Page 1 of 9
CHAPTER 7 v- instantaneous value of voltage ,
vm- peak value of voltage, ω - Angular
ALTERNATING CURRENT frequency.
AC Voltage and AC Current RMS Value (effective current)
A voltage that varies like a sine function r.m.s. value of a.c. is the d.c. equivalent
with time is called alternating voltage (ac which produces the same amount of heat
voltage). energy in same time as that of an a.c.
It is denoted by Irms or I.
The electric current whose magnitude Relation between r.m.s. value and peak
changes with time and direction reverses value is
periodically is called the alternating iN
current (ac current). IrNc =
√2
Advantages of AC: The r.m.s voltage is given by
Easily stepped up or stepped down using vN
transformer
VrNc =
√2
Can be regulated using choke coil without Phasors
loss of energy
A phasor is a vector which rotates about
Easily converted in to dc using rectifier (Pn the origin with angular speed ω.
- diode)
The vertical components of phasors V and
Can be transmitted over distant places I represent the sinusoidally varying
quantities v and i.
Production of ac is more economical
The magnitudes of phasors V and I
Disadvantages of ac
represent the peak values vm and im
Cannot used for electroplating - Polarity
of ac changes
ac is more dangerous
It can't store for longer time
Representation of ac
The diagram representing alternating
voltage and current (phasors) as the
rotating vectors along with the phase
angle between them is called phasor
diagram.
AC Voltage applied to a Resistor
An ac voltage can be represented as
Page 2 of 9
The ac voltage applied to the resistor is
Using the trigonometric identity,
Applying Kirchhoff’s loop rule
sin2 ωt = 1/2 (1– cos 2ωt )
< sin2 ωt > = (1/2) (1– < cos 2ωt >)
Since < cos2ωt > = 0
Since R is a constant, we can write this
equation as
Where peak value of current is Thus
Thus when ac is passed through a resistor In terms of r.m.s value
the voltage and current are in phase with
each other.
Or
AC VOLTAGE APPLIED TO AN INDUCTOR
Phasor diagram
Let the voltage across the source be
Using the Kirchhoff’s loop rule
Instantaneous power
The instantaneous power dissipated in
the resistor is
Where L is the self-inductance
Thus
Average power
The average value of p over a cycle is Integrating
or
Page 3 of 9
Since the current is oscillating , the
Thus a comparison of equations for the
constant of integration is zero.
source voltage and the current in an
Using
inductor shows that the current lags the
voltage by π/2 or one-quarter (1/4) cycle.
Where
Instantaneous power
Or
The instantaneous power supplied to the
inductor is
Where XL- inductive reactance
Inductive reactance (XL)
The resistance offered by the inductor to
an ac through it is called inductive
reactance.
It is given by
Average power
The average power over a complete cycle
The dimension of inductive reactance is in an inductor is
the same as that of resistance and its SI
unit is ohm (Ω).
The inductive reactance is directly
proportional to the inductance and to the
frequency of the current.
since the average of sin (2ωt) over a
Phasor Diagram
complete cycle is zero.
We have the source voltage
Thus, the average power supplied to an
inductor over one complete cycle is zero.
The current
Page 4 of 9
AC VOLTAGE APPLIED TO A CAPACITOR
Where XC – capacitive reactance
Capacitive Reactance
It is the resistance offered by the
capacitor to an ac current through it.
The dimension of capacitive reactance is
A capacitor in a dc circuit will limit or
the same as that of resistance and its SI
oppose the current as it charges.
unit is ohm (Ω).
When the capacitor is connected to an ac
source, it limits or regulates the current, Phasor Diagram
but does not completely prevent the flow
of charge. The applied voltage is
Let the applied voltage be
The current is
The instantaneous voltage v across the
capacitor is
Thus the current leads voltage by π/2.
Where q is the charge on the capacitor.
Using the Kirchhoff’s loop rule
Therefore
Using the relation
Instantaneous power
The instantaneous power supplied to the
capacitor is
Where
Average power
The average power is given by
Or
Page 5 of 9
Thus the average power over a cycle
when an ac passed through a capacitor is
zero.
AC VOLTAGE APPLIED TO A SERIES LCR CIRCUIT
The length of these phasors VR, VC
and VL are:
Let the voltage of the source to be
v = vm sin ωt
Also
From Kirchhoff’s loop rule:
Since VC and VL are always along
Where , q - the charge on the capacitor the same line and in opposite
i – current directions, they can be combined
Using, i=dq/dt into a single phasor (VC + VL) which
has a magnitude |vCm – vLm|
Since V is represented as the
hypotenuse of a right-traingle whose
Phasor-diagram solution sides are VR and (VC + VL), the
pythagorean theorem gives:
Since the current through resistor,
inductor and capacitor is the same as
they are in series.
