Measurement of the Characteristics of High-Q Ceramic Capacitors

7/28/2019 Measurement of the Characteristics of High-Q Ceramic Capacitors

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Measurement of the Characteristics of High-Q Ceramic

Capacitors

Mark Ingalls and Gordon Kent, Senior Member, IEEE

See Reference 9

Abstract - The quality factor, equivalent series resistance, and the frequency of self-resonance are parts of the specifications of high-Q ceramic capacitors. These quantitiesare obtained from measurements on transmission lines with the capacitor in series orshunt. Part A; Resonant structures designed to extend the Electronics IndustriesAssociation (EIA) standard RS-483 downwards below 10 MHz and upwards above 3GHz are discussed. For the low-frequency lines, a rule from extrapolating Q valuesoutside the range of data is proposed. The rule is based on the frequency-dependence ofthe input/output coupling. The high-frequency line incorporates a method of tuning whicheliminates the need for interpolations or extrapolation. It is particularly suited tomeasuring small parallel-plate capacitors which can be mounted on a flat shorting plate.

Part B: It is shown that the first self-resonance, when viewed in terms of a series R, L, Cequivalent circuit, is a poorly defined quantity. It is not always observable; it may notexist; and it may be of minor importance to design applications. It is proposed that theresonance specification of capacitors should be the first parallel resonance, defined as thefirst maximum in dissipation loss.

INTRODUCTION

In addition to the capacitance value and operating voltage, the electrical specifications oflow-loss ceramic capacitors normally include the equivalent series resistanceRs, thequality factorQ, and the lowest self-resonant frequencyfs. BothRs and Q are frequency-dependent, and the characteristics of the dependence are affected by the proximity of thefrequency tofs.Rs andfs are directly measurable, at least in principle, and Q is derived

from the relation

Q = |Xc| / Rs (1)

where |Xc| is the magnitude of the series reactance of the capacitor, also a directly

measurable quantity. However, asfs is approached, (1) cannot be used.

The measurement techniques forXc and Q and the techniques for determiningfs are basedon the characteristics of a transmission line with the capacitor mounted in series or shunt.

When the line ends are lossless terminations, the perturbations of the structure's Q and itsresonance's determine the values ofRs, andXc of the test unit. When measuring fs , at

least one end of the line is a matched load (or generator impedance), andfs is determined

from scattering parameter data.

The interrelation ofRs, Xc, and fs, as well as the similarities of the structures used for the

measurement, provide the threads that unify the otherwise disparate parts of this paper.Part A deals essentially with resonant lines as standard fixtures for determiningRs and

|Xc| of capacitors varying from 1-1000 pF. Although the frequency range is on the order


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of 10-3000 MHz, these structures are useful principally for the low-frequency end of thespecifications. Part B deals with the problems of self-resonances: definition,

measurement, and significance. These questions belong to the high-frequency end of thespecifications.

II. PART A: RESONANT STRUCTURES FOR MEASURINGR ANDX

A. Introduction

The genesis of resonant structures for measuring dielectric materials and capacitors may

predate the Laboratory for Insulation Research, but the published work of its staff [1]-[2]comprises the foundation for much of the current use of such devices. Since that period,

few new ideas have been proposed, but the advances in electronics have facilitatedgreater precision, ease of measurement, and a commensurate refinement of calculation

procedures.

Recent advances of resonant structure techniques [3], [4] have focused on the need forstandards to apply to the characterization of high-Q ceramic chip capacitors, i.e., the

measurement ofRs andXc at frequencies above the limits of bridge performance. Inparticular, the Boonton 34A resonant coaxial line with associated procedures and

calculations has been adopted by the Electronic Industries Association (EIA) as standardRS-483 and the American Society for Testing and Materials as standard F752-82. The

measurement frequency range is 25 MHZ to 1.25 GHz. Other work [5]-[7] has beendirected towards extending the technique down as low as 4.5 MHz and upwards to 3000

MHz.

The essential requirement in extending the standards in frequency range is that allstructures produce the same results in the frequency ranges that are common. One

difficulty in satisfying this criterion is size and style of the test capacitor that is requiredfor the overlap. Accurate data may be obtainable at the low-frequency limit of one

structure, but if that frequency is close to the upper limit of a second structure the datafrom this structure may not be reliable.

We describe in this part two low-frequency resonant lines, 1.5 and 4.5 m long, and one

high-frequency line 10 cm long. The latter is intended for the measurement of smallparallel-plate capacitors.

