S-Parameters-Characteristicsof PassiveComponents

This article describes a cost-effective method ofdetermining the true characteristics of passivecomponents using measurements of S-para-meters. A more precise description of thesecomponent parameters can aid understandingof the arithmetic path, and the calculatedvalues; from which equivalent circuit diagramscan be determined, both qualitatively andquantitatively, providing a basis for carrying outcircuit simulations.

Understanding the true behaviourof passive components at frequen-cies above 100 MHz has becomemore crucial as the number ofapplications which operate in thisfrequency range increases. Undernormal circumstances the com-ponent analysers and LCR equip-ment required for such measure-ments are seldom available and,where available, they are usuallylimited in their frequency range toa maximum of 3 GHz. However,the vectorial network analyseroften found in HF developmentlabs can provide a useful alterna-tive. The following describes how,using such equipment, the "reallife" parameters of resistors,inductors and capacitors can bedetermined through theapplication of measured S-para-meters.

Daily Difficulties forCircuit Designers

Where passive components suchas resistors, capacitors and induc-tors are required as part of acircuit design they are usuallyselected and applied based ontheir respective values. And shouldthe designer be unsure exactlywhich nominal value shouldactually be used, the value can bedetermined quickly and simplywith the use of a digital voltmeteror LCR meter. This works well upto the lower megahertz range,particularly in light of the widerange of instrumentation availabletoday.It becomes more difficult when

dealing with applications in thegigahertz range. Passive compo-nents then cease to behave asexpected: capacitors becomeinductive, coils become capaci-tive, and even the value of resis-tors becomes frequency depen-dent. Consequently, verification ofthe actual component values be-comes more difficult; with increa-sing frequency the range of suit-able measurement equipmentshrinks, and prices increase accor-dingly.

Various HF Measure-ment Procedures

Where the characteristics ofpassive components are to bedetermined beyond the order of100 MHz special RF-LCR equip-ment can be used - recently evenup to 3 GHz. The advantages ofsuch instruments are their compa-ratively large impedance measure-ment range spanning around 1 W

to 1 kW at 100 MHz, and thedirect display of impedance,inductance, purity, capacitance,dissipation factor etc. Wheremeasurements have to be madeabove 3 GHz, or money investedin a more generally applicablevector network analyser, thisequipment can be used to deter-mine the high frequency proper-ties of passive components.Whereas the LCR meter computesthe impedance directly from themeasured current and voltage, thenetwork generator permits onlymeasurements of the S-parame-ters. Impedance and additional R,L and C characteristic propertiescan, however, also be determinedfrom the reflection factor, S

11.

Measurements are carried outideally with the sample compo-nent grounded. Alternatively if, forexample, the impedance of thesample is too low at the givenfrequency, or the connection toground is of in-sufficient quality,the sample can also be tested inseries with an ideal (or as close aspossible) 50 W load resistor. Theresult must then be corrected bysubtraction of the additionallymeasured 50 W.The only requirements additionalto the analyser are a spreadsheetapplication such as Excel, and aprogramme to represent results ina diagramm (e. g. Excel or Origin).Should models for circuit simu-lation be created, the demonstra-tion version of PSpice is adequate.A limiting factor in the use of anetwork analyser is the reducedimpedance measurement range ofaround 5 to 500 W due to the

Fig. 1: The impedance of a component.The diagram comprises a real andimaginary element, described incartesian form by Z = R + jX or, in polarform by .

S-Parameters - Characteristics ofPassive Components

Thomas Bluhm

3 2001 Nov 20

jX

- jX

Z

R

d

j

equipment being designed andoptimised for measurements in50 W systems.

Parameters of Pas-sive Components

Before discussing in detail thecalculations used to determinecomponent properties from theS-parameters measured, these pro-perties should be addressed moreclosely. It is advantageous to knowqualitatively both the ideal andtypical characteristic plots in orderto judge the plausibility of thecalculated values.

ImpedanceFigure 1 illustrates the impedanceof a component. The diagramcomprises a real and imaginaryelement, described in cartesianform by Z = R + jX or, in polarform by .

ResistorsIn case of an ideal resistor Z=R,hence the impedance is indepen-dent of frequency. In practice, how-ever, the resistor's constructioncauses parasitic elements. A purely

ohmic resistor does not exist inreality, it is always a complex mix-ture, sometimes more so thanothers, in which parasitic induc-tance(s) and capacitance(s), orboth, can be present alongside theactual resistor.Experience shows that frequencydependency can be observed inwirewound resistors down in thelow megahertz range (short wave),and in film resistors from around100 MHz.

