IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013 843

New Solutions of the Corona Discharge Equationfor Applications in Waveguide Filters in SAT-COM

Primo Alberto Calva Chavarría and Isaac Medina Sánchez

Abstract— The microwave device designers in the industry ofsatellite communications systems use analytical solutions of thecorona discharge equation, employing the concept of charac-teristic diffusion length, to determine whether its intensity ina particular device, such as waveguides and filters, is withinestablished margins. The analytical solutions provide the lowestpossible breakdown threshold. Until now, the difference of thegap between the electrical breakdown values obtained experi-mentally and provided by analytical solutions has not yet beenexplained sufficiently. This paper shows how it is inappropriateto use the characteristic diffusion length (�) rather than theeffective diffusion length (�eff ), although in principle electricfields are considered in terms of its geometry. The presence ofan ionic space charge generated during the time evolution of theelectron avalanche alters the medium properties in waveguidefilters, occasioning signal absorption and reflection anomalies thatmodify the condition of homogeneous fields in inhomogeneousfields.

Index Terms— Breakdown threshold, characteristic diffusionlength, corona, effective diffusion length, filters, satellite commu-nications systems (SAT-COM), waveguide.

I. INTRODUCTION

F ILTERS and waveguides production in satellite communi-cations is a controversial topic since the industry accepts

design margin ranges from 0 to 3 dB related to the minimumbreakdown power threshold [1]. Microwave devices designersuse the analytical solutions of the corona discharge equationto determine whether the power operation is inside the estab-lished margins. In the international literature, [2] showed thatthe breakdown of experimental data differs from the analyticalones obtained through the characteristic diffusion length. Inthis paper, it is demonstrated that a better approach is obtainedthrough the effective diffusion length. Little variations inbreakdown power in filters have a significant effect on thedata rate (system capacity); per each watt augmented 70 Gb/sare increased. In the literature, only comparisons betweenanalytical results and those obtained by numerical methodsare found. Experimental data in the literature are very scarce,nevertheless the authors with access to it make comparisonsand show that the criterion used hitherto for the Ku-type filtersdesign can be improved if the effective diffusion length isemployed instead of the characteristic diffusion length.

The equation that describes the time evolution of free

Manuscript received June 13, 2012; revised November 21, 2012; acceptedJanuary 24, 2013. Date of publication February 26, 2013; date of currentversion April 6, 2013.

The authors are with the Instituto Politécnico Nacional, Colonia Bar-rio la Laguna Ticomán, Mexico 07340, Mexico (e-mail: pcalva@ipn.mx;ismesa@gmail.com).

Digital Object Identifier 10.1109/TPS.2013.2246585

electrons is [3]–[5]

∂n

∂ t= ∇ · (D∇n) − �v · ∇n + (vi − va) n − βn2 + P (1)

where vi and va are the ionization and attachment frequencies,respectively, D is the diffusion coefficient, β is the recombi-nation coefficient, and P is the electron production rate byexternal sources. The time derivative provides the evolution ofthe electrons density, ∇ · (D∇n) is the diffusion term, whichis space dependent, and �v · ∇n is the convective parameter.To analyze the corona effect and the prebreakdown state, therecombination can be neglected as it is only of relevance whenthe electron density is high enough, which only occurs whenthe discharge has already started. Besides, the convective termis omitted too because a stationary medium is assumed, i.e.,the gas particles are not moving. At high altitudes, during thebeginning of the breakdown process, the main mechanismsresponsible for the electron losses are the diffusion from high-electric field intensity regions to low-intensity regions and theneutral molecules attachment. Thus the breakdown equation is

D∇2n + (vi − va) n = 0. (2)

For rectangular waveguides and filters, the solution is foundas (π

a

)2 +(π

b

)2 = v

D(3)

where v = vi − va is the effective ionization coefficient, anda and b are the width and height of the structure, respectively.In [4], the characteristic diffusion length � is defined as

v

D= 1

�2 . (4)

