The green path is the current flow from the AC input, through D1, charging C1. That current keeps C1 charged, and from that point on, it's simplest to think of the rest of the circuit as being powered from C1. The blue arrows show the first part of the current flow from C1, and the return current from the relay coil and the LEDs flows back to the negative terminal of C1.
Sorry, I must have lost track of your questions in post #125.
I am a little further in my thoughts on AC - phase and neutral are 180 deg. out of synch with neutral carrying the opposite of phase (and vice versa). Then technically speaking if the above is true, shouldn't one get shocked if they completed the circuit between neutral and ground? It certainly is the case when you ground the phase wire!
I think there's something fundamental wrong in your understanding of voltage. You need to first understand that voltage is a potential DIFFERENCE. It is ALWAYS measured BETWEEN TWO POINTS.
You can say that any point in a circuit has "a voltage on it", but that voltage is always measured RELATIVE to something else. In a typical circuit, this reference point is the 0V ("zero volt") rail, or sometimes (and inappropriately, in my opinion), "earth", "ground" or GND.
I wish I could do animations. I could explain this so much better. But you'll have to use your imagination.
Imagine a see-saw sitting in a ditch, so each end can be above or below the ground level. We're only interested in one end. Now, if that end is moving up and down, so it's sometimes above ground, and sometimes below ground, that is analogous to the Phase or Hot or Live connection on the wall socket. The voltage at that point is alternately positive then negative relative to ground.
But if you sit on that see-saw and focus at a point on the ground, from your point of view it appears to be going up and down as well; sometimes it's above you, and sometimes it's below you. And when someone at ground level says the see-saw is above them, someone at the see-saw level will say the ground level is below them, and vice versa.
This is what I meant when I was trying to explain the 180° phase difference between the voltage at Phase measured relative to Neutral, and the voltage at Neutral measured relative to Phase. They are both AC, they both have the same amplitude; in fact they ARE the same voltage, but they are half a cycle (180°) out of phase from each other.
The point of that explanation is to try to show how important it is to understand WHAT you are measuring a voltage RELATIVE to, or WITH RESPECT TO, i.e. the 0V REFERENCE POINT against which the voltage is measured.
In the case of mains voltage, Neutral is connected to Earth, and Earth is the default reference point. If you touch the Neutral wire, you won't get a shock, but you will if you touch the Phase wire. The reason for this is that you are standing on a floor which is (roughly) at earth potential. There is a resistive and capacitive path from the floor to the soil under your house, and if you touch the Phase wire, current will flow through that path.
To be exact, the complete path for this current will be:
From the Phase side of the winding on the pole transformer that provides AC to your house; through the wiring to your house; through the main switch on your switchboard; through the fuse for that circuit; through the Phase wire in the wall wiring; through the switch on the outlet; through the Phase connection on the socket; through your skin; through your finger, hand, arm, trunk, and legs; through the soles of your shoes; through the floor and other wooden parts of your house; through the soil under and around your house; through the earth stake; through the earth stake wire; through the earth busbar in your switchboard; through the link that links the Neutral busbar to the Earth busbar; through the Neutral busbar; through the Neutral wire from your house to the pole transformer; and into the other side of the secondary winding on the pole transformer.
The mains voltage coming from the pole transformer is actually connected between Phase and Neutral. Earth is connected to Neutral and is essential for safety, of course, but current is not supposed to flow through it. The current consumed by a load always flows between Phase and Neutral; if any current flows through Earth this is normally a fault or a problem, and will trip an ELCB/RCD or whatever you call it.
So Phase is the dangerous terminal on the wall socket. Now, if you built a second wooden room inside your room, and insulated from it, and put a wall socket on that room, and connected all of that wooden construction to the Phase side of the mains, when you stood inside that room, you wouldn't be aware of anything unusual. Your floor, walls and ceiling, and your shoes and your body, would all be at the same potential. Just like birds that perch on live wires, you wouldn't feel anything unusual. But to you, the Neutral and Earth pins on the wall socket would be dangerous. If you touched either of them, you would get a shock. And if you touched the Phase connection, you wouldn't feel anything. So it all depends on your reference point.
