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Titlem.g.say ch01
TagsAlternating Current Electricity Physics & Mathematics Physics
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Total Pages5
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Page 1


1.-Alternating Current Machines. The synchronous generator, con-
verter and motor, the transformer, and the induction motor, form
the subjects of study in this book. They -are of sufficient
importance to account between them for almost all of the output
of manufacturers of electrical machinery, -apart from d.c. motors.

2. The Alternating Current System. The wide application of a.c.
machines may be said to be due to

(a) The heteropolar arrangement.
(b) The transformer.

Electrical energy is particularly useful by reason of the ease with
which it can be distributed and converted to other desired forms.
At present, the chief source of electrical energy is the mechanical

. form, either direct (as in hydraulic storage reservoirs or rivers) or
from chemical energy via heat. A convenient direct method of
converting mechanical to electrical energy is by use of the mechan-
ical force manifested when a current-carrying conductor is situated
in a magnetic field. The movement of a conductor through a polar
magnetic field having consecutive reversals of polarity results in
the production of an induced e.m.f. which changes sign (i.e. direction)
in accordance with the change in the magnetic polarity: in brief,
an alternating e.m.f. Thus all ordinary electromagnetic machines are
inherently a.e. machines.

The wider use of electricity has necessitated the development of
systems of transmission and distribution. Transmission systems are
required (a) to enable sources of natural energy (e.g. waterfalls or
storage lakes or coal fields), far distant from the centres of energy
demand (e.g. large works, industrial areas or cities), to be economic-
ally utilized; (b) to enable generating plant to be concentrated in a.
few large, 'favourably-situated stations; and (c) to permit of the
interconnection of networks or distributing areas to ensure reliability,
economy and continuity of supply.

Frequent reference is made to the companion volume, The Performance and
Design of Direct Ourrent Machines, by Dr. A. E. Clayton. For brevity, this
is contracted to [A] in the text. Since references are valuable if they do not
involve unnecessary searching, such references are confined chiefly to a few
easily-obtained textbooks, as follows- .

[A] Performance and DeBign of D.O. Machines. A. E. Clayton
and N. N. Hancock (Pitman).

[B) Electrical,Engineering Design Manual. M. G. Say (Chapman
& Hall).

[CL Performance and Design of A.O. Oommutator Motors.
E. O. Taylor (Pitman).


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A high voltage is desirable for transmitting large powers in order
to decrease the I2R losses and reduce the amount of conductor
material. A very much lower voltage, on the other hand, is required
for distribution, for various reasons connected with safety and
convenience. The transformer makes this easily and economically

Thus the fact that generation is inherently a production of
alternating e.m.f.'s, coupled with the advantage of a.c. transforma-
tion, provides the basic reason for the widespread development of
the a.c. system. The collateral development of a.c. motors and other
methods of utilizing a.c. energy has been a natural step.

It must by no means be assumed that the a.o. system represents a
finality. Apart from the superiority of d.c. motors in certain cases,
particularly where speed-control is desired, there are signs that a
high-voltage d.c. transmission system may evolve in 'the near
future. However, while changes, small or great, in methods of
electrical production and use may be confidently expected, it is
probable that a.c. machines will remain important components of
electrical generation, application and utilization.
3. Circuit-Behaviour. A voltage v, applied to a circuit of series

resistance R and inductance L, produces a current such that
v = Ri + L(di/dt)

at every instant. If v is a constant direct_voltage V, the steady-state
current is 1= VIR, because di/dt = 0 in the absence of changes;
but whenever the conditions are altered (e.g. by switching or by
changing the parameters) the new steady state is attained through a
transient condition. The instantaneous current can then be written
i = is + it, with is the final steady-state current and it the temporary
transient component. The time-form of it depends only on the
parameters, and for the RL circuit considered is the exponential

it - ios-(RIL)t = ioe~p (- R/L)t.
Fig. 1 (a) shows the steady-state and transient components and the

total resultant current for an initiation transient when the drive-
voltage is V. As the current is zero at the switching instant (and
cannot rise thereafter faster than V /L), then i, must cancel is
initially: hence
i= is + it = is - isexp (- R/L)t = (V/R)[l - exp (- R/L)t].

