Lecture Notes - Microwaves

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy - 2009

These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

Microwaves

Lecture Notes

Dr. Serkan Aksoy

v.1.3.4

2009

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

Content

1. LUMPED CIRCUIT MODEL -----------------------------------------1

2. FIELD ANALYSIS of LINES -----------------------------------------1

2.1. Wave Propagation along the Line --------------------------------------------------------------------- 1

2.1.1.

Lossless Transmission Lines ------------------------------------------------------------------------------ 2

2.1.2.

Lossy Transmission Lines --------------------------------------------------------------------------------- 3

2.2. Smith Chart -------------------------------------------------------------------------------------------------- 3

2.3. Slotted Line -------------------------------------------------------------------------------------------------- 4

2.4. Generator & Load Mismatches ------------------------------------------------------------------------- 4

3. MICROWAVE NETWORKS -----------------------------------------5

3.1.

Voltage, Current and Impedance ---------------------------------------------------------------------- 5

3.2. Impedance & Admittance Matrices ------------------------------------------------------------------- 5

3.2.1.

Scattering Matrix -------------------------------------------------------------------------------------------- 5

3.2.2.

Transmission (ABCD) Matrix ---------------------------------------------------------------------------- 6

3.3. Equivalent Circuits for 2 Port Networks ------------------------------------------------------------- 6

3.4. Signal Flow Graphs --------------------------------------------------------------------------------------- 6

4. IMPEDANCE MATCHING ------------------------------------------8

4.1. L Networks Matching ------------------------------------------------------------------------------------- 8

4.2. Quarter Wave Transformer ------------------------------------------------------------------------------ 8

4.3. Single Stub Tuning---------------------------------------------------------------------------------------- 8

4.4. Double Stub Tuning -------------------------------------------------------------------------------------- 9

4.5. Tapered Lines ----------------------------------------------------------------------------------------------- 9

5. POWER DIVIDERS & COUPLERS ------------------------------ 10

5.1. Three Port Networks (T Junction) -------------------------------------------------------------------- 10

5.1.1.

T Junction Power Divider ------------------------------------------------------------------------------- 10

5.2. Four Port Networks (Directional Coupler) --------------------------------------------------------- 10

5.2.1.

Waveguide Directional Coupler ----------------------------------------------------------------------- 10

5.2.2.

Wilkinson Power Divider ------------------------------------------------------------------------------- 11

5.2.3.

Hybrid Coupler -------------------------------------------------------------------------------------------- 11

5.2.4.

Hybrid ------------------------------------------------------------------------------------------------ 11

5.2.5.

Coupled Line Directional Coupler -------------------------------------------------------------------- 11

5.2.6.

Lange Coupler --------------------------------------------------------------------------------------------- 11

5.3. Other Couplers -------------------------------------------------------------------------------------------- 11

6. NOISE & ACTIVE COMPONENTS ----------------------------- 12

6.1. Noise Figure, -------------------------------------------------------------------------------------------- 12

6.2. Dynamic Range & Intermodulation Distortion--------------------------------------------------- 13

6.3. RF Diode Characteristics -------------------------------------------------------------------------------- 14

7. MICROWAVE AMPLIFIER DESIGN --------------------------- 15

7.1. Two-Port Power Gains ---------------------------------------------------------------------------------- 15

7.2. Stability ----------------------------------------------------------------------------------------------------- 15

7.3. Single Stage Amplifier Design ------------------------------------------------------------------------ 16

7.4. Broadband Amplifier Design-------------------------------------------------------------------------- 16

7.5. Power Amplifiers ----------------------------------------------------------------------------------------- 16

7.5.1.

Large Signal Characterization -------------------------------------------------------------------------- 16

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

5These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

3.

MICROWAVE NETWORKS

At low frequencies for electrically small circuits, lumped activeand passive circuit elements are enough for analyzing the circuitleading a type of a Quasi-Static solution (assumption of negligible phase change in any where of the circuit) of Maxwell's equations and to Kirchoff Current and Voltage Lawswith impedance concept. Moreover fields are considered asTEM type. But this way is not possible to analyze microwavecircuits. The circuit concept should modify and apply tomicrowave network theory developed in MIT in 1940. Thereasons of using it are as follow

Much easier than field theory,Calculations are performed at terminals, not everywhere,Easy to modify and combine different problems,The field solution of Maxwell's equation gives moreinformation at the every time and place of the network, butdifficult. At microwave frequencies, although thedefinition of the terminal pairs for line is relativelyeasy, the terminal pair for line does notstrictly exist.

