Page 232 - Fluid Power Engineering

Basics of Electricity and Generators 199

Basic Principles of Alternating Current
An alternating current is a sinusoidal variation in amplitude of cur-
rent.

i = i m sin ωt (10-1)

where i is alternating current, i m is the peak current, ω is the frequency,
and t is time. If the frequency is 60 Hz, then ω = 2π.60 = 120π radians
per second.
A method to get an alternating current is to apply an alternating
electromotive force (same as voltage) across a pure resistor R.

e e m sin ωt
i = = (10-2)
R R
The power in such a circuit is the product of voltage and current.

1
2
P = e.i = e m i m sin ωt = e m i m (1 − cos 2ωt) (10-3)
2
1
P av = e m i m (10-4)
2
In this circuit, current and voltage are in phase. Next, consider a pure
inductive circuit with no resistance in which the current and voltage
are at 90 phase difference. A pure inductor is closely approximated by
◦
winding a copper coil on a laminated iron core. When alternating cur-
rent is passed through such a circuit, an alternating magnetic field is
created. This magnetic field cuts the copper coil and, therefore, creates
a self-induced electro-motive force (EMF). The self-induced EMF op-
poses the applied EMF. If i = i m sin ωt is the current, then the self-

induced EMF e is:
di
e =−L =−Lωi m cos ωt =−e (10-5)

dt
where e is the applied voltage. Note the current and voltage are 90 out
◦
of phase. The power consumed in this circuit is:
1
P = e.i = e m i m sin ωt cos ωt = e m i m sin(2ωt) (10-6)
2
P av = 0 (10-7)

The power alternates between positive and negative at twice the fre-
π
quency. When the current is rising (ωt ∈ (0, )), circuit supplies power
2
π
to create a magnetic field, when the current is falling (ωt ∈ ( ,π)),
2