An
a.c current is said to be leading with respect to a.c. voltage, if it reaches
its maximum value earlier than that of voltage in every cycle.
LAGGING
An
a.c. current is said to be lagging with respect to a.c. voltage, if it reaches
its maximum value later than that of voltage in every cycle.
OPERATOR
(j)
Symbol
j is used to indicate the counter clockwise rotation of a vector through 90
degree. It is assigned a value of -1. Any quantity multiplied by j means that
the quantity is rotated through an angle of 90 degree in the counter clockwise
direction without altering its magnitude.
IMPEDANCE
(Z)
Z
is the ratio of rms voltage to rms current and is known as impedance. Its units
are Ohms.
Impedance
has two components, i.e. the real component R and imaginary component ωL. Z = R
+ JωL
[OR]
Impedance
is the factor which limits current in an a.c. circuit. Its value is equal to
the ratio of the voltage applied to the circuit element to the current in the
circuit element.
It
is denoted as Z and its unit is ohm.
ADMITTANCE
(Y)
Admittance
of a circuit is defined as the reciprocal of impedance.
In
any quarter-cycle, the positive power and the negative power are equal and thus
they cancel each other. Consequently, in a purely inductive circuit the average
power over a complete cycle is always zero and it varies at double the
frequency of the supply voltage.
CAPACITIVE
CIRCUIT
The
average power over a complete cycle is zero and the instantaneous power varies at
double the frequency of the supply voltage.
In
an inductive circuit the current lags behind the voltage by 90 degrees.
CAPACITIVE CIRCUIT
In
a capacitive circuit the current leads the voltage by 90 degrees.
INDUCTIVE
REACTANCE
When
an alternating voltage is applied across an inductor, the induced emf produced
opposes the flow of current through it. The opposition or resistance to the
flow of current offered by the inductance is called inductive reactance.
It
is denoted as Xl.
CAPACITIVE
REACTANCE
Capacitive
reactance is the ratio of the voltage applied to the capacitor and current
through the capacitor. It
is denoted as Xc.
Some physical quantities are described
completely by a single number with a unit, examples are temperature, density
and quantity of electric discharge.
VECTOR
Many other quantities have both
magnitude and direction, examples are velocity, force and displacement.
PHASORS
In the theory of electric circuits,
voltages and current can be represented in the complex plane by radius vectors
characterized by a magnitude and phase with respect to a reference angle. Such
radius vectors representing complex numbers are called phasors.
[OR]
A sinusoidal quantity may be
represented by a line fixed at one end and rotating counterclockwise at a
velocity equal to the angular velocity (ω rad/s) of the sinusoidal quantity.
This rotating line is called the phasor.
Phasor is a complex quantity while a
vector is a simple quantity. Phasors are actually moving with time. By
convection, phasors are assumed to rotate in a counterclockwise direction.
REFERENCE PHASOR
Since the phase difference remains
constant, any phasor may be drawn along the convenient direction. This phasor
will be called a reference phasor. The
position of other phasors relative to the reference phasor becomes fixed.
PHASE – The orientation of a rotating vector in space at any
particular instant is called its ‘phase’.
If we are considering a single rotating
vector, we are not concerned with its phase; but if there are two or more
rotating vectors then the difference between their orientations is extremely
important.
The phase angle is taken positive when
measured counter clockwise and negative in the clockwise direction. The angular
position of the phasor represents a position in time, not space.
PHASE DIFFERENCE – It is an angular displacement of ɸ will remain constant as the two
vectors rotate in space. The angle ɸ is called phase difference.
PHASOR
DIAGRAM
The
graphic representation of the phasors of sinusoidal quantities taken all at the
same frequency and with proper phase relationships with respect to each other
is called phasor diagram.
It
is a common practice to draw the phasor diagrams in terms of effective value
(RMS value) rather than maximum values.
LIMITATIONS
OF PHASOR DIDAGRAMS
1.
A phasor represents only one position (per cycle) of the waveform and therefore
it does not give a complete description of a sinusoidal quantity.
2.
A phasor diagram is drawn to represent phasors at one frequency only.
1.
