Some important DEFINITIONS: -
Cycle: -
One complete set of positive and negative half alternation of an AC quantity is
known as cycle. It can be specified as a complete cycle from 0o to
360o or 2Õ radians.
Time PERIOD: - It is the total time taken by an AC to complete its one cycle. For example if an AC complete 100 cycles /sec. Then its time period is T=1/100 =10 m sec or 0.01 sec.
FREQUENCY: -
The total numbers of cycles completed by an AC quantity in one second is called
its freq. Its unit is cycles/sec or Hertz (Hz). If an AC completes 100
cycles/sec then its freq is 100 Hz. F=1/t
Amplitude: - The maximum value in positive or negative half alternation is called its amplitude.
Phase
By
phase of an alternating current is meant the fraction of the time period of
that AC that has elapsed since it last passed through the zero position of
reference. For example the phase of current at point A is T/4 Seconds where T
is the time period or expressed in terms of angle, it is Õ/2
radian.
Characteristics of AC sine WAVE:-
a) Its one cycle spread over 360o or 2P radians
b)
Its polarity reverses every half cycle
c)
It has zero value at 00 and 1800 and
maximum value at 900 and 2700.
d)
Fastest rate of change when crosses zero axis and slowest
rate passes its maximum value +ve or –ve
INSTANTANEOUS VALUE:
-
It is the value of an AC at any time instant measured from any reference point. The AC sine wave is made up of instantaneous values zero to maximum.
i = Im Sin ωt= Im
Sin 2P ft or Im Sin 2P
t/T (T = Time period)
v = Vm Sin ωt = Vm Sin 2P Ft or Vm Sin 2P t/T
ROOT MEAN SQUARE (RMS) VALUE:-
The RMS value of an alternating quantity is given by that steady current ( D.C.) 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. It is also known as effective or virtual value of AC .
It
is called root mean square value because if we take some instantaneous values
of an AC and square each of them and add these squared values, and then divide
this by the nos of values taken we will get mean value. Now the square root of
this mean value will be the RMS value as .707 of its max value for a
symmetrical sinusoidal current/ voltage.
I
rms = Imax X
0.707
V
rms = VmaxX 0.707
AVERAGE VALUE:
-
It is the value of an AC which when flowing
through a circuit, can transfer same charge as is transferred by an amount of
steady DC current.
In a symmetrical AC, the average
value over a complete cycle is zero; hence the average value is obtained by
adding or integrating the instantaneous values of half cycle only. But in case
of an unsymmetrical AC (like o/p current of half wave rectifier), average value
must always be taken over for the whole cycle.
Iav = Imax x 0.637
Vav
= Vmax x 0.637
FORM FACTOR: -
It is the ratio of RMS value to the average value
.707 Im
.637 Im
CREST OR PEAK FACTOR:
-
It
is defined as the ratio of maximum value to the RMS value.
PF
=Maximum value/ RMS Value= 0.707/ 0.637
=1.414
(for a sinusoidal AC only)
POWER
IN AC CIRCUITS: -
In
general power is the rate of doing work and is independent of the total work
done. The rate of doing work can be found by dividing the total work done by
the time taken to do it.
Therefore
electric power = Electric work done/time taken
Work
done electrically in time t seconds = VIt
joules
Then
electric power = VIt/t= VI
If
V is in RMS and I is in R M S then product VI will be in Watts.
Watt=Volts
x Amps
One
watt is defined as the rate of doing one joule of work per second.
In
an AC circuit the product of V RMS and I RMS gives volt ampere which is not
true power in watts but it is apparent power in Volt ampere.
TRUE
POWER: -
In
inductive or capacitive circuits, there is no any inductance or capacitance is
free from ohmic resistance due to which these dissipate some power and the
phase angle does not remains exactly 900. To get the true power in
inductive or capacitive circuits phase angle between voltage and current must
be considered.
True power = V I Cos f
Where
Cos f
is the power factor and when angle f
is equals to zero it means that the circuit is purely resistive
True power = V x I x Cos 00
(\
Cost f0=1)
Watt=V
x I (apparent power)
When
Cos f=90
then true power=VI Cos 900 = Zero (\Cos
900=0)
The
circuit is purely inductive or capacitive and true power becomes zero.
POWER FACTOR: -It may be defined as the ratio of true power to the apparent power = Watt/Volt Amp or VI Cosf/VI=Cos f
Or
Cosine of the angle leads or lags
Or
ratio of resistance to the impedance = R/Z
AC THROUGH PURE RESISTANCE:-
The
applied voltage has to overcome the ohmic voltage drop. Hence for equilibrium
V=IR
or Vm Sin ωt=IR
or Vm Sin ωt=I R
or
I = Vm Sin ωt/R
Hence current ‘I’ will be maximum
when ‘Sin ωt’ becomes one then
Im= Vm/R it
shows ‘I’ is maximum when ‘V’ is maximum
\ i=Im Sin ωt
Hence
we found that voltage and current are in phase with each other
AC
THROUGH PURE INDUCTANCE:-
Whenever
an AC is passed through a pure inductance, back emf is produced due its self
inductance. This back emf opposes the rise or fall of current at every step. As
there is no ohmic resistance drop, the applied voltage has to overcome this
self induced emf only.
The
equation of current is
i=Im
Sin (ωt-P/2)
The
applied voltage is represented by v=Vm Sin ωt and the current
flowing in purely inductive circuit is given by i=Im Sin (ωt-P/2)
which shows the current lags behind the applied voltage by 900 or
quarter cycle.
Inductive
Reactance: -
The opposition offered by an inductor to the
flow of currents is inductive reactance, denoted by ‘XL’. Its unit
is ohm.
‘XL=ωL=2Pfl’
where ‘l’ is in hennery and ‘ω’ is in radian/sec, then ‘XL’ in ohm
and proportional to the frequency.
AC
THROUGH PURE CAPACITANCE:-
v = Vm Sin ωt
When an AC is applied to the plates
of a capacitor, it is charged in one direction during half cycle and then in
the opposite direction during the next half cycle
v=Vm Sin ωt then the current
i=Im Sin (ωt+P/2)
Hence
we find that the current in a pure capacitor leads its voltage by a quarter cycle or
phase difference is P/2 .
Capacitive
reactance
apacitive reactance ‘Xc’ is total opposition offered by the cap to flow of charge or current. Its unit is ohm.
If ‘c’
is in Farads and ‘ω’ is radians/sec then, ‘Xc’ will be in Ohm and inversely
proportional to frequency.
Xc=
1/ωc= 1/ 2Pfc
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