Empirically-derived capacitor characteristic formulasfrom distributed modelling

R.W. Brown

Abstract: Diffusion equation modelling is used to develop formulas for the normally fixed valuesof capacitance and resistance of the traditional capacitor equivalent circuit. The formulas define thedependence of the equivalent circuit values on metal film resistivity, capacitance per unit area, arealdimensions of the metallisation and on frequency. A multilayer capacitor topology, having bothcapacitor plates connected at the same end, is used for the derivation, but it is shown that theresults are also representative for the more standard double-end connected topologies with somerestrictions above the typical self-resonance frequency of these capacitors. The formulas allowaccurate prediction of dissipation factor and input impedance according to the design parametersused in constructing the capacitor, thus providing powerful tools in capacitor design. The algor-ithms also facilitate the determination of internal voltages, currents and power distributionwithin the capacitor, thus exposing the effects, for example, of partial edge disconnection. The for-mulas may potentially provide a better capacitor equivalent circuit with dependent variables forcircuit emulation. In the paper, the derivation process is described and the formulas testedagainst experimental results. A simple addition to the equivalent circuit is also included tomodel dielectric loss which dominantly determines the dissipation factor at low frequency.

1 Introduction

Low-voltage (,400 volt AC) power metallised polypropy-lene (MPP) power capacitors, as depicted in Fig. 1, arewidely used for motor start capacitors, fluorescent lightfilters, noise filtering and general purpose AC power appli-cations. MPP capacitors are compact, with excellent powerdensity, low energy loss and good frequency performance.The capacitor plates are commonly formed by a thinvacuum-evaporated metallic layer on each of two thin poly-propylene film strips. The metallic layer is made thin so thatthe capacitor can recover from an electrical short throughthe dielectric film. The short results in a momentary heavycurrent flow which removes the metal film around the break-down point, thus terminating the discharge and isolating thefault with only a minor reduction in effective capacitor platearea. This ‘self-healing’ property of MPP capacitors is amajor advantage responsible for their widespread usage.The universally used construction method of cylindrical

capacitors involves co-winding two metallised strips ofpolypropylene film onto an insulating mandrel. The twometallised strips each have a margin, along oppositeedges, that is free of metallisation, as shown in Fig. 2.This clear edge ensures that only one strip is connected toone end of the capacitor roll when the ends of the cylindricalroll are arc-sprayed with a zinc layer. This sprayed-on met-allic layer is called a schooping. Wires are soldered to eachschooping, finishing the connection to the plates of thecapacitor.

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-cds:20060013

Paper first received 11th January and in final revised form 24th September 2006

The author is with the RMIT University, PO Box 2476V, Melbourne,Victoria 3001, Australia

E-mail: rwbrown@ieee.org

IET Circuits Devices Syst., 2007, 1, (2), pp. 117–125

As shown in Fig. 3, this popular and low cost constructionmethod provides a short electrical path to the metallic filmarea, ensuring low ohmic losses and excellent frequencyperformance in the finished capacitor. Distributed circuitmodelling of this topology typically involves elementalcapacitor squares series-connected from the schooping tothe opposite edge. As the size of these elements becomesvanishingly small, modelling approaches a continuum bestrepresented by diffusion equations. However, diffusionequation modelling of the top and bottom layer is difficultas the two input connections are not both at one end, as iscommonly the case for transmission-line modelling, butare at opposite ends to each other.

Single-end connection can occur in a capacitor if, forsome reason, direct connection to the schooping is lostand the current in both layers is forced to flow circumferen-tially back to a common point instead of axially as shown inFig. 4 and Fig. 3, respectively. For experimental purposes,single-end connection is realised if the capacitor strips areconnected at the inner or outer periphery, as shown inFig. 2 where the two lightly shaded dots on the two layersrepresent electrical connection points. It is a matter forinvestigation in this and other studies, whether modellingresults for single-end connection can also be representativeof double-end connection. The single-end connection modelshown in Fig. 4 correctly represents lengths of the capacitormetallisation that have become disconnected from directcontact with the schooping, due to removal of the metallayer as corrosion progresses inward from the schoopingends. Such corrosion also tends to progress from the outerlayers toward the core because the outer layers are lesstightly wound, allowing atmospheric moisture to entermore freely [1, 2]. This progression means the remainingcentral metal film may be disconnected from the schoopingimmediately adjacent, but connected to the schoopingfurther into the roll via a circuitous circumferential path.Such a connection results in much greater ohmic losses

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and degraded frequency performance. Other studies indicatethat excessive current density and heating within the capaci-tor can arise from this corrosion–induced disconnection [3].

