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ALTERNATING CURRENTS
- Alternating EMF and Current
- Average or Mean Value of Alternating EMF and Current
- Root Mean Square Value of Alternating EMF and Current
- A C Circuit with Resistor
- A C Circuit with Inductor
- A C Circuit with Capacitor
- A C Circuit with Series LCR – Resonance and Q-Factor
- Graphical Relation between Frequency vs X
L
, X
C
- Power in LCR A C Circuit
- Watt-less Current
- L C Oscillations
- Transformer
- A.C. Generator
Created by C. Mani, Principal, K V No.1, AFS, Jalahalli West, Bangalore
Alternating emf:
Alternating emf is that emf 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.
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
0
π/2 π 3 π/2 2 π 5 π/2 3 π 7 π/2 4 πθ = ωt
E ,I
E
0
I
0
E = E 0
sin ωt
I = I 0
sin ωt
E , I – Instantaneous value of emf and current
E
0
, I
0
- Peak or maximum value or amplitude of emf
and current
ω – Angular frequency t – Instantaneous time
ωt – Phase
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
0
π/2 π 3 π/2 2 π 5 π/2 3 π 7 π/2 4 πθ = ωt
E ,I
E
0
I
0
E = E 0
cos ωt
I = I 0
cos ωt
Symbol of
AC Source
Root Mean Square or Virtual 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 = I
2
R dt = I 0
2
R sin
2
ωt dt
H = ∫ I
0
2
R sin
2
ωt dt
0
T
H = I
0
2
RT / 2
If I v
be the virtual value of AC, then
I
v
= I
rms
= I
eff
= I
0
/ √ 2 = 0.707 I
0
= 70.7 % I
0
Root Mean Square or Virtual or Effective Value of
Alternating emf:
E
v
= E
rms
= E
eff
= E
0
/ √2 = 0.707 E
0
= 70.7 % E
0
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.
2. Do not use the above formulae if the time interval under the consideration is less than
one period.
(After integration, ω is replaced with 2 π / T)
H = I
v
2
RT
0
π 2 π 3 π 4 π
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
π/2 3 π/2 5 π/2 7 π/2 θ = ωt
E 0
E v
E m
Relative Values Peak,
Virtual and Mean Values of
Alternating emf:
E
m
= E
av
= 0.637 E
0
E
v
= E
rms
= E
eff
= 0.707 E
0
Tips:
- The given values of alternating emf and current are virtual values unless
otherwise specified.
i.e. 230 V AC means E v
= E
rms
= E
eff
= 230 V
- AC Ammeter and AC Voltmeter read the rms values of alternating current
and voltage respectively.
They are called as ‘hot wire meters’.
- The scale of DC meters is linearly graduated where as the scale of AC
meters is not evenly graduated because H α I
2
x
AC Circuit with a Pure Inductor:
L
E = E
0
sin ωt
E = E
0
sin ωt
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
π/2 π 3 π/2 2 π 5 π/2 3 π 7 π/2 4 πθ = ωt
E ,I
E
0
I
0
E = E 0
sin ωt
I = I 0
sin (ωt - π / 2)
E
0
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)
E
0
sin ωt = L (dI / dt)
dI = ( E 0
/ L) sin ωt dt
I = ∫ ( E
0
/ L) sin ωt dt
I = ( E
0
/ ωL) ( - cos ωt )
I = I
0
sin (ωt - π / 2)
(where I 0
= E
0
/ ωL and X L
= ωL = E 0
/ I
0
X
L
is
Inductive Reactance. Its SI unit is ohm.
I
0
y
Current lags behind emf by π/2 rad.
0
π/
y
AC Circuit with a Capacitor:
E = E
0
sin ωt
E = E
0
sin ωt
T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
0
π/2 π 3 π/2 2 π 5 π/2 3 π 7 π/2 4 πθ = ωt
E ,I
E
0
I
0
E = E 0
sin ωt
I = I 0
sin (ωt + π / 2)
E
0
ωt
q = CE = C E 0
sin ωt
I = dq / dt
= (d / dt) [C E 0
sin ωt]
I = [ E
0
/ (1 / ωC)] ( cos ωt )
I = I
0
sin (ωt + π / 2)
(where I 0
= E
0
/ (1 / ωC) and
X
C
= 1 / ωC = E
0
/ I
0
X
C
is Capacitive
Reactance.
Its SI unit is ohm.
I
0
π/
(^0) x
Current leads the emf by π/2 radians.
C
AC Circuit with L, C, R in Series
Combination:
E = E
0
sin ωt
C
L R
V
L
V
C
V
R
- In R, current and voltage are in
phase.
- In L, current lags behind voltage by
π/
- In C, current leads the voltage by
π/2 V R
V
L
V
C
I
π/
π/
- V
C
V
L
V
R
I
π/
V
L
- V
C
V
R
I
E
Φ
E = √ [V
R
2
V
C
2
]
The applied emf appears as
Voltage drops V
R
, V
L
and V
C
across
R, L and C respectively.