By way of example, Figure 2shows the impedance character-istics of various MINI-MELFresistors. A rising plot indicates inthis case the inductive behaviourof the resistor, a sinking character-istic indicates capacitive behavi-our. The turning point (zero gra-dient) marks a resonance.Intersection of the plot with theZ/R = 1 plane does not indicaterenewal of the true resistor; alsohere the impedance remainscomplex. (This can be illustratedin a three-dimensional represen-tation. However, for the purposeof this article it will not beaddressed in any more detail).

InductancesInductances are usually describedby the parameters inductance, L,quality factor, Q, and impedance,Z, over a given frequency range.Figures 3, 4 and 5 show thequalitative characteristics of ameasured inductance.The impedance of an ideal coil isgiven by , the impe-

Fig. 2: Impedance characteristics of various MINI-MELF resistors related to frequency.

Z=R

2001 Nov 20 4

Resistor Coil Capacitor

Impedance,ideal

Impedance,real

Quality of thecomponent

Thomson’svibrationequation

Impedance fromthe reflectionfactor

Q-factor: Dissipation-factor:

Basic Equation for Passive Components

10

1

0,3

3

Frequency f

0,2 1 3 10GHz

inductive

resonant

capacitive

inductive

capacitive

Impe

danc

e

Z

R

150 W

330 W

470 W

dance increases linearly with fre-quency. In reality, however, aparallel resonant circuit is formedbetween the inductance and theparasitic capacitance whichresults from the construction of thecomponent. The circuit is charac-terised by its Self Resonant Fre-quency, SRF. Around the SRF thechange in impedance is non-linear. (Incidentally, where theinductance, L, and SRF of a coilare known, Thomson's vibrationequation can be used to calculateits parasitic capacitance (seeinset). This can be helpful in deter-mining the equivalent circuitdiagram). Above the SRF theimpedance becomes smaller asfrequency increases. This, how-ever, is the characteristic behavi-our of a capacitor, as will beshown later. Ideally, the induc-tance of a coil is independent offrequency. Around the SRF, theparasitic capacitance previouslydescribed introduces an apparentincrease in the inductive value -rendering the coil useless.The ratio of reactive impedance(= impedance of the ideal coil)to loss resistance is described bythe quality factor, Q. The greaterthe Q-factor, the closer thecomponent under consideration isto the ideal. The loss results fromthe ohmic resistance and additio-nal high frequency losses. Underthe assumption that theimpedance increases linearly withfrequency, theoretically, the Q-factor also increases linearly withfrequency. In reality the Q-factorbegins to drop as the parasiticcapacitance emerges, since thecapacitive-Q opposes theinductive-Q.

CapacitorsCapacitors are described by theircapacitance, C, their dissipation

Indu

ctan

ce L

H

Cap

acita

nce C

F

10m

1m

10 n

1 n

100 n

100 p

10 p

1 p0,2 0,5 1 3GHz

100 n

10 p

Frequency f

Frequency f

Q-f

acto

r

100

Q

80

60

40

20

00,1 0,3 1 3GHz

5 2001 Nov 20

Fig. 3: Typical frequency-dependent characteristics of an inductor L and capacitor C.

Fig. 4: Typical Q-factor characteristics with respect to frequency.

Fig. 5: Typical impedance characteristics Z of an inductor and a capacitor with respect tofrequency.

Frequency f

Impe

danc

e Z

1 k

10 k

10

100

10,2 0,5 1 3GHz

100 n

10 p

W

factor, D (= tan d) and theirimpedance, Z, related to frequen-cy. The qualitative characteristicsare shown in figures 3, 5 and 6.The impedance of an ideal capaci-tor is given by ,Z sinks with increasing frequency.But the capacitor also has aresonant frequency resulting from

a parasitic inductance. Above thisresonant frequency the impedanceincreases again; the capacitorbecomes inductive.Ideally the value of a capacitor isindependent of frequency.However, around the resonantfrequency, the parasitic induc-tance discussed above results in

an apparent increase in the valueof the capacitor; the component isthen useless. The quality ofcapacitors is described, not by aQ-factor, but rather by itsreciprocal; a dissipation factor,tan d. The dissipation factorrepresents the ratio between lossresistance(s) (Equivalent Series