2 Capacitor equivalent circuit models

The traditional electrical model of capacitors depicted inFig. 5 is composed of an equivalent series inductance(LES), resistance (RES), capacitance (CES) and conductance(GES). Conductance in MPP capacitors is virtually zero andis usually ignored. Distributed inductance within the capa-citor is neglible and inductance is principally due to wireconnections and the physical length of the capacitor [2, 3].Thus inductance may be represented by a lumped inductor.

The traditional model is good for many applications atlow frequency, but fails at higher frequencies because thecomponent values are frequency-dependent and not fixedas assumed in the model. The variability is principallydue to several factors: frequency-dependent filter effectswhich progressively isolate metal film most distant fromthe input connections, skin effects and eddy currentlosses. Many models of greater complexity have beendeveloped to better characterise capacitors, but these aretypically aimed at high-frequency applications [4–6] andare not useful for steady-state power capacitor studies.

Metal-film skin effects are insignificant in low frequencyMPP power capacitors where the metal film thickness, oftypically 10 nm, is very much less than the skin depth inaluminium. Eddy current loss and nonuniform current dis-tribution are difficult to characterise and are unlikely to

Fig. 1 Typical metallised polypropylene capacitors

The two lower capacitors are degraded and have been removed fromtheir casings

Fig. 2 Traditional construction of spiral-wound capacitor usingtwo aluminium-metallised polypropylene strips

Electrical connection is made at each end of the finished roll

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impact low frequency power capacitors [5], although thetopic is worthy of further research.Modelling using transmission-line analogues, as shown in

Fig. 5, produce much improved prediction of capacitor beha-viour [7–10]. However, the computation is complex and thelinkages between build parameters and capacitor character-istics are obtuse and untreated in the mainstream disport-ment of power capacitor studies [11]. In addition, diffusionequation describing functions and distributed equivalent cir-cuits are generally not convenient for in-circuit modelling.

3 Distributed model

Assuming as a starting point that both overlaying capacitorfilms are connected at the same end, a distributed capacitormodel may be composed of a multitude of series-connectedcapacitor equivalent circuit elements as shown in Fig. 6.Limit analysis using diffusion equations, for current andvoltage, result in the classical telegrapher’s equations forvoltage and current down the length of the strips. Thevalues of both voltage and current may be determined at arbi-trary points down the length of the metallised strip. In thiscase, the units for R, L, C and G are for each square. Thespreading resistance is in ohms per square (independent ofthe size of the square) but the capacitance square must bedefined in each model. For a normal, fully-connected capaci-tor as shown in Fig. 3, the long edge of thewound strip is con-nected directly to the schooping. The capacitor may then bemodelled as a number of parallel capacitors, each of onesquare length. To obtain higher resolution cross-strip valuesfor voltage and current, the width may be broken into an arbi-trary number of strips, each composed of series-connectedsmall squares, depending on the resolution required. In oneinvestigation, the author has used ten squares across thewidth [12]. Other conceptual models are equally valid, butthese are the simplest and are used in this paper.As previously described, edge corrosion can result in a

gap developing between the metallisation and the schoop-ing. The gap may be narrow resulting in most of the metal-lisation remaining intact. The corrosion gap tends toprogress with time, from the outer layers where moisture

Fig. 3 Electrical connection of metallised strip in traditionalspiral-wound capacitor

The second layer under the one shown above is a mirror image butreversed vertically

Fig. 4 Series or distributed connection of elemental capacitorswith input connection at the same end