This way the diffusion processes become only geometrydependent. When a microwave field is applied, the energytransfer depends on the field frequency and on the environ-mental conditions (pressure and humidity). An effective fieldis defined in [4] as

Eeff = Erms(

1 + ω2

v2c

) 12

(5)

where Erms is the root mean square field, ω is the radianfrequency (2π f ), and vc is the collision frequency betweenelectrons and molecules given by [4]

vc = 5 × 109 p [s−1] (6)

and the electric field independent diffusion coefficient for airis

D = 106

p[cm2s−1]. (7)

0093-3813/$31.00 © 2013 IEEE

844 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013

The ionization frequency is [3]

vi = 5.14 × 1011 pexp(−73ϕ−0.44

)[s−1]. (8)

Withϕ = Erms

p(

1 + ω2

v2c

) 12

≡ Eeff

p(9)

and the two body attachment frequency is

va2 = 7.6 × 10−4 pϕ2(ϕ + 218)2[s−1

](10)

so it the electric field independent three-body attachment is

va3 = 102 p2 [s−1] (11)

for electrostatic electric fields

Eb = V

a(12)

with V as the voltage. Finally, the operation power is givenby

P = V 2

Z(13)

where Z is the characteristic impedance, which, for filters andwaveguides is [2]

Z =√

μ0ε0√

1 −(

12 f a

√μ0ε0

)2(14)

where μ0 and ε0 are the magnetic permeability and electricpermittivity in vacuum, respectively, and f is the operationfrequency.

II. DIFFUSION

Fig. 1 shows the analytical, measured, and simulated val-ues of breakdown power for a low-pass Ku-waveguide filteroperating at 12.5 GHz [2]. The analytical results are below themeasured data and even from the numerical simulation results.In the Paschen curve minimum, the difference between theexperimental and analytical value is of 16%.

The diffusion losses are determined according to the vari-ation on the free electron density along certain length. Whenhomogenous electric fields are used, this length is determinedonly by the considered geometry. The results shown imply thatit is necessary to consider the inhomogeneity of the electricfields, not due to geometry but due to the diffusion processthat occurs at low pressures. Then, instead of using the char-acteristic diffusion length, the effective diffusion length �effis used. In [6], it is determined that the diffusion length, in thepresence of no homogenous fields, depends only intrinsicallyof pressure, because D is the inverse function of p, as in (15),which is obtained using numerical methods [6]

1

�2eff

= a−2

√π4 (1 + 0.3677β) + π2βq

2+

(π

b

)2(15)

where

q =(

a

La

)2

+(πa

b

)2(16)

Fig. 1. Low-pass Ku-filter operating at 12.5 GHz [2].

Fig. 2. Experimental and analytical results using � and �eff for a Ku-filteroperating at 12.5 GHz.

β is a parameter that depends on the used gas, which for airis β = 5.33, and the value for La is calculated by means of

La =√

D

va. (17)

It is noticeable that if pressure increases, the effective diffusionlength decreases, and the breakdown threshold are the same asif using the characteristic diffusion length [7]. In Figs. 2 and 3,breakdown power comparison between the results obtainedwith �, �eff , and the experimental results obtained in [2].

These results can be explained because microwave break-down in an RF device is manifested by an avalanche-likegrowth in time of the free electron density in the gas fillingthe device. The difference between these power thresholdresults resides not only in their operating frequency but intheir geometries and the number of irises the filters contain.A bigger amount of irises contributes to generate more inho-mogeneity on the electric field. Fig. 4 shows the transversalconfiguration of each filter [2].

CHAVARRÍA AND SÁNCHEZ: NEW SOLUTIONS OF CORONA DISCHARGE EQUATION 845

Fig. 3. Experimental and analytical results using � and �eff for a Ku-filteroperating at 12.2 GHz.

