Progress In Eectromagnetics Research, PIER 30, 73 84, 001 CONCHOID OF NICOMEDES AND LIMACON OF PASCAL AS ELECTRODE OF STATIC FIELD AND AS WAVEGUIDE OF HIGH FREQUENCY WAVE W. Lin and Z. Yu University of Eectronic Science and Technoogy of China Chengdu, 610054, China E. K. N. Yuang and K. M. Luk City University of Hong Kong Hong Kong, China Abstract Groove guides have sharp corners in the guide proper so are not ow oss and high power carrying wave guides. Proposed has been a groove rectanguar waveguide with rounded interna anges [1]. However, the atter is not practica in having the infinite arms. Let us try the Conchoid of Nicomedes and Limacon of Pasca to see how can they be boundaries of static fieds or high frequency waves. Aso has been proposed the pentagon waveguide [] for the high power transmission in high frequency. Yet we fee that we have very imited forms of waveguides, so we study the existing curves to see whether we can find some better guides such as rounded corner rectanguar waveguide and others by starting from the boundary curves of ρ = a/ cos ϕ + and ρ = a cos ϕ +. 1. Conchoid of Nicomedes. Limacon of Pasca 3. The Theory 4. Appications 5. Concuding Remarks References
74 Lin et a. curves Xmax Ymax 1 1.5-0.31446 0.1788.0-0.58031 0.45031 3 3.0-1.05364 1.101 4 4.0-1.54969 1.8736 5 5.0-1.9389.66870 6 7.0 -.75078 4.33455 7 10. -3.53989 6.94745 y 0 M M1 Mx,y) x=a Figure 1. Locus of Mx, y). 1. CONCHOID OF NICOMEDES We now see how to construct the Conchoid of Nicomedes, given a poe 0 and a straight ine, the base ine x = a, Fig. 1, and a ine segment, > 0, we can write down how to describe M1 and M ρ = a cos ϕ + 1a) x = a + cos ϕ 1b) y = a tan ϕ + sin ϕ 1c) cos ϕ = x x + y 1d) So that x a) z = x ) We can write from 1a), by the mutipication of e jϕ z = x + jy = e jϕ + a ejϕ cos ϕ now we construct a new equation containing ω = u + jν,byrepacing ϕ by bω, b is a constant to be fixed ejbw z = e jbw + a cosbw) and then take jb = 1 for simpicity we have then e w z = e w + a coshw) = eω + a1 + tanhw) 3)
Conchoid of Nicomedes and Limacon of Pasca as eectrode 75 where bw = ju + ϕ,sofor u =0,ϕ= ν. Equation ) describes curve in Fig. and Fig. 3, which can be the boundaries of waveguide of groove type and cose type of good quaity. The ta arms of the former wi go up rapidy to meet each other at x = a.. LIMACON OF PASCAL Now in Fig. 4, we have a poe o and a base curve of a circe x a/) + y = a /4 instead of a base ine, so the points M1 and M obey the equation in poar coordinates ρ = a cos ϕ + 4a) so that x = a + a cos ϕ + cos ϕ 4b) y = a sin ϕ + sin ϕ 4c) now 4b) gives x a = a cos ϕ + cos ϕ so together with 4c) we have x a ) + y = a + a cos ϕ + 4 that is x + y ax = a cos ϕ + ) =ρ by 4a), we finay have then x + y ax ) = x + y ) 5) as the equation of curves in Fig. 5, which are then boundaries of a new famiy of rectanguar waveguides of rounded inner corner, so the cut-off waveength of the first mode wi be approximatey λ c =4y max 6)
76 Lin et a. Figure. Groove guide the ta arms of which do not open up to infinity but asymptoticay meets the base ine x = a. We may take a =1,they may be eectrode of static fied or guide of surface waves. The equation of the curves are x = a+ cos ϕ and y = a tan ϕ+ sin ϕ. Figure 3. The cose waveguides, when changed from 1.5 to 10 and a =1. The equations of the curves are x = a + cos ϕ, y = a tan ϕ + sin ϕ, x max = a) 1/3 ; y max =a 1/3 a 1/3 ): /3 + a /3 ) 1/.