B. Review of Analytical Results

The prototype model of the structure is a coaxial line terminated at its ends in some

combination of open and short circuits. The capacitor to be tested (DUT) is inserted into agap in the center conductor. The gap, assumed symmetrical about its center, is

represented by a network of discontinuity susceptances: b1 in shunt and b0 in series.Both susceptances are normalized to the characteristics admittance. The DUT may be

included in b0, but it is assumed to have no effect on b1.


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For high-Q lines and devices, the resonant frequencies are essentially independent oflosses. The equations giving resonant frequencies are shown in Table I and Fig. 1 for

various lengths and end conditions. The conclusion indicated by these equations is that b1and b0 are easily separable only when the line is symmetric with respect to the gap center.

In this case, b1 can be determined from one set of frequency measurements, and it can be

subtracted from the combination (2b0 + b1) that is determined from another set ofmeasurements. If1 = 0 and the termination is a short, as is approximately the case withthe Boonton 34A line, the combination (b0 + b1) is measured. For many applications |b1|


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The low-loss assumption permits the calculation of (1/Q) by using the fields and currents

that would exist in the absence of losses. For the unloaded Q0 ()(the Q without coupledlosses),

1/Q0 = [R0 /L0 ] + [end losses + gap losses] / [(L0 / 2) II* dz]. (2)

whereR0 andL0 are the resistance and inductance per meter of the line and is theresonant frequency. It is assumed that there is no stored magnetic energy in the gap.

In general, the resonant frequency and current distribution depend on the gap admittance

so that the second term of (2) also depends on the gap admittance. Thus the use of (2) toextract the contribution of device losses to the reciprocal Q is complicated in a non-trivial

fashion. In several special cases, however, the device losses can be cleanly separated.

In the first case, assume a short circuit atz= 0 and an open circuit atz= l2. When no testunit is in place and the gap spacing is large, the second term of (2) is negligible. Then at a

resonant frequency 0,

1/ Q0c (0)=R0/0L0 (3)

When the gap is shorted either by a dummy conducting test unit or by closing the gap, at

the new resonance 1.

1/ Qsc=Ro/1L0 + [fixture losses] / [(1L0 / 2) II* dz]. (4)

The term (Ro/1L0 ) is (1/Q0c) for a line of somewhat different length that would produce

an open circuit resonance at 1. SinceR0 is determined by the skin effect, in principle, we

have

1/ Qsc (1 ) - 1/ Q0c (0 ) [0 /1 ] (1/Q). (5)

Knowing the current distribution for the shorted line, one can calculate a fixtureresistance from (5), i.e.,

R(fixture) =Z0 (4/)(1/Q) (6)


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By a similar procedure, the resistance of the DUT plusR(fixture) can be obtained, and (6)can be subtracted out.

Other simple cases are the symmetric lines either open or shorted on both ends.Essentially, the same procedure applies.

The simplest Q-measurement to perform is the measurement of the bandwidth of the

transmission through the resonator. The observed loaded QL is less than Q0, according tothe relation

QL = Q0 /(1+ 1+ 2) (7)

where 1 and 2 are the coupling coefficients at the two ports. With equal inductivecoupling and skin effect, the frequency-dependence of (7) is, in principle,

QL = {Q0(0 ) [/0 ]}/{1+2 (0 ) [/0 ]

3/2}. (8)

Unless the coupling is very small and the frequency range is limited, (5) fails to providesufficient accuracy. To take the coupling into account, one may fit any smooth curve to

the observed Q0c values and used that curve as an interpolation rule between data points.When it is necessary to extrapolate beyond data points, there must be a technical

argument to justify the curve-fitting procedure. The consideration leading to (8) providethe necessary rationale for its use as the extrapolation rule.

C. Structures for Testing in the Range 5-200 MHz

A principle objective in extending downwards the resonant line technique was to enable

measurements of capacitors at or near 10 MHz, as required by the Defense ElectronicsSupply Center (DESC). A second objective was to cover the frequency range upwards

past the lower limit of the Boonton 34A. The lowering of the fundamental resonance byincreasing line length also lowers the unloaded Q, but some of this loss can be retrieved

by increasing line radii while maintaining the optimum ratio of radii.

The design specifications for two low-frequency lines are shown in Table II. Except forthe fixture and short, commonly available materials were used. The input and output

ports, both located at the shorted end, employ inductive coupling. The loops can be easily

rotated for coupling adjustment.

