Microwave Circuits

1.1 INTRODUCTION

The term microwave circuits means different things to different people. The prefix micro comesfrom the Greek fiiKpog (micros) and among its various meanings has the meaning of little orsmall. Little and small are relative words. Microwave means little or small waves. This oftenmeans that the wavelength is small with respect to the physical variations in a circuit or smallwith respect to a component size. To many people, microwave means those frequencies in thel-to-10-gigahertz range. This is the range over which the wavelength in air varies from 30centimeters to 3 centimeters. In dielectric materials, the wavelength is reduced from its value inair by a factor of the square root of the dielectric constant. In the l-to-10-gigahertz frequencyrange, components with linear dimensions of 3 to 30 millimeters or approximately 1 to 20millimeters for common dielectric materials become a tenth of a wavelength long. Other peo-ple like to think about microwave circuits as those for which distributed effects are important.For those circuits, wave propagation effects are important. Whether wave propagation effectsare important depends again on the physical size or fabrication technology used to fabricate acircuit that is being described or characterized. Between the middle and the end of the twen-tieth century, microwave circuit techniques included printed circuit board (PCB), microwaveintegrated circuit (MIC), and monolithic microwave integrated circuit (MMIC) technologiesin addition to coaxial and waveguide technologies. The circuit designer uses packaged devicesroughly one centimeter in size for PCB circuits, beam-leaded devices roughly one millimeterin size for MIC circuits, and monolithic devices roughly ten to a hundred microns in size forMMIC circuits.

In this book, the term microwave circuits includes those circuits fabricated on printedcircuit boards (PCB), microwave integrated circuits (MIC), or monolithic microwave integratedcircuits (MMIC) for which the circuit parasitic elements form a significant portion of the circuitelement values. A parasitic element might arise due to distributed circuit effects or arise fromfabrication constraints. This definition includes but is not limited to those circuits for whichdistributed effects need to be considered. Printed circuit boards and microwave integratedcircuits are in use in most avionics and wireless circuit applications. The relative amountof a parasitic element depends on the dielectric constant of the medium in which the circuitelement is embedded, the distance a signal needs to travel from a source to a load, or the

1

Chapter 1 • Microwave Circuits

distance between a component and a ground conductor or region. This concept can be used todescribe circuits that are analog in nature, digital in nature, or mixed mode (analog and digital)in nature. Both parasitic elements and distributed effects need to be considered in microwavecircuit design. For the digital designer, these parasitics affect the time delay and wave shapeassociated with voltage or current pulses traveling along a wire or conductor.

A model of a circuit element accounts for the electric, magnetic, and dissipated energiesin the circuit element. A combination of lumped-constant elements consisting of resistors,capacitors, or inductors can often be used to model a circuit element. A resistor is used tocharacterize the power or energy lost in a circuit, while a capacitor and an inductor are usedto characterize the electric and magnetic energy stored in a circuit respectively. In those caseswhere the circuit is large with respect to a wavelength, a transmission line model consisting ofresistors, capacitors, and inductors will be used to characterize the distributed parameter effectfor the circuit elements.

Giving the reader the necessary tools needed to analyze or synthesize microwave circuitsis the objective of this book. Some of these tools will be developed in a sequence and othertools are developed in specific chapters. Some general analysis tools are developed and thenthe tools needed to design an amplifier, the tools needed to design an oscillator, and the toolsneeded to design filter circuits are developed. The scattering matrix formalism is developedand then measurement methods for obtaining the scattering parameters are described. Forthose readers who are interested in differential circuit characteristics, scattering parametersof active differential circuits are described before stability, gain, and match are described.For those readers who are interested only in single-ended circuits, Chapter 4 can be skippedand the reader can go directly to Chapter 5. Readers who are not interested in measuringscattering parameters and who wish to use scattering parameters directly may go from Chapter2 to Chapter 5. Chapter 6 contains material that will give the reader several methods tomatch lumped- and distributed-constant circuits. Different modes of power amplification arediscussed in Chapter 7 before oscillators are discussed in Chapter 8, in order to give the readerthe design information needed to design oscillators for high power-conversion efficiency.

Chapters 2, 3, 5, 6, 7, and 8 provide the material necessary for designing an oscillatorincluding any necessary phase shifters, an attenuator, an amplifier, and a filter. Material fromthis set of chapters has been used for over a decade in a one-semester beginning microwavecircuits course for senior and first-year-graduate students at Iowa State University. The thrust ofthat course has been to look at the circuit's aspect of microwave circuit design. The distributednature of the design is incorporated using de-embedding techniques and the scattering matrixfor transmission line sections. The laboratory portion of the course consists of the design,fabrication, build, and test of a 1-gigahertz oscillator, followed by an attenuator, an amplifier,and a two-pole filter using microstrip printed circuit boards and leaded as well as chip partsfor the discrete passive components. During the time the students were fabricating and testingthose parts of a single-frequency power source, selected material from Chapters 10 and 11 wasincluded in the lecture material.

