An Introduction To Broadband Impedance Transformation For .
From January 2009 High Frequency ElectronicsCopyright 2009 Summit Technical Media, LLCHigh Frequency DesignBROADBAND MATCHINGAn Introduction to BroadbandImpedance Transformationfor RF Power AmplifiersBy Anthony J. BichlerRF Micro Devices, Inc.This paper discussesbroadband impedance-transforming techniques specific forradio frequency poweramplifiers. Single andmultiple Q matchingtechniques are demonstrated for broadband performance; here thereader will understand the importance of aload impedance trajectory relevant to loadpull contours.This tutorial article reviewsimpedance matchingprinciples and techniques,as they are applied topower device matchingin amplifier circuitsIntroductionWhen analytically defining radio frequency circuits, a common approach incorporatesadmittance or impedance. Admittance, whichis symbolized by Y, is defined in terms of conductance G and an imaginary susceptancecomponent, jB. Admittance is often usefulwhen defining parallel elements in a networkand is expressed by the complex algebraicequation Y G jB.Impedance, the mathematical inverse ofadmittance, is symbolized by Z and consists ofa resistive component R in units of ohms and areactive or imaginary component jX. Togetherin a series complex expression they defineimpedance as Z R jX. Impedance in thisrectangular form is often used in industry todefine a power device’s optimal source or load.For linear systems, the condition for maximum power transfer is obtained when theimpedance of the circuit receiving a signal hasan equal resistance and an opposite reactanceof the circuit sending the signal. In the mathematics of complex variables, this relationshipis known as the complex conjugate. The complex conjugate of a complex number is34High Frequency Electronicsobtained by simply reversing the sign of theimaginary part. Here Z* denotes the complexconjugate of Z; thus, for linear systems thecondition for maximum power transfer iswhen ZLoad ZSource*, or: ZL ZS*.As the frequency of operation changes forZS, relative to its parasitics, the value of theresistive component can substantially changeas well as the value of the imaginary component. Transforming a standard systemimpedance to present a driving point loadimpedance ZL that maintains a complex conjugate relationship to the source impedancechange over frequency is the most challengingaspect of broadband design.Note: The linear condition for maximumpower transfer is often traded for other performance parameters such as efficiency orgain. For this tradeoff the load impedance willnot hold a conjugate relationship; however, thechallenge of maintaining a load for this performance parameter over a broadband willgenerally remain the same.A Review of Smith Chart FundamentalsPhilip H. Smith introduced the SmithChart in Electronics Magazine on January1939, revolutionizing the RF industry [1, 2].This chart simplified complex parallel toseries conversions graphically and, for thefirst time, provided intuitive transmission linesolutions.The Smith Chart is a graphical reflectioncoefficient system with normalized conformalmapping of impedance or admittance coordinates, as shown Figure 1 and 2, respectively.Reflection coefficient is often referred to asgamma and is symbolized by the Greek letterΓ. Gamma in its simplest form is defined as
High Frequency DesignBROADBAND MATCHINGFigure 1 · The Impedance SmithChart.Figure 2 · The Admittance SmithChart.the ratio of the root of the incidentpower wave versus the root of thereflected power wave [3]:will conveniently reference to 50ohms or another value as defined bythe user.The Smith Chart’s horizontal centerline is known as the resistive line.It is scaled left to right, zero to infinity, with the normalized impedance(Zo) centered in the middle of thechart. On the lower half of the chartbelow the resistive line, are thecapacitive coordinates and above theline are the inductive coordinates.The circles that are tangent to theright side of the impedance SmithChart are the