Chapter-19

Alternating Currents:
Direct Current: A current whose flows in constant magnitude with time in the same direction is called
direct current(DC). A battery is a source of emf in dc circuit.
Alternating Current: The current whose magnitude is changing continuously and reverse its direction
periodically with time is called alternating current(AC).
Its magnitude is at any instant of time t, is given by
𝐼 = 𝐼𝑜 sin 𝜔𝑡
Where 𝐼𝑜 is maximum or peak value of ac also called amplitude.
2𝜋
ω = 2𝜋𝑓 = Where ω is angular velocity, f is frequency and t is time period of AC.
𝑇

prepared by Resham Chapagain 1

 Average or Mean value of alternating current:
Average or mean value of AC over half cycle is that value of steady current(DC) which
when passed through a circuit would send the same amount of electric charge in
certain time interval as is done by AC in the same circuit in the same interval of time.
Let the instantaneous AC be given by
𝐼 = 𝐼𝑜 sin 𝜔𝑡
The charge sent by the alternating current (𝐼) in time dt is given by
dq = 𝐼 dt
dq = 𝐼𝑜 sin 𝜔𝑡 dt ------------1
If q is the total charge sent by positive half cycle of ac then

𝑇
𝑇/2
q= 0
2 𝑑𝑞 = 0
𝐼𝑜 sin 𝜔𝑡 𝑑𝑡

prepared by Resham Chapagain 2

prepared by Resham Chapagain 3
 𝐼0 𝑇
∴ q= -------------2
𝜋
If 𝐼 m be the mean value of AC over positive half cycle then the charge sent by it in time interval
𝑇 is given by
2
𝑇
∴ q = 𝐼m . --------3
2

from eqn 2and we get

𝐼0 𝑇 𝑇
= Im .
𝜋 2
2𝐼0
∴ Im = ----------------4
𝜋
∴ 𝐼 m = 0.637 𝐼0
Thus, average value of current over positive half cycle of AC is 63.7% of maximum value of
current.
Similarly , for next negative half cycle ,average value of AC is 63.7% in the opposite direction.
Hence, average value of AC over a complete cycle is zero.
A similar relation for average value of alternating emf is for half cycle is
2𝐸
∴ Em = 0 = 0.637 Eo
𝜋
prepared by Resham Chapagain 4
 Root-Mean square(RMS)value of AC:
The root mean square (Irms ) value of AC is defined as that value of steady currents(DC) which when
passed through a resistance would produce the same amount a heat in a given time as is done by
the ac when passed through the same resistance for the same time.it is also called virtual value or
effective value of AC.

Let the instantaneous value of AC is given by

𝐼 = 𝐼𝑜 sin 𝜔𝑡
If this ac flows through a resistance R for a small time dt, then
small amount of heat produced dH is given by
𝑑𝐻 = 𝐼2 𝑅 𝑑𝑡
𝑑𝐻 = 𝐼𝑜2 sin2 𝜔𝑡 𝑅 𝑑𝑡

Total amount of heat produced in one complete cycle o to T is given by

𝑇 𝑇 2 2 𝜔𝑡 𝑅
H= 0
𝑑𝐻 = 0
𝐼𝑜 sin 𝑑𝑡

𝐼𝑜2 𝑅 𝑇 prepared by Resham Chapagain 5

∴ H = ------------- 1
prepared by Resham Chapagain 6
 if Irms is the rms value of ac then heat produced in the same resistance R in the same
time is given by
H = I2rms R T -------------2
from eqn 1 and 2
2 Io2 R T
I rms R T =
2
𝐼𝑜2
𝐼2𝑟𝑚𝑠 =
2
𝐼
∴ 𝐼𝑟𝑚𝑠 = 𝑜 = 0.707 𝐼𝑜
2
Hence, the rms value of AC is equal to 70.7% of peak value of AC.
Similarly, the rms value of alternating voltage is given by
𝐸𝑜
∴ 𝐸𝑟𝑚𝑠 = = 0.707 E𝑜
2

moreover, AC and voltage continuously vary with time and their average value of ac
over full cycle is Zero .But the rms value of current and voltage is not zero for complete
cycle. So, ac ammeters and ac voltmeter record rms value of alternating currents and
voltage respectively.
prepared by Resham Chapagain 7
Phasor diagram:

A vector diagram in which alternating voltage or current are treated as vectors and
represented by straight lines with arrows and difference of phase angle between them
are called phasor diagram.

The straight line representing the current while rotates in the phase diagram with the
same angular frequency (w) as the displacement of the current in the AC current is
called phasor.

