Gap capacitance of a coaxial resonator usingsimplified mode matching

C.G. Wells and J.A.R. Ball

Abstract: The gap capacitance of a single coaxial resonator is found using a simplified rigorousmode matching method to calculate its resonant frequency. A method of finding the coefficients ofthe modal field expansions is also shown. The capacitance results for this type of structure arepresented graphically and also as a normalised cubic polynomial. The results agree very closelywith those obtained from measurements.

1 Introduction

Coaxial filters are used in wireless and mobile communica-tion applications due to their small size, low cost andrelatively high Q factor. Traditionally these filters have beendesigned using filter theory based on TEM mode transmis-sion line structures [1].

In this paper, a coaxial line in the form of a cylindricalcavity with a centre conductive rod (single coaxialresonator) as shown in Fig. 1 will be studied. The coaxialline is less than l/4 in length, with open- and short-circuitends. The inductance provided by the line at theopen circuit, and capacitance due to the gap (Cg), provideconditions for a resonant circuit. The resonant frequencycan be adjusted by altering the gap capacitance with theaid of a tuning screw. Size constraints may require thecoaxial line length to be reduced to l/8 or less, which meansthat a substantial capacitance is required to bringthe structure to resonance. This in turn means the gapmust be quite small. However, calculation using a parallelplate capacitance model will only give an approximateresonant frequency for the structure, as the total capaci-tance will be larger due to fringing effects around the open-circuit end of the centre conductor. If a tuning screw isadded, as shown in the Figure, it will further complicate thecapacitance evaluation.

This problem can be overcome by the use of a rigorous(in the sense that the solution is obtained in the form ofsuccessive approximations converging to the exact solution)mode matching method [2, 3] to compute the resonantfrequency of the cavity, and hence the gap capacitance. Thecavity is partitioned into two cylindrical regions, as shownin Fig. 2, and the fields in each region are represented as alinear combination of the radial basis functions. Since onlythe lowest order quasi-TEM resonant frequency is required,only radial modes having no circumferential variation needbe included. The transverse fields are matched at theboundaries between regions to ensure that they arecontinuous, and a set of simultaneous equations is

produced. The resonant frequency is found by equatingthe determinant of this system to zero. Resonant frequenciescomputed in this way show excellent agreement withmeasured results. Once the resonant frequency has beenfound, the gap capacitance can be calculated. For filterapplications, a requirement is to maximise the resonator Q-factor. This is accomplished by choosing the coaxial lineradius ratio a/ro to be 3.591 ([4], page 24), which minimisesthe conductor losses in the coaxial surfaces. This corre-sponds to an air-spaced characteristic impedance of 76.7O.

inner conductor

outer conductor gap capacitance (Cg)

short circuit open circuit

tuning screw

Fig. 1 Single coaxial transmission line resonator

I

II

Z

b

b1

b2

a r0

gap (g)=b1-b2 ρΦ

Fig. 2 Single coaxial resonator coordinate system

The authors are with the Faculty of Engineering and Surveying, University ofSouthern Queensland, Toowoomba, Queensland 4350, Australia

r IEE, 2004

IEE Proceedings online no. 20040727

doi:10.1049/ip-map:20040727

Paper first received 22nd October 2003 and in revised form 10th May 2004.Originally published online: 9th July 2004

IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 5, October 2004 399

Values of gap capacitance are provided for this optimumsituation.

The mode matching method was used, rather than afinite element software package, as it requires less CPU timedue to its inherent analytic pre-processing, and also allows abetter understanding of the field structure.

2 Analysis using the radial mode matchingmethod

The mode matching method used in this paper representsthe fields within the cavity in terms of radial waves, and issimilar to that used by others [5–7] to solve the generalproblem of cylindrical posts in waveguides. Anothermethod, called the longitudinal mode matching method,has been used by Risley [8] and Zaki [9]. It entails thematching being performed at a plane at right-angles to thecylindrical axis, and so uses cylindrical waveguide basisfunctions. This method was not used as it requires theintegral of the product of Bessel functions, which makesthe problem more complex and increases the computationtime [10].

