The Wave Reaches a Joint. How Much Gets Through?

01 — Where we are

The wave reaches a joint. How much gets through?

Week 7 ended with a question. We now understand how fields propagate through a transmission structure: above cutoff the pattern travels, at cutoff it stands still, and below cutoff it dies away. What happens when RF power encounters real interfaces, loads and components?

A real path is not one long uniform guide. It is a source, a cable, a connector, a transition, a length of guide, perhaps a filter or an amplifier, and a load at the end. At every one of those places the structure changes, and at every change some of the power can go somewhere other than forward.

1. where the power goes
2. characteristic impedance
3. reflection
4. S-parameters
5. the VNA
6. connectors
7. the RF path

Week 8 builds the language for that question and the instrument that answers it. It ends at a measurement: how effectively does RF power pass through a real component or interface, and how would we know?

02 — Where the power goes

Loss, reflection and transmission are three different things

The notes open their treatment of waveguide technology with an argument about loss. Above about \(1\ \text{GHz}\), two-conductor transmission lines become impractical because of their high conductor loss. A single-conductor hollow guide has relatively low loss by comparison, and hollow air-filled guides can handle much higher power than coaxial cables or printed lines. Therefore waveguides are very important for satellite communications.

That argument uses the word loss in one sense: power turned into heat in the metal and the dielectric. But power reaching a component can meet three different fates, and only one of them is a loss.

The vocabulary of loss

Several loss words are in daily use, and they are easy to confuse. Fix them now, because the rest of the week depends on them.

Decibels

Powers in microwave work span many orders of magnitude, and cascaded components multiply their effects, so engineers work in decibels, where multiplication becomes addition. A power ratio is \(10\log_{10}(P_2/P_1)\) decibels. A field or wave amplitude, whose square is a power, is \(20\log_{10}\) of its ratio.

Want practice with dB, dBm and 10 log versus 20 log? The auxiliary dB Gym is not part of any week.

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

03 — Characteristic impedance

What does a “50 Ω cable” actually mean?

Beyond the notes. The notes use two-conductor lines without defining a characteristic impedance; this section supplies it, because \(S_{11}\) is a reflection measured against it.

Put an ohmmeter across a \(50\ \Omega\) coaxial cable and it will not read \(50\ \Omega\). Along the inner conductor it reads a fraction of an ohm; between inner and shield it reads an open circuit. So the number on the jacket is not a resistance you can measure with a meter.

It is a property of the wave the cable carries. Send a signal into a long cable and, before any reflection has had time to return, the line draws a definite current for the voltage applied. The ratio of those two, for the wave travelling forwards, is fixed by the geometry:

The practical meaning follows directly: a source expecting \(50\ \Omega\) sees a matched load if the line it drives has \(Z_0=50\ \Omega\) and is terminated in \(50\ \Omega\). Everything about reflection, matching and \(S_{11}\) below is measured against that reference.

From geometry to a number

For a coaxial line with a homogeneous dielectric of relative permittivity \(\varepsilon_r\), inner radius \(a\) and outer inner-radius \(b\):

Three things are worth reading straight off it. Only the ratio \(b/a\) matters, so a thick cable and a thin cable can share an impedance. The dependence is logarithmic, so large changes in geometry make modest changes in \(Z_0\). And a denser dielectric lowers \(Z_0\), because it raises the capacitance per unit length.

This expression is standard transmission-line theory rather than one of the notes' numbered equations; it is included because \(Z_0\) is the reference impedance every S-parameter below is defined against.

Why 50 Ω, of all the numbers?

For an air-filled coaxial line the two things an engineer wants do not peak at the same place:

That compromise was adopted in the 1930s and 1940s for high-power radio and radar hardware, and then it became self-reinforcing: once connectors, attenuators, loads, amplifiers and instruments were all built to the same reference, the cost of departing from it outweighed any remaining advantage. Standardisation, not physics, is what fixed the number.

