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1. RFbehaviourofPassivecomponents

TheoreticallyaccircuitanalysistellsusthatheresistanceR,capacitanceCandinductanceLis

independentoffrequency. Asareminderthevoltageofacapacitorcanbedescribed

1 1

Orasaphasor

1

2

Andforaninductor

3

Sofor1pFcapacitorand1nHinductorat50Hz

|| =3.18*109

|| 2 50 1 3.14 10Bothnegligible

However,Thewiresandplatesthatconnectthecomponentswillhave

frequencydependentpartsevenprintedcircuittracks.

Nowletsseewhathappenwhenthefrequencyincreases

1

4

AtDCthecurrentflowsuniformlydistributedovertheentireconductorcrosssectionalarea.When

thecurrentbecomestimevariantthealternatingchargecarrierflowhasamagneticfieldthat

inducesanelectricfield(Faradaysinductionlaw)whosecurrentdensityopposestheinitialcurrent

flow. Theeffectisstrongestasthecentre,r=0,sothe resistanceisincreased. Theresultisthatas

currentflowmigratestothesurface, andtheradiusofthiseffectintheconductorincreaseswith

frequencytherebyreducingthecurrentflowandincreasethecircuitresistance. Itcanbeshown

thatthecurrentdensityalongthewireaxis,zis

2exp 1

5

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Where andIisthetotalcurrentflow. Thekeycomponentistheexponent,knownastheskindepth.

1

6

Whichdescribesthespatialdropoffincurrentdensityasafunctionoffrequencyf,permittivity

and theconductivityatDC. Theexponentialdecreaseofcurrentdensitywithdepthappliestomostgeometries. Thusshowingtheskindepthdecreaseswithfrequency.Theskinlayer

thickness isusefulforresistancecomputations,howeverthenormalisedresistance,R,canbe

simplifiertothus

2 7

Forfrequenciesabove500Mhz. Noteaistheconductorradius,assumingitisround. Theinternal

(totheconductor)inductanceimpedanceisthus

2 8

Theinternalinductanceisduetothemagneticfieldinternaltothewire. Theseequationsarebased

ontheassumptionthat

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Theskindepth hasaphysicalmeaninginthatitdenotesthereductionincurrentdensitytoe1or

37%foitsvalueatthesurface. Figure2givestheproportionofcurrentdensityrelativetoa

cylindricalconductorradius.

Figure2Normalisedcurrentdensityvsradiusatdifferentfrequencies

Althoughtheinternalinductancecanbesignificantitistheexternalinductance,Lex resultingfrom

thefieldemanatingfromtheconductorsurfacethatdominates. Theexternalinductanceofa

cylindricalwireofradiusaandlengthlcanbeapproximatedby

2ln 2

1 9

Wherel isthelength. Soletsmakeacomparisonbetweeninternalandexternalinductance

Wire(AWG26)hasaradiusof0.2mm. Theinternal inductancefromequation8is

2

And 2.39 10

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Andusingequation9tocalculatetheexternalinductanceatablecanbeconstructedshowingthe

rationofinternaltoexternalinduceatdifferentfrequencies

F(GHz) (m) Lex(nH) Lin(nH) Lin/Lex

0.1 6.266 17.1 0.0617 3.60X103

2 1.4 17.1 0.0138 8.05X104

5 0.886 17.1 0.00872 5.09X104

Resistorsathighfrequency.

Physicalresistorscomeinvariousforms

1) Carboncomposition

2) Wirewound3) Metalfilm

4) Thinfilm

Photoinsert.

OfthesethethinfilmchiporsurfacemounteddevicesareusedforRFandMWcircuitsbecauseof

theresmallsizeandgoodRFperformance. Resistorsarenecessarilyconnectedbywires,which

haveinductanceandinternalcapacitancecausedbychargeseparationintheresistormaterial. A

linearnetworkcircuitcanbedrawnthus:

Figure3TopLlaretheleadinductances,CarepresentsthematerialcapacitanceandRisdcresistance.Bottomistheequivalentcircuitforawirewoundresistance. Notetheextraininductanceresultsfromtheselfinductanceofthe

resistancewirewoundtocreatetheresistance

Takingtheequivalentcircuitfortheametalfilmresistor ofnominalvalueof2k usingthecircuit

model infigure3wherethereare2.5cmofwireconnectionsof0.203mmradiusandthestray

capacitanceandtakingtheconductivityofcoppertobe64.5X106/Sm. Usingequation9

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Lengthistwicetheleadedconnections. Notethisformulaassumesthattheskindepthis

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Soonlyatlowfrequencies,belowabout2Mhztheresistanceisthedesignvaluebutabovethatthe

straycapacitancefirstreducesitsvaluetothatatresonanceandthenincreaseduetothestray

inductance.

