transmission and radiation properties of a surface wave transmission line with a bend

Electronics and Communications in Japan, Part 1, Vol. 77, No. 5, 1994 Translated from Denshi Joho Taushin Gakkai Ronbunshi, Vol. 76-B-U, No. 1 , January 1993, pp. 45-52

Transmission and Radiation Properties of a Surface Wave Transmission Line with a Bend

Takashi Nakamura, Shinji Kamiya, and Senji Yokokawa, Members

Faculty of Engineering, Gifu University, Gifu, Japan 501-1 1

SUMMARY

As a cause of generation of unwanted electromagnetic radiation, the bend of a surface wave transmission line is investigated. The transmission and radiation characteristics at the bend are studied analytically and experimentally. First, by applying the traveling wave mode method, it is shown that the transmission characteristics of the surface wave transmission line can easily be obtained. The change of the equivalent voltage near the bend is observed.

Next, the discontinuity of the traveling wave current is considered as a radiating source and the radiation field from the bend in a surface wave transmission line is derived by means of the principle of pattern multiplication. It is found that there exist an electric field component which strongly radiates into the traveling directions of the current in a bent transmission line and the component which radiates into the intermediate direction. It is found that the power is radiated strongly into the intermediate direction. Further, by experiments which agree well with the theoretical results, the validity of the method is presented.

Key words: Surface wave transmission line; bend; transmission; radiation; traveling wave mode method.

1. Introduction

Recently, with the advance of electronic technology and the diversified use, many electrical devices have

come to be used. Therefore, many different electrical appliances are installed in the same space so that the unwanted electromagnetic waves generated and radiated by these appliances cannot be ignored.

One of the causes and factors of radiation of unwanted waves generated by various electrical appliances is the existence of the electromagnetic wave propagating along the external skin of the cables connecting the appliances and supplying the power. This wave can be treated as a transmission mode of the surface wave transmission line with a center conductor. The fundamental modes are the HE,, mode and the TM,, mode with a zero cutoff frequency. The HE,, mode is a hybrid mode of a dielectric waveguide and exists even without a center conductor. Often it is called the dipole mode since the field profile resembles that of the dipole. On the other hand, the TM,, mode is a fundamental mode requiring a center conductor. The electric field is uniform along the circumferential direction of the line. Hence, depending on the electromagnetic environment of the cable, either mode can be excited. In this paper, the TM,, mode is analyzed which is closely related to the current flowing on the cable.

The surface wave transmission line with a T&, mode is called the G-line and was invented by Goubau in 1950 [I]. Th~s is a single-wire conductor coated with a dielectric, it has a simple construction and can be fabri- cated at a low cost as a transmission line for TV commu- nal reception. However, in practice, the electromagnetic leakage of this line is significant and prevents practical applications.

44 ISSN8756-662 1 /94/0005-0044 @ 1994 Scripta Technica, Inc.

This electromagnetic leakage phenomenon was understood theoretically and experimentally to a certain extent [2-4.1. In the theoretical analysis, the integral on the vector potential was derived by a successive approximation from which the reflection and transmission coeffi cients at a far distance were derived. However, the voltage and current distributions near the bend and the radiation field have not been derived. The theory is not sufficient for comparison with the measured voltage and current distributions and the radiation field.

To find the radiation field of the unwanted electromagnetic field, it is necessary to know the current distribution. However, it is difficult to know the current distributions of the entire cable bent in a complex manner. Hence, derivation of the radiation field also is difficult. This implies that the conventional treatment of the radiation mechanism for deriving the radiation field from the integral of the entire current distribution cannot cope with the electromagnetic interference.

On the other hand, the analysis of the surface wave antenna and the discontinuous surface wave transmission line maintains the discontinuity radiation viewpoint in which the cause of the radiation is considered to be the discontinuity [ S ] . This concept is extremely clear as an explanation for the radiation mechanism and its effective- ness is understood. However, if the treatment is rigor- ous, the computation becomes extremely difficult. Hence, the object of the analysis has been limited to a simple case such as two-dimensional structures.