The structure to be analysed in this paper is shownin Fig. 2, and is composed of two air-filled regions. Region Iis the cylindrical gap between the inner rod and the tuningscrew, assumed to be of the same diameter, whereasthe remainder of the cavity is region II. All metal surfaceswill be considered to be perfect electric conductors (PECs).The fields within the structure are represented by super-positions of radial waves, which form standing waves.Their form can be derived using the boundary conditionsand the radial waveguide field equations as describedby Balanis [11]. In this paper these equations are greatlysimplified by removing the circumferential variations.This can be justified by realising that for the radialmodes to describe the field patterns of the TEM transmis-sion line, only those parts describing the radial andlongitudinal variations are necessary. This means thatthe TEmodes can be neglected as there is no Ez component,and Er and Hf are zero as they are dependent on f.The simplified transverse magnetic (TM) radial modefields are derived from a magnetic vector potential of theform:

Azðr;f; zÞ ¼ðC1JiðbrrÞ þ D1YiðbrrÞÞ�ðC2 sinðbzzÞ þ D2 cosðbzzÞÞ

ð1Þ

2.1 Radial TM modesFor region 1, using the appropriate boundaries, themagnetic vector potential field equation for the gap betweenthe conductive rod and the tuning screw can be shown to be

Az1ðr; zÞ ¼ ATMik J0ðbrrÞ cosðbzðz� b1ÞÞ ð2Þ

where i and k are the mode indices and bz ¼ ðkp=b2 � b1Þdue to the top and bottom PEC boundaries at the rod andtuning screw ends. The wave number in the radial directionbr is related to bz, and to the wave number of the mediumb, by the equation:

br ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � b2z

qð3Þ

For region II, Ez will be zero at the tangential outercylindrical boundary (r¼ a). This field component is relatedto the potential function by:

Ez ¼ �j1

omE@2

z2þ b2

� �Az

Working backwards from the outer wall, the potentialfunction for region II must have the form:

AzIIðr; zÞ ¼BTMmn ðY0ðbraÞJ0ðbrrÞ

� J0ðbraÞY0ðbrrÞÞ cosnpb

z� � ð4Þ

where m and n are the mode indices assigned to region II,and bz¼ (np/b) due to the top and bottom PEC boundariesat the outer cylinder end plates. The other TM fieldcomponents can then be derived from the respectivepotential function for each region using the relationshipsgiven by Balanis [11].

If the radial wave number br is imaginary, then themode is non-propagating, and the fields will decayexponentially in the radial direction. This can occur in bothregions, and with either mode type. In this case, the Besselfunctions Jx and Yx for that particular mode will have to bechanged to modified Bessel functions Ix and Kx, respec-tively.

2.2 Mode matching at the boundarybetween regionsTo find a field solution to the current problem thetransverse E and H fields in both regions must be matchedat the cylindrical boundary r¼ ro. Each field component isrepresented by a summation of radial modes. Matching thetransverse electric fields across the boundary betweenregions I and II leads to:

E ¼ ATMik

Xik

ETMI ¼ BTM

mn

Xmn

ETMII ð5Þ

Similarly, the transverse magnetic fields will be matched ifthe following condition holds:

H ¼ ATMik

Xik

HTMI ¼ BTM

mn

Xmn

HTMII ð6Þ

The above pair constitutes a doubly infinite set of linearequations for the modal coefficients Aik and Bmn. Tosimplify these, and provide good convergence, the followingorthogonality relation (inner product), as applied by Yao[12], was used:

oEik;Hmn4 ¼ZS

ðEik �HmnÞds ¼ 0 if ik 6¼ mn ð7Þ

This is applied to both (5) and (6). Firstly, form the cross-product of (5) and a testing function from the magnetic fieldof region II and integrate over the inner cylindrical surfaceof region II:

oE; hmnII4 ¼Zb

0

Z2p

0

ðE� hmnII Þrrodfdz ð8Þ

Secondly, form the cross-product of (6) and a testingfunction from the electric field of region I and integrate overthe outer cylindrical surface of region I:

oeikI ;H4 ¼Zb2b1

Z2p

0

ðeikI �HÞrrodfdz ð9Þ

The infinite set of linear equations so formed isthen reduced, by truncating the number of modes used,to a value that will give a desired solution accuracy.This is possible because the modes tend to taper indominance from lower to higher order, providingconvergence to near the exact value for relatively few

400 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 5, October 2004

modes used. The resultant equations in matrix form for theelectric field are:

a11 . . . a1q

..

. . ..

TMITMII...

ap1� � � apq

0BB@

1CCA

ATM1

..

.

ATMq

0BB@

1CCA

¼

b11 . . . 0

..

. . ..

TMIITMII...

0 � � � bpp

0BB@

1CCA

BTM1

..

.

BTMp

0BB@

1CCA

ð10Þ

or in an abbreviated form:

½W�½A� ¼ ½X�½B� ð11ÞThe magnetic field equations are similar and in abbreviatedform are:

½Y�½A� ¼ ½Z�½B� ð12ÞThe elements of [W] and [Y] are the result of the innerproducts on the left-hand side of (5) and (6), respectively,and those of [X] and [Z] are the results of the inner producton the right-hand side of the same equations. [A] and [B] arethe unknown coefficients. The TMik mode in region I isidentified by q and the TMmn mode in region II is identifiedby p.