E = √ [V
R
2
– V
C
2
]
I =
E
√ [R
2
– X
C
2
]
Z = √ [R
2
– X
C
2
]
Z = √ [R
2
2
]
tan Φ =
X
L
– X
C
R
tan Φ =
ω L – 1/ωC
R
or
V
C
tan Φ = or
X
L
– X
C
R
tan Φ =
ω L – 1/ωC
R
Special Cases:
Case I: When X
L
> X
C
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 X L
< X
C
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 X
L
= X
C
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.
Power in AC Circuit with L, C, R:
Instantaneous Power = E I
= E
0
I
0
sin ωt sin (ωt + Φ)
= E
0
I
0
[sin
2
ωt cosΦ + sin ωt cosωt cosΦ]
E = E
0
sin ωt
I = I
0
sin (ωt + Φ) (where Φ is the phase angle between emf and current)
If the instantaneous power is assumed to be constant for an
infinitesimally small time dt, then the work done is
dW = E
0
I
0
[sin
2
ωt cosΦ + sin ωt cosωt cosΦ]
Work done over a complete cycle is
W = ∫ E
0
I
0
[sin
2
ωt cosΦ + sin ωt cosωt cosΦ] dt
0
T
W = E
0
I
0
cos Φ x T / 2
Average Power over a cycle is P av
= W / T
P
av
= ( E
0
I
0
/ 2) cos Φ
P
av
= ( E
0
/√2) (I
0
/ √2) cos Φ
(where cos Φ = R / Z
= R /√ [R
2
2
]
is called Power Factor)
P
av
= E
v
I
v
cos Φ
E
v
Power in AC Circuit with R:
In R, current and emf are in phase.
P
av
= E
v
I
v
cos Φ = E
v
I
v
cos 0° = E
v
I
v
Power in AC Circuit with L:
In L, current lags behind emf by π/2.
Φ = - π/
P
av
= E
v
I
v
cos (-π/2) = E
v
I
v
Power in AC Circuit with C:
In C, current leads emf by π/2.
Φ = + π/
P
av
= E
v
I
v
cos (π/2) = E v
I
v
Note:
Power (Energy) is not dissipated in Inductor and Capacitor and hence they
find a lot of practical applications and in devices using alternating current.
P
av
= E
v
I
v
cos Φ
Wattless Current or Idle
Current:
I
v
I
v
cos Φ
I
v
sin Φ
90 °
The component I v
cos Φ
generates power with E
v
However, the component
I
v
sin Φ does not
contribute to power along
E
v
and hence power
generated is zero. This
component of current is
called wattless or idle
current.
P = E
v
I
v
sin Φ cos 90° = 0
q
q 0
q
q 0
Undamped Oscillations Damped Oscillations
If q be the charge on the capacitor at any time t and dI / dt the rate of
change of current, then
or L (d
2
q / dt
2
) + q / C = 0
or d
2
q / dt
2
Putting 1 / LC = ω
2
d
2
q / dt
2
2
q = 0
The final equation represents Simple
Harmonic Electrical Oscillation with
ω as angular frequency.
So, ω = 1 / √LC
or
L dI / dt + q / C = 0
t
t
2 π √LC
f =
Transformer:
Transformer is a device which converts lower alternating voltage at higher
current into higher alternating voltage at lower current.
S Load
Principle:
Transformer is based on
Mutual Induction.
It is the phenomenon of
inducing emf in the
secondary coil due to
change in current in the
primary coil and hence the
change in magnetic flux in
the secondary coil.
Theory:
E
P
= - N
P
dΦ / dt
E
S
= - N
S
dΦ / dt
E
S
/ E
P
= N
S
/ N
P
= K
(where K is called
Transformation Ratio
or Turns Ratio)
For an ideal transformer,
Output Power = Input Power
E
S
I
S
= E
P
I
P
E
S
/ E
P
= I
P
/ I
S
E
S
/ E
P
= I
P
/ I
S
= N
S
/ N
P
Efficiency (η):
η = E
S
I
S
/ E
P
I
P
For an ideal
transformer η
is 100%
P
- Iron Losses:
a) Eddy Currents Losses:
When a changing magnetic flux is linked with the iron core, eddy
currents are set up which in turn produce heat and energy is wasted.
Eddy currents are reduced by using laminated core instead of a solid iron
block because in laminated core the eddy currents are confined with in
the lamination and they do not get added up to produce larger current.
In other words their paths are broken instead of continuous ones.
Solid Core Laminated Core
b) Hysteresis Loss:
When alternating current is
passed, the iron core is
magnetised and demagnetised
repeatedly over the cycles and
some energy is being lost in the
process.
- Losses due to vibration of core: Some electrical energy is lost in the
form of mechanical energy due to vibration of the core and humming
noise due to magnetostriction effect.
This can be minimised by using suitable material with thin hysteresis loop.
S
A.C. Generator:
A.C. Generator or A.C. Dynamo or Alternator is a device which converts
mechanical energy into alternating current (electrical energy).
N
P
Q
R
S
R
1
R
2
B
1
B
2
Load
S
R
R
1
R
2
B
1
B
2
Load
N
Q
P
S
