feedback.

INTRODUCTION

The design of amplifiers for signal frequencies in the microwave bands involves more detailed considerations than at lower frequencies. One simply has to work harder to obtain the requisite performance when approaching the inherent limitations of the devices themselves. Additionally, the effect of ever-present parasitic capacitances and inductances can impose serious constraints on achievable performance. Indeed, parasitics are so prominent at RF that an important engineering philosophy is to treat parasitics as circuit elements to be exploited, rather than fought.

Having evolved during an era where modeling and simulation capabilities were primitive, traditional microwave amplifier design largely ignores the underlying details of device behavior. Instead, S-parameter sets describe the transistor's macroscopic behavior over frequency. In doing so, vast simplifications can result, but at a cost. By effectively insulating the engineer from the device physics, it is difficult to extrapolate beyond the given data set. Furthermore, real transistors are nonlinear, so the S-parameter characterizations are strictly relevant only for the bias conditions used in their generation.

Because simulation and modeling tools have advanced considerably since that time, we will consider the design of both broadband and narrowband amplifiers from a device-level point of view, rather than with the more traditional Smith-chart–based approach. Thus, we will not spend time examining stability and gain circles, for example. Readers interested in the classical approach are directed to any of a number of representative texts that cover the topic in detail.

HISTORY AND OVERVIEW

With the growing sophistication in semiconductor device fabrication has come a rapid expansion in the number and types of transistors suitable for use at microwave frequencies. At one time, the RF engineer's choices were a bipolar or possibly a junction field-effect transistor. The palette of options has since grown to a dizzying collection of MOSFETs, VMOS, UMOS, LDMOS, MESFETs, pseudomorphic and metamorphic HEMTs (MODFETs), and HBTs, all offered in an ever-expanding variety of materials systems. We'll attempt to provide a description of these types of devices, starting with a deciphering of their abbreviations. Then we'll focus on a small subset of these devices in an expanded discussion of modeling.

The bipolar transistor was discovered – not invented – in December of 1947 while the Bell Labs duo of John Bardeen and Walter Brattain was attempting to build aMOS field-effect transistor at the behest of their boss, William Shockley. Their repeated failures led them to suspect that the problem lay with the surface, where the neat periodicity of the bulk terminates abruptly, leaving unsatisfied bonds to latch onto contaminants. To verify this “surface state” hypothesis, they undertook a detailed study of semiconductor surface phenomena. One of their experiments, designed to modulate the postulated surface states, itself happened to exhibit power gain. It wasn't the MOSFET they had been trying to build; it was a germanium point-contact bipolar transistor. Its behavior was never quantitatively understood, and repeatability of characteristics was only a fantasy.

INTRODUCTION

We asserted in Chapter 15 that tuned oscillators produce outputs with higher spectral purity than relaxation oscillators. One straightforward reason is simply that a high-Q resonator attenuates spectral components removed from the center frequency. As a consequence, distortion is suppressed, and the waveform of a well-designed tuned oscillator is typically sinusoidal to an excellent approximation.

In addition to suppressing distortion products, a resonator also attenuates spectral components contributed by sources such as the thermal noise associated with finite resonator Q, or by the active element(s) present in all oscillators. Because amplitude fluctuations are usually greatly attenuated as a result of the amplitude stabilization mechanisms present in every practical oscillator, phase noise generally dominates – at least at frequencies not far removed from the carrier. Thus, even though it is possible to design oscillators in which amplitude noise is significant, we focus primarily on phase noise here. We show later that a simple modification of the theory allows for accommodation of amplitude noise as well, permitting the accurate computation of output spectrum at frequencies well removed from the carrier.

Aside from aesthetics, the reason we care about phase noise is to minimize the problem of reciprocal mixing. If a superheterodyne receiver's local oscillator is completely noise-free, then two closely spaced RF signals will simply translate downward in frequency together. However, the local oscillator spectrum is not an impulse and so, to be realistic, we must evaluate the consequences of an impure LO spectrum.

First, it was called wireless, then radio. After decades in eclipse wireless has become fashionable once again. Whatever one chooses to call it, the field of RF design is changing so rapidly that textbook authors, let alone engineers, are hard pressed to keep up. A significant challenge for newcomers in particular is to absorb an exponentially growing amount of new information while also acquiring a mastery of those foundational aspects of the art that have not changed for generations. Compounding the challenge is that many books on microwave engineering focus heavily on electromagnetic field theory and never discuss actual physical examples, while others are of a cookbook nature with almost no theory at all. Worse still, much of the lore on this topic is just that: an oral tradition (not always correct), passed down through the generations. The rest is scattered throughout numerous applications notes, product catalogs, hobbyist magazines, and instruction manuals - many of which are hard to find, and not all of which agree with each other. Hobbyists are almost always unhappy with the theoretical bent of academic textbooks (“too many equations, and in the end, they still don't tell you how to make anything”), while students and practicing engineers are often unhappy with the recipe-based approaches of hobby magazines ("they don't give you the theory to show how to change the design into what I actually Need").

