We Have a Spacecraft in Orbit. Now How Do We Talk to It?

01 — The link problem

One decision forces the next

Weeks 1 to 4 put a satellite in a known place at a known time. None of that work carried any information. This week we start on the part that does: the signal itself, and everything between the equipment rack and the spacecraft.

Every satellite link is the same chain, and each stage is a piece of engineering we have to account for:

1. transmitter
2. transmission structure
3. antenna
4. space channel
5. satellite

This lecture is a single chain of engineering decisions. Nothing in it is chosen for mathematical convenience — each step is forced by the one before it. We pick a frequency; the ionosphere sets a floor on it; the need for a narrow beam pushes it much higher; that lands us in the microwave region; microwave hardware then makes the connection between source and antenna a design problem of its own; and that problem is answered with waveguide, whose behaviour is governed by modes and cutoff frequencies.

02 — Choosing a frequency

The first decision: what frequency should we use?

Before power, before modulation, before antennas: a carrier frequency. It is the one choice that constrains everything else, and the first thing that constrains it is what the signal has to travel through — the troposphere with its weather and water vapour, the stratosphere and mesosphere, and then the layer that actually decides the answer.

03 — The ionosphere

Sunlight makes a conductor out of thin air

The ionosphere extends roughly from \(80\ \text{km}\) to \(300\ \text{km}\) above sea level, and its detailed characteristics change from day to night and from season to season. Up there the gas is extremely thin and completely exposed to the Sun, so ultraviolet and X-ray photons carry enough energy to knock electrons off the atoms and molecules they strike:

1. solar radiation
2. electrons stripped from atoms
3. free electrons & ions
4. a plasma
5. radio propagation changes

The result is a region containing free charge. Everywhere else on the path the wave simply travels; here it meets charges that can move in response to its own electric field, and a medium that responds is a medium that changes how the wave behaves.

From the original lecture notes · Topic 3, p. 3 →

Reflected, or passed through?

One threshold decides it, and it is set by how many free electrons there are. The notes give it as:

Below \(f_{\text{critical}}\) the electrons can follow the wave's field, the layer acts as a reflector and the wave is turned back towards Earth. Above it the field reverses faster than the electrons can respond, and the wave passes through.

Why the notes say well above 30 MHz

The notes are carefully worded: a wave sent from Earth to outer space must have a frequency well above \(30\ \text{MHz}\) — not simply above it. There are two reasons for the margin. First, the threshold moves: \(f_{\text{critical}}\) follows \(N_{\max}\), which rises and falls with day and night, with season, and with the solar cycle. Second, passing through is not ignoring: above the threshold the wave is not reflected, but the ionosphere is still a region of free charge and still acts on the wave as it crosses, and those residual effects weaken as frequency rises.

Historically, this layer is why Marconi's signal crossed the Atlantic at all. His experiments began in 1894, and six years later he received in St John's, Newfoundland a signal sent from England. Marconi believed the wave had followed the curvature of the Earth, as Sommerfeld also explained in 1909; others did not accept that a wave could follow the curvature over such a distance. In 1927 Appleton proved the existence of the Kennelly–Heaviside layer — later called the ionosphere — and it became widely accepted that the early success of Marconi's experiment was due to reflection of the electromagnetic waves on it.

04 — Directivity

Is getting through the ionosphere enough?

No. A frequency above \(30\ \text{MHz}\) will leave the atmosphere — and then spread its power over the whole sky. A geostationary satellite is a small target \(36{,}000\ \text{km}\) away, so an antenna that radiates equally in all directions sends almost all of its power somewhere the spacecraft is not, and interferes with everyone else while doing it.

What we actually need is directivity: the ability to concentrate radiated energy into a narrow cone aimed at the spacecraft. That requirement, not the ionosphere, is what pushes the frequency much higher than \(30\ \text{MHz}\).

From the original lecture notes · Topic 3, p. 6 →

Why shorter waves buy directivity

For an aperture antenna — a dish, a horn, an array — directivity scales with the aperture area measured in square wavelengths:

That is the whole argument. A \(1\ \text{m}\) dish at \(3\ \text{GHz}\), where \(\lambda=100\ \text{mm}\), is ten wavelengths across. The same dish at \(12\ \text{GHz}\), where \(\lambda=25\ \text{mm}\), is forty wavelengths across: roughly a quarter of the beamwidth and about sixteen times the directivity, with no change to the metal. The dish on the roof is a fixed number of metres across and we cannot keep making it bigger, so the only remaining way to make it electrically larger is to raise the frequency.

Antennas are a topic of their own later in the course; here we need only the scaling, not the full aperture theory.

05 — Into the microwave region

The physics pushes us toward microwaves

Three independent requirements, all pointing the same way. We must get through the ionosphere, which sets a floor of roughly \(30\ \text{MHz}\) and a wish to be well clear of it. We must get directivity, which needs an aperture many wavelengths across, so \(\lambda\) must be small. And we must keep the antenna buildable, since the dish has to fit on a roof, a ship or a spacecraft, so we cannot buy electrical size with metres.