Other topics in this book include noise in Chapter 10, and detection and frequencytranslation in Chapter 11. Special attention is given in Chapter 12 to the use of PIN diodes forswitching applications for the reader who is incorporating sensitive receivers with high-powertransmitters in a single antenna system. Chapter 12 contains a description of some microwavecomponents that can be used by the microwave engineer to build a system. Fully developingthat chapter in a college course could easily comprise a semester of learning and fill severalbooks. However, sufficient material is given to allow the reader to use the components in asystem. Components and connecting wires that are used in microwave circuits need to becharacterized. Methods to measure, manipulate, and verify component values are discussed.A method for time domain analysis applicable to high-speed digital and mixed-mode digital-

2

Section 1.2 • Circuit Elements 3

analog circuits is described in Chapter 13. Chapter 14 includes a discussion of bias circuitcomponent effects and nonlinear phenomena associated with bias circuit configurations.

Chapter 15 is the final chapter of the book. It contains a worked example of a 1.25-GHzamplifier, a 1.25-GHz loop oscillator, a 1.25-GHz shunt oscillator, and a couple of 1.25-GHzfilters. The reader should refer to that chapter while reading Chapters 2, 5, 6, 7, 8, and9. Appendix D contains a laboratory procedure to measure the parasitic values of /?, L,and C components. The procedures are described in terms of a test fixture with connectors;however, these procedures are applicable to MMIC versions of the components when using rfor microwave probes to connect to the test fixture.

1.2 CIRCUIT ELEMENTS

A circuit component consists of several simple circuit elements. As will be discussed inChapter 2, reciprocal passive components we call resistors, or capacitors, or inductors eachconsist of various combinations of the three circuit elements—resistance, capacitance, andinductance. When these components are used, they are placed in space somewhere in relationto a ground plane or ground reference. Transmission lines are composed of these same threeelements—resistance, capacitance, and inductance—but in a transmission line, these elementsare distributed over a region of space rather than being identified with a point in space. Inaddition to these components, controlled current and voltage sources are used to describethe performance of active elements. Equivalent circuits for transistors and diodes are givenassuming the reader has had some introduction to diode equations and transistor phenomena.

The author often tells his students that he has never seen any of the simple circuit elements,a resistor, a capacitor, or an inductor, much less a ground plane. This should become moreevident in Chapter 2 when components are described. What people have seen is a componentcalled a resistor, a capacitor, or an inductor but each of these components has varying amountsof other circuit element types associated with it.

1.2.1 Ground Planes

Typically the metal on the back of a printed circuit board or the back side metalization ona MMIC chip is called a ground plane. That is the term used in the industry and the literature.Calling something by the term, however, does not guarantee that the metal structure has themathematical characteristics of a ground plane, i.e., that the potential is the same everywhereon the surface.

What is a ground plane? A ground plane that represents the mathematical descriptionextends from minus infinity to plus infinity in both directions. It does not have any holes orcuts in it and it has zero resistance. The author has never seen one of these! The microwavecircuit designer needs to appreciate what effect a ground conductor of finite size has on amicrowave circuit and what effect drilling a hole or making a cut in the ground conductorhas on the circuit. All conductors have a finite resistance. Even ground conductors formedout of superconductors have a finite resistance at high frequencies. Normal conductors haveresistance at all frequencies. The performance of a microwave circuit depends on whether andwhere a surface of zero potential exists.

A ground plane is assumed to have a constant potential everywhere on it. Unbalancedtransmission line analysis assumes that a ground plane exists. When a ground plane does notexist as assumed for transmission line analysis, then one needs to question how the results oftransmission line analysis can be applied to real circuits. Often the ground plane is assumedto be at a potential of zero volts everywhere on it. A perfect ground plane can conductmicroamperes of current or hundreds of amps of current and in either case has no voltage

Chapter 1 • Microwave Circuits

developed across it. What is commonly called the ground plane in the circuit board industry isan approximation for what a real ground conductor is. The ground plane approximation oftengets the microwave engineer into difficulty when using the results of circuit or transmissionline analysis based on an infinite ground plane assumption. Real ground conductors do havevoltages across them even when they are formed from a continuous foil or sheet of material.This material is not infinite in extent and does have a finite resistance.