2001 Nov 20 6

Given

Resistive value R

System impedance Z0

S11_value

S11_phase

Obtaining ZS11_re

S11_im

Z_numerator_re

Z_numerator_im

Z_denominator_re

Z_denominator_im

Z_numerator_value

Z_numerator_phase

Z_denominator_value

Z_denominator_phase

Z_quotient_value

Z_quotient_phase

50

50

100000000

0,3333

30

=S.11_value*COS(S.11_phase*PI()/180)

=S.11_value*SIN(S.11_phase*PI()/180)

=1+S.11_re

=S.11_im

=1-S.11_re

=S.11_im*(-1)

=SQRT(Z_numerator_re^2+Z_numerator_im^2)

Cartesian co-ordinates

=ATAN(Z_numerator_im/Z_numerator_re)

=SQRT(Z_denominator_re^2+Z_denominator_im^2)

=ATAN(Z_denominator_im/Z_denominator_re)

=Z_numerator_value/Z_denominator_value

=Z_numerator_phase-Z_denominator_phase

Only for resistors

In degrees

Conversion into

Numerator addition

Denominator addition

Conversion into

Polar co-ordinates

Division

Z_value

Z_phase

[Z]/R

=Z_quotient_phase

=Z_value/R

=Z0*Z_quotient_value

In radians

=Z_quotient_phase*180/PI()

Z_re =Z_value*COS(Z_phase)

Z_im =Z_value*SIN(Z_phase)

Obtaining L, Q

L

Q

=Z_im/(2*PI()*f)

=Z_im/Z_re

Obtaining C, D

C

D

=1/(2*PI()*f)

=Z_re/Z_im

Multiplication by Z0

In degrees

Conversion into

Cartesian co-ordinates

Frequency f

Table 1: Calculation of the real values Z, L, Q, C, D: This variation calculates the required factors from the reflection factor, S11,using simple mathematical standard functions.

Resistances, ESR) and reactiveimpedance (= impedance of theideal capacitor).

From S-Parametersto Component Cha-racteristics

It is usual to employ specialised,and expensive, software in orderto compute component character-istics from S-parameters. This isnot specifically necessary, provi-ding one has a basic understand-ing of complex numbers andcomplex arithmetic, and a know-ledge of a standard softwarepackage with integrated functionssuch as Excel or Origin. Thissolution provides just as reliablean evaluation of the S-parametersobtained, but more cost-effective-ly. Tables 1 and 2 present twopossible ways of calculating thedesired characteristics. The firstvariation applies simple mathe-matical standard functions to thereflection coefficient, S11, in orderto arrive at the result. The secondvariation represents a more ele-gant solution, if an Excel packagewith integrated complex arith-metic functions is available. Bothvariations are suitable for calcu-lating values of R, L and C.

First Version withMathematical StandardFunctions (Table 1)Lines 2 to 5 show the values ofsystem impedance, Z0, frequency fand size and phase of thereflection factor, S11. Firstly Z iscalculated using the reflectionfactor equation with r = S11 (seeinset). S11 is then initially conver-ted into cartesian co-ordinates inlines 6 and 7 (assuming themeasurements are not already in

this form). Almost all the conver-sions which follow are carried outpurely because addition andsubtraction are simpler withcartesian co-ordinates, whereasdivision and multiplication aresimpler with polar co-ordinates.Simple computations reduce thelikelihood of error in enteringformula and aid trouble shootingwhen unexpected results arise.Following conversion, numeratorand denominator are calculated inlines 8 to 11, after which -following renewed conversion intopolar co-ordinates (lines 12 to 15)- numerator and denominator aredivided in lines 16 and 17. Theresulting value is then multipliedby Z0 in line 18. The phase angleis not affected by Z0 since Z0 isreal (line 19). Should it be desiredthat the phase information notonly in radians but also in degrees,the phase value is multiplied by180/PI() (line 20).The following calculations arecomponent-specific. Should it bethe aim to show the characteristicsof a resistor, the value alreadycalculated represents the result. Itis preferable to scale it to theoriginal nominal value of theresistor, making it easy to compare

several resistance values on thesame diagram (line 21).Calculation of the frequencydependency of the inductance andQ-factor of coils is achieved withthe following two equations:

Since, according to the definitionabove, inductance is purely ima-ginary (although both the desiredinductive element and also theundesirable parasitic capacitiveelement are represented) renewedconversion into cartesian co-ordinates separates the real andimaginary elements in lines 22and 23. After which L results fromthe division of the imaginary ele-ment of Z by the angular frequen-cy. The Q-factor is obtained bydividing the imaginary element ofZ by the real element (line 25).In this way values for the parasiticinductance of resistors, and theQ-factor of capacitors can becalculated.Analogous to inductances, the

® (1)

®(2)

7 2001 Nov 20

Fig. 6: Typical loss factor progression (tan d) for a capacitor.