In this paper, the surface wave transmission line is treated as a three-dimensional propagation structure of

Z b

2b ’

Fig. 1. Surface wave transmission line with a bend.

The scalar potential of the conductor surface of the transmission line is used as the equivalent voltage in the application of the traveling wave mode method.

The discontinuity of the traveling wave current is considered as a radiating source. The radiation field from the bend of the transmission line is derived by means of the principle of pattern multiplication.

2. Surface Wave Transmission Line Structure with Bend and Its Trans-

mission Characteristics the unwanted electromagnetic wave. The radiation characteristics from the discontinuity of the bend are derived analytically so that the electromagnetic leakage phenome- na are understood quantitatively.

The surface wave transmission line structure analyzed in this paper and the coordinate system are shown in Fig. 1. The surface wave transmission line is made of a wire conductor with whose radius is a, loaded with a

In this paper, an infuritely long surface wave transmission line with its transmitting and receiving ends at infinity is analyzed so that only the radiation components from the bend is identified.

It is difficult to derive the far field for an infinitely long current source directly from the integral representation of the current. Hence, by means of the traveling wave mode method [6, 71, the reflection and transmission coefficients of the traveling wave current at the bend of the surface wave transmission line are obtained. Since the surface wave transmission line is a single-conductor type, the voltage cannot be defined uniquely unlike the TEM mode transmission line for a two-conductor type.

dielectric layer with a relative permittivity of E,. so that the radius of the line is b. The line is located on the xy- plane. At x = y = z = 0, the line is bent from the x-axis toward the y-axis by an angle of @,,. Letting this bend be the origin, the coordinate s is taken along the conductor as shown in the figure. The incident wave is assumed to be applied from s = -00.

From the scalar potential 4 and the vector potential A, the electric field distribution E is given by

Here, the time variation is e jWr and

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(3)

where J is the current and r is the distance between the source and the observation point. Also, k is the free Space propagation constant and

k = WAC= Zx/A ( A : wavelength) (4)

If the observation p i n t is placed on the conductor surface and the unit vector along the tangent direction is j,, Eq. (1) becomes the following from the boundary condition j, - E = 0:

If the conductor type is a dielectric coated wire conductor as shown in Fig. 1, Eq. (5) becomes

where 4Js) is the potential on the conductor surface (p = a) and J(s) is the current on the conductor. Also, ro is the distance from the source to the observation point on the conductor surface.

On the other hand, from Eq. (2) the potential 4&) on the dielectric surface (p = b) is

where rb is the distance from the source to the dielectric surface.

What is needed is the potential on the conductor surface. If b is sufficiently smaller than the wavelength, the electromagnetic field in the dielectric layer a < p < b is assumed to be uniform in the circumferential direction.

Then the electric field normal to the conductor is 181

Hence, from Eqs. (7) and (8),

(9)

Note that Eqs. (6) and (9) can be considered to be equivalent to the following generalized transmission line equations:

- z(s) J ( s ) as

From Eqs. (6), (9), and (lo), the equivalent series impedance z(s) and the equivalent shunt admittance y(s) of the surface wave transmission line are

Due to the structural symmetry, the current direction vector& must be considered only for s < 0:

( - 03 < s'<O) Js'Js~'

(13)

Hence, z(s) and y(s) become

where, s1 is a point on the negative s coordinate and s2' on the positive s coordinate. Also, rla and r2a are the distances of s to s1 and s to s2' on the conductor surface while rlb and r2& are the distance of s to sI ' and s to s2' on the dielectric surface.

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Re Z f

Fig. 2. Coordinate system of source and reference points.

When Eqs. (14) and (15) are compared with the case of the transmission lines such as a parallel two-wire, we notice the lack of the contribution of the image current caused by a ground plane and the series contribution of the first term (2/er)ln(b/u) in the intermediate bracket of l/y(s). The lack of the ground plane implies an effect of a wide range of current distribution on the equivalent distributed constants. The dielectric loading adds the series capacitance.