circles of constantresistance. Above and below the rightside at R infinity are the semicircles of constant reactance.On the Admittance Smith Chart,the circles that are tangent to the leftside are the circles of constant conductance and the semicircles aboveand below to the left where R 0 arethe circles of constant susceptance.The Admittance Smith Chart issimply a mirror image of theImpedance Smith Chart where theImpedance Smith Chart can be rotated by 180 degrees to serve as anAdmittance Chart. This duality of theSmith Chart is exploited for admittance to impedance conversions bysimply rotating both the reflectioncoefficient vector by 180 degrees andthen the chart itself by 180 degrees.Note that since the Smith Chart is areflection system, 180 degrees aroundΓ WrWiwhere Wr is the reflected power andWi is the incident power. Gamma canalso be defined in terms of impedancewhereΓ Z L ZSZ L ZSFor a large impedance mismatchΓ would approach unity, and for anear perfect match Γ would approachzero.Impedances are often normalizedwhen plotted on the Smith Chart.Normalizing ZL to the center of theSmith Chart where Z 1 gives oneaccess to the chart’s maximum resolution. Note that normalizing is thedivision of an impedance by a reference [4]. For example, normalizingwith the 50-ohm system loadimpedance as the reference (Z0 50),a source impedance of ZS 100 j50would normalize to ZS /Z0 2 j1 zs, where the lower case z is used fornormalized impedances. Most computer aided design or Smith Chartprograms have simplified the normalization process. In the case ofnormalizing, the center of the chart36High Frequency ElectronicsFigure 3 · The Immittance Chartwith an SWR circle (green line)defined by [Γ] radius.the circumference is equivalent to 90degrees of wavelength rotation.A Smith Chart that has combinedthe Admittance and ImpedanceSmith Charts for simplicity is knownas the Immittance Chart [5]. Withthe Immittance Chart a network consisting of capacitors or inductors inshunt or in series can be easily cascaded without rotating the chart.In Figure 3, an Immittance Chart,the Gamma vector magnitude definesthe radius of a constant standingwave ratio (SWR) circle. Standingwaves are a phenomenon of the voltage or current waves from the summation of an incident power wave andreflected power wave on a transmission line from a mismatched load.SWR is the ratio of the maximum versus the minimum voltage or currenton a standing wave and is commonlyreferred to as VSWR or ISWR, forvoltage or current respectively. [6, 7]SWR can be expressed in terms ofGamma’s magnitude bySWR 1 [Γ ]1 [Γ ]or directlyimpedancesSWR orZLZSbythemismatch
SWR ZSZLwhere the equation choice is dictatedby which one provides a quantitygreater than unity.SWR circles are used throughoutthe following Smith Chart illustrations to quantify the mismatch overfrequency.The Importance of QualityFactorsIt is important to understand thequality factors Q, as they are integralto bandwidth. Q factors are used todefine the quality of a reactive element by its ability to store energy, tofundamentally define bandwidth,and to define the ability of a loadednetwork to store energy. To ease someof the confusion with these Q factorsthey have been assigned the termsunloaded Q, loaded Q, and Q of theload respectively [8].Unloaded Q is fundamentallydefined as the ratio of stored energyversus dissipated energy [10] orQUnloaded reactive power I 2 X 2real powerI Rwhich reduces toQUnloaded XRFor capacitors, unloaded Q isexpressed as a ratio of capacitivereactance to equivalent series resistance (ESR) [9] orQUnloaded XCRESRand for inductors unloaded Q isexpressed byQUnloaded XLRwhere R is the series resistance fromthe windings of the coil.Loaded Q is defined by the band’scenter frequency (FC) divided by the3 dB bandwidth and is expressed asQUnloaded FCBWFor simple resonant tank networks, unloaded Q can be substitutedwith loaded Q in bandwidth calculations [11]. When the resonant frequency is equal to the center frequency, then unloaded Q can define thebandwidth byBW FCQUnloadedor