Current in the second layer, beneath that above, is equal and opposite

IET Circuits Devices Syst., Vol. 1, No. 2, April 2007

enters most freely, into the inner layers. This can result inconsiderable lengths of the two wound capacitor filmsbeing only connected circumferentially at one end of thecorrosion–isolated section where the metallisation stillextends to the schooping [1]. Assuming the erosion effectis identical in both film layers, this circumferential connec-tion, corresponding to Fig. 4, represents a true transmissionline for that part of the capacitor partially isolated by cor-rosion. The long circumferential connection, of nominallength (n) in terms of metal film width, results in seriouslydegraded capacitor characteristics principally manifested asdramatic increases in dissipation factor and effective loss ofcapacitance at higher frequency [1]. The loss in capacitanceoccurs due to a filtering effect isolating metal film that ismost distant from the remaining connection of the metalli-sation to the schooping.

4 Equivalent series resistance (RES)

Acylindrical capacitor as shown in Fig. 2, with electrical con-nection to both strips at the outer periphery, is used for initialanalysis and experimental modelling. A family of curves forequivalent series resistance of such a capacitor, derived fromdistributed circuit diffusion-equation modelling with variousvalues of R, C and n, is shown in Fig. 7. The curves displaya classical Bode-plot characteristic, but the falloff isproportional to the square root of frequency and is due to pro-gressive disconnection with frequency of the metallic filmmost distant from the input terminals. The principal variablesin a polypropylene capacitor are the metal film spreadingresistance in ohms per square (R) and capacitance (C) persquare area. The square area depends on the model used.The total length of the metallised strips is also expressed inunits (n) of the metallisation strip width.Thus the total actual capacitance for the strip is nC, where

C is the total capacitance between a unit square and the twoadjacent films in the stack above and below the film inquestion.The characteristic curve, shown in Fig. 7, can be mod-

elled by an equation of the form

RES ¼a

ð jf Þ þ bð Þ0:5

�������� ð1Þ

where f is frequency, j is the imaginary operator and a and bare functions of n, R and C.

Fig. 5 Traditional capacitor equivalent-series circuit

Fig. 6 Equivalent circuit of serial cascade of traditional capaci-tor equivalent circuits

Diffusion analysis yields classical Telegrapher’s equations. L, R, Cand G are with respect to elemental length

IET Circuits Devices Syst., Vol. 1, No. 2, April 2007

It is well known [13] that, at low frequency when f � b,RES of a distributed capacitor with two uniform metallisedstrips, is given by

RES ¼2nR

3ð2Þ

Combining (1) and (2) with f � b gives

a

b0:5¼

2nR

3ð3Þ

or

b ¼9a2

4n2R2ð4Þ

Substituting (4) into (1)

RES ¼a

ð jf Þ þ ð9a2=4n2R2Þð Þ0:5

���������� ð5Þ

The problem then is reduced to finding the dependence of aon R, C and n.

For a frequency very much greater than the ‘pole’ fre-quency, (5) becomes

RES ffia

ð jf Þ0:5

�������� ð6Þ

Fig. 7 Equivalent series resistance of cylindrical capacitor withelectrical connection at the same end of the strips

Data determined by distributed circuit/diffusion equation modelling.(First number in legend designations is in ohms/square; capacitanceis per square.)

Table 1: Relationship between equivalent seriesresistance and physical parameters for frequency of1 MHz

R, V C, nF n RES, V Ratio

1 8 8 800 12.61566261

2 1.6 8 800 5.641895835 1/2.236067977

3 1.6 40 800 2.523132522 1/2.236067977

4 1.6 40 160 2.523132522 1

5 8 8 800 12.61566261 5

6 8 1.6 800 28.20947918 2.236067977

‘Ratio’ is the ratio of the value of RES in that row, to the value inthe row immediately preceding. R and C are per square of metalfilm area. A square is 35 mm

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As evident in Fig. 7, the distributed circuit modelling showsthe pole frequency to be between � 10–100 Hz implyingthat a frequency of 1 MHz is suitable for evaluation of (6)for the case under study.