As the initial value solution of the unknown current distribution J(s), the solution of the surface wave transmission without a bend is used. The solution is given as the sum of the forward wave modal current JJ(s) and the backward wave modal current Jb(s) with constant ampli- tudes and the phase constant p. Here, p is given from the characteristic equation of the surface wave transmission line [l]:

J / ( S ) = J f o e - " " , J b ( s ) = ] b O e + j s s (16)

When the foregoing are substituted into Eqs. (14) and (15), the series impedance and the shunt admittance for each traveling wave are the following definite integrals:

where for p = 1, b

-3. -2. -1. 0. 1. 2. 3.

s / x (b) Phase constant

b/X=O.O16 a/b=0.5 E r=2.3

Fig. 3. Equivalent characteristic impedance and phase constant near the bend of the line.

For t = f (forward wave), the upper sign is taken while the lower sign is taken for t = b (backward wave).

The integrals for all( p) and an( p) are reduced to the series solutions of the Hertz vector for the traveling wave line source. Since the details are given in [ 101, only the results are presented in the Appendix.

From the foregoing, the series impedance z~(s) and the shunt admittance yt(s) at an arbitrary point on the transmission line are derived. From these, the equivalent secondary parameters of the line, the characteristic

47

2.

- < In 1. v >

1 .o

- t -I 0. 2 -3. -2. -1. 0. I . 2. 3.

s / h (a ) 40=45'

2.

0 0 0 0 Measured - 2 1. In W > -

0. -3. -2. - I . 0. 1. 2. 3.

s/A ( b ) h=90'

b/X=O.O16 a/b=O.5 E r=2.3

Fig. 4. Equivalent voltage distribution near the bend of the line.

impedance Z,(s) and the propagation constant y,(s), are derived:

where Re Z,(s) > 0 and Im y,(s) > 0 are chosen.

The voltage and current distributions on the transmission line can easily be obtained by dividing this non- uniform distributed parameter transmission line into segments and replacing it with cascaded connections of uniform distributed parameter transmission lines [6, 71.

The characteristic impedance Zf and the phase constant Qf obtained in this manner are shown in Fig. 3 for $o = 0", 45", and 90". Since the distributed parameters for the backward wave are related symmetrically to those of the forward wave, only those for the forward wave are presented. The structural dimensions of the surface wave transmission line are b/A = 0.016, a/b = 0.5, and E , = 2.3 in consideration of the experiment. Then the initial value solution of 0 is Q = 1.05 k.

- - - - - Theory 2 2 :Measured

iri n A A A .- -4 -$ -? - - - -k - - ' _ - - -

O 30" 60" 90" 120" 150" d o

b/h=O.O16 , a/b=O.5 , E ,=2.3

Fig. 5. Reflection and transmission coefficients of the line.

In the case where the surface wave transmission line is an infinitely long straight line (40 = OO), both the characteristic impedance and the phase constant are real and constant. However, if a bend exists, a change ap- pears near the bend so that the real part decreases and the imaginary part increases. The change of the distributed parameters increases as the bend angle 4o is increased.

Next, the equivalent voltage distribution near the bend is shown by solid lines in Fig. 4. From this figure, it is found that the voltage distribution changes discontin- uously before and after the discontinuity and the transmitted wave decreases. Hence, it is understood that radiation takes place near the bend. When the bend angle is small, the attenuation of the higher-order modes is gradual and the fundamental transmission mode becomes dominant after several wavelengths from the bend. On the other hand, if the bend angle is 90" in (b), the fundamental transmission mode becomes dominant after only one-half wavelength.

Hence, it is possible to derive the apparent reflection coefficient r and the apparent transmission coefficient T from the voltage distributions at points sufficiently far from the bend.

The absolute values of the reflection coefficient I' and those of the transmission coefficient T obtained in this manner are shown by solid and broken lines in Fig. 5. From this figure, it is found that the transmission coefficient decreases gradually with an increase of the bend angle 4o while the reflection coefficient increases slightly.

