How should I think about "extra" impedances inserted within/across a transmission line?

Question

I'm worried this might be two separate questions — or perhaps not any coherent question at all? — but some recent thinking about transmission lines made me realize I still don't know how to think about transmission lines.

The "un-intuitions" I have are related to inline components like a feed-through terminator with a resistor from (in coaxial terms) the center conductor to the shield:

^{simulate this circuit – Schematic created using CircuitLab}

…and/or a DC-block coupler, which is basically a capacitor interrupting the center conductor:

^{simulate this circuit}

Now I get in broad terms that e.g. the feed-thru's resistor essentially terminates the coax with a "perfect load" at that point, or that the DC-block's capacitor looks like a good conductor to high frequencies and a bad conductor to low frequencies.

But when I ask myself things like:

• what would happen if, instead of placing a feed-thru terminator directly on my oscilloscope, I had an additional (long) length of coax connected to the terminator's output?
• why is it that a DC-block coupler can have a ± flat insertion loss (increasing slightly with frequency!) when the impedance of a capacitor is this dramatically inversely-proportional curve?

[^^^^ n.b. the above example questions are NOT my core question here!]

…I don't know how to think about them in terms of transmission lines!

They both seem related to the ≈resistance at a particular point in the line itself. Like normally there's approximately 0Ω resistance from one stretch of the center or outer conductor to the next — but the DC block adds some resistance at one particular spot. Or normally there's approximately ∞Ω resistance between the inner and the outer conductors — but the thru-terminator removes most of that resistance at one particular spot!

Is there any particularly helpful way to analyze these sort of random "odd spot in the coax" things?

Answer 1

To solve the first circuit:

1. Divide the transmission line into two lines, one on the left, one on the right.
2. Zoom in to the centre portion. When you are in a small ($\ll\lambda$) region, the normal rules of circuits apply. You have an impedance on the right, a parallel resistor, and terminals for connection on the left.
3. Work right to left. Assuming the Right transmission line is correctly terminated, it presents an impedance of $Z_0$ otherwise work it out from $Z_0$ $Z_L$ $\beta$ and $x$.
4. Now place a resistor of $Z_0$ in parallel, the resulting impedance is $Z_0/2$
5. This is what the Left transmission line sees as its terminating impedance.
6. Zoom out and do the normal transmission line equation if you want to know the impedance at some point on that line.

The second question can be solved in the same way. You're correct, the impedance of a capacitor is dependent on frequency, but in practice if the capacitor is large, the impedance will be effectively zero for all frequencies you're interested in. For example, 10 nf at 7 MHz is $2.2 \Omega$ and at 145 MHz is $0.11 \Omega$, neither will have any significant effect on the line if $Z_0=50\Omega$

Here's a circuit for the first one.

^{simulate this circuit – Schematic created using CircuitLab}

Answer 2

So I drew up a more generic "odd spot in the coax" and couldn't help noticing…

^{simulate this circuit – Schematic created using CircuitLab}

…that it looks like: a filter! Especially if you take the "resistors" in my diagram as actually being "impedances" i.e. capacitors/inductors at a particular frequency.

That certainly could explain both of my example questions, e.g. ideally:

• the feed-through terminator acts as a DC-to-daylight bandstop filter — and I guess continuing an additional length of coax doesn't make be a bad person but would be akin to listening for a signal downstream of an ∞dB attenuator
• the DC-block acts as a knee-at-0Hz high-pass filter — in practice the asymptotically-approaching-zero portion used of the capacitor's curve might even balance out against the usual "coax/connectors get worse at higher frequencies" curve until the latter eventually catches up

So I still don't really "get" transmission lines, and this is hardly a complete answer, but in an interesting turn of events I found these diagrams on Wikipedia:

So, perhaps, every spot in a coax is something of a tiny filter too?!