Transformer

The underlying principle. A transformer changes an alternating voltage from one value to another using mutual induction (Section 6.7 of the previous sub-topic): two separate coils are wound on a common iron core, so that a CHANGING current in one coil (the primary) sets up a changing flux that links the SECOND coil (the secondary) as well, inducing an emf in it purely through this shared, changing flux -- with no direct electrical (wire) connection between the two coils at all.

Construction. A closed iron core -- built up as a stack of thin, mutually insulated laminated sheets, for exactly the eddy-current reasons already covered in Section 6.6 of the previous sub-topic -- carries two separate windings: the primary, of $N_p$ turns, connected to the input AC supply $V_p$; and the secondary, of $N_s$ turns, connected to the external load, across which the output voltage $V_s$ appears (the figure for this section shows the layout). Because both windings are wound on the SAME closed core, essentially the entire flux produced by the primary's current also links the secondary -- the core acts as a near-perfect flux guide between the two coils.

Deriving the turns-ratio relation. Let $\Phi$ be the flux linked with EACH turn of the core at some instant (the same for both windings, since they share the same core). By Faraday's law, the emf induced in each winding is proportional to its own number of turns:

Dividing the second relation by the first (the common factor $d\Phi/dt$ cancels, since it is the same instantaneous rate for both):

Ideal power conservation and the current relation. For an IDEAL transformer (assumed to have no internal energy losses at all), the power delivered to the secondary must exactly equal the power drawn by the primary from the source, $V_pI_p=V_sI_s$. Combining this with the turns-ratio relation above gives

showing directly that whichever winding has the LARGER number of turns (and hence the larger voltage) carries the CORRESPONDINGLY SMALLER current -- a transformer can raise voltage or raise current, never both together, since the product $VI$ (the power) is what stays fixed in the ideal case.

Step-up and step-down. A transformer with $N_s>N_p$ raises the voltage ($V_s>V_p$) and is called a step-up transformer; one with $N_s<N_p$ lowers the voltage and is called a step-down transformer (the table for this section compares both cases directly). Power transmission uses exactly this pair in sequence: a step-up transformer raises the generating station's output to a very high transmission voltage (cutting the current, and hence the $I^2R$ loss along the long transmission lines, for the same power delivered), and a step-down transformer near the point of use lowers this high voltage back down to a safe level for homes and factories. …