W9_parallel_resonance.eps

Week 9: Alternating Current Circuits 315

V sin tp ω V sin ts ω R

I

I

p

s

N N

Iron Core

Flux

p s

Figure 120: A transformer transforms voltageV 1 into a new voltageV 2 , for time-varying (usually
sinusoidal) voltages only.

all of the turns in both coils. The flux is usually coupled by wrapping thecoils around e.g. a torus
of soft iron that traps flux, laminated to preventeddy currents(called the transformercore).

If we letφmbe the flux trapped in the core that passes through a single turn, then:

Vs = Ns

dφm

dt

(697)

Vp = Npdφm

dt

(698)

or (taking the ratios of these two equations, in order)

Vs

Vp=

Ns

Np (699)

Note that we omit Lenz’s law in this expression because we can wrap either coil either way around
the core so that the voltages on primary or secondary side can be “in phase” or “exactly out of
phase” as we wish.

A transformer can thusstep voltage upto higher levels orstep it downto lower ones, depending
on whetherNp< Nsor vice versa. This seems as though it would be obviously useful for many,
many things, and of course it is. Sometimes we need a high voltage anda low current in a wire;
other times we need a low voltage and a high current. Note well that we can’t magically get a
higher voltage andmorecurrent out of a transformer as this would violate energy conservation. In
fact, if we compute the power delivered by the primary voltage to the transformer and equate it to
the power consumed by the secondary circuit, then as long as the transformer itself doesn’t get hot
(removing energy from the circuit of its own accord):

Pp=VpIp=VsIs=Ps (700)

or, if we use the fact thatVs=VpNs/Npand divide a couple of times, we find that:

Is=Np

Ns

Ip (701)

When the voltage goes up (Ns> Np) the current goes down, and vice-versa.

Of course thisdoesassume that the transformer itself and all of its wiring doesn’t haveany
resistance and get hot, and the iron core of the transformer mustalsonot get hot. However, the
iron core isitselfa conductor. When the magnetic flux through it is constantly changing it induces
a voltage initthat causes a current to flow. That current, flowing in the resistance of the iron,
generates heat! This kind of inductive heating is said to be caused byeddy currents, currents
induced in any conductor by rapidly changing magnetic flux through the conductor.