3.1.

Voltage, Current and Impedance

The measurements of and at microwave frequencies aredifficult due to not easily defined terminals for non-TEM waves.Because of that the fields are measured and used asthan, the impedance can be defined as . Because thefields depend on the coordinates (like in waveguide), specialattenuation should be given for extraction of and . The way isto do thatthen, the impedance can be defined as. The impedance concept first used by O. Heaviside, and then after application to transmission lines,to electromagnetics by Schelkunoff. In this manner, types of it

Intrinsic Impedance, : It depends on only thematerial parameters.Wave Impedance, : TM, TE, TEMtypes are present which may depend on type of guide,material and frequency.Characteristic Impedance, : For TEM, it isunique, but for TE and TM, not unique because andcan not be determined, uniquely.

It can be shown that the real parts of , and areeven in , but the imaginery parts of them are odd in .and are the even function in .

3.2.

Impedance & Admittance Matrices

The and of a port microwave network having 'th terminalThe impedance matrix is in the form of Similarly the admittance matrix is in the form of Clear that . It can be shown thatThen and are known as

Input Impedance

and

Transfer Impedance

, respectively.

If the network is reciprocal (no ferrites, plasmas andactive devices inside), and are the rightrelations.If the network is lossless, and are purelyimaginary.

3.2.1.

Scattering Matrix

The form of the scattering matrix gives the

complete description of the port networks with the incidentand reflected waves asAny element of the scattering matrix

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

6These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

and are

Reflection

and

Transmission Coefficient

(from port to port ), respectively. Network analyzer is used tomeasure parameters of a network. If network is reciprocal,is symmetric. If network is lossless, is unitary means that. Using the relations asand with between and matrixesIn the , all the and are defined as a reference point at theend of every lines. If the reference point is shifted, thenwhere is the electrical length of the outward shift.

3.2.1.1.

Generalized Scattering Matrix

In the previous chapter, is defined for networks with samecharacteristic impedance for all ports. Generally for not sameimpedance for all ports, a new set of wave amplitudes as,Then, generalized matrix- In reciprocal networks, is symmetric,- In lossless networks, is unitary and satisfies the equation

3.2.2.

Transmission (ABCD) Matrix

Many microwave networks consisting cascade connection andneed building block fashion in practice. ABCD matrix is definedto satisfy this aswhere port 1 and port 2 are completely isolated in the equationmeans that cascade multiplication is possible. The currentdirection is also specially designed for ABCD matrix. Therelation between and matrix parameters are asIf the network is reciprocal (), then .

3.3.

Equivalent Circuits for 2 Port Networks

Transition between a coaxial line and microstrip line can be chosen as an example of two port networks. Because of discontinuity in the transition region, EM energy is storedin the vicinity of the transition leading to reactive effectsmean that the transition region should be modeled as black box (There is an unlimited way of equivalent circuits, butchoosing matrix equivalence), then

A nonreciprocal network can not be represented by passive equivalent circuit using reciprocal elements. If thenetwork is reciprocal (there are six degrees of freedom, sixindependent parameters), the presentations lead naturallyto and equivalent circuits asIf the network is lossless, the impedance and admittancematrix elements are purely imaginary, the degrees of freedom reduces to three and the elements of andequivalent circuits should be constructed from purelyreactive elements.

3.4.

Signal Flow Graphs

Signal flow graph is an additional technique to analysemicrowave networks in terms of reflected and transmittedwaves. Three different forms of it are given below with nodesand branches.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

8These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

4.

IMPEDANCE MATCHING

Impedance matching (or tuning) is an important issue for - Maximum power is delivered when load is matched to line(assuming the generator is matched)- Power loss is minimized.- ratio of receiver components is increased.- Amplitude and phase errors are reduced.Whenever has nonzero real part, impedance matching is possible with the factors such as- Complexity: Simpler one is prefable.- Wide Bandwidth: Match a load over a band.- Implementation: Easier one is prefable- Adjustability: Adjust to match a variable load impedance.

4.1.

L Networks Matching

The normalized should be converted to with addingimpedance, then adding

–

the impedance matching will besuccessful.

4.2.