These equations can then be used to find the resonantfrequencies of the structure or to find the unknowncoefficients of the field equations.

2.3 Resonant frequencies of the structureA homogenous system of equations may be formed from(11) and (12) by eliminating either the [A] or the [B]coefficients. It is preferable to eliminate [B], because thisleads to a smaller matrix.

½½Y��1½Z�½X��1½W�½A�� ¼ 0 ð13ÞBoth [X] and [Y] are diagonal matrices and so their inversesare easy to obtain. The eigenvalues of (13) are the resonantfrequencies of the structure and these can be determined byfinding the frequencies at which the determinant of theoverall matrix is zero. In this lossless case the elements ofthe matrix are all real.

2.4 Radial mode coefficientsTo check that the mode matching solution is physicallysensible, it is good practice to calculate and plot the fieldpatterns. Also, in high power applications, there may be arequirement to determine the peak electric field strengthwithin the structure, to check if dielectric breakdown islikely. In order to plot these patterns, the coefficients of theradial mode field expansions must first be determined. Theunknown coefficient (11) and (12) can be rearranged intothe form:

W �XY �Z

� �AB

� �¼ 0 ð14Þ

A selected coefficient is then chosen as unity, so thatequation (14) can then be written as:

AB

� �¼ W �X

Y �Z

� ��1bTMTMII

..

.

0bTMTMII

..

.

bTMTMqI

2666666664

3777777775

ð15Þ

In this paper the coefficient chosen is the first TM mode inregion II ðBTM

1 Þ and so the associated inner product valuesare bTMTM

II to bTMTMqI as shown.

The system of (15) has more equations than unknowns(i.e. over determined) but can still be solved for thenormalised values of the unknown coefficients by the use ofthe Matlab operator ‘\’ or the function ‘pinv’. Thesefunctions give a least squares solution of these modetrunctated equations, and so produce a best fit result [13].

Once the coefficients are found they can then besubstituted into the field equations, so that the fieldcomponents can be determined from the sum of the modesat a number of spatial grid points, and the resultant field inthe structure can be plotted as a superposition of thecomponents.

It was found, however, that there was a limit tothe number of modes for which a solution is feasible.Rank-deficient matrices, giving inaccurate matrix inverseresults, occurred when more than about 20 modeswere used in region II. This is due to the very largedynamic range of the element values in the matrix, whichexceed the limited numerical range of the computer. Inany case, the number of modes that can be used is notsufficient to obtain results as accurate as the resonantfrequencies of Section 2.3. However, it is sufficientto generate reasonably accurate field patterns. A magnifiedview of the electric fields in the vicinity of a 10mm gap isshown in Fig. 3 which represents the superposition of 3 TMmodes in region I, and 20 TM modes in region II. Forcalculation purposes, the ratio of region I to region IImodes should be as close as possible to the ratio of theheights of the regions. This minimises the relativeconvergence error problem [14, 15]. Some discontinuitymay be seen at the boundary between the regions in Fig. 3due to the limited number of modes used. The computationtime for the unknown coefficients and the plot, usingMatlab code on a PC running at 1.6GHz, was about5minutes.

−10 −8 −6 −4 −20

2

4

6

8

10

12

14

16

18

X, mm

Z,m

m

Ex−Ez field X−Z plane

I

II

coaxial centreconductor

Fig. 3 Ey-Ez field plot

IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 5, October 2004 401

3 Comparison of calculated and measured results

A number of mode matching calculations were performedto determine the resonant frequencies of the lowest orderTEM mode for various gap sizes (b2�b1). The tuning screwpenetration was set to zero (b1¼ 0). A sufficient overallnumber of modes were used to allow convergence of theresonate frequency to a value that changed by less than1MHz, when compared to the previous value obtained withless modes.

A single coaxial resonator was constructed, sothat measurements could be made to comparewith calculated values as shown in Fig. 4. The mechanicaldimensions were constrained by the use of readily availablematerials, so that the inner and outer radii are ro¼ 0.565cmand a¼ 1.742cm, respectively. The ratio (a/ro) is 3.0832,corresponding to a characteristic impedance of 67.5Owhich is not quite optimum ([4], page 24). Boththe measured and calculated results obtained are shownin Fig. 5. On this scale, the mode matching resultscannot be distinguished from the measuredresults. Typically, the difference between them is of theorder of 1%.

4 Discussion

It is useful to interpret the results shown in Fig. 5 in terms ofthe transmission line equivalent circuit shown in Fig. 1. Thecapacitance (Cg) for the gap (g) can be calculated from theresonant frequency and the inductive reactance of theshorted length of coaxial line. Since the capacitance isproportional to the overall size of the structure, it isconvenient to normalise the capacitance values, by dividingthem by one of the radial dimensions. In this case the radiusof the inner conductor was used. A graph of both thecalculated and measured results, presented in this format, isshown in Fig. 6. The calculated results agree very well withmeasured results, the difference being of the order of 1%.

