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range and the material (gold, silver, copper, aluminum, brass) affects the loss.
Rectangular waveguide is frequently used in radar.8 For illustration, Figure 5A.1 shows a magic T (or tee) constructed with WR-90 waveguide. The magic T is a four-port, 180°, 3-dB hybrid developed during World War II [52] and is used as both a power combiner and power divider, depending upon the ports used and is very low loss [53–55]. Copper and copper alloy are standard waveguide materials (solid or plating).9
Table 5A.1 contains the Electronic Industries Association (EIA) waveguide (WG) designations,10 inner dimensions, frequency range, and theoretical attenuation for a number of standard rectangular copper waveguides [15]. For the EIA designation, the WR number is the internal dimension in inches of the broad wall.Figure 5.2 contains plots of the theoretical waveguide loss versus frequency for several of the waveguides in Table 5A.1. We note that loss is inversely proportional to frequency. Also, the frequency boundaries are not coincident with radar designators. The general rule of thumb used to decide between multiple waveguide possibilities is to select the larger waveguide, which has lower loss.
Fig ure 5A.1 Waveguide magic T (WR-90).
Table 5A.1
Standard Rectangular Waveguide Specifications (Copper)
Fig ure 5A.2 Rectangular waveguide.
As a side note, waveguide is often pressurized, typically using dry air, nitrogen, or argon to prevent moisture buildup inside the waveguide, which can cause corrosion of the conducting surfaces, thus increasing loss.11 In addition to using dry gas as dielectric, the slight overpressure help to keep out moisture in the event of small leaks. Microwave transparent windows are used to prevent pressure loss where the waveguide would be open (e.g., feed horn).
For example, consider a copper-plated, pressurized, rectangular WR-90 waveguide, depicted in Figure 5A.2, operating at 10 GHz (X-band) filled with nitrogen. For the dominant mode,12 we want to determine the cutoff frequency in GHz and the attenuation due to conductor loss in dB/m.
From Table 5A.1, we see that WR-90 waveguide has interior dimensions of the broad and short walls of a = 2.286 cm (0.90 in) and b = 1.016 cm (0.4 in), respectively. Since the wall length ratio is ~ 2:1, the dominate mode of propagation is the TE10 mode. The cutoff
frequency for the mn mode is given by [55, p.113]
From Table 5A.1, we see that WR-90 waveguide has interior dimensions of the broad and short walls of a = 2.286 cm (0.90 in) and b = 1.016 cm (0.4 in), respectively. Since the wall length ratio is ~ 2:1, the dominate mode of propagation is the TE10 mode. The cutoff frequency for the mn mode is given by [55, p.113]
where
is the cutoff wave number. To clarify cutoff frequency as used here, for frequencies above the cutoff frequency for a given mode, the electromagnetic energy can be transmitted through the guide for that particular mode with minimal attenuation (which is backwards compared to lowpass filter cutoff terminology).
For typical gaseous dielectrics used to fill waveguides (air, nitrogen, argon), the permittivity and permeability are essentially identical to those of free space (vacuum). Recall the permittivity of free space is μ0 = 400 π ≈ 1256.637061 nH/m and the permeability of free space is ε0 = 1/μ0c2 ≈ 8.8541878176 pF/m. For the TE10 mode, (5A.1) and (5A.2) simplify to
and
The upper bound on propagation is the TE20 mode waveguide cutoff frequency calculated using (5A.1). Recall that above the cutoff frequency for a given mode, the electromagnetic energy will propagate through the guide for that particular mode with minimal attenuation.
Therefore, TE10 mode will propagate at frequencies above 6.56 GHz and below 13.11 GHz.
Digressing for a moment, we note that 6.56 GHz to 13.11 GHz does not match the operating frequency range given in Table 5A.1. We illustrate the rationale for this discrepancy by comparing the waveguide loss for both the theoretical and recommended frequency ranges, presented in Figure 5A.3. While the TE10 mode will technically propagate with up to ~3-dB loss, the amount of loss considered acceptable for waveguide is much lower.
Fig ure 5A.3 Theoretical loss for copper-plated WR-90 waveguide over theoretical and recommended operating ranges.
Returning to our loss example, the attenuation due to conductor loss (loss due to the metal of the waveguide)13 is given by [55, p. 115]
where we recall that nepers14 (Np), defined in the same Bell Labs paper as dB [56], is a unit based upon the natural logarithm, and is given, for voltage, by [55, p. 63]
and for power by
The propagation constant, β, for the TE10 mode is given by [55, p. 112]
where
is the free space wave number. The intrinsic impedance of the dielectric is
The last component of (5A.6) is the surface resistivity of the metal in the waveguide, given by [55, p. 28]
The conductivity for copper and other common waveguide materials is listed in Table 5A.2.
For this example, copper is specified, which has a conductivity of 5.813×107 mho/m [54, p.
458]. This results in a surface resistance of Rs = 0.0261Ω. Substitution into (5A.6) gives
Table 5A.2 Material Conductivity
Material Conductivity (mho/m)
Aluminum 3.816 · 107
Brass 2.564 · 107
Copper 5.813 · 107
Gold 4.098 · 107
Silver 6.173 · 107
Source: [54].
Converting this to dB/m gives
where nepers are related to dB by [55, p. 63]
For completeness, we convert the answer to dB/100 ft, since historically, many waveguide tables are presented in dB/100 ft.