with substitutionBW FCX/RThe Q of the load is often used todefine a loaded network, which typically consists of ideal (lossless) matching elements. The network is not lossless since energy is propagated to andabsorbed by the load. It is defined asbefore with the unloaded Q as a ratioof the reactance to resistanceQ XRor in terms of vectors; the imaginarycomponent magnitude versus theresistive component.Plotting Q of the load as constantratio on the Smith Chart will define aconstant Q curve. These Q curves areoften used as guideline boundariesfor broadband transformations andwill be used throughout the followingillustrations to define the transforming networks. As a rule in broadbandtransformations, maintaining a lowerQ curve for a given transformation byincreasing the number of n-sectionswill yield a higher bandwidth.For a single section transformation where the resistive line and aconstant Q of the load curve boundthe transformation, the relationshipbetween the Q of the load and theresistive transformation ratio isgiven by [12]1 Q2 RratioFor increasing bandwidth byincreasing the number of n-sectionshaving equal Q the relationshipbecomes1 Q2 (nRratio)Note: Using the guideline boundaries above in this reference does notyield the optimal broadband design.Other topologies will be discussedsuch as the Chebyshev responsetransformation, which has a significant bandwidth advantage over thesingle Q matching technique. Single Qadvantages to be considered are transformation efficiency with smaller component values and design simplicity.In practice using more than afour-section matching network willnot yield greater bandwidth.Also, Q of the load should not besubstituted with unloaded Q or loaded Q. For example, in the followingmultiple section illustrations, whichare bounded by a Q of the load curve 1.75 (for a 50 to 3 ohm transformation), yield more 3 dB bandwidththan defined by the loaded Q of 1.75.Q of the load will be referred tothroughout the remainder of this discussion as the single letter Q.Computer-Aided Design (CAD)and Other Smith Chart ProgramsSmith Chart programs such asthe early Motorola ImpedanceMatching Program (MIMP) provide auseful tool by automating the repetitive graphical computations [13].Considering the frequency point calculations required for resolution of abroadband matching network, this isa tedious task at best. Smith Chartprograms quickly and accurately plotthe required trajectories and circlesallowing the designer to focus on thedesign and not the mechanics of generating a display. Other Smith ChartJanuary 200937
High Frequency DesignBROADBAND MATCHINGFigure 4 · ZO 50, ZS 3, Q 1.75, SWR 1.4; N1-2 Series L 0.9 pH, N2-3Shunt C 25 pF; N3-4 Series L 3.8 nH, N4-5 Shunt C 6.5 pF.programs followed MIMP such aswinSmith [14], LinSmith [15], andSmith32 [16]. Although these programs leveraged an engineer’s intuitive creativity with symmetrical Qmatching solutions, they fall short ofthe sophistication that CAD systemsprovide. Systems such as AppliedWave Research’s Microwave Office[17] and Agilient’s Advanced DesignSystem (ADS) [18] offer electromagnetic simulation of arbitrary structures, complex network synthesis,and optimizers that provide a fullyautomated solution. Modern CADsystems now offer a complete simulation toolset with non-linear synthesisand layout functions.When the design challenge ismore fundamental, and when thebest solution is intuitively derived,Smith Chart programs are well suited for the task. The following demonstrations were plotted with theauthor’s preference, Smith32.Transformation and PerformanceFigure 4 illustrates a 2-section (4element) transformation from a 3ohm driving point impedance to a 50ohm load, a 16.7:1 transformationratio. Confined to a constant Q 1.75curve and the resistive line, theGamma from the 800 - 1000 MHz trajectory (in red) is quantified with aSWR circle of 1.4 (center green circle).To predict the performanceresponse from the trajectory, the38High Frequency Electronicstransformation is reversed. Transforming from the 50-ohm