If φ is the phase difference between the
Thus
voltage across the source and the current
in the circuit,
Let I be the phasor representing the
current in the circuit and VL, VR, VC, and
V represent the voltage across the Therefore
inductor, resistor, capacitor and the
source, respectively.
The phasor diagramis
Page 6 of 9
Or If XC > XL , φ is positive and the circuit is
predominantly capacitive.
If XC < XL , φ is negative and the circuit is
predominantly inductive.
Z is called impedance .
Impedance
It is the effective resistance offered by the
inductor, capacitor and resistor in an LCR
circuit.
Impedance is given by
L C R RESONANCE
For an LCR circuit the impedance
Impedance Triangle
If XC= XL , then Z = R, the impedance is
minimum and the current in the circuit is
maximum – LCR Resonance
A series LCR circuit, which admits
Since phasor I is always parallel to phasor maximum current corresponding to a
VR, the phase angle φ is the angle particular angular frequency of the ac
between V R and V. source is called a series resonant circuit.
Resonance phenomenon is exhibited by a
Thus circuit only if both L and C are present in
the circuit.
Resonant frequency
Or The angular frequency at which the
current is maximum in an LCR circuit is
called resonant frequency (ω0).
That is
Thus
Page 7 of 9
At resonant frequency, the maximum We choose a value of ω for which the
current is given by current amplitude is 1/ √2 times its
maximum value.
Graph showing variation of im with ω
Applications of resonance
In the tuning mechanism of a radio or a TV At this value, the power dissipated by
set the circuit becomes half.
In metal detectors. Thus
Tuning of radio or TV
In tuning, the capacitance of a capacitor in
the tuning circuit is varied such that the The difference ω1 – ω2 = 2Δω is
resonant frequency of the circuit becomes
called the bandwidth of the circuit.
nearly equal to the frequency of the radio
The quantity (ω0 / 2Δω) is regarded
signal received.
as a measure of the sharpness of
At this frequency , the amplitude of the
resonance.
current with the frequency of the signal of
The smaller the Δω, the sharper or
the particular radio station in the circuit is
narrower is the resonance.
maximum.
Sharpness of resonance
The amplitude of the current in the
series LCR circuit is given by
The maximum value is
Page 8 of 9
So, larger the value of Q, the smaller is the
value of 2Δω or the bandwidth and
sharper is the resonance.
Also
If the resonance is less sharp, not only is
the maximum current less, the circuit is
close to resonance for a larger range Δω
of frequencies and the tuning of the
circuit will not be good.
So, less sharp the resonance, less is the
selectivity of the circuit or vice versa.
POWER IN AC CIRCUIT: THE POWER FACTOR
Using
We have
Where
Therefore, the instantaneous power p
supplied by the source is
Therefore
The quantity cosφ is called the power
factor.
Special cases
Thus
Resistive circuit:
If the circuit contains only pure R, it is
called
Page 9 of 9
resistive. In that case φ= 0, cosφ = 1. This equation has the formfor a simple
There is maximum power dissipation. harmonic oscillator
Purely inductive or capacitive circuit:
If the circuit contains only an inductor or
capacitor, the phase difference between
The charge, therefore, oscillates with a
voltage and current is π/2.
Therefore, cos φ= 0, and no power is natural frequency
dissipated even though a current is
flowing in the circuit. This current is
sometimes referred to as wattless
The charge varies sinusoidally with time
current.
LCR series circuit: as
In an LCR series circuit the power is
dissipated only in the resistor.
Power dissipated at resonance in LCR circuit: Where qm is the maximum value of q and
At resonance Xc – XL= 0, and φ = 0. φ is a phase constant.
Therefore, cos φ = 1 and P = I 2Z = I 2 R. When φ = 0
That is, maximum power is dissipated in a
circuit (through R) at resonance.
LC OSCILLATIONS
The current is
When a capacitor (initially charged) is
connected to an inductor, the charge on
the capacitor and the current in the circuit
exhibit the phenomenon of electrical The LC oscillation is similar to the
oscillations similar to oscillations in mechanical oscillation of a block attached
mechanical systems. to a spring.
Comparison between an electrical system and a
mechanical system
According to Kirchhoff’s loop rule,
But
Therefore
*******