Fig. 2 shows the shorted end of a DLI-1.5 line. Also shown for comparison are the

Boonton 34A and a 10-cm line.


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TABLE II

DESIGN SPECIFICATION OF 1.5-m (DLI-1.5) AND 4.5-m (DLI-4.5)

RESONANT LINES

MechanicalLengthDLI-1.5 1.54 m

DLI-4.5 4.58 m

Diameter

ID outer conductor 10.16 cm

OD center conductor 2.80 cm

MaterialsConductors hard-drawn Cu pipe

Fixtures and ends OFHC copper plate

Center supports TeflonSolder Sn62

ElectricalFundamental resonanceDLI-1.5 47.99 MHz

DLI-4.5 16.21 MHzCharacteristic impedance 76.025 Coupling input and output by inductive loops

located at the shorted end; loop

area 2.8 cm2

Fig. 2. Left to Right; DLI-1.5 (shorted end showing), Boonton 34-A, and DLI-0.1.


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D. Calculations of Q and Fixture Resistance

Since both DLI-1.5 and DLI-4.5 must be used at frequencies below their fundamentalresonances, extrapolation of the open and short-circuit Q data is required. Moreover,

apparent test capacitor losses are particularly sensitive to the extrapolated values.

The frequency-dependence of the loaded Q's shown in (8), can be viewed as a function to

be fitted to the data by some criterion that determines Q0(o) and(o). The fittingprocedure that seemed most appropriate in view of the required low-frequency accuracywas the following: fit the lowest data point exactly; fit one of the remaining data points

exactly; choose from these fits the one showing the least deviation from the other data.The results of this procedure are shown in Table III.

As confirmation of the fitting procedure, the coupling factor was also calculated from the

insertion loss:

L = 20 log [2/(1+2)] (9)

For DLI-4.5, the insertion loss at 16.21 MHz was -39.5 1.5 dB, and the corresponding

(fo) = 0.0054. With this value and (8), the ratio Q0/QL atf0 was found to be 1.0109.Although the insertion loss measurement lacks accuracy, the close correspondence of

these results to the values to Table III gives additional credence to the fitting procedure.

Fixture loss was calculated from the extrapolated or interpolated Q0c values, the Qscvalues measured when the line was shorted by a copper block (dummy capacitor), and

(6). The fixture resistance, shown in Fig. 3, is approximately proportional to (f)1/2

. Thisskin effect frequency-dependence is an additional argument in support of the

extrapolation rule and the fitting procedure.

Note that extrapolation from data is a high-risk operation at best, and the problem iscompounded here by the sensitivity of the extrapolated Q to the coupling parameter

(0). Among the various fitting criteria that were explored, the one chosen produced themost plausible results, in particular, the frequency-dependence of the fixture resistance.


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TABLE III

COMPARISON OF EXPERIMENTAL Q VALUES WITH FITTING FUNCTION

A: DLI-4.5 = 0.0053 (Q0 / QL ) = 1.0109l/ Frequency (MHz) QL(Observed) QL (f) (Calculated)- 10.00 - 1162.39

1/4 16.21 1462.40 1462.40

3/4 48.65 2393.71 2419.15

5/4 81.08 2913.58 2913.58

7/4 113.50 3181.07 3143.28

B: DLI-1.5 = 0.0079 (Q0 / QL ) = 1.0161l/ Frequency (MHz) QL(Observed) QL (f) (Calculated)- 10.00 - 1118.89

1/4 48.00 2416.87 2416.87

3/4 144.07 3825.91 3908.56

5/4 240.22 4535.01 4535.04

E. A 10-cm Line for Testing Very Small Capacitors

Efforts to measure very small capacitors with the Boonton 34A have met with such

limited success that the results cannot be accepted with confidence. For example, the datafor a gap on the order of 0.1 mm without a capacitor give capacitance values, calculated

according to EIA Standard RS-483, that are appreciably below the theoretical values, andthis discrepancy increases with frequency. Moreover, the data were not well reproduced

by repeated experiments.

Fig. 3. Resistance of fixture with dummy capacitors, DLI-1.5, and DLI-4.5.

0.001

0.01

0.1

10 100 1000

F (MHz)

Rs

(ohms)

Trend

DLI 1.5

DLI 4.5


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Part of the limitation of the 34A is mechanical; it is not designed to allow precise controlof the gap spacing. The electrical problems stem from the fact that l1 is not strictly zero

and that the capacitance values to be tested are comparable to the capacitance of theempty gap. The finite value ofl1 can be accounted for by modifying the calculations of

RS-483, resulting in a procedure that is cumbersome at best, but the solution to the

second problem requires a line with smaller center conductor, less overall volume, andmicrometer control of the gap spacing.