1.2.2 Linear R, L, C Circuit Elements

Some brief comments will be made about the simple linear circuit elements, the resistor,the capacitor, and the inductor. It is appreciated that the reader will likely know these circuitelements quite well. However, with regard to microwave circuit modeling, it is helpful to reviewhow the circuit elements function in a circuit. Keep in mind that the mathematical descriptionof a circuit element and not the function of a real-world component is being considered whenthese elements are discussed in this section.

Multiple terminal networks have separate currents and voltages into and across each setof terminal pairs. In microwave circuits, a port is a region through which energy flows. A portmight be considered a terminal pair, a region between and around two wires, or it might bethe opening into a tube called a waveguide, or further, it might just be an opening into somevolume. A circuit element, either a resistor, a capacitor, or an inductor, in the absence of aground plane, is a one-port network. It has only one terminal pair and only one port. Wheneach terminal is separately connected somewhere but also in the presence of a ground planeor ground node then these elements become two-port networks.

A resistor of resistance value R is used to describe a linear circuit element for which thecurrent through the resistor is in phase with the voltage across the resistor. All energy enteringa resistor is dissipated in the resistor. The relationship between current through a resistorand voltage across a resistor is expressed as V = RI. This relationship can also be describedas a conductor or a conductance. The expression is then rewritten as / = GV. Notice thatalthough the relationships appear to be the same and R = 1/G, there is a significant differencein the equations. In the resistance expression, current, / , is the independent variable. In theconductance expression, voltage, V, is the independent variable. This does not appear to bevery important for a simple resistor, but it is quite important when the linear relationships forcircuit elements are extended to multiple terminal networks.

A capacitor of capacitance value C is used to describe the linear circuit element for whichthe current through the capacitor is said to lead the voltage across the capacitor. The capacitordoes not dissipate energy and does not store magnetic energy, but does store or release electricenergy. When current into the capacitor is the independent variable, 1 over C times the timeintegral of the current into the capacitor gives the voltage across a capacitor. When voltageacross the capacitor is the independent variable, the current through the capacitor is equal toC times the time derivative of the voltage across the capacitor.

V(t) = - l(t)dt" J — OO

An inductor of inductance value L is used to describe the linear circuit element for whichthe current through the inductor is said to lag the voltage across the inductor. The inductordoes not dissipate energy and does not store electric energy, but does store or release magneticenergy. When current into the inductor is the independent variable, L times the time derivativeof the current through the inductor gives the voltage across the inductor. When the voltage

4

I(t) = CdV(t)

dt

Section 1.2 • Circuit Elements

across the inductor is the independent variable, the current into an inductor is equal to 1 overL times the time integral of the voltage across the inductor.

dl(t)V(t) = L-

dt

lit) V(t)dt

These well-known relationships should be kept in mind when MMIC and VLSI biasnetworks are designed. The inductance of the bias line will cause the voltage across a deviceto drop when that device's current changes rapidly. The MMIC circuit designer should attemptto incorporate enough capacitance close to an amplifier module to support at least one sinusoidalor pulse cycle of current that exists in the dc bias lead to that amplifier. These formulas areused to determine the effect of inductance or the value of capacitance needed.

Nonlinear circuit elements also exist. For the purposes of this book, unless specificallystated otherwise, all circuit values are assumed to be linear or the values used for circuitelements are those derived in a piecewise linear region of operation of the circuit. Thosecircuit values are often derived by differentiation of the immittance or transfer functions ofthe device around a bias point resulting in a small signal ac analysis. In the case of PIN diodeanalysis, the circuit values are derived by large signal linear ac analysis around a small dcsignal that does not change appreciably over one large ac cycle.

1.2.3 Distributed Parameter Circuits

Transmission lines are used to model some circuits. Figure 1.1 shows a model of alossless transmission line, Figure 1.2 shows a model of a transmission line with losses, andFigure 1.3 shows various types of transmission lines. In the transmission line model, /?, L,C, and G values are given as per-unit-length values. In the metric system, transmission lineresistance has units of ohms/m, inductance has units of henries/m, capacitance has units offarads/m, and conductance has units of siemens/m. Many microwave wireless systems, radarsystems, and PCB circuits for computers use the microstrip type of construction.

8>

Transmission Line

o omi^-cwru^

— Ground Plane —

Figure 1.1 Model of a lossless transmission line.