The limit values shown in Table 1 are obtained, for RES at1 MHz, by distributed circuit diffusion equation modelling.From the first row to the second row, reduction of R by afactor of 5 results in reduction of RES by a factor of

p5 or

2.23067977. This implies that RES is proportional to R0.5

for frequencies above the pole frequency. Rows 3 and 4indicate that RES is not dependent on n at high frequency.In rows 5 and 6, R is held constant but C is reduced by afactor of 5. RES increases by a factor of

p5 indicating that

RES is proportional to C20.5 for frequencies above thepole frequency. This relationship is also exhibitedbetween rows 2 and 3. From (6) these relationships maybe expressed as

RES ffidðR=CÞ0:5

ð jf Þ0:5

���������� ð7Þ

where d is a constant. Thus, equating (6) and (7),

a ¼ dR

C

� �0:5

ð8Þ

Substituting (8) into (5)

RES ¼dðR=CÞ0:5

ð jf Þ þ ð9d2=4n2CRÞð Þ0:5

���������� ð9Þ

It is then necessary to evaluate d. Rearranging (7)

d ffi RES

Cjf

R

� �0:5�����

����� ð10Þ

Data from Table 1 are used together with (10) to generateTable 2, where frequency is 1 MHz. As can be seen, d isinvariant for changing conditions of R, C and n. It is alsonotable that 0.3989422804 is equal to (2p)20.5. The formof (9) can be changed by substituting d ¼ 2k

RES ¼2kðR=CÞ0:5

ð jf Þ þ ð9k2=n2RCÞð Þ0:5

���������� ð11Þ

where k ¼ 0.1994711402.

4.1 Fit of formula-derived values of RES withdistributed circuit results

In Fig. 8, the solid lines depict the RES curves generated by(11) superimposed over the points generated from distribu-ted circuit diffusion equation modelling. As can be seen, the

Table 2: Evaluation of constant for equivalent seriesresistance formula ( f 5 1 MHz)

R, V C, nF n RES, V d

1 8 8 800 12.61566261 0.3989422804

2 1.6 8 800 5.641895835 0.3989422804

3 1.6 40 800 2.523132522 0.3989422804

4 1.6 40 160 2.523132522 0.3989422804

5 8 8 800 12.61566261 0.3989422804

6 8 1.6 800 28.20947918 0.3989422804

R and C are per square area, and each square is 35 mm

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agreement is very good. For comparison, the solid line isalso shown for the distributed result for R ¼ 8 V/square,C ¼ 1.6 nF/square and n ¼ 800 square. Some very minormismatching between the two can be seen around the poleregion.

5 Equivalent series capacitance (CES)

The graphs of Fig. 9 depict the predicted equivalent seriescapacitances derived using distributed circuit diffusionequation modelling of capacitors with both plates drivenfrom one end. The graphs are computed for various combi-nations of R, C and n.As with the plots for RES, the CES plots display a classical

Bode plot characteristic, falling off with the square root offrequency and having the form

CES ¼p

ðð jf Þ þ qÞ0:5

�������� ð12Þ

where p and q are functions of R, C and n. For f � q,

Fig. 8 Fit of formula-predicted values (lines) of equivalent seriesresistance with distributed circuit, diffusion-equation-determinedvalues (data points)

Cylindrical capacitor with electrical connection to both strips at theouter periphery. (First number in legend designations is in ohms/square; capacitance is per square.)

Fig. 9 Equivalent series capacitance as predicted by distributedcircuit diffusion equation modelling

Electrical connection at the outer periphery of each strip of the cylind-rical capacitor. (First number in legend designations is in ohms/square; capacitance is per square.)

