IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPArlBILITY, VOL. 3 1, NO. 1, FEBRUARY 1989 69

An Approximate Solution for Coupling to a Coaxial Waveguide Which Terminates at a

Conducting Wedge

Abstract-A coaxial waveguide that terminates with a conducting flange of infinite extent is often used to model the umbilical connectors on military hardware. The placement of the actual connector, however, can be such that the use of an infinite plane will lead to a poor estimate of the aperture-short-circuited surface fields that represent source terms for the coax. In this paper, an integral equation is derived to approximate the aperture electric field for a coaxial waveguide that terminates at a conducting wedge. The integral equation is derived by retaining the standard kernel for the infinite-flange case, and adapting the solution for the two-dimensional conducting wedge for use as the excitation term. The solution of this equation gives rise to an approximation to the TEM open- circuit voltage. Because only the excitation term is modified, the equation is useful for connectors that are greater than about one wavelength away from the corner. Theoretical and experimental results are presented for the power transmitted down the coax.

I. INTRODUCTION

COAXIAL waveguide that terminates with a conducting A flange of infinite extent is often used to model the umbilical connectors on military hardware [ I]-[4]. In Fig. 1, the geometry for a coaxial waveguide with a flush-mounted center conductor is shown. A collection of well-known results for the aperture admittance and the open-circuit voltage (effective height) for this geometry can be found in [4]. Although these circuit parameters can be used to approximate TEM propagation within the coax, the results will of course only be accurate for certain geographical placement of the connector on the hardware. In particular, when the connector is electrically removed from any corners such that the aperature admittance is well approximated by an infinite- flange termination, and (obviously) when the connector is not in a shadow region with respect to the incident wave.

In this paper, an integral equation is generated so that the TEM open-circuit voltage for the connector geometry shown in Fig. 2 can be approximately determined. This problem is not known to have been studied in the open literature. Only the excitation term of the standard integral equation that describes coupling to a coaxial waveguide with an infinite-flange termination is modified to account for the presence of the conducting wedge. As long as the coaxial waveguide is greater than about one wavelength away from the corner, this

Manuscript received September 12, 1987; revised September 16, 1988. D. J. Riley is with the Electromagnetic Applications Division, Sandia

L. D. Bacon is with the Microwave Physics Division, Sandia National

IEEE Log Number 8825197.

National Laboratories, Albuquerque, NM 87 185.

Laboratories, Albuquerque, NM 87185.

i' , E'

I

Fig. 1. Flush-mounted connector terminating with a conducting flange of infinite extent

CONNECTOR 1 RIGHT-ANGLE CONDUCTING WEDGE

Fig. 2. Connector terminating at a right-angle conducting wedge.

modification should be sufficiently accurate. The implication of this approach is that the reradiated fields local to the aperture are not significantly altered from their infinite-flange value by the presence of the corner. Observe that this type of modification is similar to the common practice of determining aperture-short-circuited source terms for use with quasi-static aperture-polarizability expressions [ 5 ] . Being able to accom- modate the configuration of Fig. 2 is considered to be important, because in many realistic coupling problems the incident wave will initially encounter a leading edge before interacting with the connector.

If one models a connector with the infinite flange as shown in Fig. 1, grazing incidence will often give the greatest

0018-9375/89/0200-0069$01 .OO 0 1989 IEEE

70 IEEE TRANSACTIO1 \IS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 1, FEBRUARY 1989

TEM coupling. However, if the actual geometry is closer to that shown in Fig. 2, the greatest coupling will occur at some angle that may be far from grazing. If one naively assumed grazing gives the greatest coupling for this configuration, the answer may be in error by as much as a factor of two, because of the strong reflection induced by the leading face of the wedge. It is a minor modification to partially account for the wedge, and the techniques are outlined in the sequel. In addition, an experimental comparison is made with a moment- method solution of the integral equation to demonstrate the accuracy of the modification.

II. SURFACE CURRENT DENSITY FOR THE RIGHT-ANGLE CONDUCTING WEDGE

For a TM or TE to x incident wave (cf. Fig. 2, with the aperture short circuited), the conducting-wedge problem is separable and the solutions are well known [6]. For the problem of interest, the TE case is considered, namely, Hi = fHOe-jbO cos (Oil , where Ho denotes the peak amplitude. The surface current density Jy is given by

In this expression J,, where U is a real number, denotes the standard Bessel function of the first kind, and k denotes the wavenumber, which is taken to be real. An exp ( j u t ) time convention was assumed. This function is depicted in Fig. 3 for various angles of incidence and values of kpo. Observe that grazing incidence gives a surface current density that is approximately one-half the value for an infinite plane.