The 10-cm line, designed for this purpose, is shown in Fig.2. The fundamental resonance

with the gap shorted is 720 MHz, and the characteristic impedance is 75.056 . The DUTis placed in the center of the shorting end plate; contact with the center conductor is madewith the micrometer, which is attached to it by a dielectric rod. A Teflon disk, which

supports the center conductor, makes a smooth bearing. The location of the capacitivecouplings at the open end eliminate any possible coupling interaction with the DUT.

The flat end plate and moveable center conductor result in some simplification of the

calculations prescribed in RS-483. First of all, the last equation of Table I is strictlycorrect. When the DUT is a parallel-plate capacitor, small enough so that b1 is not

affected by its presence, the capacitance can be calculated from

C (DUT) = [Y0 /4 f0 ] [(1/d) tan d (1/c ) tan c ]-[0 A / d] (10)

wheref0 is the fundamental resonance, shorted line; c is the resonance phase with the

DUT; d is the resonance phase with gap spacing dequal to the DUT thickness; A is theDUT area. Second, the gap spacing without the DUT can be adjusted to achieve the same

resonant frequency as occurs with the DUT. The Q0c at this frequency can be measured,thus avoiding any problem of interpolation. Then the DUT conductance

G (DUT) = Y0 [(c - cosc sin c ) / 4 cos2c] (1/Q) (11)

is determined directly from experimental data.


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F. Results and Discussion

The extrapolation and curve fit represented in Table II was found to be repeatable, afterremoving and replacing coupling loops, by the simple procedure of adjusting the loop

orientation to achieve the previously measured insertion loss. This extrapolation rule was

used for the calculationRs data obtained on DLI-1.5 and DLI-4.5. The extrapolation ruleof RS-483 was used for the calculations based on data from the 34A line. Comparison ofresults from three lines, DLI-4.5, DLI-1.5, and the Boonton 34A, is presented in Fig. 4 (a)

and (b). There is evidently good agreement in the frequency ranges that overlap.

Fig. 4. (b) Series resistance of a 470-pF monolithic capacitor.

0.01

0.10

1 10 100 1000

F (MHz)

Rs(ohms)

BOONTON

34A

DLI 1.5

DLI 4.5

Fig. 4. (a) Series resistance of 43-pF monolithic capacitor.

0.01

0.10

10 100 1000

F (MHz)

Rs(ohms)

BOONTON

34A

DLI 1.5

DLI 4.5


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Fig. 5 shows results derived from the data obtained with DLI-1.5, the Boonton 34A, andDLI-0.1. The agreement between the latter two is poor near 2 GHz. The discrepancy may

be related to failure of the extrapolation rule of RS-483 at high space harmonics and to acorrectable fault in the center conductor of DLI-0.1. The parallel resonance of the

capacitor at 5.8 GHz, observed by insertion loss measurements, may account for the

variations of the DLI-0.1 results.

III. PART B: SELF-RESONANCE OF CERAMIC MONOLITHIC CAPACITORS

A. Introduction

For the circuit designer a capacitor is a device that stores electrostatic energy, and itsphysical size has no bearing on its circuit characteristics. When energy is alternately

stored and removed, there is some inertial effect that retards the transfer, and someenergy is lost to heat. These effects are accounted for by a series R, L, C equivalent

circuit, and the device appears to have a new characteristic: a self-resonance.

The validity of this concept at low frequencies is unassailable, but it is a mind set thatfails to accommodate the observed characteristics of capacitors at wavelengths in the

dielectric comparable to the physical dimensions of the device. Although the monolithicchip capacitor in particular is so small in size that the low-frequency model seems

appropriate in the microwave range, its characteristics are better described by adistributed circuit model [8]. There is not one self-resonance but a sequence of

resonance's similar to those of an open-circuited transmission line. Nevertheless, theseries resistance and series resonance are data requested by circuit engineers and required

by specification MIL-C-55681B.

The military specification lacks an operations definition of resonance that is appropriate

for monolithic capacitors. The question of definition is considered in Section III-B andrelated to the resonance behavior of monolithic capacitors in Section III-C. The

experimental problems of resonance measurement are discussed in Section III-D. Aproposal for resonance specification is contained in Section III-E.

