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## Appendix

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**A. Derivation of the main beam-pointing equation**

A linear traveling-wave array of radiating elements with a constant interelement spacing, *d*, is considered; see Fig. A. The change in phase from *a* to *c *is 2π/λ ∙ d cos *ϕ *where λ is the free space wavelength and *ϕ* the main beam pointing direction from forward endfire. Similarly, the phase change from *a* to *b* is 2π/λ* _{g}* ∙ d; λ

*is the waveguide wavelength. An additional phase shift of –*

_{g}*m*π (

*m*= 0, 1, 2, 3) is introduced with each succeeding element. The value of

*m*depends on how the elements are fed—in or out of phase (

*m*= 1 and

*m*= 3 correspond to staggered slots and

*m*= 0 and

*m*= 2 correspond to collinear or inline slots). Points

*c*and

*b*are in phase yielding.

Solving Eq. 2 for *ϕ *gives an expression for the main beam pointing direction

For the *TE*_{10} mode of the waveguide wavelength, λ* _{g}*, is related to the relative dielectric constant,

*K*, and the inside width,

*a*, of the rectangular waveguide by

The beam pointing direction is, thus

Eq. 5 represents the beam pointing direction for the array factor of an array of omnidirectional elements.

**B. Bias toward broadside**

The effective bias toward broadside of the beam pointing direction of traveling-wave arrays is caused by the variation of the magnitude of the element factor from broadside to endfire. Radiation pattern beams of non-broadside arrays are asymmetrical because the magnitude of the element factor is maximum at broadside and decreases toward forward endfire and rear endfire. In particular, the forward endfire (rear endfire) portion of the main lobe of a non-broadside beam is somewhat lower than the corresponding portion of a broadside beam from an array of the same length. Lowering the portion of the main beam closest to forward endfire (rear endfire) with respect to the portion closest to broadside effectively moves the beam peak toward broadside. The wider the beam and the closer it is to forward endfire (rear endfire), the greater the relative reduction of the forward endfire (rear endfire) portion of the main beam and the greater the effective bias toward broadside.

**C. Grating lobes**

The maximum interelement spacing, *d _{M}*, of radiating elements in a traveling-wave array (which will not allow gain reducing grating lobes to appear in the radiation pattern) is a function of the angular offset,

*ϕ*, of the main beam from endfire, as expressed in Eq. 6 for infinitely narrow beamwidths.

_{0}where λ is the operating wavelength. A more general expression applicable to arrays of various beamwidths is

where *BW _{10 dB}* is the 10 dB beamwidth or the angular width of the main beam measured at the 10 dB down points. Figs. 1 through 11 are each plotted with interelement spacings of 0.55λ and 0.65λ which correspond to minimum angular offsets of the main beam from endfire of 34.9 degrees and 57.3 degrees, respectively, (which will not allow grating lobes to appear in the radiation pattern) for infinitely narrow beams. The appropriate portions of the curves have been eliminated due to grating lobes appearing in the radiation pattern as predicted by Eq. 6. The antenna designer should use the more general expression of Eq. 7 to check their particular array for possible grating lobes.

**D. Double moding**

Eq. 5 was derived for a *TE*_{10} mode of propagation in a rectangular waveguide. Under certain conditions the *TE*_{20} mode will propagate, and those conditions (waveguide widths allowing *TE*_{20} cut off frequencies below the operating frequency) have been eliminated from the curves. The expression for *TE*_{20} cut off frequency is

where

c = velocity of light

K = dielectric constant

*a* = inside width of the rectangular waveguide

**E. Broadside beam designs**

Traveling-wave arrays are not usually designed with broadside beams. Commonly used longitudinal shunt slot radiating elements staggered (non-staggered) about the waveguide broadwall centerline are spaced odd (even) multiples of a half waveguide wavelength apart along the traveling-wave array when broadside beams are desired. However, the input admittance of the array for either odd or even multiples of half waveguide wavelength interelement spacing is the sum of the admittances of the individual shunt slot radiating elements plus the characteristic admittance of the waveguide. Since the sum of the admittances of the individual shunt elements cannot equal zero, the input admittance of the array cannot equal the characteristic admittance of the waveguide; thus, the array will not be matched without a transformer. Therefore, the more efficient standing wave arrays rather than traveling-wave arrays are commonly used to generate broadside beams.

The basic design curves, Figs. 1 through 11, include the broadside beam pointing direction. However, the engineer should not design a traveling-wave array with a beam pointing direction closer than 5 degrees from a broadside at all operating frequencies because of the matching problems.