Fig. 4. Transversal configuration of waveguide filters. (a) Operating at12.2 GHz. (b) Operating at 12.5 GHz [2].

TABLE I

CALCULATED BIT RATE FROM POWER OBTAINED BY CHARACTERISTIC

DIFFUSION LENGTH AND EFFECTIVE DIFFUSION LENGTH FOR TWO

DIFFERENT KU FILTERS

Analyzed Filter Minimum BreakdownPower [W]

Data Rate [Gb/s]

12.2 GHz � 83.5 4175�eff 86.1 4305

12.5 GHz � 97.2 4860�eff 101.4 5070

The filter operating at 12.5 GHz is affected by the electricfield inhomogeneity more than the other because of its highnumber of irises.

According to [8], the transmission capacity of satellitenetworks in terms of number of users, power, and data ratefor the downlink is

RB · N

P≤ 20 · 1018

[W−1s−1

](18)

where RB is the data rate in bits/s, N is the number of users,and P is the power in watts. Table I shows the difference ofbit rate obtained with the power from characteristic diffusionlength calculation and the one from effective diffusion lengthcalculation for the two different Ku filters.

By using (18), the resulting increase on the bit rate of thefilters, when using the effective diffusion length, is of 4.3%in the case of the 12.5-GHz low-pass filter and of 3.1% for

Fig. 5. Schematic representation of electric field distortion in a gap causedby space charge of an electron avalanche.

the 12.2-GHz low-pass filter. Therefore, a small raise in thepower, even of 3 or 4 W, is heavily reflected on the data rateand an increase of almost 200 Gb/s is achieved.

The two main processes responsible for the loss of electronsduring the initiation stage of the breakdown are: diffusion fromhigh-density regions toward regions with lower density andattachment on neutral molecules, forming essentially immobilenegatively charged ions. For high enough electron density, inthe breakdown threshold, the region saturates and the propaga-tion of the electric field is affected by reflecting or absorbingit. The most important negative ions are O−

2 and O−, andthe most important positive ions formed in air discharges atatmospheric pressures are N+

2 , N+4 , and O+

2 , where stablenegative nitrogen ions are not yet detected experimentally [9].The evolution of the avalanche is influenced by any agentaltering the space electronic charge density. Fig. 5 showsthe electric field Er around the avalanche and the resultingmodification of the applied field E0. The space charge atthe head of the avalanche is assumed concentrated within aspherical volume, with the negative charge ahead because ofthe higher electron mobility. The field is enhanced in frontof the head of the avalanche with field lines from the anodeterminating at the head, the region III. Further back in theavalanche, the field between the electrons and the ions leftbehind reduced the applied field (E0), the region II. Stillfurther back the field between the cathode and the positiveions is enhanced again, the region I [10].

The resultant field strength in front of the avalanche isthus (E0 + Er ), whereas in the positive ion region justbehind the head the field is reduced to a value (E0 − Er ).

846 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 4, APRIL 2013

Fig. 6. Applied electric breakdown E0 versus space charge electric field Erfor different values of x .

According to the results exhibited in Figs. 2 and 3, wherethe analytical values of the breakdown power are lower thanexperimental ones, this is an indication that the avalancheis mainly affected by the presence of positive charged ionsinstead of the negative charged ions. The radial field producedby positive ions immediately behind the head of the avalanchecan be calculated from the expression [10]

Er = 5.3 · 10−7 αeαx

(xp

) 12

[Volts

cm

](19)

where x is the distance in cm which the avalanche has pro-gressed, p is the gas pressure in Torr, and α is the Townsendfirst coefficient of ionization, Fig. 6 shows the behavior of Er

and E0 as function of the pressure for different values of x ,x = a = 0.25 cm is the maximum height of the Ku filter.

It can be appreciated that the influence of positive ionicspace charge field is greater as function of the development inspace of the avalanche.