Fig. 5. Series resistance of 47-pF parallel plate capacitor.

0.01

0.10

1.00

0.1 1.0 10.0

F (GHz)

Rs(ohms)

BOONTON

34A

DLI 1.5

DLI 0.1


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B. Definitions of Resonance

The definitions of resonance, or the frequencies at which resonant phenomena occur, canbe derived from three difference viewpoints. First is the experimental or operational view

that defines resonance in terms of observables in some sensible test configuration.

Second is the circuit theoretic view in which the device is conceptually treated as aknown interconnection of ideal R, L, and C elements. The field theoretic view, which isthird, defines resonance in terms of energy absorption and storage. The three viewpoints

are equivalent for some simple circuits, but a small complication may invalidate theequivalence.

Resonant frequencies may be determined from a single port measurements by one or

more of three criteria. Assuming frequency to be the independent variable, resonanceoccurs at:

1) the frequency at whichX(orB) is zero;

2) the frequency at whichR (orG) is an extremum;

3) the frequency at which |Z| (or|Y| ) is an extremum;

Clearly, if 1) and 2) are equivalent, 3) follows; if 1) and 3) are equivalent, 2) follows.

Also, 2) and 3) may be equivalent when 1) fails. However, there is no necessity that anytwo be equivalent.

A network consisting of interconnected ideal R, L, and C elements has a driving point

impedance that may depend on frequency and all elements. According to the principles ofdimensional analysis, the total number of variables can always be reduced by combining

them to form a set of dimensionless variables. In this process, one may form all possibleindependent combinations of the form (LMCn)

-1/2. One of these sets may be defined as the

characteristics (or resonant) frequencies of the network. The functions that describe theterminal characteristics will depend explicitly on some or all of these frequencies.

The circuit theoretic definition makes sense from the measurement viewpoint providing it

is possible in a noninvasive way to probe the network at points where these frequencies,or combinations thereof, can be uniquely inferred from the data. If probing is not

possible, the characteristic frequencies may be observable only by inference from anexhaustive supply of data.

As an example of the problem of relating observables to characteristic frequencies,consider the various possible interconnections of three elements, illustrated in Fig. 6. Inthe circuits shown in Fig. 6 (a)-(d), no inconsistency exists between definitions (1)-(3);

all yield the resonant frequency. In the circuits in Fig. 6 (e)-(h), only 2) yields theresonant frequency. When one of the elements is divided, as shown for two cases in Fig.

7, other complications are introduced. In the case of Fig. 7(a), the characteristicfrequency is unchanged, but there is no frequency dependence of the admittance when


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RL =Rc = (L / C)1/2

. The division of C, as in Fig. 7 (b), produces two characteristicfrequencies, the series resonance (L C0 )

-1/2and the parallel resonance

[LC0 C1 / (C0 + C1 )]-1/2

. If the criterion 1) is applied, the data yield two, one, or noresonant frequencies, depending on the relative value of R. The conditions for the three

possible results are as follows:

(L / C1 +L / C0 )1/2 (L / C0 )

1/2

>R, two resonances, =R, one resonance,


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The driving point conductance has a maximum at the series resonance, but the maximumof the driving point resistance occurs only approximately at the parallel resonance. Thus

criterion 2) yields one exact and one approximate characteristic frequency. Criterion 3),like 1), fails to give characteristic frequencies. Nevertheless, all three are good

approximations when R is sufficiently small.

From the field theoretic viewpoint the driving point impedance and admittance are

defined by

Z= [2P+j4(WH WE)] /II* (12)

Y= [2P+j4(WH WE)] / VV* (13)

where P is the average power dissipation,WH is the average stored magnetic energy, and

WE is the average stored electric energy. To calculate these averages, the volume must bedefined by a closed surface across which no energy flows except at the driving point

where the current of voltage is defined. The frequencies at which WH= WEare those

given by 1), and the frequencies of 2) are those for which P is an extremum with theconstraint of constant current or constant voltage. If the fields are known throughout thevolume, P, WH, and WEcan be calculated, and the resonant frequencies follow. These

quantities may also be calculated for a known network of interconnected R, L, Celements, and the results must contain the characteristic frequencies.