To minimise the coaxial conductor loses and maximisethe Q factor of the resonator, the optimum ratio of the radiiis (a/ro)¼ 3.591 corresponding to a characteristic impedanceof 76.6O. For this characteristic impedance, a normalisedgraph of calculated capacitance versus gap size (g) is shownin Fig. 7. The curve is the result of a least-squares cubicpolynomial fitted to the calculated points, and is of the form

Cg

ro

� �¼ ao þ

a1

ð groÞ þ

a2

ð groÞ2þ a3

ð groÞ3

ð16Þ

where ao¼ 0.1977, a1¼ 0.09551, a2¼�4.5� 10�4,a3¼ 2.5� 10�6 and the norm of the residuals is 0.0451.

coaxialprobeconnector

coaxialcentreconductor

air gap

gap screwadjustment

outercylinder

modematchingboundary

Fig. 4 Single coaxial resonator used in measurements(ro¼ 0.565 cm, a¼ 1.742 cm, b¼ 8 cm)

0 5 10 15 20 25300

400

500

600

700

800

900

1000

1100

1200

end gap, mm

reso

nant

freq

uenc

y, M

Hz

MMM ptsMMM (spline)measured pts

Fig. 5 Single coaxial resonator resonant frequencies(ro¼ 0.565 cm, a¼ 1.742 cm, b¼ 8 cm)

0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

8

9

10

11

normalised end gap − g/ro

norm

alis

ed c

apac

itanc

e −

Cf/r

o, p

f/cm

MMM ptsMMM(cubic polynomial)measured pts

Fig. 6 Normalised capacitance vs normalised end gap (b2¼ 0, endgap (g)¼ b1, a/ro¼ 3.0832)

0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14

16

normalised end gap − g/ro

norm

alis

ed c

apac

itanc

e −

Cg/

ro, p

f/cm

MMM ptsMMM(cubic polynomial)

parallel plateestimate

shieldedopen circuit

Fig. 7 Normalised capacitance vs normalised end gap (b2¼ 0, endgap (g)¼ b1, a/ro¼ 3.591)

402 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 5, October 2004

It is constructive to compare the values of capacitance forlarge and small gaps to those obtained from other sources.For small gaps, the parallel plate value of capacitancecalculated from the inner conductor end area and gapdistance can be used. The normalised value for a 0.02 cmgap (0.04 normalised) is 6.954 and this can be compared tothe calculated value of 8.348 when the fringing effects aretaken into account. A few values for the parallel plateapproximation are plotted in Fig. 7. For larger gaps Rizzi[16] states that the plane of the coaxial open circuit isapproximately 0.6(a�ro) past the end of the innerconductor. For the coaxial line used in this section thislength works out to be 0.777cm. Using the Zo of the line anormalised capacitance of 0.676 can be calculated. Thiscompares favourably to the value (0.561) from modematching for large gaps plotted in the Figure.

When the measurements were performed, the measuringequipment was connected to the cavity by means of a smallprobe, as shown in Fig. 4. Since the probe couples to theradial electric field of the coaxial TEMmode and introducesasymmetry into the cavity, it will also excite higher-ordercoaxial modes having circumferential variation. The first ofthese to propagate (TE11) has a cut-off frequency of4.14GHz, so they will all be well below cut-off over themeasured frequency range [17]. Hence although theycontribute to the local field of the probe, they will haveotherwise little effect.

5 Conclusions

The radial mode matching solution which has beendescribed provides a numerically efficient method ofcalculating the resonant frequency and gap capacitance ofa coaxial resonator. Minimum conductor loss and hencemaximum Q factor occur for a radii ratio a/ro¼ 3.591. Forthis case the gap capacitance results are well represented bya best-fit least-squares cubic polynomial, which may beuseful in design calculations.

The specific results reported can be applied to thecalculation of gap capacitance, tuning range or temperaturedrift in coaxial resonators, as used in cellular base stationmulti-cavity filters. They may also be applied to thecalculation of a coaxial open circuit end-effect correction.The general method is also applicable to the calculation ofthe equivalent circuit of a gap in a coaxial centre conductor,and hence to the design of coaxial diode mounts.

6 Acknowledgements

The prototype cavity filter was manufactured in theUniversity of Southern Queensland (USQ) mechanicalengineering workshop by Chris Galligan. Both authorswish to thank the referees for their perceptive commentsand suggestions. Colin Wells would also like to acknowl-edge the financial support provided by a USQ Scholarshipin presenting this paper.

7 References

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