Conchoid of Nicomedes and Limacon of Pasca as eectrode 77 y M1x,y) o ϕ M p B a x Figure 4. Base Circe, OB = a,locus of M1 and M. Figure 5. The Limacon Curves ρ = a cos ϕ+) a =1,=0.8; 1; 1.5; ; 3. There are two oops, when < a the inner oop shrinks with increasing. When a, there is one oop. We can have ridge waveguide and cose empty guide. x = a cos ϕ + cos ϕ, y = a sin ϕ cos ϕ + sin ϕ. where the x max,y max ) are the maximums on the Fig. 5 and are on the curve ρ = a sin ϕ cos ϕ
78 Lin et a. so together with 5) we find x Max = a +1 + y Max = x max +1 From 4a), repace ϕ by bw, w = u + jν, make jb =1,wefind, by timing by e jϕ first, z = e w acoshw + ) = a + a ) ew ew + 6a) therefore we find, as needed in 5) x + y = e u acoshu cos ν + + jasinhu sin ν = e u { coshu + cos ν)a + +acoshu cos ν } 6b) and z a = e u a ) ew + a = e u 4 eu + + ae u cos ν 6c) and z a ) a a 4 eu e u = e u sinhu + + ae u cos ν = x +y ax 6d) Now from 6a) we find a ew + e w z a ) =0 So we can sove for e w, e w = 1 a ± +a z a ) ) and then we have, w = u + jv, e u = 1 a ± a +a z a ) 7a)
Conchoid of Nicomedes and Limacon of Pasca as eectrode 79 and we have transformed 5) to 7), with u as a parameter, et m + jn = a +a z a ), then so finay where a e u = ± m + m + n [ a e u m n ] =4 m 7b) m + n = ± +ax) +4a y m = 1 { ax ± } +ax) +4a y n = 1 { ax ± } +ax) +4a y of course the curve 5) in the x-y pane of curve in Fig. 5 can be transform into curve in w = u + jν pane by means of 6b) and 6d) ) a e u 4 sinhu + + ae u cos ν = [coshu + cos ν] a + +acoshu cos ν ) 8a) so that dz dw = e u { a e u +ae u cos ν + } 8b) and for the inner region of a guide, u<0, dz dw e u,sowemay find the soution of the guide probem by starting from this approximation. 3. THE THEORY Now eq. ) is the we known concoid of Nicomedes from which we construct a compex equation 3): z in term of ω,so we can interpret ω = u + jν as a compex potentia, the rea part u and the imaginary part ν of which are both soution of the Lapace s equation, by the
80 Lin et a. Cauchy Riemahn reations between u and ν. When u = 0, ω = jν, and 3) goes back to eq. 1), so 1) is the zero potentia boundary of the static fied defined by 3). Aso we can interpret 3) as a coordinate transformation, transforming the points on the z-pane onto points on the w-pane, so that the two dimensiona Hemohotz equation ψ x + ψ y + k c ψ =0 9) is changed into that in the w-pane as ψ u + ψ ν + k dz c dw =0 10) Whie in the case where u is a static potentia, the magnitude dw dz is the fied intensity E, E = dw dz = 1 11) h From 3), we have dw dz = eω + = e ω + 1 cos u cos ν + jsinhu) sin ν ) {coshu) cosν) jsinhu sin ν} cos u + cos ν) So that 6) becomes, making a = 1 without oss of generaity h cos u cos ν +1 = eu cos ν + cosh4u + cos 4ν + cos u cos ν + 4 + sin sinhu cos ν +1 ν eu cos u cos ν + cos 4u + cos 4ν + 4 Now from 3) we find, et a =1, x = e u cos ν + eu + cos ν coshu + cos ν ) y = sin ν e u cos ν + coshu + cos ν 1) 13) 14a) 14b)
Conchoid of Nicomedes and Limacon of Pasca as eectrode 81 to observe how do u and ν vary aong the rea axia y = 0,wesee 1) y =0,ν=0,x= e u +1+tanhu 0 for u x = e ω +1+tanhu =0for u =0; ) y =0,ν= π, x 0asu,x as u 3) y =0,e u cos ν = then 14a) gives coshu + cos ν e u 1 x = 14c) coshu + cos ν which gives x = 0when u = 0ify = 0ony, whie 14a) gives x = cos ν +1. Sowehave to go back to 3) to investigate w as function of z, bychanging 3) into the foowing equation, etting a =1: e 3w z) + e ω + e ω + e ω z/ =0 15) we can sove e ω from this equation, first by soving ξ 3 + pξ + q =0 16) e ω = ξ 1 z 3 17) p =1 1 ) z 3 3 q = z 1 ) z + z 3 7 the discriminate is q ) 3 q = + 3 { ) [ z = 1 1 1 7 + z ) ) ]} z + + 14 = 1 {1 1 7 + z } η ) 3 where ) z ) z η = + + 14