The dilemma that occurs when the equivalence of the empirical criteria of resonance 1)-

3) fails is not resolved by the field-theoretic definition, but (12) and (13) serve to focusattention on stored and dissipated energy rather than element interconnections. To apply

them in this context, we assume only that the capacitor is a parallelopiped of unknowncontents, attached in some known matter to a microstrip transmission line. The current

and voltage must be defined on some transverse plane of the microstrip that has a specificrelationship to the geometry of the device. The equations motivate the interpretation of

terminal characteristics in terms of field distributions, a viewpoint suited to the higherfrequency range.

From the circuit-theoretic viewpoint the classification of resonances into series and

parallel can be decided by the known mode of connection of the elements. When thecircuit configuration is unknown, an experimental standard for this classification is

needed. If criterion 1) applies, series and parallel resonance can be distinguished by thesign of the derivative of the reactance, i.e., positive for series and negative for parallel. If

criterion 2) applies, the classification can be based on whether R at resonance is a

minimum (series resonance) or a maximum (parallel resonance). These two classificationschemes are not necessarily equivalent.

The fact that the empirical criteria for resonances and their classification are not alwaysequivalent raises at least two questions: a) which resonance criterion is a principal

significance? and b) must there be a single universal definition of resonance?


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C. Resonances of Monolithic Capacitors

The totality of measurements of monolithic capacitors supports the conjecture that thedriving-point impedance has the approximate form

Z() = {1/C} { A0 / j-1/2

1

An [jn + n ] / [(j+ n)

2

+ (n0)

2

] } (14)

The actual resonant frequencies are not strictly harmonic; they fall on a dispersion curve

that is characteristic of a folded line with periodic loading [8]. Nevertheless, the

approximation by the set [n0] is reasonable up to the point where resonances areobscured by losses. The loss parameters {n} increase slowly with frequency, and theresidues {An} are affected by boundary and excitation conditions.

When the excitation is strictly at the end of the equivalent folded line, the residues takevalues that permit the approximation of (14):

Z() = jZc(){(coscosh+ j sinsinh ) / (sincosh - j cossinh )}. (15)

HereZc() is the characteristic impedance of the folded line and is the attenuationover

the line length. Both vary slowly with frequency. The electrical length isapproximately proportional to frequency.

Criteria (1)-(3) are equivalent definitions of resonance for both (14) and (15). At

resonances,

dx /d= - (1/C) csch2< 0 dx /d= + (1/C) sech2> 0 (16)

Expression (16) provides a clear distinction between parallel and series resonances.

The date from which characteristics (14) and (15) are inferred are the resonancesobserved as well-defined peaks in dissipation loss when the capacitor is mounted in series

or as a load on a 50- microstrip transmission line. The connection of a capacitor to themicrostrip introduces, unavoidable, some additional circuit elements. These result fromthe discontinuity in the line as well as the electrode structure that couples the line to the

portion of the capacitor where (15) approximates the impedance. The impedance data thatare obtainable on the line include the effects of these lossless elements, and in general,

criteria 1) and 2) are not equivalent. The observed resonances, defined by 2), are

classified as parallel by the sign convention for the derivative of the phase, althoughX(0)0, in general. The series resonances are poorly defined by 2), but the data showthat a series resonance occurs between each pair of parallel resonances. As illustrated in

Fig. 8, this bracketing of series resonance is useful at the higher frequencies, but it isunsatisfactory for specifying the lowest series resonance.


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When the lowest series resonance is measured by application of criterion 1), the couplingand discontinuity reactances affect the measurements. One cannot say with certainty how

the resonance of the capacitor is related to the apparent resonance. The illusive nature ofthe first series resonance is the principal measurement problem.

Apart from the reactances that are not strictly attributable to the capacitor, the manner of

excitation of the active portion of the capacitor and it losses also affect the seriesresonances. To demonstrate these effects, consider the folded line model of the capacitor

when it is simultaneously driven at one end and at the center by sources in phase and ofrelative strengthsA andB. By criterion 1), the resonances occur when

[(A + 4B) cos + 4B cosh ] sin = 0 (17)

Parallel resonances occur when sin=0 unlessA = 0. Then = (2n + 1) is the set of

series resonances, and = 2n gives the parallel resonances. The A = 0 case occurs whenthe capacitor is mounted on edge, symmetrically with respect to the gap in the

microstrip.) IfB0, a shift occurs in the series resonances that depends onA, B, and thelosses. For a small shift (B/A


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(n - n0) / n0 (-1)n 4(B/A) cosh (18)

where n is the nth resonance whenB = 0. Evidently, the first series resonance is shifteddownwards and the second moves upwards. Other phase relationships between the

sources produce different shifts. In particular, ifA andB are out of phase, i.e., (B/A) < 0,

the first two series resonances tend to move close to parallel resonance that is betweenthem.

