Alternating Current
DESIGNED
By
Edustudy point
Alternating Current: An Alternating current(AC) is one whose magnitude changes
continuously with time between zero and a maximum value & whose direction reverse
periodically is known an Alternating current.
The simplest type of AC is one which varies with time simple harmonically. It is represented
by i = ioSinωt or i = ioCosωt
Where, i = instantaneous current
io = maximum current
t = time
T = time period
Alternating emf: The emf (voltage) whose magnitude changes continuously with time
between zero and a maximum value & whose direction reverse periodically is known an
Alternating emf.
• The instantaneous value of alternating emf is represented by
E = EoSinωt or E = EoCosωt
• Graphical representation of E is same as i.
Some important terms:-
➢ Angular frequency: It is the angular displacement of an element per unit time.
𝟐𝝅
ω = = 2πf where, T = time period & f = frequency
𝑻
➢ Amplitude: The AC current varies in magnitude and reverse in direction periodically, the
max value of current is called the Amplitude of the current.
• It is represented by io.
➢ Periodic time: the time taken by alternating current to complete one cycle of variation
is called the “periodic-time” of the current.
𝟐𝝅
T =
𝝎
➢ Frequency: The no. of cycles completed by an AC in one second is called the frequency
of the current. The frequency of the domestic alternating current is 50 Hz.
𝟏 𝝎
f = =
𝑻 𝟐𝝅
➢ Mean or Average value: It is the quantity which is defined as the ‘mean value’ of
alternating current. It is given by-
𝟐
imean = io = 0.637 io
𝝅
➢ Root-mean square value: The RMS is the square root of the mean square, which is the
arithmetic mean of the squares of a group of values.
𝑖𝑜 𝐸𝑜
irms = = 0.707 io Erms = = 0.707 Eo
√2 √2
➢ Phasors: A phasor is a vector which
rotates about the origin with angular
speed ω. The vertical components of
phasors V and I represent the sinusoidally
varying quantities v and i.
AC voltage applied to a Resistor: We consider a
source which produces sinusoidally varying potential
difference across its terminals. this potential difference
also called AC voltage, be given by
V = Vm Sinωt
Vm Sinωt = iR
𝑉𝑚
i = Sinωt
𝑅
So, i = im Sinωt
➢ Power:
Instantaneous Power, P = i2R = im2R Sinωt
_ 1 2
Average power, P = 𝑖 𝑅
2 𝑚
• Voltage and current are in same phase.
AC voltage applied to a Inductor: An AC source connected to an inductor. Let the
voltage across the source be
V = Vm Sinωt
𝑑𝑖
Using the Kirchhoff’s loop rule, V - L =0
𝑑𝑡
𝑑𝑖
V = L
𝑑𝑡
Ldi = Vdt
Ldi = VmSinωtdt
Integrate above equation both side, we get
𝑉𝑚 𝐶𝑜𝑠𝜔𝑡
Li = -
𝜔
𝑉𝑚 𝐶𝑜𝑠𝜔𝑡
i =-
𝐿𝜔
𝜋 𝜋
𝑉𝑚 𝑆𝑖𝑛(𝜔𝑡− 2 ) 𝑉𝑚 𝑆𝑖𝑛(𝜔𝑡− 2 )
i = = where, XL = Lω
𝐿𝜔 𝑋𝐿
𝜋 𝑉𝑚
i = im 𝑆𝑖𝑛(𝜔𝑡 − 2 ) where, im =
𝑋𝐿
➢ Inductive Reactance (XL): It is the opposition offered by inductor to the flow of
current.
XL = Lω = 2πvL
Inductive Reactance [ Resistance due to an inductor].
• Unit of XL is ohm(Ω).
• Current Lags the voltage by π/2.
➢ Power:
𝑖𝑚 𝑣𝑚
Instantaneous Power, PL = iv = - Sin(2ωt)
2
_
Average power, P = 0
AC voltage applied to a Capacitor: An AC source 𝜖 generating AC voltage
(V = VmSinωt) connected to a capacitor only a purely capacitive ac circuit. Let q be the
charge on the capacitor at any time t. The instantaneous voltage v across the capacitor is
𝑞
v =
𝐶
𝑞
Applying Kirchhoff’s loop rule, VmSinωt =
𝐶
q = CVmSinωt
𝑑𝑞 𝑑
for current, i = = CVmSinωt
𝑑𝑡 𝑑𝑡
𝑽𝒎 𝑪𝒐𝒔𝝎𝒕
i = CVmCosωt ω =
𝟏/𝑪𝝎
𝑽𝒎 𝑪𝒐𝒔𝝎𝒕
i = (Capacitance reactance) Xc = 1/Cω
𝑿𝒄
i = imCosωt
i = im 𝑺𝒊𝒏(𝝎𝒕 + 𝝅𝟐)
➢ Capacitive Reactance (Xc): Resistance due to capacitor.