systemimpedance the load trajectory is illustrated relative to a laterally diffusedmetal oxide semiconductor device(LDMOS) load pull performance contours in Figure 5.The trajectory intersects severalcontours in gain and linearity; however, the contours represent performance for single frequency operation(2-tone 880 MHz). These contourswill follow a trajectory of their ownrelative to the parasitic capacitanceof an LDMOS device. Moreover andimportant to note, the trajectory ofthese contours will track opposite(counterclockwise) to the drivingpoint load trajectory thus furtherdegrading broadband performance.Figure 6 illustrates a model of aLDMOS power amplifier with plottedcomplex conjugate load impedancepoints at 800 MHz, 900 MHz, and1000 MHz. For this model, load pullcontours would track counterclockwise as with the indicated conjugatetrajectory points [4]. An ideal counterclockwise load trajectory would bea challenge to any broadband designer; the popular compromise is a compressed and or folded trajectorydesign.The optimal output load impedance of RF transistors as generallypublished in manufacture data sheetsincludes all capacitive and packagelead parasitics. In the absence of thisFigure 5 · Trajectory relative to60WPEP LDMOS Load Pull contours[19]: Zo 3, 28VDC, 900 MHz; MaxGain 23.6 dB @ Z 1.0 j1.3; MinIMD3 -32.7 dBc @ Z 1.0 j0.0;Max Eff 66.75% @ Z 0.9 j1.8.data or when the data sheet is notapplicable to the design an approximation can be derived [20].The purely resistive component ofthe optimal load (RL) can be approximated from the operational RF output power and supply voltage fromthe equationRL V22 PoutWith the transistor biased off theoutput parasitic capacitance can thenbe measured directly with a capaci-
High Frequency DesignBROADBAND MATCHINGFigure 6 · A simple LDMOS modelwith indicated complex conjugateloads: Z0 5, RL RS 13, CS 47.5pF, LS 250 pH.tance meter. The output inductancecan be derived from package andwire-bond mechanical dimensions.In this packaged 28V, 30W,LDMOS modelRS Model RL 282 / 2 ( 30) 13ΩFor 900 MHz the capacitive parasitic (CS) and lead inductance (LS)transform RS Model to ZS Model 1 – 2jOther transistor technologies suchas gallium arsenide (GaAs) and gallium nitride (GaN) have greatlyreduced capacitance for broadbandperformance. For example RFMD’sGaN1C process having higher currentdensity is only 0.05 - 0.1 pF/W whereLDMOS has roughly 0.75 pF/W ofoutput capacitance. The bandwidthachievable is highest for the GaN followed by GaAs, and LDMOS. Thetradeoff of the lower capacitance technologies is monetary with LDMOShaving the best economical value.Transformation with the Low-PassL-NetworkFor standardization and uniformi40High Frequency ElectronicsFigure 7 · Z0 50, Q 1.75; N1-2Shunt C 6 pF.Figure 8 · Z0 50, Q 1.75; N1-2Series L 15nH.ty, the following transformations areconfined to a 25% factional bandwidth (800 MHz to 1000 MHz) with900 MHz set as the reference.In Figure 7 shunt capacitancerotates the trajectory clockwise fromZL 50 with increasing frequency orwith capacitance value following theadmittance equationIn Figure 9 a two-element lowpass network is charted on a Z0 25normalized Smith Chart. The normalized impedance of 25 ohms is calculated from the geometric mean ofthe system load and sourceimpedance, 50 to 12.5 ohms respectively [5].1jBC jωC jX CNote the trajectory from 800 MHzto 1000 MHz (in red) is co-angularwith the shunt capacitive reactance(in blue) following the constant conductance circle.In Figure 8, series inductancerotates the trajectory clockwise alonga constant resistance circle withincreasing frequency or with anincrease of inductance following thereactance equationIn summary, shunt C and series Ldisperse a trajectory with increasingfrequency. In other words when usingthese matching elements in a lowpass network, the higher frequencieswill rotate and transform more thanthe lower frequencies, which spreadsthe trajectory relative to frequency ina clockwise direction.ZGeo Z L ZSThe constant Q curve of 1.75 isderived from the resistive ratio of50/12.5 from the equation1 Q2 RratioNote that the impedance trajectory is no longer co-angular to the constant resistance arc of the seriesinductance reactance (nodes 2-3).High-Pass Lumped Elements andthe High-Pass L-NetworkShunt inductive reactance asdemonstrated in Figure 10 rotatesclockwise along a constant conductance circle with increasing frequency following the susceptance equationjBL 11 jX Lj ωL