INTRODUCTION

Designers of low-frequency analog circuits are often puzzled by the seeming obsession of microwave engineers with impedance matching. Analog circuit design textbooks, for example, almost never have a chapter on this topic. Instead, engineers working at lower frequencies usually express specifications in terms of voltage gain, for example, with little or no reference to impedance matching. In striking contrast, RF engineers are indeed frequently preoccupied with the problem of impedance matching. The principal reason for the difference in philosophical outlook is that power gain is so abundant at low frequencies that designers there have the luxury of focusing on convenience, rather than necessity. For example, a textbook transformer could provide impedance matching to maximize power gain, but electronic voltage amplifiers are more readily realized (and are certainly more flexible) than are coils of wire wound around magnetic cores. The need for an impedance transformation is usually acknowledged only implicitly (if at all) and is frequently satisfied with a crude transformation to an unspecified low value with a voltage buffer, for example. On the other hand, RF power gain is often an extremely limited resource, so one must take care not to squander it. Impedance-transforming networks thus play a prominent role in the radio frequency domain.

There are many good reasons for seeking an impedance match at RF beyond simply maximizing gain. One is that a match makes a system insensitive to the lengths of interconnecting lines.

INTRODUCTION

Both time- and frequency-domain characterizations provide comprehensive information about a system. The latter require the ability to generate and measure sinusoidal voltages and currents over a broad frequency range. The network analyzer, in either scalar or vector incarnations, is an example of such an instrument.

An alternative is to use time-domain methods to characterize a system. The principal tool of this type is the time-domain reflectometer (TDR), which is in essence a miniature radar system. The TDR launches a pulse (“the main bang”) into the device under test and then observes any echoes. The timing of a reflection with respect to the main bang indicates the location of a discontinuity, and the shape of the reflected pulse conveys important information about its nature. With a reflectometer, then, one can quickly locate and characterize both resistive and reactive discontinuities “and evaluate their fixes”. A network analyzer can also provide this information but requires considerably more labor to do so.

We consider both network analyzers and TDRs in this chapter, beginning with the latter.

THE TIME-DOMAIN REFLECTOMETER

There are two primary applications of TDRs: finding and characterizing impedance discontinuities. These capabilities translate directly into the ability to correct defects and evaluate the quality of any compensation performed

LOCATING DISCONTINUITIES

As shown in Figure 8.1, a TDR consists of just two main modules: a pulse generator and an oscilloscope. A pulse generator applies a fast risetime step to the device under test (DUT).

INTRODUCTION

A recurring theme in RF design is the need to pay careful attention to the electrical characteristics of everything along the signal path. This concern extends to printed circuit (PC) boards, so this chapter considers their high-frequency properties as well as those of numerous passive components made with PC board materials. We will focus on a particular type of transmission line known as microstrip, which is particularly suited to the realization of planar microwave circuits. In addition, numerous passive components can be made out of transmission lines, so capacitors, inductors, resonators, power combiners, and a variety of couplers – including baluns and hybrids – are presented as well.

GENERAL CHARACTERISTICS OF PC BOARDS

Just as with PC boards used at lower frequencies, those for RF applications consist of metal layers separated by a dielectric of some kind. By convention in the United States, the metal thickness is given indirectly as a certain weight of copper per square foot. Thus, “1 ounce” copper (a common value) is approximately 1.34 mil (35 μm) thick. Half-ounce and 2-oz copper are other common values. The DC resistivity of bulk copper is approximately 1.8 μΩ-cm, so the corresponding sheet resistivity of 1-oz copper is about 0.5 mΩ/square. The skin depth in copper is about 2.1 μm at 1 GHz, raising the sheet resistivity to roughly 8 mΩ/square at that frequency.

INTRODUCTION

The first stage of a receiver is typically a low-noise amplifier (LNA), whose main function is to provide enough gain to overcome the noise of subsequent stages (typically a mixer). Aside from providing this gain while adding as little noise as possible, an LNA should accommodate large signals without distortion and frequently must also present a specific impedance, such as 50Ω, to the input source. This last consideration is particularly important if a filter precedes the LNA, since the transfer characteristics of many filters (both passive and active) are quite sensitive to the quality of the termination.

