The ratio of turns tells us the expected in and out voltages, however, that's only a small part of the story. How can you determine the output current? How do you figure out the voltage drop under different loads? Also, how does the primary react when the secondary is under a heavy load?
As always, mathematical rigor would be extremely helpful in helping me really understand it all

I know I ask a lot of questions, but these are the issues I can't seem to find the answers too, at least not in a way that makes sense.

 

What you say is true. How can we help you? The topic is covered exhaustively in many textbooks.

 

The ratio of turns tells us the expected in and out voltages, however, that's only a small part of the story. How can you determine the output current? How do you figure out the voltage drop under different loads? Also, how does the primary react when the secondary is under a heavy load?
As always, mathematical rigor would be extremely helpful in helping me really understand it all

I know I ask a lot of questions, but these are the issues I can't seem to find the answers too, at least not in a way that makes sense.

Transformers, are all about balance.........in that a heavy load on the secondary will induce a matching draw on the primary, within the engineered " VA " of the unit.......anything outside those engineered parameters, will announce itself vigorously.

One can bend the laws of physics to a degree, but not outright break them. The math is your best friend.

 

Let me explain in simple language, then you can read other things elsewhere.
In a normal transformer, where there is a primary and secondary winding,
following happens.
When you power up, a current flows in to the primary windings. This causes a magnetic field to develop. Since the primary current is alternating, the magnetic field also alternates, switching between north pole to south pole. But the magnetic field is changing at a lagging time. The changing magnetic field produces a voltage which is called induced voltage. This opposes the primary (applied) voltage. the result is a very low current in the order of 5% in the primary coil.
=(applied voltage-induced voltage)/primary impedance
The changing magnetic field is also producing a voltage in the secondary.
This is proportional to the turns ratio between primary and secondary. A lower turns in secondary produce lower voltage, higher turns produce higher voltage, equal turns produce equal voltage. The secondary voltage polarity is the same as the primary supply (when the direction of winding is the same). When the secondary is loaded, the current flows in the opposite direction to that of the primary. This current also produces a magnetic field inside the transformer, alternating, but opposite direction.
This reduces the effective magntic field in the transformer core. The induced voltage in the primary falls now and the current increases to compensate for the loss in the magnetic field,
When there is a shorted secondary, the primary current is limited to how much the secondary windings can cancel the magnetic field and the primary winding resistance and the magnetic core path length.
Any further is too complicated to describe here, I suggest you read some good references in the net.
When you powerup the transformer for the first time, there is no or minimum magnetic field in the core. Until the magnetic field is developed (and induced voltage is generated) the current is very high for a first few cycles. This is called the inrush current.
The transformer is a very efficient device in transfering power into a different voltage. Primary VA = total secondary VA+ (3 to 10%) loss.
Smaller transformers are about 92 % efficient.

regards,
Maanga.