IET Circuits Devices Syst., Vol. 1, No. 2, April 2007

(12) becomes

CES ffip

q0:5

�������� ð13Þ

At low frequency, CES must equal nC, the total lowfrequency capacitance. Therefore, from (13)

q ffip2

n2C2

���������� ð14Þ

Substituting (14) into (12)

CES ¼p

ðð jf Þ þ ðp2=n2C2ÞÞ0:5

�������� ð15Þ

For frequencies very much higher than the pole frequency

CES ffip

ð jf Þ0:5

�������� ð16Þ

As evident in Fig. 9, a suitable frequency, well above thepole frequency, is 1 MHz. Following the procedure usedfor determining the relationships between RES and R, Cand n, Table 3 is derived using distributed circuit diffusionequation modelling at a frequency of 1 MHz. Rows 1, 2 and3 indicate that CES is independent of n at this frequency.Comparing row 3 and row 4 indicates that CES is pro-portional to C0.5, because a five-fold reduction in Cresults in a

p5 reduction in CES. This is reinforced by

similar change in CES between rows 6 and 7. A five–foldreduction in R from row 5 to row 6 results in a

p5 increase

in CES indicating CES is proportional to R20.5.From (16), these relationships may be expressed

CES ffimðC=RÞ0:5

ð jf Þ0:5

���������� ð17Þ

where m is a constant. Thus, equating (16) and (17),

p ¼ mC

R

� �0:5

ð18Þ

Substituting for p in (15)

CES ¼mðC=RÞ0:5

ðð jf Þ þ ðm2=n2RCÞÞ0:5

���������� ð19Þ

Although (19) may be used to determine m, it is sufficient touse a simplified equation assuming a frequency well abovethe pole frequency. The previously used frequency of1 MHz is a suitable frequency.

Table 3: Relationship between equivalent seriescapacitance and physical parameters ( f 5 1 MHz)

R, V C, nF n CES, F Ratio

1 8 8 32 1.261566 � 1028

2 8 8 160 1.261566 � 1028 1

3 8 8 800 1.261566 � 1028 1

4 8 1.6 800 5.641896 � 1028 1/2.236067977

5 8 8 800 1.261566 � 1028 2.236067977

6 1.6 8 800 2.820948 � 1028 2.236067977

7 1.6 40 800 6.307831 � 1028 2.236067977

‘Ratio’ is the ratio between the value of CES in a row with that ofthe preceding row. R and C are per square area. The square areahas a dimension of 35 mm. (Note that 2.236067977 is numeri-cally equal to

p5)

IET Circuits Devices Syst., Vol. 1, No. 2, April 2007

Equation (19) becomes

CES ffimðC=RÞ0:5

ð jf Þ0:5

���������� ð20Þ

Rearranging

m ffi CESð jf Þ0:5 R

C

� �0:5�����

����� ð21Þ

To determine the value of m, data from Table 3 is used in(21) to produce Table 4 at a frequency of 1 MHz (wellabove the pole frequencies exhibited in Fig. 9), and as canbe seen, m is not dependent on R, C or n.

As m is identical in value to d, which is equal to 2k, thesubstitution m ¼ 2k is made in (18)

CES ¼2kðC=RÞ0:5

ðð jf Þ þ ð4k2=n2RCÞÞ0:5

���������� ð22Þ

The format for the equation for CES (22) is thus similar tothat for RES (11). The pole is at a lower frequency thanthat of RES.

5.1 Fit of formula-derived values of CES withdistributed circuit diffusion equation results

Formula-predicted values for CES using (22) are plotted asthe solid lines in Fig. 10, with markers representing valuesobtained from distributed circuit diffusion equation analysis.It can be seen that the congruence between the two determi-nations is excellent. Some disparity exists in the region ofthe pole, as illustrated in the case of the upper trace, wherethe distributed circuit trace is also shown.

6 Dissipation factor

For a capacitor equivalent circuit of series resistance andcapacitance, dissipation factor (DF ) is given by

DF ¼ 2pfRESCES ð23Þ

Substituting (11) and (22) into (23)

DF ¼ 2pf2k

ðð jf Þþð9k2=RCn2ÞÞ0:5

�������� 2k

ðð jf Þþ ð4k2=RCn2ÞÞ0:5

��������

ð24Þ

Examination of (24) shows that, if the product of RCn2 isconstant, then the graphs for DF will be identical for anycombination of R, C and n.