Quarter Wave Transformer

It is used for only real load impedance. Complex loadimpedance can always be transformed to real impedance byappropriate length of transmission line. But this generally altersthe frequency dependency of the equivalent load reducing the bandwidth of the matching. The following relation has to besatisfied aswhereFor multiple reflection, total isThe condition of is alsoenough to make total reflection (multiple) is zero.If and , then and .For each frequency, has to be equal to . Thus fixed linelength is possible only for one frequency.Bandwidth performance of QWT for wide band matching: If one set the maximum value of reflection coefficient,that can be tolerated, then the fractional bandwidth isThis shows that as becomes closer to , the bandwidthincreases. This result is valid only for TEM lines. The stepchanges of reactance effect can be compensated by making asmall adjustment in the length of the matching section.Approximate behavior of reflection coefficients is shown at below.The QWT can be extended as a multisection form for matchingof broader bandwidth.

4.3.

Single Stub Tuning

A single open-circuited (or short-circuited) transmission line isconnected either in parallel or series with the feed line at acertain distance from the load. Single Sub Tuning can match anyload impedance to line, but suffer from disadvantage of requiring a variable length between the load and stub.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

9These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

Since lumped elements are not required, single stub isconvenient and easy to fabricate in microstrip form. Twoadjustable parameters and susceptance or reactance provided by stub. Although for microstrip lines open circuit is easy tofabricate since a via-hole is enough, for coax or waveguide shortcircuit is preferred since open circuit line may be large for radiation. If the impedance has the form of at thedistance . Then stub reactance can be chosen as

–

, resultingfor matching condition. For a given susceptance or reactance,the difference in lengths of open and short-circuited stub is .

4.4.

Double Stub Tuning

The double stub tuner can not match all load impedances, butload may be arbitrary distance from the first stub.Distance between the stubs should be generally chosen asor to reduce the frequency sensitivity.

Single Section Transformer

: The reflection coefficient of single section transformer can be written when discontinuities between the impedances are small asThis shows that is dominated by and .

Multisection Transformer

:If the applications require more bandwidth, multisection transformers consists of equal lengthsections of the lines can be used. The reflection coefficient can be written asThe importance of this result is that the desired reflectioncoefficients response as a function of frequency can besynthesized by proper choosing of . To obtain passbandresponses, binominal (maximally flat) and Chebysev (equalripple) multisection matching transformers can be used. In firstone: the derivatives of is settled to zero, in secondone: is equated to Chebyshev polynomial.

4.5.

Tapered Lines

The line can be continuously tapered for decreasing the effect of the step changes in characteristic impedance between thediscrete sections. The incremental reflection coefficientSince , by using theory of small reflectionswhere from can be found. Chancing type of taper (Exponential, , Triangular, Klopfenstein), different band pass characteristic may be applied. Klopfenstein yields theshortest matching section. The Bode-Fano criterion for certaintype of canonical load impedances will help us to definetheoretical limit on the minimum reflection with the upper limitof matching performance and provide a benchmark againstwhich a practical design can be compared.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

10These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

5.

POWER DIVIDERS & COUPLERS

These are passive components used for power division or power combining. In power division, an input signal is divided bycoupler in two (or more) signals, equally or not.

5.1.

Three Port Networks (T Junction)

It has two inputs with one output.

If component is passive (no anisotropic material), the network is reciprocal and when all ports are also matched, considering lossless oneUsing the features of for lossless and reciprocal network To satisfy, above equation at least two parameters have to bezero means that three port network can not be reciprocal,lossless and matched all ports.If the network is nonreciprocal with matching all port andsatisfaction of energy conservation, such a device is known as

Circulator

relies on anisotropic materials.if only two ports of the network are matched, a lossless andreciprocal network can be physically realizable.If the network is being lossy, network can be reciprocal andmatched at all ports (

Resistive Divider

or

Isolator

). As anexample,

Ferrite Isolators

are two-port device havingunidirectional transmission characteristics. Because is notunitary, the isolator must be lossy. The isolators can be used between a high-power source and load to prevent possiblereflections from damaging the source by absorbing reflected power.

5.1.1.

T Junction Power Divider

This can be used for power division (or combining).

Lossless Divider

: This suffers from the problem of not beingmatched at all ports and in addition does not have any isolation between two output ports. The fringing fields and higher order modes at the discontinuity leading stored energy can beaccounted by a lumped susceptance, . The output lineimpedances and can be selected to provide various power decision.

Resistive Divider

: Possible to match all ports simultaneously,the resistive (lossy) divider is used, but no isolation between twooutput ports due to being not lossless. Half of the supplied power is dissipated in resistors.

5.2.