Wave diagram:

It is the displacements of current and voltage in a circuit are represented in a graph by

their wave forms along with their phase difference, the diagram is called wave diagram.

prepared by Resham Chapagain 8

 Ac through Resistance only
Let us consider an ideal resistor of resistance R connected between the terminals of an
ac source E and frequency f is shown in fig.
The applied alternating emf is given by
𝐸 = 𝐸𝑜 sin 𝜔𝑡 ----------1
Where Eo is peak voltage.
Let I be current in the circuit at any instant of time t; so
Potential difference across the resistor
𝐸=𝐼𝑅
𝐸 𝐸𝑜 sin 𝜔𝑡
∴ I= = = I𝑜 sin 𝜔𝑡 -------------2
𝑅 𝑅
𝐼𝑜 is peak value of alternating current.
comparing eq1 and 2 we see that ,
The voltage and current both are proportional to sin 𝜔𝑡 , so the current and voltage
are in phase with each other.
Thus, in purely resistive circuit, there is no phase difference between voltage and
current
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prepared by Resham Chapagain 10
 AC through Inductance only :
Let us consider a pure inductor having self inductance L connected to an AC source of E volts and frequency f
as shown in fig.
The applied alternating emf is given by
𝐸 = 𝐸𝑜 sin 𝜔𝑡 ---------1
𝑑𝐼
The induced emf across the inductor, V= -𝐿 ;
𝑑𝑡
which oppose the growth of current in the circuit.
As there is no potential drop across the circuit. So, we can write;
𝒅𝑰
𝑬 + (-𝑳 ) =0
𝒅𝒕
𝒅𝑰
or 𝑳 =E
𝒅𝒕
𝒅𝑰 𝑬 𝑬𝒐 𝒔𝒊𝒏 𝝎𝒕
or, = =
𝒅𝒕 𝑳 𝑳
𝑬
𝒅𝑰 = 𝒐 𝐬𝐢𝐧 𝝎𝒕 𝒅𝒕
𝑳
integrating both sides ,we get

prepared by Resham Chapagain 11

 2
prepared by Resham Chapagain 12
Comparing eqn 1 and 2 we find that in AC circuit containing L
only, current I lags behind the voltage E by phase angle
90o. conversely, voltage across L leads the current by phase
angle of 90o.

prepared by Resham Chapagain 13

Inductive reactance (XL):
𝐸𝑜 𝐸𝑂
Comparing 𝐼𝑜 = with 𝐼𝑜 =
𝜔𝐿 𝑅
We conclude that 𝜔𝐿 has the dimension of resistance.
The term 𝜔𝐿 is known as inductive reactance
represented by XL.
The inductive reactance is the effective opposition
offered by the inductor to the flow of AC.
Thus, XL = 𝜔 𝐿 = 2𝜋𝑓𝐿 where f is frequency of ac
supply.
In d.c circuit f = 0 then XL =0 .
The unit of inductive reactance is ohm in S.I system.
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 AC through capacitor only:
Let us consider a pure capacitor having capacitance C is connected to an AC source of emf E volts and
frequency f as shown in fig.
The applied alternating emf at any instant of time is given by
𝐸 = 𝐸𝑜 sin 𝜔𝑡 --------- ( i)
Let q be the charge on the capacitor at any instant of time t due to emf E is

prepared by Resham Chapagain 15

Comparing eqn (i) and (ii) it is clear that , in a circuit containing
𝜋
capacitor only, the current leads voltage by . In other words,
2
𝜋
voltage lags behind the current by phase angle .
2

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 Capacitive Reactance( XC):
𝐸𝑂 𝐸𝑜
Comparing 𝐼𝑜 = 𝜔𝑐𝐸𝑂 = 1 and 𝐼𝑂 =
𝜔𝑐 𝑅
We conclude that 1 𝜔𝑐 has the dimension of resistance R. The term 1
𝜔𝑐
is known as Capacitive reactance represented by XC .
1 1
∴ XC = =
𝜔𝑐 2𝜋𝑓𝑐
The capacitive reactance is the effective opposition offered by a capacitor
to the flow of ac in the circuit.
Its unit is ohm in SI system.
Again,
1 1
for dc circuit, f = 0 so Xc = = =∞
𝜔𝑐 2𝜋𝑓𝑐
Hence, capacitor will block d.c.

prepared by Resham Chapagain 17

 AC through L and R in series(LR circuit):
Let a pure inductance L and a pure resistance R are
connected in series to an alternating emf. Let E be the
rms value of applied emf and 𝐼 be the rms value of the
current flowing in the circuit. Let VR and VL
be the potential drop across R and L respectively.

𝜋
The potential drop across Inductor VL = 𝐼 𝑋𝐿 , the potential leads the current by phase.
2
The potential drop across Resistor VR = 𝐼 𝑅 , the potential and current are in same phase.
𝜋
Since , the VL leads the 𝐼 by phase in inductive circuit
2 C
𝜋
so the potential has been represented by OC at of OA
2
Since, the VR and 𝐼 are in the same phase in case of
resistive circuit so they have been represented by
the same line OA.
Now From vector diagram resultant OB is given by
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 𝐸2 = 𝑉2𝑅 + 𝑉2𝐿
𝐸 = 𝑉2𝑅 + 𝑉2𝐿
= 𝐼𝑅 2 + 𝐼𝑋𝐿 2

= I2 𝑅2 + 𝑋𝐿2
𝐸 𝐸
∴ 𝐼= = which is current in the LR circuit.
𝑅2+𝑋𝐿2 𝑍
Where Z is called impedance of the L-R circuit. And is given by
Z = 𝑅2 + 𝑋𝐿2

Let ∅ 𝑏𝑒 𝑡ℎ𝑒 angle by which the current lags on the applied voltage. Then
𝑉𝐿 𝐼𝑋𝐿 𝑋𝐿
tan ∅ = = =
𝑉𝑅 𝐼𝑅 𝑅

−1 𝑋𝐿
∴ø= tan ( )
𝑅
1
Admittance of circuit is , =
𝑍

prepared by Resham Chapagain 19

AC through Inductance, Capacitance and Resistance (LCR circuit)
Let us consider an alternating source of
emf E is connected to a series combination
of a resistor of resistance R, inductor of
inductance L and a capacitor of capacitance C.
Let E be the rms value of applied emf and I be
the rms value of the current flowing in the circuit.