III. APPLICATION TO THE COAXIAL-CONNECTOR COUPLING PROBLEM

In [4], an integral equation was derived that describes external plane-wave coupling to the TM to z symmetric modes of a coaxial waveguide with the infinite-flange termination shown in Fig. 1. The basis for this equation was continuity of the azimuthal magnetic field through the aperture. For the excitation Hi = fHoe-jkp sin cos %, which has the symmetric azimuthal component jHoJl (kp cos e;), this equation is J: d p ’ p ’ ~ ~ ~ ( p ’ ) ~ % ( p , p ’ )

= j 2 t z O 4 ( k p COS e;), a c p < b (2)

where the infinite-ground-plane kernel XzO accounts for all TM symmetric modes and is given by

f m 1

1 JI ( t p ) J i (b ’1 @=P + + j So d t t

k l n (:) PP’

and E, denotes the unknown radial aperture electric field that

2.2

m 3 1.4

1.2

I 0.8 ‘ I I , , 1 I

0 10 20 30 40 50 60 70 80 90

01 (Degrees) Fig. 3. Normalized surface current density on the top surface of a right-

angle wedge (aperture short circuited).

represents the source term for the symmetric modes in the coaxial waveguide. When k > [, d v - = j v I t is noted that a similar kernel can be found in [l]. The additional terms in the expression for the kernel are defined by

QI ( ~ o ~ P ) = J I (konp) YO(k0na) - YI (konP)Jo(kona)

and the eigenvalues k,, satisfy the equation

JO(k0n b ) YO(k0na) - YO (ken b)Jo(kona) = 0. Observe that free space is assumed to occupy the coaxial waveguide and the region external to it (eo denotes the permittivity). Yo denotes the standard Bessel function of the second kind. Note that the aperture admittance can be obtained from (2) by replacing the right side with l/?rp.’

The term on the right side of (2) represents the symmetric component of the free-space incident plane wave and the aperture-short-circuited reflected wave for an infinite plane. This is the term that will be modified by the wedge solution. To accomplish this, it is necessary to introduce a shifted coordinate system as shown in Fig. 4. Note that the axis of the coaxial waveguide is located a distance yo from the edge of the wedge.

Because the integral equation is based on the azimuthal component of the magnetic field, it is necessary to write wedge solution (1) as H x ( p o ) f = - $Hx sin cp + p̂ Hx cos cp, and use the term -H, sin cp. Also, because the symmetric component of this term is required, it is necessary to average over cp.

’ The voltage V across the aperture is obtaird by forming JtdpE,. For TEM propagation, the aperture admittance for this mode is Y, = 2/ V - ’yo 111, where YO is the characteristic admittance of the coaxial line, and E, is obtained by solving (2) with right side Uup. The open-circuit voltage is obtained by forming V(Y0 + Y,)/Y,, where, in this case, V is due to the external excitation Y, is still based on l l x p .

RILEY AND BACON: SOLUTION FOR COUPLING TO A COAXIAL WAVEGUIDE 71

WILTRON

2

E' t

/ OPEN COAX

c Fig. 4. Shifted coordinate system for merging the two models.

Fig. 5. Experimental setup.

Therefore, to account for the effect of the wedge, the right- hand side of (2) is replaced by

4 -

where K , = 1 for n = 0 and K, = 2 for n 2 1, and po in (1) has been replaced by yo + p sin cp, a < p < b. This modification is appropriate for yo > X + 6, where X is the wavelength, and for A' parallel to the edge of the wedge. The integral can be evaluated as a summation over products of Bessel functions [7]; however, the numerical evaluation of the integral representation is simpler to implement and more efficient to evaluate.

IV. EXPERIMENTAL COMPARISON

The test geometry was a large coaxial aperture centered on a face of an rf-tight box that was 50 cm on a side. The coax had an outer diameter, 2b, of 6.03 cm, and an inner diameter, 2a, of 2.54 cm, for a characteristic impedance of approxi- mately 52 Q . The coax was 42 cm long, with the final 10 cm being a conical transition section that terminated in a standard type-N connector.

Fig. 5 depicts the measurement setup. A Wiltron Automated Scalar Network Analyzer system performed the transmit- receive-data recording functions. This system consists of a

Model 6647A programmable sweep generator, a Model 560A scalar network analyzer, a Model 85 controller, and controller software. To achieve the dynamic range we required, an external 1-W solid-state amplifier was used.

The results are shown in Fig. 6 for the case of TE excitation and a coaxial-waveguide termination which was matched to the characteristic line impedance. A moment-method numeri- cal procedure was used for the solution of ( 2 ) with the right- hand side given by (3) to determine the TEM open-circuit voltage for the coax. The aperture admittance was obtained by numerically solving (2) with the right-hand side l lap; thus, the aperture admittance was taken as its infinite-flange value. The details for an efficient numerical solution of ( 2 ) are described in 141. Note that the agreement in both the angular trend and the amplitude is quite good. Only a narrow bandwidth is displayed due to a suspected shunt-capacitance mismatch that became significant above 2.5 GHz at the transition from the large coax to the type-N connector. The accuracy of the hybrid model, however, should improve with increasing frequency (within the TM symmetric mode and moment-method limitations).

