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- 1. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy - 2009These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).MicrowavesLecture NotesDr. Serkan Aksoyv.1.3.42009www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 2. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 2009These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).Content1. LUMPED CIRCUIT MODEL-----------------------------------------12. FIELD ANALYSIS of LINES -----------------------------------------12.1. Wave Propagation along the Line --------------------------------------------------------------------- 12.1.1. Lossless Transmission Lines------------------------------------------------------------------------------22.1.2. Lossy Transmission Lines---------------------------------------------------------------------------------32.2. Smith Chart-------------------------------------------------------------------------------------------------- 32.3. Slotted Line-------------------------------------------------------------------------------------------------- 42.4. Generator & Load Mismatches------------------------------------------------------------------------- 43. MICROWAVE NETWORKS -----------------------------------------53.1. Voltage, Current and Impedance ---------------------------------------------------------------------- 53.2. Impedance & Admittance Matrices ------------------------------------------------------------------- 53.2.1. Scattering Matrix --------------------------------------------------------------------------------------------53.2.2. Transmission (ABCD) Matrix ----------------------------------------------------------------------------63.3. Equivalent Circuits for 2 Port Networks------------------------------------------------------------- 63.4. Signal Flow Graphs --------------------------------------------------------------------------------------- 64. IMPEDANCE MATCHING ------------------------------------------84.1. L Networks Matching------------------------------------------------------------------------------------- 84.2. Quarter Wave Transformer------------------------------------------------------------------------------ 84.3. Single Stub Tuning---------------------------------------------------------------------------------------- 84.4. Double Stub Tuning -------------------------------------------------------------------------------------- 94.5. Tapered Lines----------------------------------------------------------------------------------------------- 95. POWER DIVIDERS & COUPLERS ------------------------------ 105.1. Three Port Networks (T Junction)--------------------------------------------------------------------105.1.1. T Junction Power Divider ------------------------------------------------------------------------------- 105.2. Four Port Networks (Directional Coupler)---------------------------------------------------------105.2.1. Waveguide Directional Coupler----------------------------------------------------------------------- 105.2.2. Wilkinson Power Divider ------------------------------------------------------------------------------- 115.2.3. Hybrid Coupler-------------------------------------------------------------------------------------------- 115.2.4. Hybrid ------------------------------------------------------------------------------------------------ 115.2.5. Coupled Line Directional Coupler -------------------------------------------------------------------- 115.2.6. Lange Coupler --------------------------------------------------------------------------------------------- 115.3. Other Couplers --------------------------------------------------------------------------------------------116. NOISE & ACTIVE COMPONENTS ----------------------------- 126.1. Noise Figure, --------------------------------------------------------------------------------------------126.2. Dynamic Range & Intermodulation Distortion---------------------------------------------------136.3. RF Diode Characteristics--------------------------------------------------------------------------------147. MICROWAVE AMPLIFIER DESIGN--------------------------- 157.1. Two-Port Power Gains ----------------------------------------------------------------------------------157.2. Stability -----------------------------------------------------------------------------------------------------157.3. Single Stage Amplifier Design------------------------------------------------------------------------167.4. Broadband Amplifier Design--------------------------------------------------------------------------167.5. Power Amplifiers -----------------------------------------------------------------------------------------167.5.1. Large Signal Characterization-------------------------------------------------------------------------- 16www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 3. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20091These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).1. LUMPED CIRCUIT MODELFields Circuits: Real size of the circuit, : WavelengthCircuitTransmission Lineand will vary in magnitude and phase over its length.: Resistance per unit length ( ): Finite conductivity: Inductance per unit length ( ): Self inductance of two wires: Capacitance per unit length ( ): Proximity conductors: Conductance per unit length ( ): Dielectric lossesKirchoff Voltage LawKirchoff Current LawTaking the limit as , Telegraph equationsTime domainFrequency domain2. FIELD ANALYSIS of LINESThe idea is to find relations between and with .where , and , , , are time-averaged values calculated for length line and is the linecross-sectional area.2.1. Wave Propagation along the LineUsing the frequency domain Telegrapher equationwhere is propagation constant.The solution of Telegrapher equationThen, taking the derivation of , the calculation ofThenConverting to time domain by usingwhere , , is the phase.