How do transformers work and how do toroidal transformers differ?

[FeynmanFtw]

I have a few questions regarding general transformer workings, and a couple of questions regarding toroidal transformers as well. For ease of reading I've put them in bullet points:

1. The magnetic flux induced by the primary winding creates the voltage (and hence current) across the secondary winding, but this secondary winding produces a magnetic flux which opposes (and TOTALLY cancels) the magnetic flux which created it, as per Lenz's law. Is this correct?

2. Continuing on from this, I've heard that since the initial power source has to compensate for this loss of flux in order to drive the secondary circuit, it boosts the power accordingly. But then surely now you have a greater flux being transferred from the primary, which would mean a greater reverse flux from the secondary...ad infinitum? I know I'm missing something simple here, but I cannot see it.

3. How can you prove mathematically that the voltage across the secondary winding is opposite to the primary winding (i.e. the current flows in the opposite direction; polarity is reversed?). If you take any standard textbook transformer (typical square shaped ring with the windings on opposite sides), the positive terminals are at the same position at any given time, meaning the voltages are opposite. As an example see below.

My guess is the following, if you take vectors into account; Since V₁/N₁ = - dΦ/dt and we take the direction of an upward flux to be negative, then a downward flux (on the other side of the transformer) will be dΦ/dt, ergo dΦ/dt = V₂/N₂. Since now we end up with - V₁/N₁ = V₂/N₂, since we cannot have "negative turns", the only other conclusion is that V₁ and V₂ just have opposite signs in order for that equation to hold. please correct me if I'm wrong.4. In a toroidal transformer, the primary and secondary windings are essentially on top of each other (obviously insulated so no short circuit). When I drew up a cross section of how a toroidal transformer would work, I noticed that the directions of the magnetic flux of the primary and secondary windings were going in opposite directions (regardless of whether I looked at the inner or outer part of the transformer). Do these fields have no impact on the efficiency of the transformer? I would assume they simple cancel in the area (albeit very small area) between the primary and secondary windings?

Please forgive any seemingly mundane and trivial misunderstandings but I've never tackled these issues outside of high school. Thanks in advance for any help provided.

 

[zoki85]

1. The magnetic flux induced by the primary winding creates the voltage (and hence current) across the secondary winding, but this secondary winding produces a magnetic flux which opposes (and TOTALLY cancels) the magnetic flux which created it, as per Lenz's law. Is this correct?

WRONG.
Back to the drawing board (or good magnetics fundamentals book)

 

[berkeman]

It only totally cancels it if the impedance of the secondary winding and the load resistance are zero (which is non-physical). To the extent that the load impedance is finite, the final balance of the two fluxes is what is needed to support the output current into the load. As zoki85 suggests, take another look at the math of that balance of flux...

 

[jim hardy]

Observe that flux is pushed UP through primary
and it's going DOWN over at secondary.

It might help you to redraw secondary on same side as primary and observe direction of MMF's.

But then surely now you have a greater flux being transferred from the primary, which would mean a greater reverse flux from the secondary...ad infinitum?

Flux is pushed around the core by "magnetomotive force", MMF. measured in amp-turns.
I think it's better to speak in terms of of MMF .
Primary and secondary amp-turns indeed oppose.

The magnitude of flux in your picture is set by the applied voltage not the current
it'll be the flux necessary to produce in primary winding a voltage equal and opposite to applied voltage.
The mmf(amp-turns) to produce that much flux will be small compared to the amp-turns flowing due to load
and that's why we say "secondary MMF cancels primary MMF" because it almost does but not quite. Enough is always left over to magnetize the core to flux level determined by primary voltage.
When secondary is open, only MMF is from primary and some small current will flow in primary.
That primary current is called "magnetizing" current . It's small compared to load current and is often ignored. But that shortcut confuses students.

So in your mental picture, think of flux as proportional to voltage and you'll be on right track.
That's counter-intuitive at first because we're taught Φ=NIA/length, proportional to current with no mention of voltage.
Secret there is NI is algebraic sum of opposing primary and secondary amp-turns including their signs
and I will settle at value to produce e=NdΦ/dt, e being primary voltage. When secondary NI is zero primary NI is small.