1. all three constraints
2. short wavelength
3. GHz — the microwave region

A fourth, historical reason from the notes: by the end of the 1930s radio and television already occupied the MF, HF, VHF and UHF bands, covering from hundreds of kilohertz up to several hundred megahertz, so satellite systems developed in the 1950s had to be placed at bands higher than those.

06 — Satellite bands

Now the band chart means something

The usable window opens where the ionosphere stops reflecting and closes where the atmosphere and the available hardware make the link impractical. The notes record where the balance settled: limited by the available technologies, satellite systems stayed for decades between \(1\) and \(12\ \text{GHz}\) — the L, S, C and X bands. Later hardware opened Ku and Ka above them without making the lower bands obsolete.

Ranges follow the IEEE column of Table 3 in the Rohde & Schwarz Satellite Technologies guide held in the course reference library, which also places K band between Ku and Ka, from \(18\ \text{GHz}\) up to the start of Ka. Band boundaries differ between the IEEE, ITU and NATO conventions, so always state which convention you are quoting.

So why not simply go as high as possible?

If every argument pointed upward there would be one band and no discussion. Several bands exist because the arguments disagree:

Frequency selection is a system decision: propagation, antennas, bandwidth, hardware, interference and regulation together. No single band is "the" satellite frequency.

07 — Transmission lines

We have entered microwave engineering

Choosing gigahertz was a propagation and antenna decision. It is also a hardware decision, and from here the problem changes character. On transmit the chain is microwave source → transmission line → antenna → space channel, and on receive it runs the other way. Everything in that chain we have either covered or will meet later, except one block: the connection between the source and the antenna.

At microwave frequencies, a wire stops being a wire

In a low-frequency circuit a connecting wire is an ideal node: the same voltage appears at both ends at the same instant. That picture quietly assumes the wire is much shorter than a wavelength. At \(12\ \text{GHz}\) the free-space wavelength is \(25\ \text{mm}\), so a \(50\ \text{mm}\) connection is two whole wavelengths long. The consequences are immediate: voltage and current vary along the line, so they are functions of position as well as time; the structure guides a wave, so its geometry is part of the circuit; and loss and radiation matter, because conductor loss rises with frequency and an open structure radiates the power we were trying to deliver.

So the connection becomes a designed component in its own right: a transmission line.

Three ways to guide the energy

A two-wire line has two conductors and an open field, so it radiates. A coaxial line has two conductors with the field confined by the outer shield. A hollow waveguide has a single conductor and nothing inside but air. These ranges overlap heavily — coaxial line is used well into the microwave region and waveguide is used below it — so what changes with frequency is which one is the better engineering choice, not where a hard boundary lies. What drives that choice:

This is why the notes conclude that waveguides are very important for satellite communications: in the high-power, low-loss parts of a microwave chain, the trade comes out in their favour.

From the original lecture notes · Topic 3, p. 7 →

08 — Inside a waveguide

Can a wave travel through a hollow metal pipe?

There is no second conductor, no signal wire, no return path. Just air, surrounded by metal. It is worth pausing on how unreasonable this sounds before finding out that it works.

The answer is yes — but not just any field. The walls are conductors, so a tangential electric field cannot exist on them, and only particular field distributions can satisfy that condition. Those allowed distributions are the guide's modes.

Geometry and notation

A rectangular waveguide is described by its broad dimension \(a\), its narrow dimension \(b\) — by convention \(a>b\) — and the propagation direction \(z\) along its length. Modes are classified by what the field does along that direction:

The two subscripts \(m\) and \(n\) count how the field varies across the cross-section — \(m\) along the broad dimension \(a\), \(n\) along the narrow dimension \(b\). \(TE_{10}\) therefore means one half-cycle of variation across \(a\) and none across \(b\): an electric field that is vertical everywhere, strongest in the middle and zero at the two side walls.

From the original lecture notes · Topic 3, p. 16 →

09 — Maxwell's equations

Why can a wave propagate at all — in space, and inside a pipe?

Both answers come from the same four equations. We have needed them twice now: once for the wave that crosses the ionosphere, and once for the pattern that fits inside the guide.

Equations (3) and (4) say where fields come from, and neither contains time, so neither can by itself describe anything that travels. Equations (1) and (2) couple the two fields together, and they are the pair that makes propagation possible: a changing \(\mathbf E\) creates an \(\mathbf H\) through (2), and that changing \(\mathbf H\) creates an \(\mathbf E\) through (1). The term \(\partial\mathbf D/\partial t\) in equation (2) is Maxwell's own contribution, the displacement current, and it is the reason a satellite link is possible at all: a changing electric field makes a magnetic field with no conductor and no charge present.

