Page 233 - Fluid Power Engineering

200 Chapter Te n

magnetic field decreases. This decreasing magnetic field supplies
power to the circuit.
Consider a circuit with resistance (R) and inductance (L) in se-
ries. Assume the potential drop is e m sin ωt and the current is i =
i m sin(ωt − ϕ), then

e m sin ωt = i m R sin(ωt − ϕ) + i m Lω cos (ωt − ϕ)
= i m (R(sin ωt cos ϕ − sin ϕ cos ωt) + Lω(cos ωt cos ϕ + sin ωt sin ϕ))
= sin ωt [i m R cos ϕ + i m Lω sin ϕ] + cos ωt [−i m R sin ϕ + i m Lω cos ϕ]
(10-8)

Equating the sin ωt and cos ωt terms yields:

Lω
0 =−i m R sin ϕ + i m Lω cos ϕ => tan ϕ = (10-9)
R

e m = i m R cos ϕ + i m Lω sin ϕ => |e m | = i m R + (Lω) 2 (10-10)
2

1
P = e m i m sin ωt sin (ωt − ϕ) = e m i m (cos ϕ − cos(2ωt − ϕ)) (10-11)
2
1
P av = e m i m cos ϕ (10-12)
2
The power delivered to the circuit is lower than a pure resistance cir-
cuit.Thisisbecausethevoltageandcurrentarenotinphase.Therefore,
the sum of the product of the two quantities over time is lower. cos ϕ
is called the power factor: the ratio of the true power and the apparent
power.
Three phase alternating current is current in three conductors with
a phase difference of 120 .
◦
i 1 = i m sin ωt, i 2 = i m sin(ωt + 120), i 3 = i m sin(ωt + 240) (10-13)

Basic Principles of Electrical Machines
Electricitygeneratorsarerotatingelectricalmachinesthathaveastator
and a rotor. Stator is the stationary part of the generator. This is the
outer cylindrical casing of the generator that contains three, six, or
other multiple of three numbers of coils. In rare configurations, the
stator may be the inner cylinder. To simplify the exposition only three

coils will be considered as shown in Fig. 10-2; a-a , b-b , and c-c are

the three coils. The coils are connected to three-phase AC.