The voltage drop across the resistor is, VR = I R ; (VR is in phase with I)
The voltage across the inductor coil is VL = I XL ; ( VL leads I by π/2)
The voltage across the capacitor is, VC = IXC ; ( VC lags behind I by π/2)

VL and VC are 180o out of phase with each other and the resultant of VL and VC is (VL – VC), assuming
the circuit to be predominantly inductive. The applied voltage ‘E’ equals the vector sum of VR, VL
and VC.
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Now, from vector diagram
𝑂𝐵2 = 𝑂𝐴2 + 𝐴𝐵2
𝐸2 = 𝑉𝑅2 + 𝑉𝐿 − 𝑉𝐶 2

Or, 𝐸= 𝑉𝑅2 + 𝑉𝐿 − 𝑉𝐶 2

or, 𝐸= (𝐼𝑅)2 + 𝐼𝑋𝐿 − 𝐼𝑋𝐶 2

𝐸 = 𝐼 𝑅2 + 𝑋𝐿 − 𝑋𝐶 2

𝐸 𝐸
∴𝐼 = = which is current in LCR circuit.
𝑅2 + 𝑋𝐿 −𝑋𝐶 2 𝑍

Where Z is called Impedance of the LCR circuit and is given by

𝑍= 𝑅2 + 𝑋𝐿 − 𝑋𝐶 2

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Now ,

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 Series resonance or Electrical resonance:
The LCR series circuit is said to be in electrical resonance when the inductive reactance equals the
capacitive reactance. In this condition, the impedance of the circuit becomes minimum and the current
becomes maximum.
so for electrical resonance, we have,
𝑋𝐿 = 𝑋𝐶
1
𝜔𝑜𝐿 =
𝜔𝑜𝐶
1
𝜔𝑜2 =
𝐿𝐶
1
𝜔𝑜 =
𝐿𝐶
1
2𝜋𝑓𝑜 =
𝐿𝐶
1
∴ 𝑓𝑜 =
2𝜋 𝐿𝐶
This frequency is called resonant frequency .
The impedance of LCR circuit is given by 𝑍= 𝑅2 + 𝑋𝐿 − 𝑋𝐶 2

prepared by Resham Chapagain 23

 𝐸
Also 𝐼 =
𝑅2 + 𝑋𝐿 −𝑋𝐶 2

for electrical resonance , XL = XC , then

impedance , Z = R
𝑉
current , Ima𝑥 =
𝑅
Thus ,LCR series circuit at resonance behaves as a purely resistive circuit.

Quality factor( Q- factor):

The selectivity or sharpness of a resonant circuit is measured by the quality factor or Q factor. In
other words it refers to the sharpness of tuning at resonance.
The Q factor of a series resonant circuit is defined as the ratio of the voltage across a inductor L
or capacitor C to the applied voltage(voltage across R).

prepared by Resham Chapagain 24

 That is , Q is just a number .Thus, Q-
factor may also be taken as voltage
multiplication factor of the circuit. As R
is increased, Q factor of the circuit is
decreased.

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 Power in an ac circuit:
In an AC circuit the current and emf vary continuously with time. Therefore, power at a given instant of time
is calculated and then its mean is taken over a complete cycle. Thus, we define instantaneous power of an AC
circuit as the product of the instantaneous emf and the instantaneous current flowing through it.

The instantaneous value of emf and current is given by

prepared by Resham Chapagain 26

 ∴ Pav = apparent power × power factor
Where Apparent power = Erms Irms and Power factor= cos ∅
The average power of an ac circuit is also called the true power of the circuit.
What is Wattless current?
The current consuming no power in the circuit is called wattless current. This is possible only when
𝜋
the phase difference between the current and voltage is .
2
it can be obtained by using inductor or capacitor in the circuit. that is, if a circuit doesn't have any
resistance , no power is consumed in the circuit.
Choke coil:
Choke coil is device used in AC circuit to control the current without wastage of electrical energy in
the form of heat. In case of AC circuit, current can be reduced by using resistor, there will be
wastage in electrical energy but when an inductance is used ,no electrical energy will be wasted.
𝜋
the reason is that the alternating emf leads the current by phase angle and average power
2
consumed will be zero.
𝜋
∴ 𝑃 = 𝑉 𝐼 𝑐𝑜𝑠 = 0 27
2
prepared by Resham Chapagain 28