Transmission lines are used to model distributed parameter circuits. Capacitance isdistributed over a region of space between two conductors and inductance is distributed alongthe length of these conductors. One cannot identify a resistance, inductance, capacitance, orconductance with any single point on the conductors since the energies associated with theseelements are distributed along the length of the conductors.

Two important parameters result from solving the differential equations for the uniformtransmission line equivalent circuits shown in Figure 1.1 and Figure 1.2 [1]. If a transmission

5

h

v2

L

C

L

C

LL

Vi C C

1

L L

Chapter 1 • Microwave Circuits

RectangularWaveguide

Shielded Microstrip Line

Some Transmission Line Types

Figure 1.3 Various types of transmission lines.

Twinlineand

TwistedPairs

6

Triaxial

Coaxial

Fin LineCircularWaveguide

Stripline Coupled Stripline

Slot LineCoplanar Line

Dielectric/Air Interface

Dielectric Conductor

Inverted SuspendedMicrostrip Line

SuspendedMicrostrip Line

Microstrip Line Coupled Microstrip Line

Figure 1.2 Model of a transmission line with losses.

Ground Plane

G C G- CVi

RL

RL

V2

h Transmission Line with Loss h

Section 1.2 • Circuit Elements 7

line has an infinite length, then the input impedance looking into the line is the characteristicimpedance of the line. For a lossless line, the characteristic impedance is purely real. Considerone L-C section of the line shown in Figure 1.2. Terminate that section in a real impedanceR. Set the impedance looking into this shunt C, series L, shunt R circuit equal to the samevalue of R.

(R+ja>LAx)jcoCAx D

— = KR + jaLAx + j ^

R + jcoLAx = R + jcoCAxR2 - co2LC(Ax)2R

o [LcoLAx = coCAxR2 => R = J —

This procedure can be done for each LC section along the line. The input impedance ateach spot along the line stays equal to the same value of R. The term containing the differentiallength squared has been ignored in the equation since it is much smaller than R. The value of thetermination impedance R is called the characteristic impedance Zo of a lossless transmissionline and is given by:

The values in the equivalent circuit of a transmission line are differential values, Ldx,Cdx, etc. Therefore, the cutoff frequency of these transmission line sections becomes infiniteas dx goes to zero.

/cutoffin^LdxCdx in+jLCdx

This might imply that the transmission line functions from dc all the way up to beyondlight frequencies. If the model were correct this might be true. However, notice that thereare no capacitors shown between nodes of the transmission line model. These capacitors doexist but are not shown for a TEM (transverse electromagnetic) transmission line. The energycontained in those capacitors for transmission lines operating in the TEM mode is negligible.This results from the assumption that important voltage differences exist only between theconductor and ground. The voltage difference between each of the nodes is zero since theelectric field in the direction of propagation is zero. The small impedance of the series R-Lcircuit essentially shorts out any capacitance reactance between these nodes. When voltagedifferences between the distributed nodes become important (i.e., the wavelength along the linebecomes short with respect to the distance between the conductor and ground), the TEM modelshown is no longer valid. The capacitance that does exist between nodes on the transmissionline then begins to have significant energy. In field analysis, the line is then said to supporthigher-order modes and the TEM assumption is violated.

Each section of the transmission line has a time delay associated with it. Recall that thedimension of the product LC is equal to time squared (the resonant frequency of an LC circuitis 1 over the square root of the product LC). This differential time delay for a small sectionof the line is:

r Ax = VLAxCAx = Vic AxWhen L and C are per-meter values, the time delay is also in seconds per meter. Therefore,the phase velocity v of wave propagation along a lossless transmission line is given by:

_ 1v ~ VLC

When lumped-constant equivalents for transmission lines are given later in this book forlines that have a finite length, the time delay for those circuits cannot be determined using this

ZQ =VC

1 1

Chapter 1 • Microwave Circuits

simple formula. Integration of the differential time delay of the transmission response needsto be done from dc to the frequency of interest to get the time delay at a given frequency forthose networks as discussed in Chapter 13.

TEM transmission lines have both the electric and magnetic field lines contained ina transverse plane perpendicular to the direction of propagation. Other lines such as themicrostrip and slot line have some components of the electric and magnetic field line in thedirection of propagation. Those lines that have only a very small amount of the total energyin the line associated with these longitudinal components of the electromagnetic field areoften termed quasi-TEM lines. At low frequencies, the microstrip line is a quasi-TEM line.In general, non-TEM lines have propagation constants and characteristic impedances thatvary with frequency. This type of line is called dispersive since its propagation constant andtherefore the phase velocity of propagation varies with frequency.