We will see that one can obtain the minimum noise figure (NF) from a given device by using a particular magic source impedance whose value depends on the characteristics of the device. Unfortunately this source impedance generally differs, perhaps considerably, from that which maximizes power gain. Hence it is possible for poor gain and a bad input match to accompany a good noise figure. One aim of this chapter is to place this trade-off on a quantitative basis to assure a satisfactory design without painful iteration.

We will focus mainly on a single narrowband LNA architecture that it is capable of delivering near-minimum noise figures along with an excellent impedance match and reasonable power gain. The narrowband nature of the amplifier is not necessarily a liability, since many applications require filtering anyway. The LNA we'll study thus exhibits a balance of many desirable characteristics.

INTRODUCTION

Phase-locked loops (PLLs) have become ubiquitous in modern communications systems because of their remarkable versatility. As one important example, a PLL may be used to generate an output signal whose frequency is a programmable and rational multiple of a fixed input frequency. Such frequency synthesizers are often used to provide the local oscillator signal in superheterodyne transceivers. PLLs may also be used to perform frequency modulation and demodulation as well as to regenerate the carrier from an input signal in which the carrier has been suppressed. Their versatility also extends to purely digital systems, where PLLs are indispensable in skew compensation, clock recovery, and the generation of clock signals.

To understand in detail how PLLs may perform such a vast array of functions, we will need to develop linearized models of these feedback systems. But first, of course, we begin with a little history to put this subject in its proper context.

A SHORT HISTORY OF PLLs

The earliest description of what is now known as a PLL was provided by de Bellescize in 1932. This early work offered an alternative architecture for receiving and demodulating AM signals, using the degenerate case of a superheterodyne receiver in which the intermediate frequency is zero. With this choice, there is no image to reject, and all processing downstream of the frequency conversion takes place in the audio range.

INTRODUCTION

Electrical engineering education focuses so much on the study of linear, time-invariant (LTI) systems that it's easy to conclude that there's no other kind. Violations of the LTI assumption are usually treated as undesirable, if acknowledged at all. Small signal analysis, for example, exists precisely to avoid the complexities that nonlinearities inevitably bring with them. However, the high performance of modern communications equipment actually depends critically on the presence of at least one element that fails to satisfy linear time invariance: the mixer. The superheterodyne receiver uses a mixer to perform an important frequency translation of signals. This invention of Armstrong has been the dominant architecture for 75 years because frequency translation solves many problems simultaneously.

In the architecture shown in Figure 10.1, the mixer heterodynes an incoming RF signal to a lower frequency, known as the intermediate frequency (IF). Although Armstrong originally sought this frequency lowering simply to make it easier to obtain the requisite gain, other significant advantages accrue as well. As one example, tuning is now accomplished by varying the frequency of a local oscillator rather than by varying the center frequency of a multipole bandpass filter. Thus, instead of adjusting several LC networks in tandem to tune to a desired signal, one simply varies a single LC combination to change the frequency of a local oscillator (LO). The intermediate frequency stages can then use fixed bandpass filters. Selectivity is therefore determined by these fixed-frequency IF filters, which are much easier to realize than variable-frequency filters.

INTRODUCTION

Oscilloscopes and spectrum analyzers are ubiquitous pieces of test equipment in any RF laboratory. The reason, of course, is that it is useful to study signals in both time and frequency domains, despite the fact that both presentations theoretically provide equivalent information.

Most electrical engineers are familiar with basic operational principles of lower frequency oscilloscopes. However, an incomplete understanding of how probes behave (particularly with respect to grounding technique) is still remarkably widespread. The consequences of this ignorance only become worse as the frequency increases and so, after a brief review of a conventional low-frequency scope, our primary focus will be the additional considerations one must accommodate when using scopes at gigahertz frequencies. Also, because the sampling oscilloscopes commonly used at high frequencies have subtle ways of encouraging “pilot error,” we'll spend some time studying how they work and how to avoid being fooled by them. High-speed sampling circuits are interesting in their own right, so these types of scopes give us a nice excuse to spend a little bit of time examining how samplers function.

Another amazing instrument is the modern spectrum analyzer (with cost approximately proportional to the square of amazement), which is capable of making measurements over a wide dynamic range (e.g., 80–100 dB SFDR) and over a large frequency span (e.g., near DC to 20 GHz in a single instrument).