Table 4: Constant for equivalent series capacitanceformula ( f 5 1 MHz) evaluated from data obtained fromdiffusion equation modelling

R, V C, nF n CES, F m

1 8 8 32 1.261566 � 1028 0.3989422804

2 8 8 160 1.261566 � 1028 0.3989422804

3 8 8 800 1.261566 � 1028 0.3989422804

4 8 1.6 800 5.641896 � 1029 0.3989422804

5 8 8 800 1.261566 � 1028 0.3989422804

6 1.6 8 800 2.820948 � 1028 0.3989422804

7 1.6 40 800 6.307831 � 1028 0.3989422804

R and C are per square (a square has a dimension of 35 mm)

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The dissipation factor clearly has two frequency regimes;the lower frequency band where f � 4k2/RCn2 and theupper frequency band where f � 9k2/RCn2.

In the lower frequency band, the DF asymptotically risesmonotonically with frequency. In the upper frequency bandthe DF asymptote has the value of one. In the upper fre-quency band, (24) becomes

DF ¼ 8pk2 ð25Þ

Because the DF, as predicted by distributed circuit diffusionequation modelling, is equal to 1.0 in this region

k ¼1

ð8pÞ0:5¼ 0:1994711402 ð26Þ

This corresponds to the empirically–determined value for kin Section 4.

6.1 Fit of formula-derived values of DF withdistributed-circuit-diffusion-equation-determinedresults

Plots of DF derived, from both (24) and from distributedcircuit diffusion equations are shown in Fig. 11. As can

Fig. 10 Fit of formula-predicted values (lines) of equivalentseries capacitance, with distributed circuit diffusion equation-determined values (data points)

Electrical connection at the outer periphery of each strip of the cylind-rical capacitor. (First number in legend designations is in ohms/square; capacitance is per square.)

Fig. 11 Fit of formula-predicted values of dissipation factor withdistributed circuit diffusion equation-determined values

First number in legend designations is in ohms/square; capacitance isper square

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be seen, correspondence is very good with some limitedincongruity in the pole or inflexion region.Substituting for k and converting to standard s–plane

forms where s ¼ j2pf, equations (11) and (22) become

RES ¼ðR=CÞ0:5

ðsþ ð2:25=RCn2ÞÞ0:5

���������� ð27Þ

and

CES ¼ðC=RÞ0:5

ðsþ ð1=RCn2ÞÞ0:5

���������� ð28Þ

7 Dielectric loss

As frequency is reduced, losses in the metal conductors fallbelow losses in the dielectric, and the dissipation factor isdominantly determined by the dielectric losses [6, 7]. Thedissipation factor plateaus, typically at a value of around0.0002 for polypropylene capacitors [2, 14]. A frequencydependent resistor can be added to the traditional capacitorequivalent circuit to model dielectric losses. The value ofthe resistor has to be inversely proportional to frequencyfor the DF to remain constant [6, 15]. A more comprehen-sive equivalent circuit model of a capacitor, shown inFig. 12, incorporates such a loss element jRD/f j in serieswith RES to model dielectric loss. In effect, the additionalresistance together with RES, forms an overall equivalentseries resistance encompassing dielectric losses and metalfilm ohmic losses. The modified equivalent circuit alsodepicts RS and LS for ‘source’ or external impedance.Leakage conductance (GES) for polypropylene capacitors istypically quoted in data sheets as less than 3 � 10210 V21,for capacitors up to 20 mF, and may therefore be neglectedexcept for long term charge storage analysis. It is thereforeignored in this model.At very low frequency, jRD/f j will be very much larger

than the equivalent series resistance arising from metalfilm losses. Using this to simplify (23) and rearranging

RD ¼DF

2pCES

ð29Þ

This equation provides a ready means of calculating RD of acapacitor for a given low-frequency dissipation factor. Thelow-frequency asymptotic dissipation factor value is set towhatever is desired, in most cases around 0.0002 for highquality polypropylene capacitors.

8 Characterisation of a commercially-manufactured capacitor without schoopingconnections

An uncompleted commercial 8 mF metallised polypropy-lene capacitor of cylindrical construction, without schoop-ing metallisation, was used to model a long dual-stripcapacitor with electrical connection at the outer ends ofthe two metallised strips forming the spiral winding.