Real(Z)

Z)Imaginary(=Q

resonant

inductive

Frequency f

tan

5 . 10-4

4 . 10-4

3 . 10-4

2 . 10-4

1 . 10-4

010 30 100 300MHz

d

following is true for capacitors:

The conversion in lines 26 and 27is the equivalent of the routedescribed above for coils.

Second Version with theAid of Excel (Table 2)This approach does not differ fromthe first through the first seven

lines. From line 8 complexfunctions which are available inExcel are applied. This simplifiesthe exercise as conversions aredispensed with. The line of calcu-lation is, however, the same asthat taken in the first variation. Forthis reason only the issues relatedspecifically to the complex func-tions will be addressed here.In line 8 the function COMPLEX isused to create a complex numberfrom the previously separate realand imaginary elements. Thiscomplex number is necessary forthe application of practically allcomplex functions. The function

parameters are the values incartesian format, calculated inlines 6 and 7.Calculation of the impedance isnow straight forward using thefunction IMPRODUCT in themultiplication of complexnumbers (line 9). Functionparameters are Z0 and the result ofthe complex division (IMDIV) ofthe previously calculated num-erator (IMSUM) and denominator(IMSUB). In lines 10 to 12 followsthe reduction of the complexresult into the required compo-nents: IMABS (line 10) providesthe value of the complex number,

Cartesian co-ordinates

Only for resistors

In degrees

Conversion into

Formation of complex number

In radians

In degrees

Cartesian co-ordinates

Conversion into

Given

Resistive value R

System impedance Z0

S11_value

S11_phase

Obtaining ZS11_Re

S11_Im

Frequency f

50

50

100000000

0,3333

30

=S.11_value*COS(S.11_phase*PI()/180)

=S.11_value*SIN(S.11_phase*PI()/180)

S11_complex

Z_complex

=COMPLEX(S.11_re;S.11_im;²i²)=IMPRODUCT(Z0;IMDIV(IMSUM(1;S.11_complex);IMSUB(1;s.11_complex)))

Z_value

Z_phase

|Z|/R

=IMABS(Z_complex)

=IMARGUMENT(Z_complex)

=IMARGUMENT(Z_complex=*180/PI()

=Z_value/R

Z_Re

Z_Im

=IMREAL(Z_complex)

=IMAGINARY(Z_complex)

Obtaining L, QL

Q

=Z_im/(2*PI()*f)

=Z_im/Z_re

Obtaining C, D

C

D

=1/(2*PI()*f*Z_im)

=Z_re/Z_im

(3)

(4)

®

®

Table 2: Calculation of the real values of Z, L, Q, C, D: this variation represents a more elegant solution, assuming an Excel packagewith integrated complex functions is available.

2001 Nov 20 8

Z)Imaginary(Real(Z

=D

whereas INARGUMENT (lines 11and 12) provides the phase of thecomplex number. IMREAL (line14) and IMAGINARY (line 15)return the results of the real andimaginary components of Z,which are further processed inthe equations of lines 16 to 19.These are not different to lines 24to 27 of the first variation.Of course, it is possible to inter-lace the functions and, hence, toreduce the number of compu-tational steps. However, this bringsthe associated disadvantage thatclarity of calculation is lost, andthe likelihood of errors in theformulae increases.For the purpose of clarity thecalculation has been arranged in acolumn. This representationenables the calculation for singlefrequencies. Arranging the calcu-lation in a row permits the calcu-lation for any number of frequen-cies arranged in a column.Additionally, it is helpful to hidecolumns which simply containconversions, to apply backgroundcolours to results columns and tolock all cells in the worksheet,with the exception of f, S11-valueand S11-phase.

