AC Fundamental & Single Phase AC Circuit

AC Fundamental & Single
Phase AC Circuit
Prof. A. B. Kasar
Dept. of Engineering Sciences
International Institute of Information Technology, I²IT
www.isquareit.edu.in

International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057
Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - info@isquareit.edu.in
Definition of Alternating Quantity
RMS or Effective Value
Average Value
Phasor Representation
AC flowing through Pure resistance
AC flowing through Pure Inductance
AC flowing through Pure Capacitance
Overview of Presentation

Definition of Alternating Quantity
An alternating quantity changes continuously
in magnitude and alternates in direction at
regular intervals of time. Important terms
associated with an alternating quantity are
defined below.

Important Definitions
1. Amplitude
It is the maximum(+ve or -ve) value attained by an alternating quantity.
Also called as maximum or peak value
2. Time Period (T)
It is the Time Taken in seconds required to complete one cycle of an
alternating quantity. Denoted by T
3. Instantaneous Value
It is the value of an alternating quantity at any instant. Instatenios
values denoted by small letters
4. Frequency (f)
It is the number of cycles that occur in one second. The unit for
frequency is Hz or cycles/sec.
The relationship between frequency and time period can be derived as
follows.
Time taken to complete f cycles = 1 second
Time taken to complete 1 cycle = 1/f second
T = 1/f

Important Definitions
5. Angular Frequency (ω)
Angular frequency is defined as the number of radians covered in one
second(i.e the angle covered by the rotating coil). The unit of angular frequency
is rad/sec. ω=2πf
6.Amplitude and Peak-to-Peak Value
Amplitude of a sine wave
Distance from its average to its peak
We use Em for amplitude
Peak-to-peak voltage
Measured between minimum and maximum peaks
We use Epp or Vpp

RMS/EFFECTIVE VALUE
Definition: That steady current which, when flows through a resistor of
known resistance for a given period of time than as a result the same
quantity of heat is produced by the alternating current when flows through
the same resistor for the same period of time is called R.M.S or effective
value of the alternating current.
In other words, the R.M.S value is defined as the square root of means of
squares of instantaneous values.

Alternating Voltage:
Alternating voltage is that voltage which continuously changes in
magnitude and periodically reverses its direction.
Alternating Current:
Alternating current is that current which continuously changes in
magnitude and periodically reverses its direction.
V,I V
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
Vm
Im
θ = ωt
I = Im sin ωt
V= Im sin ωt
V, I – Instantaneous value of emf and current
Vm, Im – Peak or maximum value or amplitude of emf and current
ω – Angular frequency t – Instantaneous time
ωt – Phase

Average or Mean Value of Alternating Current:
Average or Mean value of alternating current over half cycle is that steady
current which will send the same amount of charge in a circuit in the time of
half cycle as is sent by the given alternating current in the same circuit in
the same time.
dq = I dt = Imsin ωt dt
q = ∫ Im sin ωt dt
q = 2 Im / ω = 2 Im T / 2π = Im T / π
Mean Value of AC, Im = Iav = q / (T/2)
Im = Iav = 2 Im / π = 0.637 Im = 63.7 % Im
Average or Mean Value of Alternating emf:Average or Mean Value of Alternating emf:
Vav = 2 Vm/ π = 0.637 Vm = 63.7 % E0
Note: Average or Mean value of alternating current or emf is zero over a
cycle as the + ve and – ve values get cancelled.

Root Mean Square or Effective Value of Alternating Current:
Root Mean Square (rms) value of alternating current is that steady current
which would produce the same heat in a given resistance in a given time as
is produced by the given alternating current in the same resistance in the
same time.
dH = I2R dt = Im
2 R sin2 ωt dt
H = ∫ Im
2 R sin2 ωt dt
0
T
H = Im
2 RT / 2 (After integration, ω is replaced with 2 π / T)
If Iv be the virtual value of AC, then
H = Iv
2 RT Iv = Irms = Ieff = Im / √2 = 0.707 Im = 70.7 % Im
Root Mean Square or Virtual or Effective Value of
Alternating emf: Vrms = Vreff = Vm / √2 = 0.707 Vm = 70.7 % Vm
Note:
1. Root Mean Square value of alternating current or emf can be calculated over any
period of the cycle since it is based on the heat energy produced.
.