III. CONCLUSION

The designers of waveguides and filters used the continuity(3) to obtain the lowest possible breakdown power threshold,which implied to have considered homogeneous electric fieldsas function of the geometry. However, the presence of thespace charge in the discharge process caused inhomogeneitiesin the electric field. The breakdown power differences of thevalues obtained analytically through (3) and those obtainedexperimentally for Ku filters can be explained as function ofthe influence of the positive ionic space charge.

A better approach of the breakdown power threshold ana-lytical results, compared with the experimental ones, wasobtained through the effective diffusion length �eff insteadof the characteristic diffusion length �. This implies that theelectrical field can be considered inhomogenous. Also, byusing �eff instead of the characteristic, we found an increaseof 3%–4% on the data rate, considering that the design margin

was of 3 dB lower than the minimum breakdown threshold,and this obtained higher data rate was of great importance.

REFERENCES

[1] M. Yu, “Power-handling capability for RF filters,” IEEE Microw. Mag.,vol. 8, no. 5, pp. 88–97, Oct. 2007.

[2] P. Carlos and Q. Vicente, “Passive intermodulation and corona dischargefor microwave structures in communications satellites,” Ph.D. disserta-tion, School Electr. Eng., Tech. Univ. Darmstadt, Darmstadt, Germany,2005.

[3] W. Woo and J. DeGroot, “Microwave absorption and plasma heatingdue to microwave breakdown in the atmosphere,” IEEE Phys. Fluids,vol. 27, no. 2, pp. 475–487, Oct. 1983.

[4] A. D. MacDonald, Microwave Breakdown in Gases, New York, USA:Wiley, 1966.

[5] A. D. MacDonald, D. U. Gaskell, and H. N. Gitterman, “Microwavebreakdown in air, oxygen and nitrogen,” Phys. Rev., vol. 130, no. 5,pp. 1841–1850, Jun. 1963.

[6] U. Jordan, D. Anderson, L. Lapierre, M. Lisak, T. Olsson, J. Puech,V. E. Semenov, J. Sombrin, and R. Tomala, “On the effective diffusionlength for microwave breakdown,” IEEE Trans. Plasma Sci., vol. 34,no. 2, pp. 421–430, Apr. 2006.

[7] U. Jordan, D. Anderson, V. Semenov, and J. Puech, “Discussion onthe effective diffusion length for microwave breakdown,” U.S. patent283 713, Feb. 20, 2011.

[8] M. Wittig, “Satellite onboard processing for multimedia appli-cations,” IEEE Commun. Mag., vol. 38, no. 6, pp. 134–140,Jun. 2000.

[9] S. Badaloni and I. Gallimberti, Basic Data of Air Discharges, Istitutodi Elettrotecnicae di Elettronica, Università di Padova, Tech. Rep. 5,Jun. 1972.

[10] E. Kuffel and W. S. Zaengl, High Voltage Engineering. London, U.K.:Newnes, 2000.

Primo Alberto Calva Chavarría was born in Ato-tonilco el Grande, Mexico, in 1953. He received thePh.D. degree in electrical engineering from Insti-tuto Politécnico Nacional, Mexico City, Mexico, in1993.

He has been engaged in research and educa-tion with Instituto Politécnico Nacional, since 1994,where he is currently a Professor of electrical dis-charges. He was a Part Time Consultant for researchand development work in industry for many years.His current research interests include a broad range

of basic aspects and applications of electrical discharges e.g. plasma physicswith application to microwave devices and electrical insulation.

Dr. Chavarría is member of the National Researchers System and theAcademy of Engineering of Mexico.

Isaac Medina Sánchez was born in Distrito Federal,Mexico, in 1987. He received the Masters degreein advanced technology from Instituto PolitécnicoNacional, Mexico City, Mexico, in 2009. In thesame year, he started studying the Ph.D. degree incommunications and electronics with the InstitutoPolitécnico Nacional.

His current research interests include aspects andapplications of electrical discharges in microwavedevices.