When a transmission line is propagating energy in a mode that is not TEM, the electricfield lines are not all contained in the transverse plane. Therefore, all of the electrical fieldenergy cannot be represented by a shunt capacitor. When the portion of the electric fieldtransverse to the line varies with frequency, the shunt capacitance also varies with frequency.Some of the shunt capacitive energy will be in air and some in the dielectric for an air/dielectricmicrostrip line. The phase velocity of propagation for an air-filled line is:

_ 1

where Co and Lo are the capacitance and inductance per unit length for an air-filled line andc is the speed of light. The phase velocity of propagation for a uniformly filled dielectric butnonmagnetic medium TEM transmission line is:

1 cv = , = for a TEM line

where sr is the relative dielectric constant of the dielectric material in the line. When the cross-sectional area of the transmission line contains more than one value of dielectric constant, thecapacitive energy is distributed among several dielectric constants. That distribution varieswith frequency and the resultant electric and magnetic field distributions are not TEM. In thiscase the phase velocity of propagation for the nonmagnetic, non-TEM line is not given byequation above but by:

1 cv = r _ = = , for a non-TEM line

where £reff is the effective dielectric constant of the line. The effective dielectric constantaccounts for the portion of electric energy that is distributed among the different materialswith different dielectric constants. Approximate equations for the effective dielectric constantand characteristic impedance of a microstrip line are given in Appendix A. Equations for othertransmission line configurations are given in the literature [2]. The approximate equations givenin Appendix A give an effective dielectric constant and characteristic impedance but they donot account for any variation with frequency. However, the results are quite useful when used atfrequencies for which that variation is small. Many computer-aided design programs performcomputations that determine the variation of these parameters with frequency. One shouldremember that a microstrip line is not a TEM line and at higher frequencies when the linedeviates from a TEM form, the variation of £reff with frequency should be taken into account.

One period of a sinusoidal signal of frequency / i s 1/ / . A wave with phase velocity vtravels a distance of v/f in one period of the waveform. The wave also has a phase shift oftwo pi over this distance at a given point in time. A phase shift constant beta with dimensions

^/LQCQ

8

V^reffV ^oQ) V£reff

y/sr v LQCQ >/£/'V =

Exercise

of radians per unit length is defined such that beta times one wavelength is two pi.

P " vA transmission line of length D is often referred to as theta radians long with a phase shiftconstant of beta.

1.2.4 Transmission Line Types

Figure 1.3 shows five types of microstrip lines. Whether one uses a suspended dielectricline or a dielectric line that is placed on a ground plane often depends on manufacturingrequirements. Fin line is used for placing printed circuit components and circuits insidewaveguides. Stripline is often used to seal the circuits for environmental and electromagneticprotection and for high-power operation. Coplanar circuits are often found on the top ofintegrated circuit chips. Coplanar circuits often also have a ground plane under the circuit.The top "ground" lines are often connected to the supporting ground plane. The effect of abottom conductor on the characteristics of the top coplanar circuit needs to be considered.Even if the back side of the chip or board does not have a conductor, the proximity effect of aconductor somewhere in the vicinity of the back side needs to be considered. Both coaxial andwaveguide circuitry are still used. They were used more often in the several decades beforeminiature manufacturing processes and devices were available.

Throughout this book, microstrip lines are referred to. However, the results of the variousanalyses in the text should not be limited to microstrip circuits and can be applied to differenttransmission line types. When a device, such as a transistor, is placed on a transmission line,a voltage and a current are assumed to exist on the conductor leading to the device. The typeof transmission line type is not as important in applying the results of the microwave analysisgiven here as much as the fact that there is a point at which a voltage and a current exist. Fordevices that interact with a field over a surface or a volume and for which a current or a voltagefor the device cannot be defined, the results of the analyses in this book have a more limitedapplication. However, when the structure contains multiple modes, the reader can extend theanalyses given in this book to some of those instances by using superposition of an analysesfor each of the multiple modes that exist. The scattering parameter techniques given in thisbook do apply to distributed structures, waveguide structures, and, as indicated in Section3.1.3, even to optical devices under certain conditions. The analyses are focused not on thecharacteristics of individual transmission line types but on developing techniques that apply tocircuit analysis and synthesis that have applicability independent of the transmission line typeor of whether the circuit is of a distributed nature.

EXERCISE

1-1 Calculate the characteristic impedance, phase velocity, effective dielectric constant, and phaseshift constant at 1 GHz for a line that has 35 pF/m capacitance and 750 nH/m inductance.Assume the material medium is nonmagnetic.

9

pV

f• =2TT

P =CO

V

/3D = 0