Alternating current periodically changes the direction in which it is flowing.
2.
It also changes magnitude either continuously or periodically.
3.
An alternating voltage is a voltage which produces an alternating current when
used to power circuit.
4.
Most types of alternating current, the magnitude is changing continuously.
5.
The AC power refers to power that is produced by alternating current and
alternating voltage.
6.
Alternating quantities may be represented graphically. The curve obtained by
plotting the values of voltage or current at different instants on vertical
axis and time or angle on horizontal axis is called a waveform.
7.
Most common type of waveform is sine wave. It is an alternating waveform in
which sin law is followed.
8.
Non-sinusoidal waveform is an alternating waveform in which sine law is not
followed.
9.
Periodic waveform is one which repeats itself after definite time intervals.
10.
Alternating current can be electronically produced in an almost infinite
variety of waveform.
TYPES
OF A.C. WAVEFORMS
The alternating current and voltage supplied to homes and factories is
sinusoidal.
Square wave form is used extensively in computer circuits.
Sawtooth waveform is used in television receivers, radar receivers and other
electronic devices.
Electronic music is created by producing and mixing together a wide variety of
waveforms.
ADVANTAGES
OF SINE WAVEFORM
In
any natural object which has a periodic motion such as a swinging pendulum, a
vibrating string or the rippling surface of a body of water, we find this form
of wave. The sine curve is apparently nature’s standard. Circular motion
produces a sine wave naturally.
The
sine wave greatly simplifies the theory and calculations of a.c. circuits. For
this reason, the designers of a.c. generators try always to obtain a waveform
approximating as closely as possible to that of a pure sine wave.
The sine wave can be expressed in a simple mathematical form.
The rate of change of any sinusoidal quantity is also sinusoidal.
When current in a capacitor, in an inductor or in a transformer is sinusoidal,
the voltage across the element is also sinusoidal. This is not true of any
other waveform.
The mathematical computation, connected with alternating current work, are much
simpler with this waveform.
By means of Fourier series analysis, it is possible to represent any periodic
function of whatever waveform in terms of sinusoids.
IMPORTANT TERMS
1.
Cycle – One complete alternation or repletion of a set of values of
current is called a cycle.
2.
Alternation – One-half cycle of an alternating quantity is called
alternation. An alternation spans 180 degree electrical.
3.
Period – The time required to complete one cycle is called the periodic
time or simply the period (T).
4.
Frequency – The number of cycles completed in one second is called the
frequency (f). The unit of frequency is the hertz.
One
hertz = One cycle per second.
5.
Amplitude – The maximum value positive or negative attained by an
alternating quantity is called its amplitude or peak value. The amplitude of an
alternating voltage or current is noted by Vm or Im
IMPORTANT
RELATIONS
1.
Time period and frequency [f = 1/T Hz]
2.
Angular velocity and frequency [2πf = 2π/T in radians /sec.]
3.
Frequency, speed and number of poles [f = PN/120]
f
= frequency in Hertz, T = Time period in sec.
N
= Revolution per minute and P = No. of poles.
VALUES
OF ALTERNATING QUANTITIES
1.
Instantaneous value – The value of an alternating quantity at a given
instant (time) is called instantaneous value. It varies from instant to
instant. It is denoted by small letters v or i.
2.
Maximum or crest value – This is the maximum value of the alternating
quantity attained by it in a cycle. It is the highest of the instantaneous
values. It is denoted by a capital letter and a subscript m (Vm or Im).
3.
Average or mean value – The average value of a waveform is the average
of all its values over a period of times.
Average
value
=
[Total (net) area under curve for time T] / [Time T]
Average
value of a symmetrical wave
=
[Area of one alternation] / [Base length of one alternation]
Average
value of an unsymmetrical wave
=
[Area over one cycle] / [Base length of one cycle]
4.
Effective value – The effective value of an alternating
current is that value that produces the same heat in a resistive circuit as a
direct current of the same value. Also, equal amounts of dc voltage and
effective ac voltage produce equal power across resistors of equal value.
5.
RMS value – The effective value of a waveform can be
determined by a mathematical process known as Root Mean Square (RMS) value.