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 4. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20092These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).2.1.1. Lossless Transmission LinesWhen the lossless line is terminated by a loadReflected waves occur.Reflection coefficient at the load,where and consists of a superposition of an incidentand reflected waves called Standing Waves ( ).Time Average Power Flow:This shows is constant at anywhere on the line.When the line is matched is constant.When the line is mismatched ( ReturnLossMatched load (No reflectedpower, maximum power is delivered).Total reflection, (All powerreflected).is not constant.The maximum value occurs when. The minimum valueoccurs when . When increases,increases as a measure of mismatch. Then Standing Wave Ratio( ) iswhen means matched line. In that case at , thereflection coefficient and input impedanceUsing the definition of , more useful form known asTransmission Line Impedance Equation as- Transmission Coefficient:Some part of EM wave is alsotransmitted to second region asInsertion Loss:Short Circuit:Open Circuit:The proper length of open or short circuited transmission linecan provide any desired reactance or susceptance.(No regard to ): The sameimpedance is observed at the input.(Quarter WaveTransform)- (Short circuit) Open circuit- (Open circuit) Short circuitwww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 5. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20093These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).2.1.2. Lossy Transmission LinesIn practice, finite conductivity (or lossy dielectrics) lines can beevaluated as a Lossy Line.Low-loss line , . Thenignoring the last term of In the lossy line; , & can be approximated tolossless line.Distortionless Line: For the lossy line, in fact the exactis not a linear function of frequency meansdispersive. But specifically if the following conditionholdsthen , mean that the lossy linebehaves as lossless (distortionless) line.Terminated Lossy Line: Loss is assumed small thatPower lost in the line:Perturbation Method for Calculating AttenuationPower flow along lossy line:Power loss per unit length:Attenuation constant:Wheeler Incremental Inductance Rule:: Incremental inductance: Loss per unit length: Power at the .Then, Attenuation constant:where is skin depth and .Taylor series Inductance Rule:where .2.2. Smith Chartwhere is Resistance, is Reactance, is Conductance andis Susceptance. Whenever is normalized impedanceThe apsis and ordinate of Smith chart are and .Rearranging themwww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 6. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20094These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).These are two families of circles as and . Superposition ofSmith Chart and its 180o( ) rotated version is known asCombined Impedance-Admittance Smith Chart.is the complete revolution of Smith chart. is the halfof Smith chart (180o). The images of is in Smith chart.2.3. Slotted LineThis device is used to find as first .- Measurement of on the distance from the line.- Calculate-- Using and , write- Calculate at .2.4. Generator & Load MismatchesThen, using thiswhere and .where and . Generally isfixed and three cases are considered asLoad Matched to Line: , , .Then .Generator Matched to Line: , .Conjugate Latching:Maximum power transfer . If one directlychose , it does not mean that the best efficiencydue to the phase differences. The efficiency can be improvedonly by making as small as possible.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 7. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20095These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).3. MICROWAVE NETWORKSAt 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 ofnegligible phase change in any where of the circuit) ofMaxwells 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 followMuch easier than field theory,Calculations are performed at terminals, not everywhere,Easy to modify and combine different problems,The field solution of Maxwells 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 ImpedanceThe 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 usedby O. Heaviside, and then after application to transmission lines,to electromagnetics by Schelkunoff. In this manner, types of itIntrinsic 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 MatricesThe and of a port microwave network having th terminalThe impedance matrix is in the form ofSimilarly the admittance matrix is in the form ofClear that . It can be shown thatThen and are known as Input Impedance and TransferImpedance, 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 MatrixThe form of the scattering matrix gives thecomplete description of the port networks with the incidentand reflected waves asAny element of the scattering matrixwww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 8. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20096These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).and are Reflection and Transmission Coefficient (fromport 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 MatrixIn 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 equation3.2.2. Transmission (ABCD) MatrixMany 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 NetworksTransition between a coaxial line and microstrip line canbe chosen as an example of two port networks. Because ofdiscontinuity 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 blackbox (There is an unlimited way of equivalent circuits, butchoosing matrix equivalence), thenA nonreciprocal network can not be represented bypassive 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 offreedom reduces to three and the elements of andequivalent circuits should be constructed from purelyreactive elements.3.4. Signal Flow GraphsSignal 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.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 9. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20097These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).The decomposition rules are also given belowwww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 10. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20098These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).4. IMPEDANCE MATCHINGImpedance 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 ispossible 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 MatchingThe normalized should be converted to with addingimpedance, then adding – the impedance matching will besuccessful.4.2. Quarter Wave TransformerIt 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 thebandwidth 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: Ifone 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 atbelow.The QWT can be extended as a multisection form for matchingof broader bandwidth.4.3. Single Stub TuningA 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 ofrequiring a variable length between the load and stub.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 11. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 20099These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. 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 providedby 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 forradiation. 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 TuningThe 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 ofsingle section transformer can be written when discontinuitiesbetween the impedances are small asThis shows that is dominated by and .Multisection Transformer: If the applications require morebandwidth, multisection transformers consists of equal lengthsections of the lines can be used. The reflection coefficient canbe 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 LinesThe line can be continuously tapered for decreasing the effect ofthe step changes in characteristic impedance between thediscrete sections. The incremental reflection coefficientSince , by using theory ofsmall reflectionswhere from can be found. Chancing type of taper(Exponential, , Triangular, Klopfenstein), differentband 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.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 12. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200910These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).5. POWER DIVIDERS & COUPLERSThese are passive components used for power division or powercombining. 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 networkis reciprocal and when all ports are also matched, considering lossless oneUsing the features of for lossless and reciprocal networkTo 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 asCirculator 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 usedbetween a high-power source and load to prevent possiblereflections from damaging the source by absorbing reflectedpower.5.1.1. T Junction Power DividerThis 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 isolationbetween two output ports. The fringing fields and higher ordermodes at the discontinuity leading stored energy can beaccounted by a lumped susceptance, . The output lineimpedances and can be selected to provide various powerdecision.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 suppliedpower 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 meansDirectional 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 forforward and backward waves. The ideal coupler has infinitedirectivity and isolation and also lossless.The directional property of the all directional coupler isproduced 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 CouplerBethe Hole Coupler: Couple one waveguide to anotherthrough a single small hole in thecommon wall. Types of theparallel guides and skewed guideswork properly only at the designfrequency (narrow bandwidth interms of its directivity).Multi Hole Coupler: Series ofcoupling holes are used to increasebandwidth as similar design tomulti section transformer. Makingcoupling coefficients proportionalto binominal coefficients,maximally flat response can be obtained. Using Chebysevpolynomial, different responses are possible.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 13. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200911These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).5.2.2. Wilkinson Power DividerIt 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 ofway Divider or Combiner. The even-odd mode technique is usedfor analysis.5.2.3. Hybrid CouplerIt has having types of the following.5.2.3.1. Quadrature Hybrid ( Hybrid)This is a directional coupler (knows as Branch LineHybrid) 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 below5.2.4. HybridIt is a four port network with a phase shift (2 3) betweentwo outputs (also may be in phase). It can be used as a combinerand 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 CouplerCoupled 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 isbased 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 ofappropriate 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 formultisection case and degrades coupler directivity. Increasingbandwidth can be obtained with low coupling limits.5.2.6. Lange CouplerTo increase coupling factor,Lange Coupler (severallines) with phasedifference between outputsis used as a 3 dB couplingratio in an octave or morebandwidth 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 CouplersMoreno Crossed Guide CouplerSchwinger Reversed Phase CouplerRiblet Short Slot CouplerSymmetric Tapered Coupled Line CouplerCoupler with Apertures in Planar LinesAs an example of a device uses a directional coupler isReflectometer isolate and sample the incident and reflectedpowers from a mismatch load as a heart of a scalar (or vectorial)network analyzer.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 14. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200912These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).6. NOISE & ACTIVE COMPONENTSNoise is usually generated by random motions of charges (orcharge carriers in devices and materials). Such motions can becaused by the mechanism ofThermal Noise: Thermal vibrations of bound charges.Shot Noise: Random fluctuations of charge carriers.Flicker Noise: noise.Plasma Noise: Random motions of charges.Quantum Noise: Quantized nature of charge carriers.Noise is a random process and can be passed into a system fromexternal sources or generated within the system itself. Noiselevel defining the system performance determines for minimumsignal reliability detected by a receiver.Dynamic Range and Compression Point: The linearity anddeterministic features of all components can be satisfied in arange called Dynamic Range. The floor level of noise dominatesthe output power at very low frequencies. CompressionPoint is defined as the input power for which the output isbelow that of an ideal amplifier.Noise Power and Equivalent Noise Temperature: Rayleigh-Jeans approximation results Voltage Fluctuations aswhere is Boltzmann’s constant, is temperature, isbandwidth and is resistance. Because of frequencyindependency, this is known as White Noise Source can betreated as Gaussian distributed variables.A noisy resistor can be replaced with a noiseless resistorand a voltage source of RMS . Then connecting a load resistorresults in maximum power transfer called Noise Power as- then : Cooler device, less noise power.- then : Smaller bandwidth, less noise power.- then : Use the exact definition of for .If is not strong function of frequency (White Noise), anEquivalent Noise Temperature is defined aswhere is Noise Power delivered to load .A noisy amplifier with a source of resistor at a temperatureof can be replaced with a noiseless amplifier and aresistor having Equivalent Noise Temperature aswhere is output noise power and is amplifier gain. ExcessNoise Ratio (ENR) is also used to characterize Noise Power ofactive noise generator consisting of a diode or a tube aswhere & are Noise Power & Equivalent Temperature ofgenerator.Measurement of Noise Power: factor method is applied aswhere should be determined via power measurement. Thenwhere & are temperature of hot & cold load, respectively.6.1. Noise Figure,It is a measure of the degration in ratio between the inputand output asis defined for a matched input source and for a noise sourceconsist of a resistor temperature .Noise Figure of a Noisy Network: Having the parameters ofwith input noise and signal power , theoutput noise power and the output noisesignal . Then Noise Figure isIf the network is noiseless , .Noise Figure of a Two-Port Passive and Lossy Network:Having such as attenuator (or lossy line) with a matchedsource resistor at , overall system temperature also at , noisefactorwhere is lossy factor. The equivalent noisetemperaturewww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 15. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200913These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. 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. GenerallyNoise Figure of a Passive Two Port Network: Available PowerGain does not depend on .where , then isasThe definitions of and are close as lossy lines. When thenetwork is matched , thenNoise 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 DistortionAll realistic devices are nonlinear at very low power levels dueto noise effects and also practical components became nonlinearat high power levels. A given component (or network) canoperate as desired between minimum and maximum realisticpower ranges known as Dynamic Range. In the sense ofintermodulation distortion, it is called as Spurious FreeDynamic Range. The output response of a nonlinear device(diode, transistor) by using a Taylor series expansionwhere is DC output, is Lineeroutput, others are Nonlinear outputs. Different output responsecan be obtained relating to coefficients asOnly , RectifierOnly , Attenuator or AmplifierOnly , Mixing or Frequency ConversionGain Compression:In most practical system, , then gain tends to decreasenamed Gain Compression or Saturation.Intermodulation Distortion:2the 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.