Fig. 12 Capacitor equivalent circuit incorporating dielectricloss element and connection impedance

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Measurements were repeated for various remaining lengthsof wound metallised film on the capacitor cylinder as filmwas progressively unwound and removed. Maximum striplength was 1121 squares or 39.2 m. The unit area or‘square’ was a square of dimension equal to the width ofthe metallisation of 35 mm. Spreading resistance (R) of themetallised strip was measured to be 2.92 V/square andcapacitance per unit area (C ) was calculated to be7.136 nF per 35 mm square. A small but unavoidable andvariable length of the metallised strip extended from therolled capacitor to the connections to the measuringbridge. With two strips involved and some transition effectin the first wrap of the capacitor, this effectively added asource resistance of up to around 20 V to the capacitormodel test assembly.Inductance for a capacitor configuration, fed from the

same end, is very low [2, 3, 14], and the only significantinductance is associated with packaging [10] (in this case,the wiring between the cylindrical roll and the measuringinstrument). As the test configuration is a true single-endfeed (or type ‘A’ connection topology as described byMurphy [16]) distributed inductance is ignored.Source resistance RS was set for best fit in the upper fre-

quency range. The values were 16 V, 23 V and 13 V for thecapacitor lengths of 33, 121 and 721 squares, respectively.Dielectric loss resistance RD was evaluated from (29)assuming a limit DF of 0.0002, typical of polypropylenecapacitors of this type. Values were 135 V, 36.9 V and6.19 V for capacitor lengths of 33, 121 and 721 squares,respectively. The experimental and theoretical results areshown in Fig. 13.The traces diverge from the asymptote value of 1.0, in the

upper frequency region, due to the effect of source resist-ance. With the appropriate empirically-determined valuesof source resistance in each case, good congruity betweenexperimental and formula-derived results are exhibitedover the complete frequency range shown.

9 Input impedance

Input impedance (ZIN) of the simple capacitor model isdetermined by RES in series with CES, thus

ZIN ¼ðR=CÞ0:5

ðsþ ð2:25=RCn2ÞÞ0:5

����������þ

1

s

ðsþ ð1=RCn2ÞÞ0:5

ðC=RÞ0:5

���������� ð30Þ

Fig. 13 Experimental and formula-determined values for dissi-pation factor for a commercial capacitor with both strips electri-cally connected at the outer periphery

Curvature at the bottom of the lower trace is due to dielectric lossbecoming significant as frequency falls

IET Circuits Devices Syst., Vol. 1, No. 2, April 2007

For jsj � 1/RCn2, (30) becomes

Zin ¼2

3nRþ

1

snCð31Þ

This corresponds to the classical input impedance of twotimes one third the serial resistance of one strip [13] plusthe reactance of the sum of the distributed series capaci-tance. For jsj � 2.25/RCn2, (30) becomes

ZIN ¼ð1� jÞ

v0:5

R

C

� �0:5

ð32Þ

where v is scalar frequency (in radians per second).It can be seen from (32) that the magnitude of the input

impedance falls with the square root of frequency in theupper frequency band. The corresponding asymptoticphase angle for the input impedance for a distributed R–Ccircuit is 245 degrees (represented by the (12 j) term)and is independent of the values of R and C. However,the addition of any external nondistributed resistancebetween the measuring point and the distributed circuit,changes the phase angle asymptote to zero degrees andthe ultimate input impedance to be equal to the externalnondistributed resistance rather than zero.

10 Experimental and predicted inputcharacteristics of a model capacitor

Measured and theoretical plots of the magnitude of the inputimpedance of the previously described commercial capaci-tor roll fed from a common end, is shown in Fig. 14. Thecapacitor parameters were

† metallised film length (n) 33.1 squares,† capacitance (C) 7.136 nF per square,† sheet resistivity (R) 2.92 V per square,† dielectric loss resistance (RD) 135 V and† empirically-determined ‘source’ resistance (RS) 16 V.

