Open access peer-reviewed chapter

Characteristics of Radiation of a round Waveguide through a Flat Homogeneous Heat Shield

By Viktor F. Mikhailov

Submitted: November 27th 2019Reviewed: March 6th 2020Published: April 17th 2020

DOI: 10.5772/intechopen.92036

Downloaded: 88

Abstract

The problem of obtaining an analytical description of the radiation characteristics of a circular waveguide closed by a flat homogeneous dielectric plate is solved. The radiation characteristics include: the radiation field; the conductivity of the aperture radiation; and the fields of surface, flowing, and side waves, as well as energy characteristics. In such a statement, a strict solution of Maxwell’s equations is required. The paper uses the method of integral transformations and the method of eigenfunctions. In this case, the assumption is used that the electrical parameters of the dielectric plate (thermal protection) and the geometric dimensions do not depend on time. The relations describing the directional diagram of a circular waveguide with dielectric thermal protection and taking into account the electrical parameters of thermal protection and its thickness are obtained. Expressions are also obtained for the fields of lateral, surface, and outflow waves, from which it is possible to calculate the power taken away by these fields. Numerical calculations were made for some of the obtained relations. The results showed that the power of the side waves is zero. It also follows from the calculations that the radiation field of surface and flowing waves is absent, that is, their contribution to the directional diagram is not.

Keywords

  • circular waveguide
  • a flat
  • uniform thermal protection
  • the radiation characteristics

1. Introduction

The onboard antennas of the returned spacecraft are subjected to intensive aerodynamic heating when the spacecraft passes through the dense layers of the atmosphere [1]. In these conditions, radio-transparent heat-resistant thermal protection is used to protect the antennas from external influences. The open ends of the transmission lines are used as the emitter to obtain a wide directional pattern. The most offer used radiation from the open end of the round waveguide. In the first approximation, we consider a flat uniform thermal protection. Under the conditions of aerodynamic heating, the electrical parameters of thermal protection significantly change (relative permittivity ε and tangent of the dielectric loss angle tgδ). These changes lead to a noticeable increase in absorption losses in the heat shield, reflection from its boundaries, as well as to the appearance of surface and side waves. Together, all this leads to a change in the directional pattern and a decrease in the efficiency of the onboard antenna.

Evaluation of these changes is absolutely necessary to determine the radio technical characteristics of the onboard radio equipment. The paper solves the problem of determining the characteristics of the radiation of a circular waveguide through a heat shield subjected to aerodynamic heating.

The problems of calculating the interaction of the onboard antenna with a heat-shielding dielectric insert are very difficult and poorly developed. The development of mathematical models of antenna windows is reduced to solving an external problem of electrodynamics—electromagnetic excitation of bodies or diffraction of radio waves. At the same time, we will use well-known analytical methods of solution. The radio technical characteristics of the antenna window, for which we obtain an analytical description, include a radiation pattern, radiation conductivity, antenna temperature, and a number of other characteristics that describe more subtle electrodynamic effects, as well as energy characteristics.

In theoretical terms, the electrodynamic problem in general can be formulated as follows. There is a radiating, open antenna a, located on an infinite screen, in front of which is a dielectric layer of thickness d with a complex permittivity-bridge εα(x, y, z, t) (see Figure 1).

Figure 1.

Electrodynamic model of radiation from a circular waveguide with thermal protection.

In this general formulation, the solution of the electrodynamic problem is associated with significant mathematical difficulties, the main problem being the need to solve the Maxwell equations for an arbitrary law of change in the parameters of media in space and time. With some simplifying assumptions, the problem was solved in the ray approximation. In [2], the wave front method is used to analyze the radiation diagram of an antenna covered by a dielectric layer. In [3, 4, 5], the method proposed in [6] is used to find the radiation diagram, according to which the antenna radiation diagram in the presence of an infinite flat dielectric layer is simply multiplied by the diagram in the free space by the flat wave transmission coefficient for the flat layer, taking into account the corresponding angle of arrival and the plane of polarization of the wave.

At their core, all these methods are close to each other and are essentially based on the approximation of geometric optics, which is true in the quasi-optical domain. In relation to the problem under consideration (the resonance region), a strict solution of the Maxwell equations is required. From analytical methods of the solution, it is possible to apply the method of integral transformations and the method of eigenfunctions. Both of these methods will be used in the future. In this case, we assume that the parameters of thermal protection do not depend on time, that is, in fact, the slow-moving processes of heating of the dielectric thermal protection are considered. This approach and methods of solution were used in a number of works, for example [7].