Four Port Networks (Directional Coupler)

It has two inputs and two outputs. After considering using thefeatures of matrix for reciprocal, matched and losslessnetwork, the possible solutions are means

Directional Coupler

. Using different phase references,

Symmetrical

or

Anti-symmetrical

Directional Coupler

may bedefined. The design parameters of directional coupler areThe coupling factor shows the fraction of input power to theoutput. The directivity is a measure of isolation ability for forward and backward waves. The ideal coupler has infinitedirectivity and isolation and also lossless.The directional property of the all directional coupler is produced through the use of two separate waves or wavecomponents, which add in phase at the coupled port, and cancelin phase at the isolated port.

5.2.1.

Waveguide Directional Coupler

Bethe Hole Coupler

: Couple one waveguide to another through a single small hole in thecommon wall. Types of the parallel guides and skewed guideswork properly only at the designfrequency (narrow bandwidth interms of its directivity).

Multi Hole Coupler

: Series of coupling holes are used to increase bandwidth as similar design tomulti section transformer. Makingcoupling coefficients proportionalto binominal coefficients,maximally flat response can be obtained. Using Chebysev polynomial, different responses are possible.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

11These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

5.2.2.

Wilkinson Power Divider

It is a network with the useful property of being lossless whenthe output ports are matched, that is, only reflected power isdissipated. It is known that a lossy three port network can bemade having all ports are matched with isolation between theoutput ports. Wilkinson Power Divider can be made inmicrostrip or stripline form with arbitrary power division of way Divider or Combiner. The even-odd mode technique is usedfor analysis.

5.2.3.

Hybrid Coupler

It has having types of the following.

5.2.3.1.

Quadrature Hybrid ( Hybrid)

This is a directional coupler (knows as

Branch Line Hybrid

) with a phase difference in outputs (2 3). Even-odd mode technique can be applied for analysis. matrix has ahigh degree of symmetrymeans any port can be usedfor input as given below

5.2.4.

Hybrid

It is a four port network with a phase shift (2 3) betweentwo outputs (also may be in phase). It can be used as a combiner and has unitary symmetric scattering matrix asIt may be produces as the form of ring hybrid (rate race),tapered matching lines and hybrid waveguide junction (MagicT, (Rate Race)) in which symmetrically (or antisymmetrical) placed tuning ports (or irises) can be used for matching.

5.2.5.

Coupled Line Directional Coupler

Coupled lines of two (or more) transmission lines are closedtogether, power can be coupled between the lines. GenerallyTEM mode is assumed rigorously valid for striplines, butapproximately valid for microstrips. Coupled Line Theory is based on types of excitations as even mode (strip currents areequal in amplitude with same directions) and odd mode (stripcurrents are equal in amplitude with opposite directions).Arbitrary excitation can be treated as a superposition of appropriate even and odd modes amplitudes. Moreover designgraphs are present for coupled lines.

Design Considerations:

- Although a single section coupled line has limited bandwidthdue to requirement, the bandwidth can be increased usingmultiple sections coupled line having close relations tomultisection QWT.- The assumption of the same velocity of propagation for evenand odd modes in design, generally not satisfied for a coupledmicrostrip or non TEM lines. This gives poor directivity. Byusing more effective dielectric constant (smaller phase velocity)for even mode, phase differences should be minimized. Thisalso produces problems as the mismatching phase velocities for multisection case and degrades coupler directivity. Increasing bandwidth can be obtained with low coupling limits.

5.2.6.

Lange Coupler

To increase coupling factor,Lange Coupler (severallines) with phasedifference between outputsis used as a 3 dB couplingratio in an octave or more bandwidth can be achieved.The main disadvantage of it(a type of quadraturehybrid) is difficult tofabricate due to verynarrow lines. Folded Langecoupler is also used for more easily analysis to model equivalentcircuit.

5.3.

Other Couplers

Moreno Crossed Guide Coupler

Schwinger Reversed Phase Coupler

Riblet Short Slot Coupler

Symmetric Tapered Coupled Line Coupler

Coupler with Apertures in Planar Lines

As an example of a device uses a directional coupler is

Reflectometer

isolate and sample the incident and reflected powers from a mismatch load as a heart of a scalar (or vectorial)network analyzer.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

13These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

If the line is at the temperature , .

Noise Figure of Cascaded System

:A cascaded system of twocomponents (), () andThe output of first stage :The output of second stage:where and . The first stage isdominant due to reduces to second stage. Generally

Noise Figure of a Passive Two Port Network

:

Available Power Gain

does not depend on .where , then isasThe definitions of and are close as lossy lines. When thenetwork is matched , then

Noise Figure of a Mismatched Lossy Line

:Previously, NoiseFigure of a lossy line is calculated under assumptions of line ismatched to its input and output circuits. Consider that lineis mismatched to input circuit asThenWhen the line is matched same asthe matched lossy line.

