Cycle
One complete set of positive and negative values of alternating quantity is
known as cycle. Hence, each diagram of Fig. below represents one complete
cycle.
In general, however, an alternating current or voltage is one the circuit
direction of which reverses at regularly recurring intervals
Time Period
The time taken by an alternating quantity to complete one cycle is called its
time period T. For example, a 50-Hz alternating current has a time period of
1/50 second.
Frequency
The number of cycles/second is called the frequency of the alternating quantity.
Its unit is hertz (Hz).
It may be noted that the frequency is given by the reciprocal of the time period
of the alternating quantity.
f = PN/120 where N = revolutions in r.p.m. and P = number of poles
Amplitude
The maximum value, positive or negative, of an alternating quantity is known as
its amplitude.
Different Forms of E.M.F. Equation
The standard form of an alternating voltage, as already given by
e = Em sin θ = Em sin ω t = Em sin 2 π f t = Em sin (2πt)/T
By closely looking at the above equations, we find that
(i) the maximum value or peak value or amplitude of an alternating voltage is
given by the coefficient of the sine of the time angle.
(ii) the frequency f is given by the coefficient of time divided by 2π..
For example, if the equation of an alternating voltage is given by e = 50 sin 314t
then its maximum value of 50 V and its frequency is f = 314/2π = 50 Hz.
Phase
By phase of an alternating current is meant the fraction of the time period of
that alternating current which has elapsed since the current last passed through
the zero position of reference. For example, the phase of current at point A is
T/4 second, where T is time period or expressed in terms of angle, it is π/2
radians (Fig.). Similarly, the phase of the rotating coil at the instant shown in
Fig. below is ω t which is, therefore, called its phase angle.
Phase Difference
Now, consider three similar single-turn coils displaced from each other by
angles α and β and rotating in a uniform magnetic field with the same angular
velocity
The three equations for the instantaneous induced e.m.fs. are (Fig. Above)
eA = Em sin ω t ...reference quantity
eB = Em sin (ωt − β)
eC = Em sin [ωt − (α + β)]
‘lag’ or ‘lead’.
quantity B leads A by an angle φ. Hence, their equations are eA = Em sin ω t
...reference quantity eB = Em sin (ωt − φ)
A plus (+) sign when used in connection with phase difference denotes ‘lead’
whereas a minus (–) sign denotes ‘lag’.
Root-Mean-Square (R.M.S.) Value
The r.m.s. value of an alternating current is given by that steady (d.c.) current
which when flowing through a given circuit for a given time produces the
same heat as produced by the alternating current when flowing through the
same circuit for the same time.
The direct current will equal Im/ 2 which is called r.m.s. value of the sinusoidal
current.
r.m.s. value of current = 0.707 × max. value of current
The r.m.s. value of an alternating current is of considerable importance in
practice, because the ammeters and voltmeters record the r.m.s. value of
alternating current and voltage respectively.
Average Value
The average value Ia of an alternating current is expressed by that steady
current which transfers across any circuit the same charge as is transferred
by that alternating current during the same time.
average value of current = 0.637 × maximum value
Note. R.M.S. value is always greater than average value except in the case of a
rectangular wave when both are equal.
Form Factor
Crest or Peak or Amplitude Factor
Vector Diagrams of Sine Waves of Same Frequency
A.C. Through Resistance, Inductance and Capacitance
The circuit is shown in Fig. 11.56. Let the applied voltage be given by the
equation. v = Vm sin θ = Vm sin ωt ...(i)
Let R = ohmic resistance ; i = instantaneous current Obviously, the applied
voltage has to supply ohmic voltage drop only. Hence for equilibrium
v = iR;
Comparing (i) and (ii), we find that the alternating voltage and current are in
phase with each other as shown in Fig. It is also shown vectorially by vectors
VR and I in Fig. .
