sin(t + 180°) = sin(t) * -1

So a 180 degree phase difference between two waves can still be represented as a scalar transformation of a single phase vector. However since we are only concerned with the angle of the vector, negative scalars are the only ones to produce phase angle differences, and conversely, 180 degrees is the only phase angle difference a scalar can produce.

The RMS of a sin wave is independent of it’s phase. So both 120V legs of the circuit have the same voltage relative to the the neutral. The instantaneous DC voltage at any point in time is the inverse of the voltage of the other leg (sin(t) * -1). It would be misleading to say one leg has 120V and the other the opposite voltage of -120V relative to neutral since that is nonsense for AC voltage, so instead it is referred to by the phase angle between the two voltages.

Just as 3 points on the same line cannot make a polygon, the 3 wires from a split phase system, whose voltage differentials are scalar transformations of each other cannot make a polyphase system.

A 204vac(rms) CT xfmr is one (single) coil, they add a tap in the center. The single coil has only one dot on it.

It's like taking a tension spring, adding a dot to one end to designate DC coil direction, then stretching the coil in the middle (elastic deformation) enough to clip on a tap wire.

A tap is a voltage divider, nothig more than that. A CT just so happens to be in the center, which yields two 120v and one 240v source.

Add more taps and you get more voltages, the issue for AC is, a tap that is not in the middle will have voltage less than 120v and then on flip cycle more than 120. The Diff at any tap location across 1 full sine cycle ALWAYS = the xrfmr ends, and in this case, 240vac.

(+120v) - (-120v) = 240
(+60) - (-180v) = 240v
(+30v) - (-210v) = 240v

A tap is nothing special, but a CT is kinda the only tap that is user "friendly" when it comes to power delivery to a home.

Europe uses mostly 240c 50Hz, no taps.

120v is "safer" than 240v.

120v has big disadvantage in terms of efficiency, because at same kVA the 120 is wasting a lot more power than 240v is. Eletric batt EV's are way more efficient than gas vehicles, and if we wanted to save on wasted power we should do away with these 240v CT xfrmers and just use 240vac. Toasters, microwaves, etc, would all waste less power if they were using 240v instead of 120v.

Pros & Cons.

The +120 -120 is only seen on the scope because 'you' the scope user is not flipping the probes over at 60Hz to follow the same "side" of Sine wave relative to your probe leads.

This is no different then putting a V meter on a 9v batt and then just flipping the batt (or leads) over, the meter reads +V and then -V.

When you push neg charged e's away from one side, one side becomes + and the other side -

Voltage is nothing more than a potential energy. The neg charge (that can move) wants to run home to the pos side (which cannot move).

And yes, true, if you take an isolated CT coil, L1///////CT/////L2 and take your V meter leads and connect -meter to L1 and +meter to CT, you get the same voltage reading (at that time) if you connect a 2nd meter -meter to CT and +meter L2. In this configuration, both meters read the exact same voltage magnitude and sign, and will flip sign at 60Hz.

The grounded CT, causes more confusion than not.

Yes, except AC voltages can't be negative as the AC voltages are the RMS amplitude of the time dependent voltage and thus always positive, instead they are said to be either in phase or 180 degrees out of phase with respect to another AC voltage.

Where you say "(+30v) - (-210v) = 240v" I would say (30v ?0°) - (210v ?180°) = 240v .

Why not 30v 30° - 210v 210°  ??
Angles don't really make that much sense in 1ph, the magnetic field cuts the full secondary coil at any given time (all at the same time).


all neg charge here (-------------------) and over there there is missing neg charge (+++++++++++++++++), technically voltage is only one potential, it's always gonna be "here" is neg and "there" is pos, like a DC batt, but then flip the charge, making "here" pos and "there" neg. If you keep your probe neg lead on the neg end at all times, the voltage will always read the same regardless of "60Hz".

RMS is, as you know, root- mean-square, which boils down to DC equiv, but this does not address sign of magnitude, because we know in AC the amps flow one way, then back the other way, at 60Hz. RMS is also just an arbitrary thing, it was to create a context for comparing AC to DC. I could do the same with avg voltage too.

Those differential probes are not a good fit for the kind of work I do due to the 150V rating and the excessive bandwidth.  I frequently have anywhere from 340V to 430V offset channel to channel.  When I use active probes, I have a few sets of Pico probes rated to 700V and 100MHz.  Pico makes the probes for most of the OEMs, including Keysight and Teledyne LeCroy.  My work usually allows me to use fairly slow scopes as the inverters are switching at 16kHz (high enough that most adults cannot hear it and low enough that the switching losses do not cause the IGBT junctions to get too hot).  Same reason for voltage doubling.  I do use one of the LeCroy Waverunner 6Z units with the 16-bit ADC as I see less error for some signals.

The phase angle of AC voltage is a relative measurement, just like Voltage itself is. If you stated the two as 30° and 210° it would stand to reason that these phase angles are relative to a third AC voltage that you use as the 0° reference.

Average voltage is useless for AC as it will be 0v. Average of the absolute value will always be positive. Peak voltage is by definition positive. Using a sign on an AC voltage is only useful for indicating that it is 180 degrees out of phase relative to another AC voltage.

I just drew a line, said that was 30°, then added 180° to that to get the 2nd line at 210°, their diff is 180°. It's completely arbitrary and I did not to define anything to be "zero" (0).

The CT is referenced as zero because someone said "lets make earth the zero reference point, look what we can do with this".