The theoretical data is based only on TEM propagation; however, above 2.28 GHz the TEl l mode is above cutoff. The good agreement above this point clearly indicates the domi- nance of the TEM mode relative to TEII when the incident wave excites both of these modes. It is also noted that the experimental setup involves a three-dimensional box, whereas the theoretical model uses a two-dimensional wedge. The good agreement indicates that diffraction off the two side edges and

72 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. I , FEBRUARY 1989

-56.0

-58.0

-60.0

-62.0

3 n 0

-64.0

-66.0

2 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.5

Frequency (GHz) Fig. 6. Comparison for measurement and theory for the time-averaged received power with respect to 1 W. 20 = 2.54 cm, 2b =

6.03 cm, Ho = 1/377 A h (Eo = 1 Vh). The incident wave was TE to x , and the axis of the connector was 25 cm from the X- directed edge. The theoretical results are for the TEM power.

L Q) - 10.0 i

8.0 a U

Q t a 4.0 Q) > 0 2.0 Q)

I CI

w

0-0 iz

0 1 2 3 4 5

Frequency (GHz) Fig. 7. Further theoretical results for the configuration described in Fig. 6.

The effective area based on TEM propagation with a line-matched load is shown. This area was calculated by forming 270P/(E0)z, where 70 denotes the impedance of free space and P denotes the time-averaged TEM power.

the back edge is not significant compared to the interaction of the plane wave with the leading edge and front face. As a final comment, had the hybrid approach not been used to compute the theoretical results, the 30 and 0 degree curves would have essentially been reversed.

Additional theoretical results are shown in Fig. 7.

V. CONCLUDING REMARKS

In this paper, a simple technique was presented for generating an integral equation that can be used to approxi-

mate the TEM open-circuit voltage for a connector that is situated on a conducting wedge. The equation is useful for connectors that are at least one wavelength away from the corner. Within the framework of the approximation, the aperture admittance for the TEM mode is taken to be its infinite-flange value. As previously noted, the implication of this approach, which is certainly not novel, is that the radiated fields local to the aperture are not significantly altered from their infinite-flange value by the presence of the corner. The results obtained were found to be consistent with experimental

RILEY AND BACON: SOLUTION FOR COUPLING TO A COAXIAL WAVEGUIDE 1 3

observations. More rigorous (complicated) techniques, such REFERENCES as a hybrid moment-method/GTD technique, be adopted to accommodate the case when the aDerture is electrically

[I] C. W. Harrison and D. C. Chang, “Theory of the annular slot antenna based on duality,” IEEE Trans. Electromagn. Compat., vol. EMC-

close to the edge of the wedge [8]. Standard EMP references, for example [9], often depict

results for the normal electric field or surface current density on the flat ends of finite cylinders by exclusively using a broadside incident wave. From the results presented here, it is obvious that such excitation does not generally give rise to the largest surface fields and hence the greatest coupling to a connector.

For a given frequency and aperture offset from the edge of the wedge, one can obtain a quick approximation for the angle of incidence that will give rise to the greatest coupling by using the results for the surface current density for the conducting wedge presented in Fig. 3.

ACKNOWLEDGMENT

Appreciation is given to R. P. Toth and J. F. Aurand for their assistance in performing the experimental measurements.

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13, no. 1, pp. 8-14, Feb. 1971. D. C. Chang and C. W. Harrison, “On the pulse response of a flush- mounted coaxial aperture,” IEEE Trans. Electromagn. Compat., vol. EMC-13, no. 1, pp. 14-18, Feb. 1971. D. E. Merewether, R. C. Cook, and R. Fisher, “The receiving properties of the small annular slot antenna,” EMA Rep. Sandia Lab. Contract #13-0765, Albuquerque, NM, Sept. 29, 1978. D. J. Riley, “Analysis of the receiving characteristics of the circular diffraction antenna with both a flush-mounted and a recessed center post,” SAND-86-2398, Sandia Labs, Albuquerque, NM, Nov. 1986. K. S. H. Lee, Ed., EMP Interaction: Principles, Techniques And Reference Data, AFWL EMP Interaction 2-1, Albuquerque, NM, 1980. R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, pp. 238-242. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. 4th ed. New York: Academic, 1980. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981, pp. 500-510. D. E. Merewether, J . A. Cooper, and R. L. Parker, Eds., Electromag- netic Pulse Handbook For Missiles And Aircraft In Flight, Sandia Lab., Albuquerque, NM, 1972.