EE 499 · Satellite Technologies

Week 7

What Is the Field Actually Doing Inside the Waveguide?

Weeks 5 and 6 gave us cutoff, modes and sizes: numbers on a page. This week we watch the field itself — travelling along the guide above cutoff, standing still at cutoff, and dying away where it cannot propagate.

By the end: you should be able to describe what the TE₁₀ field does above, at and below cutoff, calculate the guided wavelength and decay rate for a real guide, explain why a field that decays below cutoff has not been absorbed, and say why operating above the next mode's cutoff risks multimode propagation.

Primary source: Weeks 5 and 6, and the lecture notes, Topic 3, section 1.3.2, equations (8)–(10).

01 — Recap from Week 6

We know whether a mode can travel. What does it do?

The dominant mode	\(TE_{10}\): the lowest-cutoff mode of a rectangular guide with \(a>b\).
Cutoff	\(f_{c,10}=\dfrac{c}{2a}\). A mode travels only if \(f>f_c\).
The single-mode window	One mode only between \(f_{c,10}\) and the next cutoff; the notes' design range is \(1.2f_{c,10}
The guide on the bench	WR-340: \(a=86.36\ \text{mm}\), \(b=43.18\ \text{mm}\); \(f_{c,10}=1.74\ \text{GHz}\), \(f_{c,20}=f_{c,01}=3.47\ \text{GHz}\).

Those answer whether a signal can get through. They do not say what the field is doing when it does, when it cannot, or in between.

02 — Objectives

What you should understand by the end

Above cutoff: the pattern travels, with a guided wavelength longer than the free-space one.
At cutoff: nothing advances; the same pattern fills every cross-section.
Below cutoff: the field falls off exponentially and nothing travels.
Near cutoff: how the guided wavelength stretches, and why it matters for design margin.
The TE₁₀ pattern: strongest at the centre, zero at the side walls.
Phase and group behaviour: the pattern can move faster than \(c\); energy cannot.
Decay is not dissipation: a field that dies away below cutoff has not been absorbed as heat.

Part 1 — the physics

One field, three behaviours

The same \(TE_{10}\) pattern across the width, and three different things happening along the guide, depending on where the frequency sits relative to cutoff.

03 — The field in three regimes

Across the width, and along the guide

\[\mathbf E=\hat a_y\,E_0\sin\!\left(\frac{\pi x}{a}\right)e^{j(\omega t-\beta z)}\ \Longrightarrow\ E_y=E_0\sin\!\left(\frac{\pi x}{a}\right)\cos(\omega t-\beta z)\]

Equation (8) in the notes, and its physical (real) form.

The factor \(\sin(\pi x/a)\) is fixed by the walls: zero at both side walls, strongest at the centre. The factor \(\cos(\omega t-\beta z)\) is what happens along the guide — and that is where the three regimes come from.

From the original lecture notes · Topic 3, pp. 10–11 →

03 — The field in three regimes

\(\gamma=\alpha+j\beta\): one constant, three cases

Along the guide the field varies as \(e^{-\gamma z}\), with \(\gamma^{2}=k_c^{2}-k_0^{2}\), \(k_c=\pi/a\), \(k_0=\omega/c\).

	Condition	\(\gamma\)	What the field does
Above cutoff	\(f>f_c\)	\(j\beta\), \(\ \beta=k_0\sqrt{1-(f_c/f)^{2}}\)	The pattern travels; amplitude constant.
At cutoff	\(f=f_c\)	\(0\)	Nothing advances; every cross-section identical.
Below cutoff	\(f	\(\alpha\), \(\ \alpha=k_c\sqrt{1-(f/f_c)^{2}}\)	Amplitude falls as \(e^{-\alpha z}\); phase does not advance.

04 — Above cutoff

The pattern travels

\[\begin{gathered}v_p=\frac{\omega}{\beta}=\frac{c}{\sqrt{1-(f_c/f)^{2}}},\qquad v_g=\frac{c^{2}}{v_p}=c\sqrt{1-(f_c/f)^{2}}\\ \lambda_g=\frac{2\pi}{\beta}=\frac{\lambda_0}{\sqrt{1-(f_c/f)^{2}}}\end{gathered}\]

\(v_p>c\) and \(\lambda_g>\lambda_0\) at every frequency above cutoff; \(v_g

A crest carries no information, so \(v_p>c\) breaks no law: energy and information travel at \(v_g\). Above cutoff the electric and magnetic fields rise and fall together, and that combination carries power along the guide.

05 — At cutoff

The pattern stops

At exactly \(f=f_c\): \(\beta=0\) and \(\gamma=0\). The field no longer depends on \(z\): every cross-section carries the same pattern, and it pulses up and down together, \(E_y\propto\sin(\pi x/a)\cos\omega t\). The wave bounces between the side walls without making progress — a transverse resonance.