Relative Values Peak,
Virtual and Mean Values of
Alternating emf:
Vav = 0.637 Vm
Vrms = Veff = 0.707 Vm
0
π 2π 3π 4π
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
π/2 3π/2 5π/2 7π/2 θ = ωt
Vm
Tips:
1. The given values of alternating emf and current are virtual values unless
otherwise specified.
i.e. 230 V AC means Vrms = Veff = 230 V
2. AC Ammeter and AC Voltmeter read the rms values of alternating current
and voltage respectively.
They are called as ‘hot wire meters’.
3. The scale of DC meters is linearly graduated where as the scale of AC
meters is not evenly graduated because H α I2

AC Circuit with a Pure Resistor:
Vm
I = V / R
= (Vm / R) sin ωt
I = Im sin ωt (where Im = Vm / R and R = Vm / Im)
Emf and current are in same phase.
y
x
0
Im
y
0
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
π 2π 3π 4ππ/2 3π/2 5π/2 7π/2
V = Vm sin ωt
I = Im sin ωt
VmIm
R
V = Vm sin ωt
Vm

AC Circuit with a Pure Inductor:
v = Vmsin ωt
Induced emf in the inductor is - L (dI / dt)
In order to maintain the flow of current, the
applied emf must be equal and opposite to
the induced emf.
e = - L (dI / dt)
V= - e = - L (dI / dt
Vm sin ωt = L (dI / dt)
dI = (Vm / L) sin ωt dt
v = Vm sin ωt
I = ∫ (Vm / L) sin ωt dt
I = (Vm/ ωL) ( - cos ωt )
I = Imsin (ωt - π / 2)
(where Im = Vm / ωL and XL = ωL = Vm / Im)
XL is Inductive Reactance. Its SI unit is ohm.
Current lags behind emf by π/2 rad.
Vm
v = Vm sin ωt
I = Im sin (ωt - π / 2)
0
π 2π 3π 4ππ/2 3π/2 5π/2 7π/2
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
Im
x
0
Vm
Im
π/2
y

AC Circuit with a Capacitor:
v = Vmsin ωt C
q = C = CVm sin ωt
I = dq / dt
= (d / dt) [CVm sin ωt]
I = [Vm / (1 / ωC)] ( cos ωt )
I = Im sin (ωt + π / 2)
(where Im = Vm / (1 / ωC)
XC = 1 / ωC = Vm / Im)
XC is Capacitive Reactance.
Its SI unit is ohm.
Current leads the emf by π/2 radians.
π 2π 3π 4ππ/2 3π/2 5π/2 7π/2
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
I = Im sin (ωt + π / 2)
v = Vm sin ωt
Vm
0
I m
Vm
V,I
x0
Im
Vm
π/2
y
ωtωt

Variation of XL with Frequency:
Im = Vm / ωL and XL = ωL
XL is Inductive Reactance and ω = 2π f
XL = 2π f L i.e. XL α f
XL
f0
Variation of XC with Frequency:
Im = Vm / (1/ωC) and XC = 1 / ωC
XC is Inductive Reactance and ω = 2π f
XC = 1 / 2π f C i.e. XC α 1 / f
XC
f0

SERIES RLC CIRCUIT: C
L R
The applied emf appears as
Voltage drops VR, VL and VC
across R, L and C respectively.
1) In R, current and voltage are in
phase.
2) In L, current lags behind voltage by
π/2
3) In C, current leads the voltage by
π/2
π/2
VRIπ/2
VC
0
VL
π/2
VCV = √ [VR
2 + (VL – VC)2]
I =
E
√ [R2 + (XL – XC)2]
Z = √ [R2 + (XL – XC)2]
Z = √ [R2 + (ω L – 1/ωC)2]
tan Φ =
XL – XC
R
tan Φ =
ω L – 1/ωC
R
VL - VC
VRI
E
Φ
V = √ [VR
2 + (VL – VC)2]
VL
VC
VR

tan Φ =
XL – XC
R
or tan Φ =
ω L – 1/ωC
R
Special Cases:
Case I: When XL > XC i.e. ω L > 1/ωC,
tan Φ = +ve or Φ is +ve
The current lags behind the emf by phase angle Φ and the LCR
circuit is inductance - dominated circuit.
Case II: When XL < XC i.e. ω L < 1/ωC,
tan Φ = -ve or Φ is -ve
The current leads the emf by phase angle Φ and the LCR circuit is
capacitance - dominated circuit.
Case III: When XL = XC i.e. ω L = 1/ωC,
tan Φ = 0 or Φ is 0°
The current and the emf are in same phase. The impedance does
not depend on the frequency of the applied emf. LCR circuit
behaves like a purely resistive circuit.

RESONANCE IN SERIES RLC CIRCUIT
In series RLC circuit, XL & XC are frequency component element, when
frequency varied both XL & XC also varied. At a certain frequency, XL
becomes equal to XC called as series resonance.
XL = XC ……….at resonance
i.e. ω L = 1/ωC, tan Φ = 0 or Φ is 0° and
Z = √ [R2 + (ω L – 1/ωC)2] becomes Zmin = R and Imax = Vm / R
i.e. The impedance offered by the circuit is minimum and the current is
maximum. This condition is called resonant condition of LCR circuit
and the frequency is called resonant frequency.
At resonant angular frequency ωr,
ωr L = 1/ωrC or ωr = 1 / √LC or fr = 1 / (2π √LC)

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AC Fundamental & Single Phase AC Circuit

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