6.2.

Dynamic Range & Intermodulation Distortion

All realistic devices are nonlinear at very low power levels dueto noise effects and also practical components became nonlinear at high power levels. A given component (or network) canoperate as desired between minimum and maximum realistic power ranges known as

Dynamic Range

. In the sense of intermodulation distortion, it is called as

Spurious Free Dynamic Range

. The output response of a nonlinear device(diode, transistor) by using a Taylor series expansionwhere is DC output, is Lineer output, others are Nonlinear outputs. Different output responsecan be obtained relating to coefficients asOnly , Rectifier Only , Attenuator or Amplifier Only , Mixing or Frequency Conversion

Gain Compression:

In most practical system, , then gain tends to decreasenamed

Gain Compression

or

Saturation

.

Intermodulation Distortion:

2

the output is the combination of two inputfrequencies are called

Intermodulation Products

.Passive Intermodulation: Connectors, cables, antennas, andevery metal-metal contact can cause passive intermodulationdue to poor mechanical contact, oxidation, contamination etc.,and also thermal effects of high power source. This hasgenerally lower power levels.

Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy

–

2009

16These lecture notes are heavily based on the book of

Microwave Engineering

by David M. Pozar. For future versions or any proposals, please contact with

Dr. Serkan Aksoy

(saksoy@gyte.edu.tr).

If device is unconditionally stable, the stability circles must becompletely outside (or totally enclose) the Smith chart. This can be stated mathematically asfor for If or , the amplifier can not beunconditionally stable because a source (or load impedance)leading to (or ) can cause (or ). If transistor is only conditionally stable, and must bechosen in stable regions. Specially unconditionally stability can be tested with the methods of

Rollet’s Condition

or

Parameter

.

7.3.

Single Stage Amplifier Design

Maximum gain with stability can be realized when input andoutput sections provide a conjugate match between source andload impedance, but generally as a narrowband. To perform thisconditions simultaneously have to be satisfied means that also by maximizing the transducer gain, first of all , then should be solved by considering stability conditions. It is also preferable to design for less than the maximum obtainable gain,to improve bandwidth (or to obtain a specific amplifier gain). Todo that,

Constant Gain Circles

on the Smith chart to representloci of and that give fixed values of gain are used. Besidesstability and gain, the Noise Figure of the amplifier should beminimized by using

Constant Noise Figure Circles

.

7.4.

Broadband Amplifier Design

The bandwidth can be improved with designing for less thanmaximum gain will improve bandwidth, but the input andoutput ports will be poorly matched. Common approaches tosolve this problem are listed below-

Compensated Matching Network,-

Resistive Matching Network,-

Negative Feedback,-

Balanced Amplifiers,-

Distributed Amplifiers.

7.5.

Power Amplifiers

This is used to increase power level with consideration of efficiency, gain, intermodulation and thermal effect.

Amplifier Efficiency

is defined aswith the effect of input power,

Power Added Efficiency, PAE

where is power gain.

PAE

drops quickly with frequency.Another parameter is

Compressed Gain

defined as the gain of amplifier at 1 dB compression gain aswhere is small signal (linear) power gain. Class A amplifierswith theoretically maximum efficiency are inherentlylinear that transistor is biased to conduct over entire range of input signal cycle (low-noise amplifier). Class B amplifiers withtheoretically maximum efficiency are biased to conductonly during one-half of input signal cycle (Push-Pull amplifier).Class C amplifiers with efficiency near are operated withtransistor near cut-off for more than half of the input signalcycle (in a resonant circuit, constant envelope modulation).Higher classes such as D, E, F and S are also used with highefficiency.

7.5.1.

Large Signal Characterization

If input power is small enough, parameter is independent frominput power and linear small signal model is suitable for modeling. But for high input powers, transistor behaves as anonlinear device (

Large-Signal Characterization

) and moredifficult to design. The following methods are possible for LargeSignal Characterization-

Measure the output power as a function of source and loadimpedances and produce tables, then determine large signalsource and load reflection coefficients to maximize power gainfor a particular output power.-

Plot contours(

Load-Pull Contours

) of constant power outputon a Smith chart as a function of load reflection coefficient withconjugately matching at input and design for a specified gain.-

Use nonlinear equivalent of the transistor circuit.Especially for designing Class A amplifiers, the stability can bechecked by using small signal model because instabilities beginat low signal levels.

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