2. Main part

The problem can be formulated as a boundary with respect to the tangent magnetic field in the material, and with respect to the tangent electric field. The second method is more convenient because of the simple form of boundary conditions for z = 0 [8].

The magnetic component of the electromagnetic field Hyat z ≥ 0 must satisfy the following wave equation in a Cartesian coordinate system x, y, z:

2Hyx2+2Hyy2+2Hyz2+k2εHy=0,E1

where ε= ε1 for 0 ≤ z ≤ d,ε= 1 for z > d, and k is the wave number.

We apply the Fourier transform for x and y coordinates to equation (Eq. (1)). We get

2Ĥyz2+k2εk2xk2yĤy=0,E2

where is the direct Fourier transform,

Ĥy=Hyxy0expjkxx+kyydxdy.

Solution of equation (Eq. (2)) satisfying the radiation conditions (for zd) has the form for area 1, that is, the area occupied by the dielectric plate (0 < z < d),

Ĥy1=Dexpjkz1z+Lexpjkz1z.

For region 2, that is, the region behind the plate (z ≥ d), we get

Ĥy2=Mexpjkzz,

where kz1=k2εkx2ky2;kz=k2kx2ky2.

Reasoning in a similar way, with respect to the tangent magnetic component of the field, we obtain the following equations for the spectral component:

Ĥ1x=Aexpjkz1z+Bexpjkz1z,
Ĥ2x=Cexpjkzz.

Satisfying the equations arising from the Maxwell equation, we obtain expressions for the spectral components of the electric field

Êx1=kz1ωε0ε1Dexpjkz1zLexpjkz1z;
Êx2=kzωε0Mexpjkzz;
Êy1=kz1ωε0ε1Aexpjkz1zBexpjkz1z;
Êy2=kzωε0Cexpjkzz.

Unknown functions A, B, C, D, L, and M are determined from the boundary conditions for z = 0 and z = d. In this case, the boundary conditions for radiation from a circular waveguide, when the field in the aperture is determined by H11 waves, have the form

kz1ωε0ε1DL=Êx0;E3
kz1ωε0ε1AB=Êy0;E4
Dexpjkz1dLexpjkz1d=ε1kzkz1Mexpjkzd;E5
Aexpjkz1dBexpjkz1d=ε1kzkz1Cexpjkzd;E6
kxkyDexpjkz1d+Lexpjkz1d+k12kx2Aexpjkz1d+Bexpjkzd==ε1k2kx2CexpjkzdkxkyMexpjkzd;E7
k12ky2Dexpjkz1d+Lexpjkz1dkxkyAexpjkz1d+Bexpjkz1d==k2ky2MexpjkzdkxkyCexpjkzdε1E8

where

Êx0=ПExxy0expj(kxx+kyy)dxdy.E9
Êy0=ПEyxy0expj(kxx+kyy)dxdy.E10

Here, П is the area of integration on the opening of the waveguide and x,yare the coordinates counted in the opening of the studied waveguide.

We present in full the expression of the coefficients included in the solution of the wave equations of spectral components. In this case, we use the following variable replacement:

kx=βcosα,ky=βsinα,kz=k2β2,kz1=k2ε1β2.

Since the coefficient expressions are very cumbersome, we use the following notation to obtain a compact form of the record:

k2β2=a,k2ε1β2=b,k2ε1β2d=c,
β2sinαcosα=e,k2β2cos2α=f,k2ε1β2cos2α=g,
k2β2sin2α=h,k2ε1β2sin2α=l,k2β2d=p.

As a result

A=ωε0ε1bexpjcΔ[Êx0eabk21ε1coscjsinc++Êy0e2a+bjbsinc+acosc+bf+agjsinchnbcoscla];E11
B=AÊy0ωε0ε1b;E12
C=baexpjp+cÊy0ωbj2Aε1sincexpjc;E13
D=ωε0ε1expjcbΔÊx0[e2a+bjbsinc+acoscbh+al××jfbsinc+agcosc+Êy0aebk21ε1jsinccosc];E14
L=D+Êxoωε0ε1b;E15
M=baexpjp+cÊx0ωε0b+j2Dε1sincexpjc.E16

In the expressions (Eqs. (11)(16))

Δ=2k2ε1β2k2β2k2jk2β2sin(k2ε1β2d++k2ε1β2cosk2ε1β2d[ε1k2β2cosk2ε1β2d++jk2ε1β2sink2ε1βd].