Figure 9 · Z0 25, Q 1.75; N1-2 ShuntC 6.1 pF; N2-3 Series L 3.8 nH.Figure 10 · Z0 50, Q 1.75; N1-2Shunt L 5 nH.This element is different than thetwo matching elements discussedpreviously such that shunt inductance susceptance decreases withincreasing frequency.Series capacitance is similar; however, its reactance is plotted on a constant resistance circle in Figure 11following the reactance equationlow-pass L-network, the higher frequencies are transformed less thanthe lower frequencies. If the low-passtrajectory of Figure 9 were overlaidonto Figure 12, the two trajectorieswould form the letter X. Exploitingthis relationship by combining thesedispersion effects can leverage abroadband transformation.jX C 1j ωCSeriescapacitivereactancerotates clockwise with increasing frequency and decreases with increasing frequency.Shunt L and series C disperse animpedance trajectory in a clockwisedirection with frequency, but thereactance will be decreasing with frequency. Hence, high-pass matchingnetworks consisting of shunt inductors and series capacitors will transform the lower frequencies more thanthe higher frequencies.In Figure 12, a two-element highpass L-network transformation from50 to 12.5 ohms is demonstrated on a25-ohm normalized Smith Chart.Note that the trajectory is no longerco-angular to the constant resistancecircle of (nodes 2-3) and that unlike aFigure 11 · Z0 50, Q 1.75; N1-2Series C 2 pF.Compressing TrajectoryDispersionA broadband band-pass networkis illustrated in Figure 13, a 50 to 3ohm transformation similar to theone in Figure 4. With the SmithChart normalized to the geometricmean, it is easy to see that low passnodes 1-2-3 are symmetrical in Q tothe high pass nodes 3-4-5. Combiningthese two networks’ halves folds andcompresses the trajectory into a condensed 3-ohm driving point load.Compare this transformation,which has a mismatch SWR of 1.08,to that of Figure 4 where the mismatch SWR is 1.4.A Chebyshev broadbanding technique is illustrated in Figure 14. Asdiscussed earlier, when using lowpass networks the higher frequenciestransform and rotate more. Here thefrequencies higher than 800 MHz areFigure 12 · Z0 25 Ω, Q 1.75; N12 Shunt L 5.1 nH; N2-3 Series C 8.2 pF.over-rotated well beyond the resistiveline at node 3, which compresses theupper frequency dispersion. Again,compare this network of Figure 14 tothat of Figure 4; a 3-ohm SWR bandwidth of 1.12 versus 1.4.The transformation is mostlysymmetrical with two Q curves, anouter curve (Q1 green) and an innercurve (Q3 magenta). However, node 5January 200941
High Frequency DesignBROADBAND MATCHINGFigure 13 · 50 to 3-ohm transformation; Z0 12.5, Q 1.75; N1-2 Shunt C 6.10pF; N2-3 Series L 3.85 nH; N3-4 Shunt L 1.32 nH; N4-5 Series C 32.3 pF.Table 1 · Q curves per transformation ratio (2-section network). The Qcurves are numbered from theouter most Q1 towards the inner Q3.Figure 14 · Z0 12.3, SWR 1.12 @ ZS 3; N1-2 Shunt C 6.9 pF; N2-3 SeriesL 4.4 nH; N3-4 Shunt C 30.4 pF; N4-5 Series L 0.99 nH.falls at a higher impedance than the3-ohm target to center the fishshaped trajectory at Z 3 j0 and sotherefore a third Q curve (Q2, cyan) isdefined at node 4.With the complexity of the multiple Q curve network, deriving adesign from a Smith Chart alonewould not be an intuitive process. TheQ curves are in overlapping fractionsof the resistive transformation anddo not hold the relationship with thetransformation ratio as before withsingle Q networks. This network andTables 1 and 2 were derived by optimization with an ADS simulator utilizing a gradient optimizer.Note that the inner Q curve (Q3)as a function of the