US2623122A - Individually matched bead supports for coaxial cables - Google Patents

Description

Patented Dec. 23, 1952 UNITED STATES PATENT OFFICE INDIVIDUALLY MATCHED BEAD SUPPORTS FOR COAXIAL CABLES Application January 2, 1947, Serial No. 719,808

1 Claim.

This invention relates to bead supports for coaxial cables, and it is especially concerned with the design of such supports to provide the individual matching of each bead with the cable to reduce or prevent wave reflection.

Heretofore, reflection in bead supported coaxial cables has been reduced or eliminated in two principal ways. First, the beads have been arranged in spaced pairs whereby reflection from one bead is cancelled by reflection from the other head. Secondly, individual beads have been compensated by undercutting or overcutting the adjacent conductors, but this has been done simply by taking into account the nominal impedance of the bead, and without taking into consideration the capacitance introduced by the sharp corners of the cuts which are formed in the metallic conductors.

The simplified design principles discussed above are stated in Sperrys book on Microwave Transmission Design Data, pages 46 to 53.

There are certain deficiencies in the prior methods of obtaining reilectionless transmission. First, the spaced pairs are frequency sensitive depending upon the thickness of the beads used, and neglect of the thickness results in a VSWR beyond that expected from the compensation. Also, it has been found that undercut or overcut beads which were designed without regard to the capacitance resulting from the sharp corners did not perform as expected, especially in the high frequency ranges of 3000 megacycles per second and above.

The present invention provides a bead structure which is individually matched to the characteristic impedance of the cable so as to take into consideration the capacity introduced by the sharp corners of the undercutting or overcutting. This results in much lower VSWR in the high frequency ranges, andalso produces a broadband characteristic. Furthermore, the bead supports of the present invention are individually compensated or matched, and pairing of the heads is not necessary.

The present invention will be explained in connection with the accompanying drawing in which Figures 1, 2 and 3 are longitudinal sectional views of coaxial cables showing, respectively, an

undercut and overcut thin bead support, a

spool type of bead, and an overcut quarter- 'wave type of head.

Figures 1a, 2a and 3a are the equivalent twoconductor transmission lines of Figures 1 to 3.

Figure 4 is an admittance diagram used in explaining the invention,

Figure 5 is a graph showing two curves or plots used in designing the bead supports of the pres ent invention.

In each of Figures 1 to 3, the outer conductor I and the bead 2 are shown in section, while the center conductor 3 is shown in elevation.

It can be shown that any discontinuity in the coaxial cable having axial symmetry may be represented by a lumped capacity at the place of discontinuity. Thus in each of Figures 1a, 2a and 3a lumped capacity elements C are shown at each point of discontinuity in Figures 1, 2

and 3 respectively.

The electrical performance characteristics of the beads shown in Figures 1 to 3 are shown by means of the admittance diagram of Figure l giving the admittance loci of the three bead types. 7 H

Consider first the simplest case of the so-called quarter wavelength bead of Figure 3. Looking at the two wire equivalent in Figure 3a, the discontinuities are represented by capacitances C, C connected in shunt on the transmission line. A condenser in the chart of Figure 4 is represented by a vertical line (susceptance plus a'cC). With such a diagram, a connection in shunt means merely a vertical addition to the input admittance of the coaxial line behind the bead (load end). Thus, for a matched line behind the bead, with input admittance equal to the characteristic admittance Y0, one starts with that value in the diagram and adds it to the susceptance y'wC. This leads to the point (a) in the chart, as the total input admittance of the two-wire line at the point (a). To obtain the locus of the input admittances proceed in a clockwise direction around the circle about Yo. A point (b) in this circle may be found which is directly under Y0. By symmetry the addition of the susceptance due to the discontinuity at the front of the bead will cary us back to Yo resulting in a match. If this procedure is followed on an exact transmission line chart the arc (ab) will be less than a quarter wavelength in the dielectric for exact cancellation at one frequency. At neighboring frequencies, the needed length or width of the dielectric bead will differ slightly and satisfactory performance extends over a rather broad wavelength range. The final width of the bead may be calculated exactly for mid-band wavelength using transmission line equations given hereinafter.