Using the obtained expressions of the angular spectrum of plane waves and applying the inverse Fourier transform, taking into account (Eqs. (9) and (10)), we write

Ex12=ПFx112xyzxy0Exxy0dxdy++ПFx212xyzxy0Eyxy0dxdy;E17
Ey12=ПFy112xyzxy0Exxy0dxdy++ПFy212xyzxy0Eyxy0dxdy;E18

The functions Fx in these expressions are determined based on equations (Eqs. (11)(16)). After a series of transformations, you can write

Fx12=14π202π{j2sink2ε1β2dΔ[β4sin2αcos2α(k2β2++k2ε1β2)jk2ε1β2sink2ε1β2d+k2β2cosk2ε1β2d++k2ε1β2k2β2sin2a+k2β2(k2ε1β2sin2a][jk2β2cos2ak2ε1β2××sink2ε1β2d+k2ε1β2cos2ak2β2cosk2ε1β2d1}××exp[j(kzkz+kz1dexpxxcosa+yysina]βda==14π202πφx2expjkzzexpxxcosa+yysina]]βdβda;E19
Fx22=14π202πj2sink2ε1β2dΔβ2sinαcosα(k2ε1β2k2××k2β2expjk2β2dexp[jkzzkz+kz1d××expxxcosa+yysina]]βda==14π202πφy2expjkzzxxcosa+yysina]]βdβda;E20
Fy12=14π202πj2sink2ε1β2dΔβ2sin2αcos2α(k2β2××k2ε1β2k21ε1cosk2ε1β2djsink2ε1β2d××expjkzzkz+kz1dexpxxcosa++yysina]βda=14π202πξx2expjkzz××expxxcosa+yysina]βda;E21
Fy22=14π202πj2sink2ε1β2dΔ[β4sin2αcos2α(k2β2++k2ε1β2)][(jk2ε1β2sink2ε1β2d++k2β2(k2ε1β2cos2a)][jk2ε1β2k2ε1β2sin2a××sink2ε1β2dk2β2k2ε1β2sin2a××cosk2ε1β2d]1}exp[jkzzkz+kz1d××expxxcosa+yysina]βda==14π202πξy2expjkzzexpxxcosa++yysina]βda.E22

The obtained expressions (Eqs. (19)(22)) together with (Eqs. (17) and (18)) determine the radiation field of a circular waveguide with uniform thermal protection through the tangent components of the electric field in the opening of the waveguide.

To calculate the integrals (Eqs. (20) and (22)) by the saddle method, it should be taken into account that, when the integration contour is deformed, it is necessary to bypass the branch points of the integrand and that the saddle path intersects the poles of the integrand. We determine which branch points and poles of the integrands must be taken into account in the said approximation. Integrals (Eqs. (20) and (22)) can be written as

I=02πfβexpjdzk2β2expjβρcosαφ0dβdα==2πfβexpjdzk2β2J0βρ.

where φ0=arctgxx/yy,ρ=xx2+yy2,

J0βρis the Bessel function.

We use the asymptotic expression of the Bessel function for large arguments

J0βρ2ππβρexpjβρπ4+expjβρπ4.

Then, passing to the plane of the complex angle τ by replacing β = ksinτ, we obtain

I=AfβexpjkLcosτ±φE23

where A=2k2πβρcosτexpjπ4,dz=Lcosφ,ρ=Lsinφ.

Considering the obtained expression of the integral I, we find the analytical expression of the saddle path from the following equation:

Jmjcosτ±φ=const,sciliset Jmjcosτ±φ=j. Means cosτ±φ=1.. Because theτ=τr+jτj,

then cosτr±φ=cosτr±φchτjjsinτr±φshτj,cosτr±φ=1chτj=schτj.

Finally, we get the expression of the saddle path

τr±φ=arccosschτj.E24

Satisfying the radiation condition at infinity, we obtain regions on the complex plane τ in which the saddle path defined by (Eq. (24)) lies:

  1. 0<τr<π, τi>0;

  2. π<τr<0,τj<0;

moreover, the second area of determination of the transit path in our case makes sense, since from (Eqs. (20) and (22)) it is clear that the lower limit of β is 0. Denoting the coordinates of the branch points τв and the pole τр, we obtain from (Eq. (24)) the condition that determines the branch points and poles intersected during deformation of the initial path of integration into the saddle,

τzв,zр±φ>arccosschτjв,jр.