D. Resonant Frequency Measurements

Three connections of a capacitor to a 50- microstrip transmission line are practical forthe measurement of resonances. First is the series connection between source and load,

second is the shunt connection from line to ground at a point between source and load,and third is the connection from line to ground to serve as load. The first two, duals of

each other are suitable for measurement of the scattering parameters, S11 and S21. Theload connection is suitable for measuring only the reflection coefficient. These

connection are illustrated in Figs. 9 and 10.


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From measurements of return loss and insertion loss, using either the series or loadconnections, well-defined peaks in energy absorption can be observed. In almost every

case the frequencies of absorption can be classified as parallel resonances by using theminimum-maximum criterion for the series connection or the derivative of the phase

criterion for the load connection. Also in almost every case, the series resonances cannot

be observed by criterion2).

Equation (15) can be used to derive the characteristics of a capacitor near a resonance.

Near a series resonance where = 0,

R =Zc [1+(- 0)2

/ cosh2] tanh (19)

G = (1/Zc)[1-(- 0)2 / sinh2] coth (20)

Near a parallel resonance, also represented by = 0,

R =Zc [1-(- 0)2

/ sinh2] coth (21)

G = (1/Zc)[1+(- 0)2

/ cosh2] tanh (22)

These results show that when is small, the series resonance is well defined by (20) butpoorly defined by (19), and the parallel resonance is well defined by (21) but poorlydefined by (22). With the series or load connection, (19) and (21) apply, and only parallel

resonances are readily observable.

In principle, criterion 2) suffices for determining all resonances, providing the dual seriesand shunt connections can be achieved. The possible shunt connection illustrated in Fig.

9 (b) shows the difficulty in satisfying this requirement. Near the resonant frequencies thecapacitor has dimensions that are comparable to the wavelength in the dielectric of

microstrip substrate and in the dielectric of the capacitor. This fact eliminates anytransformation from the series to shunt connection that preserves the reference plane.

Accordingly, the well-defined resonances that are measurable with the shunt connectionmay not correspond to the series resonances that occur with the series connection.

When criterion 2) is impractical for the determination of resonances, a vector analyzer isneeded to obtain the phase information that permits the use of criterion 1). With thecapacitor in the load connection and the analyzer adjusted to shift the reference plane to

the measurement port, the series resonance occurs when the polar plot of the reflectioncoefficient crosses the negative horizontal axis. Essential to an accurate measurement is

the correct analyzer adjustment and a well-defined reference plane.


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The wide gap on which the capacitor is mounted in Fig. 10 (a) tends to distribute thecharacteristics of the capacitor over a significant length of the microstrip, and the

measurement may be invalid at the reference plane located at the center. Evidence of thiseffect is the observed change of the reflection coefficient when the capacitor is rotated

180 about the vertical axis, a change which does not occur when the narrow gap in Fig.

10 (b) is used. Nevertheless, the length l1 from the reference plane to the short in Fig. 10(a) provides a convenient variable for discussing the problem of calibrating the analyzerand correctly transferring the reference plane to the measuring port. With the HP-8510

analyzer the transference of the reference plane is accomplished by introducing a time

delay , preferably equal to the transmission time over the distance l0. The reflectioncoefficient observed at the measurement port is then

S() = ej2

[R -Z0 + jX1] / [R +Z0 + jX1 ] (23)

where (R + jX) is the impedance of the capacitor, X1 is the sum of the capacitor reactanceand that introduced by the shorted line of length ll,Z0 is the characteristic impedance of

the microstrip, and = ( 0) represents the deviation of the time delay from the

correct value o. The frequencies at which the imaginary part ofS() is zero are solutionsof

X/Z0 = -tan - cot 2+ [1 + cot22- (R/Z0)

2]

1/2(24)

Where = (l1 / Vph). If it is assumed that (R /Z0)


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The narrow gap connection of Fig. 10 (b) removes much of this uncertainty but notwithout cost. The increased capacitance across the gap is a shunt element across the test

unit, and the result is the equivalent circuit of Fig. 7 (b) near the resonance. Thefrequency of zero reactance is shifted upwards; it is possible, though unlikely, that the

reactance remains negative as the frequency is swept well beyond the series resonance of

the capacitorper se.