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 16. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200914These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).6.3. RF Diode CharacteristicsShottky Barrier Diode Detectors: This is a nonlinear deviceconsisting of semiconductor-metal junction resulting lowerjunction capacitance can be used frequency conversion(rectification, detection, mixing). It has awith a Small Signal ModelThese diodes are used as rectifiers, detectors and demodulationof an AM modulated RF carrier.PIN Diode: This is used to construct an electronic switchingfor control circuits such as phase shifters and attenuators. Theseare preferable because of small size, high speed and inerrabilitywith planar circuits. Especially single-pole PIN diode switchescan be used in either a series or a shunt configuration to form asingle pole RF switch. Insertion Loss of switcheswhere is diode impedance asVaractor Diode: Junction capacitance varies with bias voltageused for electronically frequency tuning.Impatt Diode: Similar to PIN diode, but based on avalancheeffects exhibiting negative resistance over a broad frequencyrange, therefore used to directly convert DC to RF power.Gunn Diode: It exhibits a negative differential resistancebased on Gunn effect and used to generate RF power to DC.Baritt Diode: Similar to junction transistor without a basecontact and useful for detector and mixer applications withadvantages of lower AM noise.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 17. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200915These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).7. MICROWAVE AMPLIFIER DESIGN7.1. Two-Port Power GainsThe gain and stability of a general two-port amplifier in terms ofparameters of transistor will be investigated for amplifier andoscillator design. Three types of power gain can be derived aswhere is independent of . is defined with an assumptionthat conjugate matching of both source and load depend onbut not . depends on both and . Whenever input andoutput are both conjugately matched, gain is maximized and.The average power delivered to networkThe power delivered to loadThen, the power gainwhereIf , then .If , then , Unilateral Transducer Power Gain,More generally, most useful power definition is TransducerPower Gain account for both source and load mismatchwhere, ,If transistor is unilateral, ; , , then, ,Similar relation can be obtained by Equivalent circuit parameter.7.2. StabilityThere are necessary conditions for a transistor amplifier to bestable based on the possible oscillation for input and outputimpedance has a negative real part as a two-sub group:- Conditional Stability: If and , network isstable for a range of passive source and load impedance.- Unconditional Stability: If and for allpassive sources and loads, network is unconditionally stable.The stability condition is usually frequency dependent sincematchings generally depend on frequency (stability may bepossible for a frequency but not possible for others). Rigoroustreatment of stability requires parameters of network have nopoles in the right-half complex plane in addition toand . If device is unilateral , more simplyresults and are enough for stability.Stability Circles: Applying the above requirement forunconditional stability, following conditions have to be satisfiedThese conditions define a range for and where amplifierwill be stable. Finding this range by using Smith chart, plottingthe input and output Stability Circles are defined as loci in the(or ) plane for which (or ), then defineboundaries between stable and unstable regions. The equationsfor input and output stability conditions can be extracted aswww.jntuworld.comwww.jntuworld.comwww.jwjobs.net
- 18. Microwaves - Lecture Notes - v.1.3.4 Dr. Serkan Aksoy – 200916These lecture notes are heavily based on the book of Microwave Engineering by David M. Pozar. For future versions or any proposals, please contact withDr. Serkan Aksoy (saksoy@gyte.edu.tr).If device is unconditionally stable, the stability circles must becompletely outside (or totally enclose) the Smith chart. This canbe stated mathematically asforforIf 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 canbe tested with the methods of Rollet’s Condition orParameter.7.3. Single Stage Amplifier DesignMaximum 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 alsoby maximizing the transducer gain, first of all , then shouldbe solved by considering stability conditions. It is alsopreferable 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 DesignThe 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 AmplifiersThis is used to increase power level with consideration ofefficiency, gain, intermodulation and thermal effect. AmplifierEfficiency is defined aswith the effect of input power, Power Added Efficiency, PAEwhere is power gain. PAE drops quickly with frequency.Another parameter is Compressed Gain defined as the gain ofamplifier 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 ofinput 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 CharacterizationIf input power is small enough, parameter is independent frominput power and linear small signal model is suitable formodeling. 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.www.jntuworld.comwww.jntuworld.comwww.jwjobs.net

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