Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Rewrite $τ^{2} = k^{2} - h^{2}$ with $k = ω / v$ and $h = ω_{c} / v$ , where $ω_{c} = 2 π f_{c}$ :

Dispersion relation of a guided mode

ω^{2} = ω_{c}^{2} + v^{2} τ^{2} . (7.1.19)

In plain words

In the $ω$ – $τ$ plane every mode is a hyperbola. It starts at $ω = ω_{c}$ with $τ = 0$ and approaches the light line $ω = v τ$ of the filling, which is the TEM wave.

Reading the diagram. At an operating point on a mode curve, $v_{p} = ω / τ$ is the slope of the line from the origin to the point, and $v_{g} = d ω / d τ$ is the slope of the tangent there (Figure 7.1.6). Every mode curve lies above the light line, so the line from the origin is steeper than the light line and the tangent is less steep: $v_{p} > v > v_{g}$ whenever $f_{c} > 0$ .

The \omega–\tau diagram of the parallel-plate guide of Section 7.2, where f_{cn}=nf_{c1} and \tau b/\pi=\sqrt{(f/f_{c1})^2-n^2}. The straight line is the TEM wave (the light line). The dot marks n=1 at f=1.5f_{c1}: the dashed line from the origin has the slope v_p=1.342v, the tangent the slope v_g=0.745v. — **Figure 7.1.6.** The $ω$ – $τ$ diagram of the parallel-plate guide of Section 7.2, where $f_{c n} = n f_{c 1}$ and $τ b / π = (f / f_{c 1})^{2} - n^{2}$ . The straight line is the TEM wave (the light line). The dot marks $n = 1$ at $f = 1.5 f_{c 1}$ : the dashed line from the origin has the slope $v_{p} = 1.342 v$ , the tangent the slope $v_{g} = 0.745 v$ .

The tool below draws the same diagram, with $f$ in place of $ω$ , for the simplest guide, the air-filled parallel-plate guide of Section 7.2, here with $f_{c 1} = 10 GHz$ and the operating point $f = 1.5 f_{c 1}$ . Change $f$ and watch the secant ( $v_{p}$ ) and the tangent ( $v_{g}$ ) follow the dot along the TM $_{1}$ curve.

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7.1.6 The dispersion relation and the ω–τ diagram