transformationratio holds an inverse relationship to44High Frequency Electronicsthat of the other Q curves.In Figure 15, a three-sectiontransformation, the trajectory fitsinto a 3-ohm 1.01 SWR circle. ThreeQ curves are adquate for defining thethree section network since the trajectory is small and circular in shape,unlike in Figure 14; here noimpedance offset is needed at node 7.As mentioned above, Table 2 wasderived from optimization. Here Qcurves are provided for resistivetransformation ratios of 1.67:1 (50ohms to 30 ohms) to 100:1 (50 ohmsto 0.5 ohms).Complex TransformationsAll transformations discussedpreviously have been purely resistiveto resistive (50-ohm to 3-ohm) trans-Table 2 · Q curves per resistivetransformation ratio (3-section network). The Q curves are numberedfrom the outer most Q1 towards theinner Q3.formations. Figure 16 demonstratesan immediate approach to a high Qtransformation from a purely resistive impedance of Z0 50, to a loadimpedance with where ZL 20 j50.A broadband match in this case isseemingly impossible to design, especially when considering the sourceimpedance dispersion from the largecorresponding parasitics. However,the transformation can be forbearing;Figure 17 includes an additional
ConclusionFigure 15 · Z0 12.2, SWR 1.01 @ ZS 3; N1-2 Shunt C 4.30 pF, N2-3Series L 6.22 nH; N3-4 Shunt C 16.57 pF, N4-5 Series L 2.42 nH; N5-6Shunt C 42.05 pF, N6-7 Series L 0.63 nH.Multiple frequency point load pullcontours demonstrate the necessityfor a compressed and/or foldedimpedance trajectory for optimizedbroadband power amplifier design.Single Q matching where the resistive line and Q curve serve as guideline boundaries are too often presented as the mainstream broadbanddesign technique. Here we haveshown that multiple Q curve transformations although more complex intheir derivation have superior bandwidth over the single Q matchingtechnique although the single Qmatching technique is easily demonstrated we recommend that designersconsider a multiple Q transformation. Furthermore, where device andpackage parasitics disperse thesource impedance counter to a broadband transformation, the use of multiple Q curve transformations is perhaps a categorical.AcknowledgementsThe author is indebted to manyfriends who reviewed this documentfor accuracy.Special thanks to John B Call formany broadband network discussions. Special thanks to Kal Shallalfor the LDMOS load pull contours,and for the device modeling discussion.References:Figure 16 · Z0 50 Ω, Q 2.5.shunt capacitor that is proportionedfor the complex target impedance.This additional element re-orders thedispersion effects of the transformingnetwork; hence improving broadbandperformance. It is another example ofFigure 17 · Z0 50 Ω, Q 2.5; N1-2Shunt C 4.0 pF, N2-3 Series L 2.8nH; N3-4 Shunt L 5.0 nH, N4-5Series L 7.0 nH.how the dispersion effects fromlumped elements can be leveraged tocompress and fold the trajectory.1. “Philip H. Smith: A BriefBiography” by Randy Rhea, NoblePublishing 1995.2. Smith Chart is a registeredtrademark and is the property ofAnalog Instrument Company, NewProvidence, NJ.3. Michael Hiebel, Fundamentalsof Vector Network Analysis, Rohde &Schwarz 2007, pg. 14.4. Chris Bowick, RF CircuitDesign, Newnes imprint of Butterworth-Heinemann, 1982, Ch. 4 - 5.5. Herbert L. Krauss, Charles W.Bostian, Fredrick H. Raab, SolidState Radio Engineering, ZhuyiJanuary 200945
High Frequency DesignBROADBAND MATCHINGPublishing of Taiwan 1980.6. Donslf W. Dearholt, William R.McSpadden, Electromagnetic WavePropagation, McGraw-Hill Inc. 1973 ,Ch. 5.4 - 5.5.7. Joseph F. White, HighFrequency Techniques / An Introduction to RF and Microwave Engineering, John Wiley & Sons 2004.8. Randy Rhea, “Yin-Yang ofMatching: Part 2- Practical MatchingTechniques,”HighFrequencyElectronics, April 2006.9. The RF Capacitor Handbook,American Technical Ceramics Corp.1994.10. 