In the second case, that of the spool bead of Figure 2, the procedure is initially the same. At some point (0) in the bead as shown in both diagrams the bead is reduced in diameter. This lowers the characteristic admittance to the value Yb which is lower than the characteristic admittance Yo of the line. This results in an increase in the characteristic impedance (the reciprocal of characteristic admittance) of the head section over and above the normal characteristic impedance of the line. Actually there is some field distortion occuring at place where the vertical cut in the bead appears and some capacitance. This capacitance however is known to be quite small and is neglected for this case. Proceed now about Yb to the point ((1) the place Where the bead diameter returns to that of the normal characteristics admittance Yo. Proceed about Yo to (b) and return to Yo by way of the susceptance at the bead face.

In Figure l, for the thin bead we proceed in the same manner as for the quarter-wave bead except that the input admittance locus is now in a circle about Yb instead of Y0. With this procedure a, much smaller portion of the circle is required.

To test for the relative frequency sensitivity of the beads we note that the chest of the capacittance measured in susceptance is given by wC (see Figure l), a quantity proportional to frequency. But the length of circular path between (a) and (b) is also proportional to frequency. When the frequency is changed. we note that the path length for the thin bead most nearly approaches the length 2wC. Thus for the lengths varying together this thin bead gives the best approximation to the design condition when the frequency is varied.

From the electrical point of view of broad bandness the bead given in Figure 1 is the best. That of Figure 2 is next best and that of Figure 3 is the worst. The bead of Figure 3 depends least upon the exact evaluation of the capacitance at its face.

The design principles underlying the present invention will now be discussed under two headings, that is, the general bead formula will be developed first and then the formula for thin beads.

I. The general bead formula The outer and inner diameters of the bead are, respectively, D and (1', such that the character'- istic admittance of the bead section is given by where s is the dielectric constant of the bead ma-- terial and Yig may be called a geometric characteristic admittance. For the configuration shown above, most of the field distortion occurs in the dielectric. In this case the capacity, C,

4 caused by this distortion, may be either calculated or measured and may be represented as the thickness of the bead, Z, and the dielectric constant, c, for which matched beads result. A general formula illustrating this fact can readily be derived. In order that the bead be matched, the input admittance to the head which is terminated in Y0 should have an input admittance equal to Yo. Analytically this is written hen: E C Yo Ym aw 1 'eml 4) Y +jwC+Y where e is the base of natural logarithms. This can be reduced to the formula wC' tan fil= are where Yo, Y, and C are defined above and If the diameter ratio is arranged to give a bead whose characteristic impedance is equal to that of the air coaxial line, then Y'=Yc. For this Equation 5 reduces to 21rl 1 2Y As a rule,

and the equation then specifies that 2w 21rl T where n=0, 1, 2, 3, etc. that is, the bead shall be an odd multiple of a wavelength in the dielectric. A more exact formula for the length is given by Equation 7.

Using Equations 2 and 3 in 5, a relationship is obtained which clarifies the interrelationship 0 various factors 92: t 5/ M Xi l ea II. Thin bead formula that is, approximately for the arguments of 16 or less. Further,

where Co is the capacitance per unit length, L0 is the inductance .per unit length and V1 is the velocity of propagation in free space. Substituting these values in yields for 2 2 E i e) ix) which is usually the case. Simplifying, one has It is noted that for the approximations made, the match condition (5') is independent of frequency. As the frequency becomes smaller the approximation becomes better. If an attempt is made to go higher in frequency, however, the approximation is gradually invalidated and other means, such as spacing, may be applied to reach satisfactory matching conditions.

The degree of the approximation determines the frequency sensitivity of the bead and to this extent governs the mechanical thickness that can be used for a given frequency. In order to reach higher frequencies, the dielectric constant of the bead should be made as small as possible. An attempt should be made to always attain maximum thickness which permits the assumption if mechanical strength at the higher frequencies is desired.