Guided wavelength	\(\lambda_g\to\infty\)
Phase velocity	\(v_p\to\infty\)
Group velocity	\(v_g=0\): no energy carried along the guide
Power	No net power flows; the fields are a quarter-cycle apart in time

06 — Below cutoff

The field dies away, and nothing travels

\[E_y=E_0\sin\!\left(\frac{\pi x}{a}\right)e^{-\alpha z}\cos\omega t,\qquad \alpha=\frac{\pi}{a}\sqrt{1-(f/f_c)^{2}}\]

The shape across the width is unchanged; the amplitude falls by \(e\) every \(1/\alpha\); the phase does not advance with \(z\).

Far below cutoff \(\alpha\to\pi/a\): the field falls by \(e\) in about a third of the guide width. Just below cutoff \(\alpha\) is small and the decay slow, which is what makes the transition at cutoff continuous.

06 — Below cutoff

Decay is not dissipation

Evanescent decay	Field falls with distance because the mode cannot propagate. No power absorbed. Reflection.
Dissipative loss	Finite wall conductivity and dielectric. Happens above cutoff too, small in a good guide, measured in dB per metre. Not modelled here.

Week 8 asks, for any component, how much power is transmitted, reflected and dissipated.

07 — Guided wavelength

Near cutoff the wave stretches without limit

\(f/f_c\)	1.01	1.05	1.2	1.5	2	3
\(\lambda_g/\lambda_0\)	7.1	3.3	1.8	1.34	1.15	1.06
\(v_g/c\)	0.14	0.31	0.55	0.75	0.87	0.94

Close to cutoff, \(\lambda_g\), \(v_p\) and \(v_g\) all change quickly with frequency, so a small change in frequency changes the electrical length of a run of guide and the speed of the signal a great deal. Operating at \(1.2f_c\) or above keeps that modest — the design margin of Week 6.

08 — Multimode

Above the next cutoff, more than one pattern can travel

For WR-340 the next cutoff is \(f_{c,20}=f_{c,01}=3.47\ \text{GHz}\): above it \(TE_{20}\) and \(TE_{01}\) can travel alongside \(TE_{10}\). The next pair, \(TE_{11}\) and \(TM_{11}\), appears at \(3.88\ \text{GHz}\).

Modes with different patterns have different phase constants at the same frequency, so they travel at different speeds. A signal shared among them arrives smeared in time — intermodal distortion, the notes' reason for designing a guide to carry one mode in its band.

Part 2 — guided activity

The WR-340 on the bench

From a pair of dimensions to a verified prediction: examine, calculate, predict, check in the Waveguide Field Explorer, then interpret. Use \(c=3\times10^{8}\ \text{m/s}\) for hand calculations; the explorer uses the exact value, so third-figure differences are expected.

09 — The WR-340 activity

Using the explorer for WR-340

The Waveguide Field Explorer draws a WR-90 guide (\(a=22.86\ \text{mm}\), \(3\)–\(13\ \text{GHz}\)). WR-340 is exactly \(86.36/22.86=3.778\) times larger, and the field depends only on \(f/f_c\):

\[\text{WR-340 at } f\ \equiv\ \text{explorer at } 3.778\,f\]

Lengths the explorer reports (\(\lambda_g\), decay length) are \(3.778\times\) too small for WR-340; \(\beta\) and \(\alpha\) are \(3.778\times\) too large.

WR-340	1.50 GHz	2.50 GHz	3.20 GHz	3.60 GHz
Explorer	5.67 GHz	9.44 GHz	12.09 GHz	beyond the slider: use the engineering view

The engineering view has a WR-340 preset and a list of the modes that can propagate.

Open the Waveguide Field Explorer Engineering view

09 — The WR-340 activity

Steps 1–3: examine and calculate

Examine WR-340. Measure \(a\) and \(b\); confirm \(a/b=2\). What does that say about the next mode up?
Cutoffs. \(f_{c,10}=\dfrac{c}{2a}=1.74\ \text{GHz}\); \(f_{c,20}=\dfrac{c}{a}=3.47\ \text{GHz}\); \(f_{c,01}=\dfrac{c}{2b}=3.47\ \text{GHz}\) — degenerate, because \(b=a/2\).
Window. Theoretical \(1.74\) to \(3.47\ \text{GHz}\); the notes' conservative range \(2.08\) to \(2.78\ \text{GHz}\); published band \(2.20\) to \(3.30\ \text{GHz}\).

Try each step yourself before reading the answer in the lecture notes.

09 — The WR-340 activity

Step 4: predict before you look

For each frequency: travel, decay or stand still? Which modes? \(f/f_c\)? \(\lambda_g\), or \(\alpha\) below cutoff? Inside the conservative window?