The takeaway is not that \(50\ \Omega\) is optimal — it is not optimal in any single respect. It is that once a microwave system is designed around a reference impedance, its connectors, cables, instruments, loads and S-parameter measurements are all referenced consistently to that same value. That shared reference is what makes a number like \(S_{11}=-20\ \text{dB}\) mean the same thing on two different benches.

04 — Matching and reflection

What happens if the load is not 50 Ω?

Beyond the notes. The reflection coefficient is the one-port version of \(S_{11}\), which the notes define next.

A transmission line carries a wave towards a load. If the load does not absorb everything the line offers, the remainder has nowhere to go but back. That reflected wave is the single most common fault in a microwave path — and the easiest one to measure.

The reflected wave is proportional to the incident one, and the constant of proportionality at the load is the reflection coefficient:

\(|\Gamma|=1\) is the ideal limiting case of a lossless, purely reactive termination. A real short or open has some loss, so a measured \(|\Gamma|\) is always slightly under one. Keep \(\Gamma\) in view: it reappears below, under the name \(S_{11}\).

Two other ways to say the same thing

\(|\Gamma|\) is rarely quoted directly. Datasheets use one of two derived numbers, both carrying exactly the same information:

Return loss is a positive number that gets bigger as the match gets better. VSWR is a positive number that gets closer to \(1\). Both are just \(|\Gamma|\) rescaled, so quoting one is enough — but you have to be able to move between them. The same reflection measured at the input of a terminated two-port is called \(S_{11}\), so return loss is equally often written \(\mathrm{RL}=-20\log_{10}|S_{11}|\).

05 — S-parameters

At 12 GHz, how do we characterise a component at all?

From the notes — section 1.3.4, equations (16) and (17).

We now have a way to describe one reflection at one load. A real component has an input and an output, and we need a description that covers both, in both directions, at every frequency of interest.

Why do we prefer travelling waves at microwave frequencies?

It is worth being precise here, because the usual shorthand overstates the case. Voltage and current do not stop being meaningful at \(12\ \text{GHz}\). On a two-conductor transmission line such as coax they remain perfectly well defined quantities. What changes is how convenient they are to measure, and how naturally they describe a structure that has no second conductor.

Incident and reflected travelling waves, by contrast, exist in both structures, and a measuring instrument can separate them directly. That is why they became the practical characterisation language at these frequencies — not because the older quantities are wrong, but because the wave description is the one the hardware and the instrument both agree with. The notes put the waveguide half of it in one sentence: it is clear that in waveguides we cannot define potential, as the structure has only one conductor. Instead, we define waves.

Four waves, before any matrix

Draw the component as a two-port network. At port 1 there is a wave going in, \(a_1\), and a wave coming out, \(b_1\); at port 2 there is \(a_2\) going in and \(b_2\) coming out. The notes define \(a_i\) and \(b_i\) as the square roots of the incident and reflected wave powers, so \(|a|^{2}\) and \(|b|^{2}\) are powers and every ratio built from them is dimensionless.

A scattering parameter is then the ratio of a wave leaving the network to a wave sent into it, with no wave arriving at any other port:

For a two-port that unpacks into exactly four measurements, two with port 2 terminated and two with port 1 terminated:

Collecting the four combinations gives the matrix form the notes write as equation (17):

Read a row at a time: what leaves port 1 is whatever bounces off port 1, plus whatever travels back through the network from port 2.

The four entries, in plain language

\(\Gamma\) and \(S_{11}\) are the same quantity

These are not two unrelated ideas that happen to describe reflection. For a one-port, or at the input of a two-port whose output is properly terminated, they are identical under the chosen reference impedance:

The change of name reflects a change of context rather than of physics: \(\Gamma\) is what we write when thinking about a load on a line, \(S_{11}\) is what the instrument reports when the same reflection is measured at the input of a component. The reference impedance must be stated either way, because both numbers are defined against it.