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2 * Z f /

Z f /

# n

I , ,

X LY 'A Y Fig. 6. Vectors and coordinate system for the

radiation field.

3. Radiation Field Representation by the Traveling Wave Current Discontinuity

Point

From Fig. 4, it was found that the radiation takes place within several wavelengths after the bend. For simplicity, let us assume that the radiation takes place only at the bend. Then the bend at s = 0 becomes the discontinuity point of the traveling wave current in the surface wave transmission line in Fig. 1. The current distribution at this point can be regarded as the discontinuity where the three traveling wave currents, forward, reflected and transmitted currents, vanish or are generated. In general, the composite radiation electric field E:& by the N traveling wave current discontinuities with a phase constant of p are given as follows if the coordi- nates are as shown in Fig. 6 [9]:

where I, and j , are the traveling wave current and its unit vector at the point #n and d, is the distance vector from the origin as the phase center to the point #n. Also, r is the distance from the origin to the observation point and j,. is the unit vector in the r direction. Also, V, =

- + 1 with the positive sign for the generation of the traveling wave current and the negative sign for the vanishing.

Let the traveling wave current discontinuity of the surface wave transmission line be #1 for the incident wave, #2 for the reflected wave, and #3 for the transmitted wave. Then the vectors and the coefficients for Eq. (23) are

Fig. 7. Radiation patterns of component fields from the bend point of the line.

h = j X cos 4 0 + j , s in 4 0

j T = j , sin B cos d + j , sin 0 sin d + j , cos e dn=O I , = I ~ , I ~ = no, A= TI,,

u1=-1, u2=u3=1 (24)

The foregoing results are substituted into Eq. (23) to obtain the radiation electric field E:&:

where for a = 8, (0

From the foregoing, the radiation field of the surface wave transmission line with a bend can be represent- ed by simple element radiation patterns fe((O') and

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z z

( a ) 40=45' ( b ) b0=90' b/X=O.O16 a/b=0.5 E r=2.3

Fig. 8. Radiation patterns of total fields from the bend point of the line.

f,(4'). Also, it is found that the reflection coefficient l' and the transmission coefficient Tat the bend are important elements to determine the radiation pattern.

Figure 7 shows the three-dimensional view of the radiation pattern from the bend of the transmission line for 4, = 45" and 90". Also, Fig. 8 shows the sum of the 0 and 4 components; and, hence, the radiation pattern I D 1 of the total radiated electric field for &-, = 45" and 90".

The increase of each curve is 5". The following values derived from the voltage distribution of the transmission line are substituted into Eq. (26) for I' and T. For 4, = 45", l" = 0.024 L 54.7" and T = 0.859 L -4.3" while for 4, = 90", I' = 0.019 L -96.2", and T = 0.717 L -4.6". These values were obtained at a location several tens of wavelengths away so that accura- cy of more than three digits is obtained.

As seen from Fig. 7, the 9 component of the radiation electric field radiates strongly toward the traveling direction of the current or 4 = 0" and 4, whereas the 4 component is strong in the direction of (0 = 4,/2. Also, the radiation pattern is almost symmetric with respect to 4 = 4,/2. This symmetry is degraded as the transmission is decreased. Especially, IDd[ becomes small in the traveling direction of the transmitted wave. Also, since II'l < 0.1, the effect of the reflected wave is small.

From Fig. 8, it is found that the radiation pattern of the total radiated electric field as a combination of the 0 and 4 components has characteristics of both and is radiated mainly between + = 0" and 4 = 4w Therefore,

z Monopole Y a a i ant. A

Road=

Fig. 9. Measurement system.

the range of the main radiation increases as 4, is increased.

4. Experiments

To study the validity of the theoretical results in sections 2 and 3, some experiments were carried out. The operating frequency was chosen at 3 GHz. The transmission line is a dielectric-coated conductor with b = 1.6 [mm], a = 0.8 [mm], and E , = 2.3. The measurement setup is shown in Fig. 9. The transmitting end is excited with a circular horn connected to a coaxial cable while the receiving end is made to be reflectionless by installation of an absorbing material. The measurement was carried out in an anechoic chamber.