Equivalent CircuitDiagrams - and theirLimitations

In HF development it is a commonnext step to calculate and simulatea circuit with the aid of acomputer. Since the simulationcan only ever be as good as themodel upon which it is based, it iswell worth paying close attentionto the model. Models should besimple, but also precise and, inaddition, be valid over a widefrequency range. The equivalentcircuits for resistors, coils andcapacitors depicted in figures 7

and 8 are commonly used inliterature and, hence, generallyknown. It can be seen that there isno qualitative difference betweeninductance and resistance. Whenthe construction of these compo-nents is considered, this comes asno surprise: when a wire is woundaround a body the result, depen-dent upon the properties of thewire, can be either a resistor or aninductor. The quantitative, andthus decisive, difference is in thefact that a resistor should have thesmallest possible parasiticreactance, and a coil should havethe smallest possible resistanceand low parasitic capacitance.The equivalent circuits for bothcomponents can be simplifiedthrough the basic conditionsdetailed alongside figure 7.When the impedance equation isapplied to the illustrations tocalculate the progression of impe-dance against frequency, thediagrams for R, L and C arearrived at. It should be noted that,beyond around 3 GHz, the simpleequivalent circuits illustratedbecome increasingly inaccurate;additional elements are necessaryin order to provide an accuratedescription of the high frequencycharacteristics. Whereby thecircuit simulation also becomesmore complex. How, then, can theequivalent circuit values bedetermined without purchasingexpensive software? The answer isvery simple: the free demo versionof PSpice enables the calculation

of the equivalent circuit diagram.The measured and calculatedcharacteristics merely have to bebrought into line with those calcu-lated using PSpice. This route maybe slightly laborious, but has ad-vantages, including the fact thatone is obliged to consider eachindividual element of the equiva-lent circuit in detail.The base circuit for the calculationof the equivalent circuit diagram isshown in figures 7 and 8. Thesymbols IPRINT and VPRINT writethe values calculated for currentand voltage to a file. In this way thevalues can be processed further inother applications (e.g. Origin,Excel). This enables the simul-taneous depiction of the measuredvalues and those calculated withPSpice.

Advantages andDisadvantages ofsome Methods ofDepiction

A simple yet comprehensivemethod is required to representthe frequency response of resis-tance, or impedance. Below theadvantages and disadvantages ofvarious possible options arediscussed.

a) b) c)

9 2001 Nov 20

Fig. 7: (a) Equivalent circuit diagram for a resistor and inductor respectively; (b) variationsfor resistors < 100 W and inductances with value << SRF and (c) variations for resistors> 200 W .

Fig 8: Equivalent circuit diagram for acapacitor.

Smith DiagramThe Smith diagram is one of themethods of depiction mostfrequently used in HF develop-ment. It allows the qualitativevalue and phase characteristics tobe shown against frequency, aswell as any resonant frequencies.The non-linearity of the represen-tation, however, makes a quantita-tive estimation comparatively. TheSmith diagram is particularlyuseful where resistors of the samevalue, but different designs, are tobe compared: the longer theresponse in the frequency rangeconsidered, the worse the highfrequency characteristics of thecomponent (fig. 9).

Depiction of MagnitudeThis is another common represen-tation, often using a logarithmicscale. In the case of resistors auniform representation can provedeceptive with regard toresonances.

Standing Wave RatioThe standing wave ratio diagramcan be used to show resonances

1

0,5

0,2

-0,2

-0,5

-1

-2

-5

5

2

0 0,2 0,5 1 2 5

as well as changes in the resistor'svalue. However, the method ofcalculation of the standing waveratio has the associated disadvan-tage that it is not possible todifferentiate between inductiveand capacitive behaviour.

Depiction of ImpedanceRepresentation of impedanceagainst frequency provides asimple representation of thequalitative and quantitativechanges in resistance, includingany resonances. Scaling thecalculated impedance to the(nominal) value of the resistor alsoenables the comparison of diffe-rent resistor values on the samediagram.

The Author

2001 Nov 20 10

Fig. 9: Smith diagram: The longer thecharacteristics is in the frequency rangeunder consideration, the worse the highfrequency behaviour of the resistor.

Thomas Bluhm studied informa-tion technology at IlmenauTechnical University. Followinggraduation his first professionalposition was as a developmentengineer in industrial control.Since 1995 he has worked for theresistor manufacturer,BCcomponents BEYSCHLAG,where he has been involved inresistor applications, providingtechnical customer support and incomponents standardisation.