Frequency	Region	Modes	\(f/f_{c,10}\)	\(\lambda_g\) or \(\alpha\)
1.50 GHz
2.50 GHz
3.20 GHz
3.60 GHz

09 — The WR-340 activity

Step 4: the predictions

Frequency	Region	Modes	\(f/f_{c,10}\)	\(\lambda_g\) or \(\alpha\)
1.50 GHz	below cutoff: evanescent	none	0.864	\(\alpha=18.3\ \text{Np/m}\)
2.50 GHz	dominant region, inside the conservative window	\(TE_{10}\)	1.44	\(\lambda_g=167\ \text{mm}\)
3.20 GHz	dominant region; above \(2.78\ \text{GHz}\), inside the published band	\(TE_{10}\)	1.84	\(\lambda_g=112\ \text{mm}\)
3.60 GHz	above \(f_{c,20}\): higher-order propagation possible	\(TE_{10}\), \(TE_{20}\), \(TE_{01}\)	2.07	\(\lambda_g=95.1\ \text{mm}\)

At 1.50 GHz: \(\alpha=\frac{\pi}{a}\sqrt{1-(1.50/1.74)^{2}}=18.3\ \text{Np/m}\); the field falls by \(e\) every \(54.5\ \text{mm}\).

09 — The WR-340 activity

Steps 5–7: verify and investigate

Verify. Set the explorer to \(5.67\), \(9.44\), \(12.09\ \text{GHz}\); read the status and \(f/f_c\) in the Advanced panel. For \(3.60\ \text{GHz}\) use the engineering view's WR-340 preset and read the mode list.
Below-cutoff decay. At \(5.67\ \text{GHz}\) switch to Cross-section and move the slice in. After one broad dimension the field has fallen to \(e^{-\alpha a}=e^{-1.58}=0.205\): about a fifth. The arrows pulse in place and do not move.
Near cutoff. Predict \(\lambda_g\) at \(1.85\ \text{GHz}\): \(\lambda_g=162/\sqrt{1-(1.74/1.85)^{2}}=471\ \text{mm}\), about \(5.5a\). In the explorer (\(6.99\ \text{GHz}\)) read \(\lambda_g\approx124\ \text{mm}\) and scale by \(3.778\). Then approach cutoff and use the cutoff marker.

09 — The WR-340 activity

Steps 8–9: interpret

Why \(3.60\ \text{GHz}\) is different. Above \(3.47\ \text{GHz}\) the \(TE_{20}\) and \(TE_{01}\) patterns have a real \(\beta\). Different \(\beta\) means different speeds, so a shared signal is smeared. Whether they are excited depends on the source and on discontinuities. The visual explorer draws \(TE_{10}\) only; the engineering view's mode list shows the rest.
Engineering interpretation. Operating point: \(2.50\ \text{GHz}\), inside \(2.08\)–\(2.78\ \text{GHz}\). At \(1.50\ \text{GHz}\): nothing propagates, the signal is reflected, not heated — use coax. At \(3.60\ \text{GHz}\): potentially multimode.

10 — Check yourself

Check yourself

1 · Where does the power go?

A lossless waveguide is driven below its cutoff frequency. The field falls off exponentially along the guide. Where has the power gone?

It has been absorbed by the guide walls and turned into heat It has been reflected back towards the source; nothing is absorbed It is carried to the far end, but at a reduced amplitude

10 — Check yourself

Check yourself

2 · Approaching cutoff

The frequency in a guide is lowered towards the cutoff frequency from above. What happens to the guided wavelength \(\lambda_g\)?

It shrinks towards the free-space wavelength It grows without limit, and is infinite exactly at cutoff It stays equal to the free-space wavelength

10 — Check yourself

Check yourself

3 · A phase velocity above c

In a waveguide above cutoff the phase velocity \(v_p=\omega/\beta\) is greater than \(c\). Does this violate relativity?

Yes, so the formula must be wrong No, because energy travels at the group velocity \(v_g=c^{2}/v_p No, because the speed of light inside a guide is higher than \(c\)

11 — Week summary

Carry these forward

Three regimes

\(\gamma=\alpha+j\beta\): above cutoff \(\gamma=j\beta\); at cutoff \(\gamma=0\); below cutoff \(\gamma=\alpha\).

Above cutoff

The pattern travels. \(\lambda_g=\lambda_0/\sqrt{1-(f_c/f)^{2}}\); \(v_p>c\) is a crest, \(v_g=c^{2}/v_p

At cutoff

\(\beta=0\), \(\lambda_g\to\infty\), \(v_g=0\): the same pattern in every cross-section, pulsing in time.

Below cutoff

Amplitude \(\propto e^{-\alpha z}\), \(\alpha=\frac{\pi}{a}\sqrt{1-(f/f_c)^{2}}\); the phase does not advance and no net power flows.

Decay is not loss

The power is reflected, not absorbed. Dissipation is a separate, small effect measured in dB/m.

Multimode

Above the next cutoff more than one pattern can travel; different speeds smear the signal.

EE 499 · Week 7

Week 7 complete

above cutoff
at cutoff
below cutoff
guided wavelength
multimode
WR-340

Week 5 asked why microwaves and how a waveguide can carry the signal; Week 6 asked which structure and which size; Week 7 asked what the field is actually doing inside it.

We now understand how fields propagate through a transmission structure. What happens when RF power encounters real interfaces, loads and components? That is Week 8.

Read the Week 7 article