Capacitorsathighfrequency.

Chipcapacitorsarewidelyusedinfrequencydependentcircuits,filters,oscillatorsanddecoupling.

ADCthecapacitanceis

WhereAtheareaofthecapacitanceelectrodesanddtheseparation. Ifthedielectricisanything

otherthanairthenthepermittivityoffreespaceismultipliedbyaunitlessfactortherelative

permittivityusually>1. Asthefrequencyincreasesthedielectriccanbecomelossysoitsimpedanceisexpressedastheparallelcombinationofconductance,GeandsusceptanceC

1

Theconductance Ge=dielectricA/dwheredielectric istheconductivityofthedielectric. Theratioofthisconductivityand isconventionallycalledthelosstangent.

tan tan

Thelosstangentisfrequencydependentitself. Sonowwecanconstructtheequivalentcircuit

Figure5Equivalentcircuitofcapacitanceathighfrequency,Liistheleadinductance,RstheleadresistanceandRethe

dielectriclossthatisfrequencydependent

WhereLiistheleadinductanceandRsistheleadresistanceReistheresistancerepresentingthe

dielectriclossRe=1/Ge. Sotheoverallimpedancecanbesetdown

11

Foraspecificexample:whatistheresponseofa47pFcapacitorwithdielectricmediumof

aluminiumoxide,Al2O3,whichhasalosstangentto104

. Theconnectleadsare1.25cm,0.203mm(

26AWG)andtheconductivityofcopperis64.5X106/Sm. Usingequation9

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Plottheresponseofthemodelinfigure5weseetheequivalentcapacitance,liketheresistancehas

resonantfrequency.

Figure6Thefrequencydependenceofarealcapacitor.Noticeresonantnadircausedbytheleadinductanceresonanting

withcapacitance.

Itisassumedinthisdemonstrationthattanfrequencyindependent. Thelosstangentcanalsobe

definedasanequivalentseriesresistance,ESR

Zc t( ) j L Rs+ 1

j C 1

R+

+:= RsRs

Ideal

ca acitor

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Type equation here.

Wherelaccountsforbothleads. Theseriesresistancemustdeterminedbytakingskindepthintoaccount.

ThelossresistanceinthecapacitorisRe=1/Ge

At 100Mhz.

Realcapacitorsareconstructedtominimisecontactimpedanceandmaximumcapacitance/volume

H

0 4 10 7

:= l 0.02:= a 0.20310 3

:=

Lex

0l

2 ln

2 l

a

1

:=

Lex 2.253 10 8

=

Rsl

2 a cu f 0 cu:= Rs 0.048=

Rsl

2 a

0f

cu:=

Rs 0.048=

Tan 1 10 4

:= Cc 47 10 12

:=

Re1

2 Cc f Tan:=

Re 3.386 105

=

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Figure7Constructionofaceramicchipcapacitor

Inductorsathighfrequency

InductorsareusedinRFcircuitsforblockRFinbiasingandgivingfrequencyselection. Physically

thoughadjacentwindingofthecoilsarelinkedbycapacitance andthecoilconductor willhave

resistancewhichitselfisfrequencydependentduetotheskineffect.

Figure8Physicalcoilwithseriesresistanceandwindingcapacitance

Theimplicationofthismodelisthatatsomefrequencyitwillresonate,callsenseresonance. As

beforeanequivalentcircuitcanbedefinedbasedontheseobservations

Figure9Equivalentcircuitofarealinductor

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Andits Admittiane=1/Z describedthus

1

Againasinthecapacitorasanexampleacoilisconstructedwith3.5turnsof127umdiameterwirewitharadiusof1.27mm. Theinductanceofshortcoilisgivenapproximatelyby

Givingavalueof32.3nH. IftheskineffectisneglectedthenRsissettotheDCvalueofLwire/

Givingavalueof0.034ohms. Itsresponsewouldbe foracoilselfcapacitanceof40pF

Figure10 Frequencydependenceofarealinductor

L10 r

2 0 N

2

9 r 10 l+:=

Lwire 2 r N:=

Rdc

Lwire

cu a

:=

Ideal

inductor