This is a well-known case of practical interest in the switching of
d.c. field circuits.

If v is the sine voltage Vm sin cot, the steady-state current is the
sinewave i.= (vm/Z) sin (wt - e), where Z = y'(R2 + w2L2) =
y'(R2 + X2), and e = arc tan (X/R). Suppose the voltage to be
switched on at some instant in its cycle corresponding to t = to: the
steady-state current for this instant would be (vm/Z) sin (wto - e).
But-the current is actually zero, so that again it initially cancels i.8

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Subsequently it decays exponentially as before.' The current
following circuit switching is therefore i = is + it, giving
i = (vmIZ) [sin (wt - 0) - sin (wto - 0). exp (- RIL)(t - to)]'
In machine circuits R is often much smaller than X so that

Z ~ X and 0 ~ 90°. For these conditions, approximately,
i ~(vmIX)[ - cos cot + cos wto' exp (- RIL)(t - to)].

If the voltage is switched on at a zero (to = 0, 7Tlw, 27Tlw ... ), the
transient term has its greatest value vml X, and the resultant current


t=O Time, t


starts from zero with complete asymmetry, Fig. 1 (b). In contrast,
if the switch is closed on peak voltage, the resultant current attains
steady state instantaneously without any transient, Fig. 1 (c). The
current in (b) has an amplitude nearly double that in (c), an example
of the doubling effect. Intermediate switching instants give partial
asymmetry, with smaller transient components.

TRANSIENTANDSTEADYSTATES. It is usual to develop circuit
theory on a steady-state basis, using complex algebraic treatment
with complexor or "vector" diagrams. The consideration of tran-
sient conditions demands a return to more basic concepts.
, STEADY-STATECONVENTIONS.A complexor* voltage V, of mag-

nitude V and drawn at an arbitrary angle IX to a horizontal datum-
can be variously described as

V = VIIX = VI + jV2 = V (cos IX + j sin IX) = V . exp (j1X).
* Voltages and currents are not vectors in the true physical sense, and are

here called "complexors." However, in deference to common usage the term
"vector" is also occasionally used in the text,

Page 4


Fig. 2A shows a circuit of series resistance R and inductive reac-
tance jwL = jX. An r.m.s. voltage V is applied, and an r.m.s.
current I flows such that V = I(R + jX) = IZ. The impedance
operator Z = Z/() = v'(R2 + X2) . /arc tan (X/R): it does not
stand for a sinusoidally-varying quantity. If 1= l/(J, then 1= V/Z
and () = (X - p. ' -

The voltage can be regarded as the complex sum of components
IR and IX. The latter is the component opposing Ex (the e.m.f. of

,---,. V=IZ IX--£[ R ( x;¢ . - x

~._) I 8:· JR'P J
LJR V=/Z IX=:j


self-inductance in L) whose instantaneous value is ex = - L(di/dt).
The magnetic flux <I> of the coil, proportional to the current if satura-
tion effects can be ignored, is represented by a complexor in phase
with the current. Then Ex lags on it by 7T/2. The r.m.s. flux could
quite properly be employed, but the peak value is usual as a prac-
tical measure of the degree of magnetic saturation.

~er Power

Tlme, t.

POWER. The instantaneous power in a single-phase circuit pul-
sates at double supply frequency. In general it is asymmetric about
the time axis so that it cannot be represented in a diagram containing
V and 1. The average of the power/time curve is the mean power
dissipation P watts, about which the remaining alternating part has
an amplitude Q vars (reactive volt-amperes), Fig. 2B.

If V = V/(X and I = I/(J, the product VI = VI/(X + (J is unrelated
to the powerp. But the power can be derived from the voltage V.
and the conjugate of I, namely 1* = 11- (J: for

VI* = V/(X. 1/- (J = VI/(X - (J = VII!!.
= VI (cos ()+ j sin () = P + jQ.

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