In the general case, in accordance with the Cauchy theorem, the integrals for functions can be represented in the following form:

F=14π2l+UCвlв+UCplp,E25

where UCв,pis the only Heaviside function; Cв,Cpare values determined on the basis of Eq. (24) as follows on the complex plane:

Cв,p=Rearcsinβв,pk±φarccosschJmarcsinβв,pk,E26

The first integrals over the circuit l are calculated by the saddle method. Finally, by the saddle method, we get

Fx1sad2=jkexpjkr2πr2zφx2kx=kx'ky=ky',E27
Fx2sad2=jkexpjkr2πr2zφy2kx=kx'ky=ky',E28
Fy1sad2=jkexpjkr2πr2zξy2kx=kx'ky=ky',E29
Fy2sad2=jkexpjkr2πr2zξy2kx=kx'ky=ky.E30

In the expressions (Eqs. (27)(30))

kx'=kxr,ky'=kyr,r=xx2+yy2+z2.

After a series of transformations for even E modes and odd H modes, we obtain

Fx1E2=i2πi=1nUCpi02πφx2βiaNβiadaψ1βiψ1'βiβi=βiE,E31
Fx2E2=i2πi=1nUCpi02πφy2βiaNβiadaψ1βiψ1'βiβi=βiE,E32
Fy1E2=i2πi=1nUCpi02πφx2βiaNβiadaψ1βiψ'βiβi=βiE,E33
Fy1E2=i2πi=1nUCpi02πφy2βiaNβiadaψ1βiψ1'βiβi=βiE.E34

In these expressions

N=2k2ε1β2k2β2k2expjkzz×
×expxxcosa+yysina]]β;
ψ1β=jctgk2ε1β2dk2ε1k2β2k2k2ε1β2,

Cpi is calculated according to (Eq. (26)). The expressions for the H modes will be characterized by equations similar to equations (Eqs. (31)(34)), in which ψ1 (β) is replaced by ψ2 (β) and ψ1 (β) by ψ2 (β) and which will be calculated for the values of β corresponding to the poles, and

ψ2β=k2β2jk2k2ε1β2ctg(k2ε1β2d.

Expressions Fx1E1,Fx2E1,FxyE1,Fy1E1are written similar to expressions with replacement, φxN2by φxN1, φyN2by φy1, ξxN2by ξxN1, and ξyN2by ξy1, respectively.

where

φxN1=φx1NN+N+;E35
φyN1=φy1NN+;E36
ξxN1=ξx1NN+;E37
ξyN1=ξx1NN+N+.E38

In expressions (Eqs. (35)(38)), the expression N (−) corresponds to the expression N with the replacement of exp (−jkzz) by exp (−jk1z);

expression N (+) corresponds to expression N replacing exp (−jkzz) with exp (−jk1z),

φx1=expjcΔe2a+bjbsinc+acoscbh+aljbsinc+agcosc;
φ1=ξx1aebk2ε11Δ;
ξy1=expjcΔbf+agjbsinc+alcoscl2a+bjbsinc+acosc.

The relation for functions includes expressions of spectral components. Considering the main type of oscillations in the waveguide H11, we get

Êx1=πωμ0H0a6kxky01+65+103a234+kx2+ky2+145103a4kx2+ky2;
Êy1=0,85πωμ0H0a2[11,25a2b2+kx2+05ky2+3125102×
×a4033kx2ky2+033b2kx2+05b2ky20,5104a6b2kx2ky2];

The second integral (Eq. (25)) is calculated along the contour lв, which covers the cut and is carried out so that the integrand is unique. From the form of the integrands, it is obvious where the first-order branch points are located. From the form of the integrands of the integrals (Eq. (25)) it is obvious that the first-order branch points are located at βв1=±kand βв2=±kε1. From these branch points, it is necessary to take βв1=kand βв2=kε1, in order to satisfy the radiation conditions. A calculation according to (Eq. (26)) shows that Cв > 0 for named branch points and, with the exception of the case when ε1 is complex and the losses are sufficiently large (tanδ > o.5). Thus, we find that the side wave can contribute to the radiation field, and this should be taken into account. An analysis of the integrands (Eqs. (20) and (22)) showed that the developed and well-known methods for the asymptotic estimation of integrals along the banks of a section covering branch points that are valid for a saddle point turn out to be inapplicable in this case. For this reason, the contribution of the side wave to the radiation diagram can be determined only by numerical integration (Eq. (25)) along the contour lв. Moreover, from the analysis of integrands, it follows that it is advisable to choose the section so that it is a straight line parallel to the imaginary axis of the complex plane β. Then the integral along the banks of the sectionlвwill take the form

Rek+jkv1β+kRek+jv2β,

where v1βand vβ2are sub-integral expressions with signs in front of k2β2.

