Electronic – The math and the physics behind POTS phone lines

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Since high school you learn something about Fourier Analysis and you can easily grasp the idea that a given signal that is variable over a given timeframe, is composed of multiple sub-signals with different frequencies.

Lately I learned that POTS lines work, at the same time, with both AC and DC power, the DC signal is used to feed the device, the phone, and the AC signals are the "inputs" from/to the user or the ISP ( dialing, ringtone, dtmf tones ).

This is kinda of a black magic to me, at this point, I really can't grasp how the so called multiplexing of the signal is happening.

It's true that POTS signals typically travel over RJ plugs and separate copper cables, usually there are 4 cables, so things are much simpler in practice since the 2 AC and DC are probably always separated from each other.

In modern times a phone cable also carries a DSL signal, but I'm just going to assume that this is just an "addon" to the several AC signals that a POTS line already delivers.

Anyone can explain the math/physics principles behind the multiplexing of POTS lines or phones and ADSL lines in general ?

Best Answer

You claim to be familiar with the Fourier analyses.

Take a look at this signal:

DC+AC

Here we have a sinusoidal signal summed together with a DC signal. The AC signal has voltage of 1 V peak to peak, while the dc component is 2 V. For each point on the graph, what happened is that the value of the cosinusoidal signal was summed with the 2 V DC level.

If we do Fourier transform of the signal, we'd get a value at the frequency of 0, which represents the DC component and the value at the frequency of the AC signal itself.

FFT of AC+DC

A "pure" AC signal goes from Vpeak to -Vpeak and is centered around zero. If the signal's average value is not zero, then it has "DC bias", that is to say that it can be represented as a sum of a DC signal and an AC signal that is centered around zero.

Next, a bit about phone lines and its cabling: Traditional POTS uses only two lines, one of which is ground and the other is traditionally said to be -48 V DC. The negative voltage then summed with the actual signal which carries the data.

ADSL "basically" works same way as fax modems and voice calls: The -48 V DC is summed together with the AC signal which carries the data. At both telephone central office and at subscriber's location filters are used to separate the voice traffic from ADSL signalling. This is possible since ADSL and telephone work in different frequency bands, so they do not overlap. You can then "simply" filter out different bands and send them to different devices.

EDIT: The way right to think about this is that in reality there's no "DC" and "AC" voltage (in fact, terms DC and AC aren't really all that well defined and the conventions depend on region). Those terms are simply abstractions that are used to simplify circuit analyses.

Let's say that you have grounded line G and line with voltage A. Now imagine that it's possible to measure voltage of the A line extremely quickly and that you associate one voltage measurement with one point on time axis. You'd get something that looks like this: AC+DC stem

Each measurement point tells you "exact" voltage at the time of the measurement. You don't get two lines because there's only one actual voltage and its the voltage that's shown.