As for the 28VAC example, I am having difficulty envisioning the pathways each phase should take. I outlined the pathway if the indicated input is the phase in blue and what I think would happen if the phase was on the other pin (the ground plane) in green. Where there is a solid colored line, either blue or green that is where I think power stops, either is blocked by a diode or because of polarity.
Within that circuit, everything is exactly like I just described except the voltage is a lot lower. So you don't need to worry about things getting dangerous if the AC is connected "the wrong way". In fact there may not even BE a wrong way, since most probably, neither side of the secondary of the wall adapter for the irrigation controller is connected to Earth. That whole lot of circuitry, including everything inside the controller and all external wiring to the pumps, could be fully floating relative to earth. Kind of like a see saw suspended below a hot air balloon.
The important fact is that there is 28V AC RMS ACROSS or BETWEEN the two AC input connections to that circuit. If you look at the voltage at one terminal relative to the other terminal, it is varying between about 40V positive and about 40V negative. Looking with the terminals swapped, the voltage is the same but 180° out of phase from what it was before. That's irrelevant because we're just converting it to DC and using the DC.
So just look at the subcircuit consisting of (a) the 28V AC RMS supply, coming in on two connection points which I've called L (left) and R (right), (b) diode D1, and (c) electrolytic C1. I've drawn that circuit in two forms, which are electrically identical except for where I've put the 0V triangle marker.
Here's version A, which is as drawn on the schematic. In this version, L is the 0V reference for the circuit, and the groundplane on your layout.
The voltage between R and L (measured EITHER way round) alternates smoothly from +40V to -40V. At a certain point in time, R is becoming more positive than L. Let's assume that C1 has discharged to about 38V. Once R becomes about 38.6V more positive than L, D1 will start to conduct. R will continue to get more positive than L, and current will flow through D1, increasing the voltage across C1.
Eventually that half-cycle will reach its peak instantaneous voltage of 40V (positive at R, relative to L). C1 will have been charged to about 39.4V. Now R will start to become less positive relative to L, and D1 will stop conducting. The voltage at R, relative to L, will become less and less positive, until it crosses zero, and R is now negative relative to L. During this time, D1 will be reverse-biased and no current will flow. The load current will gradually discharge C1 until the next positive half cycle comes around with R reaching its positive peak relative to L.
So version A charges C1 to about 38V and produces a DC supply across the output terminals marked + and - which can be used by the rest of the circuit. The rest of the circuit is fully isolated from everything, so the way in which it connects to the 28VAC supply voltage is irrelevant; all it cares about is that it gets its 38V DC supply.
Here's version B, which is shown with the right hand pin, R, as the 0V reference for all voltages within the power supply circuit. I will be describing all voltages as being measured relative to R.
The voltage between R and L (measured EITHER way round) alternates smoothly from +40V to -40V. At a certain point in time, L is becoming more negative than R. Let's assume that C1 has discharged to about 38V. Once L becomes about 38.6V more negative than R, D1 will start to conduct. L will continue to get more negative than R, and current will flow through D1, increasing the voltage across C1. (The polarity of this voltage always matches the polarity of C1.)
Eventually that half-cycle will reach its peak instantaneous voltage of 40V (negative at L, relative to R). C1 will have been charged to about 39.4V. Now L will start to become less negative relative to R, and D1 will stop conducting. The voltage at L, relative to R, will become less and less negative, until it crosses zero, and L is now positive relative to R. During this time, D1 will be reverse-biased and no current will flow. The load current will gradually discharge C1 until the next negative half cycle comes around with L reaching its negative peak relative to R.
So version B also charges C1 to about 38V and produces a DC supply across the output terminals marked + and - which can be used by the rest of the circuit. The rest of the circuit is fully isolated from everything, so the way in which it connects to the 28VAC supply voltage is irrelevant; all it cares about is that it gets its 38V DC supply.
No matter which side of the 28VAC supply you regard as your reference side, the circuit behaves the same. It produces 38V DC at its output to supply the rest of the circuit. It uses only one half of each cycle of the incoming AC supply (it's a half-wave rectifier) and it doesn't matter whether that corresponds to the half-cycle where mains Phase is positive relative to mains Neutral, or mains Phase is negative relative to mains Neutral, because the mains phase, relative to this circuit, is irrelevant.
Does that help?
and my not catching it, again.. 