Why they are almost always quoted in decibels

The notes record that S-parameters are usually measured in dB. The magnitudes span many orders, and cascaded components then add instead of multiplying:

Note that \(S_{11}=-20\ \text{dB}\) and “return loss \(20\ \text{dB}\)” are the same statement with opposite sign conventions: \(\mathrm{RL}=-S_{11,\mathrm{dB}}\) for a passive component. The notes' remark that a value of \(-10\ \text{dB}\) is considered low, however this depends on the application refers to the reflection: \(-10\ \text{dB}\) means only a tenth of the power comes back.

From the original lecture notes · Topic 3, pp. 17–18, equations (16) and (17) →

06 — Components and the VNA

What the numbers look like for real hardware

From the notes — equations (18) and (19) and notes (i)–(iii), with worked component examples added.

The ideal section, and the rules every S-matrix obeys

The notes work the ideal case explicitly: a lossless length \(\ell\) of waveguide reflects nothing and transmits everything, delayed by its electrical length.

The second rule is the most useful diagnostic in this lecture: add up the reflected and transmitted power fractions of a real component, and the shortfall is the power that was dissipated or radiated.

The instrument that measures all four

A vector network analyzer sweeps a known signal across the band and separates what returns from what passes through, recording both magnitude and phase. Drive port 1 with port 2 terminated and it reports \(S_{11}\) and \(S_{21}\); drive port 2 with port 1 terminated and it reports \(S_{22}\) and \(S_{12}\). The notes add the practical footnote that VNAs are usually much more expensive than oscilloscopes and spectrum analyzers.

The reference plane — the point in the setup beyond which the numbers describe the device rather than the cables — is established by calibration. SOLT, TRL and their relatives are the standard procedures, and they are a topic of their own; for now it is enough to know that an uncalibrated VNA measures your cables as enthusiastically as it measures your component.

From the original lecture notes · Topic 3, pp. 18–19, equations (18) and (19) and notes (i)–(iii) →

07 — RF connectors

How do we physically connect microwave hardware?

Every component in this lecture has to be joined to the next one. At low frequency that is a mechanical question. At \(12\ \text{GHz}\) it is an electromagnetic one, because the joint is a length of transmission line in its own right.

Upper frequencies are indicative. Ratings differ between manufacturers and between grades of the same family, and a connector's usable range also depends on the cable behind it and on how it was fitted — so read the specific datasheet rather than memorising this column.

At microwave frequencies, a connector is part of the transmission line

At \(1\ \text{MHz}\) a connector is a way of joining two cables. At \(12\ \text{GHz}\) it is \(10\ \text{mm}\) of transmission line with its own geometry, its own dielectric and its own discontinuities — and the wave does not distinguish it from the cable it joins.

08 — An RF path

One complete RF path, end to end

Every idea in Weeks 5 to 8 appears somewhere in a real transmit chain:

1. source
2. SMA
3. 50 Ω coax
4. transition
5. WR-75
6. circular feed
7. antenna

Each stage exercises a concept from these weeks, and the middle column names it:

The mode changes twice along this path — TEM in the coax, \(TE_{10}\) in the rectangular run, \(TE_{11}\) in the circular feed — and each change happens inside a component that has to be designed and measured. Every interface is a possible mismatch and so a possible \(\Gamma\); the whole chain answers to one measured \(S_{11}\) at the input and one \(S_{21}\) end to end. That is the conceptual payoff of Week 8: how effectively RF power passes through real components and interfaces is answered, component by component, by two measured numbers.

09 — Week summary

How effectively power passes through real components

Week 7 asked what the field does inside a guide. Week 8 asked what happens when RF power meets real interfaces, loads and components, and how we describe and measure the result.

Next: We can now characterize individual RF components and interfaces. Next we combine antenna gains, propagation losses and system losses to determine whether the complete satellite link closes.

Study this week as slides → On to Week 9 →

10 — Check yourself

Three quick checks

11 — Week 8 exercises

Practice problems

The lecture ends above. What follows is a separate set of four problems to work through afterwards. Each one hides its worked solution behind a disclosure — attempt the problem first, then open it. Use \(c=3\times10^{8}\ \text{m/s}\) throughout, as the lecture notes do.

These four problems assess impedance, reflection and S-parameters. The waveguide design problems are in Week 6.