The conditions for the existence of surface and leaky waves are determined from the location of the poles of the integrands (Eqs. (20) and (22)), and the poles correspond to the equality of the denominators of the marked expressions to zero.

By the Cauchy theorem, integral (Eq. (25)) along the contour lр is defined as follows:

lр=i=1nResβi,

where βi are the roots of the denominators (Eq. (25)).

In order to separate the singular points into poles corresponding to surface and outgoing waves, it is advisable to again go from the complex plane β to the plane of the complex angle τ. Moreover, from the analysis of the exponent of expression (23), it follows that the surface wave will take place at

τr=π2,τj>0

and the outgoing wave corresponds to the following region of complex angles:

0<τ<π,except forτr=π/.2,τji>0.

Moreover, the region 0 < τr<ϕ determines the backward wave that does not satisfy the conditions at infinity.

In the presence of losses in the dielectric plate k1 = kr-jkj, analysis of the exponent (Eq. (23)) shows that the poles corresponding to the relations for the complex angle τ

τr=arccosεtgδshτj,ε1ε2tg2δsh2τjchτj<0

determine the surface wave field.

If

cosτrshτjεtgδ,

then the poles that satisfy the last relation determine the field of the leaky wave.

The conductivity of the aperture is determined by the following expression:

Ya=Ex1Hy1dxdy.

Applying the Parseval theorem, after a series of transformations, we obtain the expression for the conductivity referred to as the conductivity of the open end of the waveguide in the form

Yап=ωε02π2a00πβÊx02expjcΔ((eÊx0+eÊy0eabk21ε1××coscjsinceÊx0eÊy0e2a+bjbsinc+acosc++Δ2expjc+eÊx0hb+lajfbsinc+agcosceÊy0bf+ag)da=ωÊ02π2b002πGβaΔda.E39

The contribution of surface and leaky waves to the conductivity of a circular waveguide with thermal protection is determined using the Cauchy theorem. As a result, we get

Ysur=jωε02π2ai=1n02πGβiαΔ.E40

We will conduct a quantitative assessment of various loss mechanisms by means of transmission coefficients χ, attenuation υ, and reflection ρ, which are defined as follows:

η=η1+η2+η3+η4=РradРf+Рrad.surРf+Рrad.resРf+Рrad.sidРf,ν=ν1+ν2+ν3+ν4=РradРrad.surРf+РradРrad.resРf+РradРrad.sidРf,R=1Yn1+Yn.E41

These expressions take into account the orthogonality of surface, leaky, and side waves and Pf is the power incident (supplied to the emitter), Prad-radiated power, Prad.sur, Prad.res, Prad.sid. waves, respectively, Рsur, Рres, and Рsid are the power of surface, leaky, and side waves, respectively; υ1 is the attenuation coefficient of radiated power, determined by losses in dielectric thermal protection.

ν1=1ην2ν3ν4R2.

In expression (Eq. (41))

Yn=YanYsur.

3. Conclusion

Expressions (Eqs. (17), (18), (27)(30), (39), and (40)) characterize the radiation field of a circular waveguide with uniform thermal protection and its input conductivity. The theoretical relations obtained, along with an estimate of the apparent loss of radiated power due to absorption in and reflection from thermal protection, allow us to estimate more subtle effects, such as losses on surface, leaky, and side waves. It is also possible to assess the influence of these waves on the radiation pattern.

Further development of the obtained theoretical relations should consist of taking into account the probable inhomogeneity of the heat-shielding layer in the direction of the z axis. For this, the method of approximate solution of the wave equation can be applied—the WKB method (Wentzel, Kramer, Bruellen) [9]. If we consider a round1 waveguide in the form of an onboard antenna of the returned spacecraft, then due to aerodynamic heating, a melt layer appears on the outer surface of the heat shield, which has electrical characteristics different from the characteristics of the material in the solid phase. Then the radiation from the waveguide should be considered through a two-layer dielectric thermal protection [10].

Unfortunately, all the obtained relations turned out to be very cumbersome and their use becomes only with numerical integration. For some of the obtained ratios, numerical calculations were performed. The results showed that the power of the side waves is zero. It also follows from the above calculations that the radiation field of surface and leaky waves is absent, that is, their contribution to the radiation pattern is not. Further research in this area should be directed to the development of computer calculation programs for the basic radiation characteristics.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Viktor F. Mikhailov (April 17th 2020). Characteristics of Radiation of a round Waveguide through a Flat Homogeneous Heat Shield, Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering, Patrick Steglich, IntechOpen, DOI: 10.5772/intechopen.92036. Available from:

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