Design procedure for thin beads The design procedure for thin beads according to the present invention will now be explained in a number of steps. The calculated values of ca pacity cannot be ascertained with sufficient accuracy to give the final specifications for the bead structure, but the theoretically predicted bead performance can be realized by making a few final check measurements, as will be explained later.

Step 1.For a given bead material, a bead thickness, Z, should be chosen that is short compared to the shortest wavelength (highest frequency) at which the bead is expected to be used.

6. Specifically, this length is specified by the statement dm Z3 36 where Mm is the wavelength in the dielectric corresponding'to the highest frequency. If this condition for the length or thickness is met, Equation 5" assures that the desired electrical properties can be attained for all longer wavelengths. It is permissible to exceed this thickness if a certain amount of reflection is not objectionable. In some microwave applications the thickness may be .as much as /18 of the wavelength in the dielectric corresponding to the highest frequency.

Step 2.-With the length of bead and the dielectric constant of the bead material chosen as above, it is now possible to plot the value of n ie by utilizing the relationship between these quantities given by Equation 5". The values of Y0 and Co are constants known from the dimensions of the standard coaxial line for which the bead is to be used. This curve then gives the corresponding values of for which the bead is will be matched. Curve 4 of Figure 5 shows these values for a specific example given hereinafter.

Step 3.For a given proportioning of the overcutting and undercutting as discussed in section I above, and for a given bead thickness, 1, the capacity, Cg, may be calculated as a function of Yig. By this means an independent plot of may be obtained. This curve, one example of which is shown at 5 in Figure 5, now represents the as calculated from the ratio it is necessary merely to determine the amount of overcut.

as a function of and as a function of Step 4'.--It will be found that the "two curves plotted in Steps 2 and 3 will intersect at a. certain point, and at this point the condition for match is satisfied, that is Equation is satisfied by the capacity Cg as calculated from the dimensions. Also, the resultant value Yig gives reasonable dimensions which can be used for design purposes.

Step 5.-Since the capacity values, Cg, will not be quite accurate, but generally can be given within limits not exceeding 115%, the final reflectionless bead must be obtained by testing a series of beads having thickness varying il5% from the thickness originally chosen. A series of 3 or 4 beads, however, should be sufficient to obtain the condition stipulated by Equation 5. At the higher frequencies, making the head thicker may cause violation of the condition of Step 1. This is not serious in general, since the departure from operation stipulatedby Equation 5 occurs only at a slow rate as the condition is violated. 7

The above method of design may be used for other non-characteristic impedance beads by using the exact Equation 5 in the place of (5"') The electrical performance of these beads, however, will be limited to given bands of frequency and do not look as promising electrically as the thin head.

The curves shown in Figure 5 are for an overcut thin bead of the type shown in Figure 3 in which the inner diameter of the outer conductor is 0.8125 inch and the diameter of the inner conductor is 0.375 inch. The bead is formed of polystyrene having a dielectric constant of 2.56, a thickness of 0.178 inch and an outside diameter of 1.5 inches. The crossover point of the curves 4 and 5 occurs at 1.793 of the ratio Y0/Yig.'

We claim:

A coaxial cable in which the center conductor wherein Z is the length of the part having the characteristic impedance Zig, e is the dielectric constant of the bead section, Cg th efiective additional capacitance introduced by the groove and Co is the normal capacitance per unit length for the standard transmission line of characteristic impedance Zo;

ERNST WEBER. v JOHN W. E. GRIEMSMANN.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 1,859,390 Green May 24, 1932 2,000,679 Walter May '7, 1935 2,165,961 Cork et a1. July 11, 1939 2,267,371 Buschbeck Dec. 23, 1941 2,406,945 Fell Sept. -3, 1946 OTHER REFERENCES Proceedings of the I. R. vol. 32, No. 11, November 1944, pages 695 to 704.

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