The measured equivalent voltage distribution shown by 0 in Fig. 4 was obtained by moving an infinitesimal monopole along the transmission line and plotting the output voltage proportional to the z-directed electric field on the conductor surface. The results are normalized by the incident voltage Vm. Except for the vicinity after the bend, theoretical and measured results agree well. The region after the bend with discrepancy is due to the fact that the higher-order modes are dominant and the z-directed electric field on the conductor surface does not correspond to the equivalent voltage.

In Fig. 5, the absolute values of the transmission and reflection coefficients derived from the measured values of the equivalent voltage distribution are indicated by 0. From the figure, good agreement is seen between the theoretical and measured values. For +o > 90", the measured values of the reflection and transmission coefficients tend to be somewhat larger than the theoretical

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270"

0" 180"

90" (a) IDel, $Q=45", 8 =45" (b) IDQI , $Q=45", 8 =45O

2 70" n

180" 0"

( c ) IDel, m=90", 8 =45" 90"

(d) IDQ1 , $Q=gO", 8 =45"

o o o o Measured - Theory

b/X=O.O16 a/b=0.5 r=2.3

Fig. 10. Radiation patterns from the bend point of the line.

values. This indicates that the approximation using the uniform traveling wave current as the initial value solution is not sufficient as the coupIing between the adjacent conductor becomes strong.

The measured radiation pattern in Fig. 10 was derived as the outputs of a sixelement detecting Yagi antenna placed on the conical surface with a constant 8 (45"). The gain of the Yagi antenna is about 10 dBi and the distance from the bend to the antenna is 50 cm (five wavelengths). By performing measurement with the antenna aimed at the bend, the radiating wave from the bend is detected and other unwanted electromagnetic

180"

180"

waves are eliminated. In Figs. 10(a) and (b), the absolute values ID@ I of the 8 component of the electric field and ID, I of the @ component are shown for a bend angle of

Cb0 = 45". Also, in Figs. 1O(c) and (d), ID81 and ID,/ for a bend angle of 4o = 90" are shown.

From Fig. 10, it is found that the theoretical and measured radiation patterns agree well. Slight errors are observed in the sidelobes and in the direction of @o = 60" in Fig. 10(a). The latter is considered to be due to the error in the theory used in which the radiation is from the discontinuity only, although the current distribution changes gradually near the bend as seen in Fig. 4.

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If these points are taken into consideration, the theoretical and experimental results of the transmission and radiation characteristics agree well so that the validity of the present paper is confirmed.

5. Conclusions

In this paper, as a cause of generation of unwanted electromagnetic waves, the bend in a surface wave transmission line was treated. The transmission and radiation characteristics at the discontinuity were analyzed and measured.

Previously, the transmission characteristics of a three-dimensional surface wave transmission line such as a surface wave transmission line with a bend have not easily been derived. In the present paper, it was shown that the results can be obtained rather easily by the application of the traveling wave mode method. The transmission behavior near the bend was studied.

Next, according to the discontinuity radiation viewpoint, the discontinuity of the traveling wave current was considered as the radiating source and the radiation field from the bend of the surface wave transmission line was derived by applying the principle of pattern multiplication. It was pointed out that the reflection and transmission coefficients at the end are important elements for determination of the radiation pattern. It was found that there exists an electric field component which strongly radiates into the traveling directions of the bent transmission line and a component radiating into the intermediate direction. As the power, the radiation is strong in the direction in the middle of the bend angle.

Further, an experiment was camed out. Since the measured results agree well with the theoretical results, the method is considered valid.

When the bent angle cb0 is larger than 90°, the coupling to the adjacent conductor becomes strong and the uniform traveling wave current is no longer sufficient

as the initial value solution. A method effective even for such cases is a topic for the future.

Acknowledgement. The authors thank Professor Risaburo Sato, Dean of Engineering, Tohoku Gakuin University, for his useful advice.