As shown, the match between the formula–determinedplot and the measured plot is excellent over the wholefrequency range. The singularity frequency is representedby 1/RCn2 ¼ 6971.5 Hz. The corresponding phase angle ofthe input impedance of this capacitor, depicted in Fig. 15,shows good correlation between experimental and formula-derived results. It is interesting to note that, if this capacitor

Fig. 14 Experimental and formula-determined values for inputimpedance magnitude for commercial capacitor with both stripselectrically connected at the outer periphery

123

had normally edge connection of each square of metallisationdirect to the schooping, the singularity frequency would be7.64 MHz. This is typically more than an order of magnitudeabove the self-resonance frequency for these capacitors.

11 Standard commercial capacitor having filmmetallisation connected at both ends of the roll

A degraded capacitor was measured and characterised. Itscritical parameters were

† outside diameter of roll 24.4 mm,† inside diameter of roll 9.4 mm,† metallisation width 53 mm,† capacitance 1.74 mF,† metal strip length 422 square (calculated from dimen-sions) and† lumped inductance 30 nH (best fit).

A plot of the measured dissipation factor is given in Fig. 16.As can be seen, the graph is not of the standard form and thevalues of DF are much higher than expected. This indicatespartial disconnection of the schooping as previouslydescribed.

Fig. 15 Experimental and formula-determined values for inputimpedance phase for commercial capacitor with both strips elec-trically connected at the outer periphery

Fig. 16 Experimental and formula-determined DF of normally-constructed, degraded 1.74 mF MPP capacitor

R ¼ 2.865 V/sq, metallised width ¼ 53 mm, length (n) ¼ 422,RD ¼ 391 V, RS ¼ 0.11 V, L ¼ 30 nF, schooping disconnectionpart ¼ 10%. Electrical connections to each end of the capacitor roll

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The capacitor thus consists essentially of two capacitors (ormore) in parallel: the first fully connected to the schooping ateach end, the other, series or circumferentially connected tothe schooping ends. Calculating the parallel admittance foreach and adding them together for an assumed percentageof disconnection, and deriving the dissipation factor fromthe aggregate, the match between theoretical predictionsand experimental results is shown in Fig. 16. As depicted,the match is excellent over the whole frequency range foran assumed degree of disconnection of 10%. This matchattests to the efficacy of the empirical formulas in modellingdouble-end connected, as well as single-end connected,capacitors over the whole operating frequency range up toself-resonance.Values of RD and RS of 391 V and 0.11 V, respectively,

were determined by trial and error for best fit.

12 Conclusion

Capacitor formulas quantitatively linking equivalent seriesresistance and equivalent series capacitance, to the physicalparameters of a capacitor and to frequency, closely matchthe results obtained from more complex distributed circuitdiffusion equation modelling and from experimentalmeasurement. Matching is excellent over the whole fre-quency range for the single-end connected capacitor top-ology, which commonly occurs inside capacitors due topartial disconnection from the schooping edge. Matchingis also very good for double-end connected capacitors,which is the standard capacitor connection format, at leastin the frequency band below the singularities or pole fre-quencies for RES and CES. These singularities are typicallyan order of magnitude higher in frequency than the self-res-onance frequency in MPP power capacitors. Matchingbetween predicted and measured characteristics above thepole frequencies for double-end connected capacitors isless accurate, but provides useful insight into the effect ofchanges in capacitor parameters. For typical operatingregimes, the formulas provide accurate means of predictingcapacitor characteristics from build parameters. In addition,the effects of internal changes (particularly in the case ofpartial disconnection of the metallic film from direct con-nection to the schooping, resulting in a compound capacitorconnection topology), can be modelled to deduce the effecton overall capacitor characteristics.The formulas provide insight into how capacitor par-

ameters are affected by changes in metal film conductivity,per-unit capacitance and build geometry. An addition of afrequency-dependent resistance to the traditional capacitorequivalent series resistance to represent dielectric loss isdemonstrated. A simple formula is derived relating thevalue of this resistance to total capacitance and nominallow-frequency dissipation factor due to the dielectric.Further work to develop diffusion equation modelling

of the double-end connected capacitor topology, may bebeneficial for more accurate characterisation above thepole frequencies of capacitors connected in this way.

13 References

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