8 Lin et a. Here η is given by prescribed z and, usuay, prescribed is z = x+jy. Then 16) is soved, and soved is e ω in 17), so that we find the static fied E, 11) and the scae factor h, 13), enabing us to study the fied ψ in 10). From 13), e u = e w = ξ 1 z = F x, y), so that ) is transform into an equation with u of 3) as a parameter. Of course ) can be transformed into an equation of u and ν, a curve in u-ν-pane. To do so, we find from 3) x a = e u asinhu cos ν + cos u + cos ν ) z = e u 4acoshu cos ν + coshu + cos ν and write ) in the from x a) z =x a + a) =x a) +ax a)+a so that we can sove for x a) : a ± a z x a) = z 1 18a) 18b) and to remove the ± sign, we have to write a a z ) x a z ) = z ) ) 1 1 then putting in 15a) and 15b) we have an equation in u and ν competey, for curves in u-ν pane or w-pane of ) of x and y variabes. Now from 3) we find Making use of dz dw = ew +1 tanh w) d dw tanhw = tanh w 1 ) = cos ν +jcoshu sin ν coshu + cos ν),
Conchoid of Nicomedes and Limacon of Pasca as eectrode 83 so that it is neary true that dz dw e u and is free of ν so is separabe in variabe u and ν. 4. APPLICATIONS A pot eq. ), with >0asaparameter, x = a is the asymptote in a the cases of different.sothe vertica arms wi be cosed for arge y, not open up to infinity. Each curve of Fig. can be made to act as an eectrode when charged and/or to act as waveguides, in three difference ways: a) The part in the eft haf pane, shown aso the maximum y and its positions can be some forms of ridge wave guide to give wide operating frequency ranges; b) On the right are ta waveguides good for high microwave power and high frequency transmissions which ca for high cearance for high microwave fied and for ow waveength; c) The whoe structure is a guiding structure to a surface wave, i.e., the whoe is artificia dieectric wave guide. If the configuration of Fig. is charged with a charge Q per meter the fied intensity can be found by 14), at the surface u =0,itcan be cacuated from 14), { E = ε dz 1 dw = Q 1+ ) } 4 1 π x 1 + 19) x 1 We pot Fig. 5 simiary as another famiy of smooth wave guides, so we have very arge choices of guide figure in appications. Detai studies wi foow in coming papers. 5. CONCLUDING REMARKS Nowadays we have trianguar, rectanguar, pentagon, circuar, eiptica and paraboic waveguides. In this paper we propose two famiies of waveguides whose boundaries are described by simpe equations and ρ = a/ cos ϕ +. ρ = a cos ϕ + respectivey. They are shown in Fig., Fig. 3 and Fig. 5. They ook ike reguar rectanguar waveguide with smooth, rounded inner corners, so they
84 Lin et a. shoud be more efficient in transmission of high frequency power. Further study, numericay and experimentay wi be carried out and wi be reported shorty. ACKNOWLEDGMENT This work is supported by Chinese NSF under the grant 6977107 and the State Key Lab. on Miimeter Wave, South East University, Nanjing, Jiangsu, China. REFERENCES 1. Lin, W., E. K.-N. Yung, and K.-M. Luk, Groove rectanguar waveguide with rounded interna anges, MWSYM, 1843 1844, 1996.. Lin, W., H. Zhou, Z. Yu, E. K.-N. Yung, and K.-M. Luk Theoretica and experimenta study of a new famiy high-power waveguides, Journa of Eectromagnetic waves and Appications, Vo. 13 No. 8, 1013 100, 1999. 3. Vygodsky, M., Mathematica Handbook, 4th ed., Mir. Pubishers, Moscus, 1984 or Bayer, CRC Stan. Math Tabe, 8th Ed.