An invasive experiment that provides good values of series-resonant frequencies requires

removing the top surface of the capacitor and shorting the end of the active part. Theshort transforms series resonances to parallel resonances, which are readily observable by

criterion 2). The data indicate that series resonances occur almost exactly midwaybetween parallel resonances. Although this result is of technical interest, the unknown

effect of the destructive operation on the resonances should be ignored.

E. Resonances Specification

Both theoretical and experimental considerations make the determination of parallel andseries-resonant frequencies problematic. The series resonances are unobservable by

criterion 2) in the series connection, an equivalent shunt connection is unachievable.Criterion 1) may fail to give series resonances which do, in fact, exist; when applied to

parallel resonances, the frequencies may differ significantly from the frequencies ofmaximum energy absorption. Moreover, the use of different criteria for different

resonances is a questionable method for the establishment of standards. Destructivetesting is equally unacceptable.

Monolithic capacitors are commonly used as dc blocks in a series connection, as RF by-

pass elements in a shunt connection, and as elements in a filter. For the first two

applications the impedance measure is usually 50, and the transmission system ismicrostrip or stripline. Filter applications are too diverse to permit generalization.

In series or shunt connections, parallel resonances are usually a detriment to the operationof the circuit. They may be the cause of unacceptable insertion loss or parasitic

oscillations of amplifiers.

Series resonances, by contrast, are frequencies at which both insertion and dissipationlosses are so low as to be on the threshold of measurement capability. Moreover, they

tend to be so broad that their precise values are of no import. The circuit designer maychoose to center the band at the series resonance to take advantage if the low loss, but the

limits of the band are set by the adjacent parallel resonances.

The change of phase near a series resonance is important in the filter application.However, the designer must be aware that the configuration of the capacitor mounting

can alter the apparent value of the resonance by a significant amount. It is commonpractice to incorporate as an integral part of a filter design some means of phase

adjustment or tuning.


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In view of these considerations, specifications of monolithic capacitor self-resonantcharacteristics in terms of parallel resonances would prove satisfactory both to the

applications engineer and the manufacturer of capacitors.

Resonant-frequency measurements should be made on a 50- microstrip line, mounted in

the series connection. Parallel resonances should be defined as those frequencies at whichthe dissipation loss is a maximum. Capacitors should be mounted with the internalelectrodes parallel to the ground plane, and the gap length should not exceed 20 percent

of the length of the capacitor. The capacitors should be centered on the gap. Excellentparallel resonance data can be obtained with a relatively primitive network analyzer

system. Thus the manufacturer would not be burdened with the necessity of makingquestionable measurements on a very costly vector analyzer.

Better performance of monolithic capacitors is obtained when the hidden electrodes are

normal to the ground plane. In this mounting the odd-ordered parallel resonancesobserved with the horizontal mounting are transformed approximately to series

resonances. Whatever the mounting, series resonances, as defined by criterion 2), mustoccur between adjacent parallel resonances.

REFERENCES

1. W. B. Westphal,Dielectric Materials and Application, A. R. VonHippel, Ed.

Cambridge, MA:MIT Press, 1959, 4th printing, ch. 2, p. 63.

2. A. R. VonHippel,Dielectrics and Waves. New York: Wiley, 1954, 3rd printing, ch. 23,p. 74.

3. R. E. Lafferty, "Measuring capacitor loss,"Electronic Design, New York: Hayden,1976.

4. J. P. Maher, R. T. Jacobsen, and R. E. Lafferty, "High-frequency measurement ofQ-factors of ceramic chip capacitors,"IEEE Trans. Components Hybrids, Manuf. Technol.,

vol. CHMT-1, no. 3, 1978.

5.Application Note 80500, Alpha Industries, Inc, Woburn, MA, July 1976, p. 202.

6.Interim Data Sheet C35, Vitramon, Inc., Bridgeport, CT, ppl. 1, 4, 1978.

7. M. Ingalls and G. Kent, "Resonant coaxial lines for measurement of capacitors in thefrequency range: 10-100 MHz" presentation to Amer. Soc. for Testing Materials,

subcommittee FO1.12, June 18, 1986.

8. ----, "Monolithic capacitors as transmission lines,"IEEE Trans. Microwave TheoryTech., vol. MTT-35, pp. 964-970, Nov. 1987.

9. Document copied to best original by Dielectric Laboratories for internet use. 6/13/02.

Documents

Measurement of the Characteristics of High-Q Ceramic Capacitors