𝟏 𝟏
Xc = =
𝝎𝑪 𝟐𝝅𝒗𝑪
• Unit of Xc is ohm(Ω).
• Current leads voltage by π/2
➢ Power:
𝑖𝑚 𝑣𝑚
Instantaneous Power, Pc = iv = Sin(2ωt)
_ 2
Average power, P = 0
AC voltage applied to a series LCR circuit: we
know that current and voltage remain in same phase in
case of resistor & in case of inductor and capacitor there
is an phase difference of π/2 in voltage and current.
Here, VR = iR , VL = iXL , VC = iXC
From phasor diagram
V2 = VR2 + (VL – VC)2
V2 = (iR)2 + (iXL – iXC)2
𝑉
= √𝑅 2 +(𝑋𝐿 − 𝑋𝐶 )2 = Z
𝑖
𝑉
= Z (Impendance) where, Z = √𝑅 2 +(𝑋𝐿 − 𝑋𝐶 )2
𝑖
Here, the potential difference leads the current by an angle, So
V = VmSin(ωt + ϕ)
𝑉𝐿 −𝑉𝐶
tan ϕ = | |
𝑉𝑅
𝑉𝐿 −𝑉𝐶
ϕ = tan-1 | |
𝑉𝑅
➢ Resonance: A LCR series circuit is said to be in the resonance condition when the
current through it has its maximum value.
Condition for resonance is: XL = XC
𝟏
ωL =
𝝎𝑪
𝟏
ω2 =
𝑳𝑪
𝟏
ω = ω = 2πf
√𝑳𝑪
𝟏
f = where, f = frequency
𝟐𝝅√𝑳𝑪
• The frequency at which the current amplitude I0 attains a peak value is called Natural
or Resonant frequency.
• At resonance frequency, Z = R (Due to XL = XC)
• Resonance occurs only in Series LCR circuit.
➢ Q–factor: The Sharpness of Resonance is
measured by a coefficient called the quality
or Q-factor.
It is defined as the ratio of the voltage
drop across the inductance (or capacitance)
at resonance to the applied voltage.
1 𝐿
Q = √
𝑅 𝐶
Power in AC circuit (Power factor): The rate of dissipation of energy in an electrical
circuit is called the power. The power of an AC circuit depends upon the phase difference
between the voltage and the current.
• The instantaneous Power in the circuit,
𝑣𝑚 𝑖𝑚
Pinst = [𝐶𝑜𝑠𝜙 − 𝐶𝑜𝑠(2𝜔𝑡 + 𝜙)]
2
• The Average Power in the circuit,
Pavg = Vrms Irms Cosϕ
Cos ϕ is known as the power factor of the circuit and its value is depends on the
nature of the circuit.
𝑅
Cos ϕ = where, R = resistance & Z is Impendance.
𝑍
Transformer: It is a device which is either used to increase or decrease the voltage in AC
circuits through mutual induction. It works only on AC.
• A transformer consists of two coils wound on the same core.
• The Coil connected to input is called primary while the other connected to output is
called secondary coil.
• An Alternating current passing through the primary creates the continuously changing
flux through the core. This changing flux includes an alternating emf in the secondary.
• As magnetic field lines are closed curves, the flux per turn of primary must be equal
to flux per turn of the secondary.
𝜙𝑃 𝜙𝑠
Therefore, =
𝑁𝑃 𝑁𝑠
1 𝑑𝜙𝑃 1 𝑑𝜙𝑠
=
𝑁𝑃 𝑑𝑡 𝑁𝑠 𝑑𝑡
𝑑𝜙
As, V is directly proportional to
𝑑𝑡
𝑉𝑠 𝑁𝑠
So, = …1
𝑉𝑃 𝑁𝑃
In an ideal transformer, there is no loss of power. Hence
𝑉𝑠 𝐼𝑃
= …2
𝑉𝑃 𝐼𝑠
From Eqn 1 & 2
𝑉𝑠 𝐼𝑃 𝑁𝑠
= =
𝑉𝑃 𝐼𝑠 𝑁𝑃