1.

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10.

REFERENCES

G. Goubau. Surface waves and their application to transmission lines. Jour. Appl. Phys., 21 (Nov. 1950). M. Suzuki. Surface-wave losses of transmission lines at bending points. Jour. I.E.C.E., 37, 1, pp.

R. Sato, J. Chiba, S. Park, K. Iwata, R. Koide, and S. Miyamoto. Losses due to bends in surface wave transmission lines. Jour. I.E.C.E., 46,9, pp. 22-24 (Sept. 1963). R. Sato, J. Chiba, S. Park, K. Iwata, R. Koide, and S. Miyamoto. Radiation from launching horn and bending points of surface wave transmission line. Jour. I.E.C.E., 47, 3, pp. 22-27 (March 1 964). R. E. Colin and F. J. Zucker. Antenna Theory. Ch. 21. McGraw-Hill, New York (1969). T. Nakamura, T. Sekine, and S. Yokokawa. An analytical method of a spiral line. I.E.C.E. Trans.

T. Nakamura, T. Sekine, and S. Yokokawa. An analysis of a meandering line. I.E.C.E. Trans. (B), J68-B, 2, pp. 244-250 (Feb. 1985). J. H. Richmond and E. H. Newman. Dielectric coated wire antennas. Radio Sci., 11, 1, pp. 13-20 (Jan. 1976). T. Nakamura, N. Okochi, S. Yokokawa, and R. Sato. Radiation from a bend of transmission line. I.E.C.E. Trans. (B), J70-B, 2, pp. 261-268 (Feb. 1987). T. Nakamura, S. Kamiya, and S. Yokokawa. A series solution of Hertz vector for traveling wave line source. I.E.I.C.E. Trans. (B-11), J74-B-11, 3, pp. 110-1 12 (March 1993).

33-37 (Jan. 1954).

(B), J66-B, 10, pp. 1207-1214 (Oct. 1983).

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APPENDIX

Series Solutions of Eqs. (19) and (20)

The functions IT,(-, x, p) and IT&, a ~ , p) representing the Hertz vector for a traveling wave line source are defined as

where r = z m p and the forward wave takes the upper sign while the backward wave takes the lower sign.

The power series representations of the foregoing are

Hence, Eqs. (13) and (20) can be derived as (I) to (VI) in the following. Here, &( * ) is the zeroth-order modified Bessel function of the second kind:

where

pS = Js2 sin2do + p2

(IV) when do h tan-'( 24E)

where En( * ) is the n-th order exponential integral and can be reduced to the well-known sine integral and co- sine integral by recurrence formula.

where

AUTHORS

Takashi Nakamura graduated in 1972 from the Department of Electronic Engineering, Toyama University, and received a Dr. of Eng. degree from Tohoku University in 1977. In that year, he became an Assistant at the Research Institute of Electrical Communication, Tohoku University. In 1978, he became a Lecturer in the Department of Electronic and Computer Engineering, Gifu University, where he was promoted to an Associate Professor and in 1991 to Professor. He has been engaged in research on the transmission and radiation of the electromagnetic waves. He is a member of IEEE. He is the co-author of Concept of Electrical Communication Engineering iI.

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AUTHORS (continued, from left to right)

Shiqji Kamiya graduated in 1990 from the Department of Electronic Engineering, Gifu University, where he received an M.S. in 1992. In that year, he joined Nippon Denso. In school, he was engaged in research on transmission and radiation of the surface wave transmission lines.

Senji Yokokawa graduated in 1955 from the Department of Communication Engineering, Shinshu University, and received a Dr. of Eng. degree from Tohoku University in 1961. In that year, he joined the Faculty of Engineering, Yamagata University. Since then he has been engaged in research on high-frequency transmission circuits. Presently, he is a Professor in the Department of Electronic and Computer Engineering, Gifu University. He is the author of Transmis- sion Engineering.

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transmission and radiation properties of a surface wave transmission line with a bend