1989 ARRL Handbook for theRadio Amateur, American RadioRelay League, pp 2-27 thru 2-29.11. Thomas L. Floyd, ElectronicsFundamentals: Circuits, Devices, andApplications, 2nd ed. McMillanPublishing Co. 1991, Chapters 14.4 14.712. J. F. White, High FrequencyTechniques / An Introduction to RFand Microwave Engineering, JohnWiley & Sons 2004, pp. 70-71.13. DanMoline, MotorolaImpedanceMatchingProgram,Motorola Inc., April 6, 1992.46High Frequency Electronics14. Agilent Technologies, winSmith 2.0, Noble Publishing 1998.15. linSmith, John Coppens, 19992008, www.jcoppens.com/soft/linsmith.16. Pederson, Ib F., Smith32,Denmark 2002.17. Applied Wave Research, Inc.Microwave Office , El Segundo, CA.18.AgilentTechnologies,Advanced Design System.19. Load Pull contours courtesy ofKhalid Shallal, RFMD.20. B. Becciolini, ImpedanceMatching Networks Applied To R-FPower Transistors, Motorola AN-721,Motorola Inc., 1974.Author InformationAnthony Bichler is a design engineer with RFMD in ChandlerArizona. His 25years of RFexperienceincludes poweramplifier designfor RFID, cellularhandsets,and base stations. Presently,Tony is working on quad-band transmit modules for GSM, PCS, and DCShandsets. Interested readers maycontact him at tbichler@rfmd.comArticle ArchivesRemember, all of our technical articles and columns areavailable as PDF files for download from our Web site!Articles in the current print and Online Edition becomeavailable in the Archives upon publication of the next issue.www.highfrequencyelectronics.com
Impedance Smith Chart where the Impedance Smith Chart can be rotat-ed by 180 degrees to serve as an Admittance Chart. This duality of the Smith Chart is exploited for admit-tance to impedance conversions by simply rotating both the reflection coefficient vector by 180 degrees and then the chart itself by 180 degrees. Note that since the Smith .
Odd-Mode Impedance: Z d Impedance seen by wave propagating through the coupled-line system when excitation is anti-symmetric (1, -1). Common-Mode Impedance: Z c 0.5Z e Impedance seen by a pair of line and a common return by a common signal. Differential Impedance: Z diff 2Z d Impedance seen across a pair of lines by differential mode signal .
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Differential Impedance Differential Impedance: the impedance the difference signal sees ( ) ( ) 2 2( ) Z 0 small I V I V diff Z diff one one Differential impedance decreases as coupling increases 1v -1v I one x I two How will the capacitance matrix elements be affected by spacing? C 12 C 11 C 22 Eric Bogatin 2000 Slide -18 www .File Size: 1MBPage Count: 25
DiffZ0 (ohm) - Calculated differential impedance. Like Single Impedance you can change the value for impedance to the needed value. The tool calculates the necessary width. You can change all values of the white boxes to calculate your impedance. Note: If you want to change the material disable “Show Diff Impedance”.
2.2.3. Electrochemical impedance spectroscopy The electrical properties of the bigels were studied using computer controlled impedance analyzer (Phase sensitive multimeter, PSM1735, Numetriq, Japan) The impedance parameters such as impedance, phase angle, capacitance an
2.4. Electrochemical impedance spectroscopy studies Electrochemical impedance spectroscopy (EIS) can provide useful information on the impedance changes of the electrode surface. Lower impedance values indicate higher conductance. Therefore, electrochemical impedance spectroscopy wa
suring acoustic impedance and calibrating impedance heads and propose a general calibration technique for heads with multiple transducers. We consider the effect of transducer errors on impedance measurements and present a technique for distributing any measurement errors over the frequency range. To demonstrate the technique we use an impedance
I. DNA, Chromosomes, Chromatin, and Genes DNA blueprint of life (has the instructions for making an organism) Chromatin uncoiled DNA Chromosome coiled DNA You have 46 chromosomes or 23 pairs in the nucleus of each body cell. o 23 from mom and 23 from dad Gene a segment of DNA that codes for a protein, which in turn codes for a trait (skin tone, eye color, etc); a gene is a stretch of .