From the original lecture notes · Topic 3, p. 8 →

The result, and the speed it predicts

This is the wave equation — the same form a vibrating string and a sound wave obey, \(\nabla^{2}f=\dfrac{1}{v^{2}}\dfrac{\partial^{2}f}{\partial t^{2}}\). Comparing the two fixes the propagation velocity, and nothing else in the equation can:

From the original lecture notes · Topic 3, p. 9 →

The free-space solution, and its parameters

Working one frequency at a time, every quantity varies as \(e^{j\omega t}\) and the wave equation becomes the form the notes write as equation (5), \(\nabla^{2}\mathbf E+k^{2}\mathbf E=0\) with \(k^{2}=\omega^{2}\mu_0\varepsilon_0\). For a wave travelling along \(z\) with its electric field along \(x\), one solution is equation (6):

\(\omega=2\pi f\) is radians of phase per second at a fixed position — the cycle counted in time, fixed by the transmitter. \(\beta=2\pi/\lambda\) is radians of phase per metre at a fixed instant — the cycle counted in space, which the notes call the phase constant. A crest keeps \(\omega t-\beta z\) constant, so it advances at \(v=\omega/\beta=f\lambda\). In free space that is \(c\), and \(\lambda=c/f\) — the relation the whole antenna argument rested on.

10 — Modes and cutoff

Maxwell plus metal walls gives modes and cutoff

The same equations, solved the same way, but now with a boundary condition: the electric field tangential to a conducting wall must vanish. Everything practical about waveguide design comes out of that one extra requirement.

Inside the hollow guide the region is still source free, so the field still satisfies \(\nabla^{2}\mathbf E+k^{2}\mathbf E=0\). What is new is that the solution must also vanish on the conducting walls. The notes give the resulting field as equation (8):

The \(\sin(\pi x/a)\) factor is zero at \(x=0\) and at \(x=a\), which is exactly what the side walls demand. Substituting back gives equations (9) and (10):

That square root is the whole story. In free space \(\beta=\omega/c\) is a straight line through the origin, so every frequency propagates. Inside the guide \(\beta\) is real only above a threshold; below it the square root is imaginary and there is no travelling wave at all — the field decays away from the source instead. That threshold is the cutoff frequency. Far above cutoff the guide behaves almost like free space; near cutoff it does not behave like free space at all.

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

Every mode has its own cutoff frequency

Every allowed field pattern has its own cutoff. For an air-filled rectangular guide the notes give equation (14):

Which mode appears first?

The mode with the lowest cutoff frequency is called the dominant mode — it is the first pattern an electromagnetic wave can take inside the guide. Putting \(m=1,\ n=0\) into the general formula, with \(a>b\):

Its cutoff wavelength is \(2a\) — the guide must be at least half a wavelength wide before anything can propagate at all.

From the original lecture notes · Topic 3, p. 12 →

11 — A 12 GHz design

We need to carry a 12 GHz satellite signal

A ground terminal operates at \(f=12\ \text{GHz}\), a frequency representative of Ku-band satellite downlinks. We have to get that signal from the equipment to the feed of the dish. What waveguide can we use? Start with the free-space wavelength:

Question 3 — so choose a smaller guide

Shrinking \(a\) raises every cutoff frequency together, sliding the whole single-mode window upward until it contains \(12\ \text{GHz}\):

The second guide, with \(a=19.05\ \text{mm}\) and \(b=9.525\ \text{mm}\), carries \(12\ \text{GHz}\) in a single mode with margin at both edges. That is the answer to the design question.

From the original lecture notes · Topic 3, p. 13 →

The wave is longer inside the guide

Rewriting the notes' \(\beta=\sqrt{\omega^{2}\mu_0\varepsilon_0-(\pi/a)^{2}}\) in terms of the cutoff frequency gives the two quantities a microwave engineer works with:

The square root is always less than one, so \(\lambda_g>\lambda_0\): the wavelength along the guide is always longer than in free space, and it stretches without limit as \(f\) approaches cutoff. For our chosen guide at \(12\ \text{GHz}\), \(\lambda_g=33.1\ \text{mm}\) against \(\lambda_0=25\ \text{mm}\), and \(\beta=2\pi/\lambda_g=190\ \text{rad/m}\). \(\beta\) is the phase change per metre along the guide — the quantity that sets the electrical length of every section of the feed, and the reason a waveguide run cannot be measured in free-space wavelengths.

12 — Check yourself

Reason across the week

13 — Week summary

From a spacecraft in orbit to a waveguide dimension

Every step in this lecture was forced by the one before it. Read the chain backwards and the waveguide on the bench is explained by an orbit.

1. pick a frequency
2. cross the ionosphere
3. get directivity
4. so: microwaves
5. feed the antenna
6. so: waveguide
7. modes & cutoff

Next: Week 6 turns the waveguide dimension into hardware — circular guides, standard sizes including WR-340, and coaxial cable. Week 7 then looks inside the guide at the field itself, and Week 8 asks what happens at the joints, loads and components along the path.

Study this week as slides →