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Alternating Current and Voltage
Alternating Current (A.C.)
(i) Electric current, which keeps on changing in magnitude and direction periodically is defined as alternating current.
(ii) It obeys Ohm’s law and Joule’s heating law.
(iii) A.C. source is represented by the symbol.
∼
Alternating Voltage
The voltage, which changes in magnitude and direction with respect to time is defined as alternating voltage.
Instantaneous Value Of A.C. (I)
It is the electric current flowing at any instant of time t in an A.C. circuit.
Amplitude Of A.C. Or Peak Value (I 0 ) Or (Im):
It is the maximum value of A.C. The value of A.C. becomes maximum twice in one cycle.
The average value of A.C. (I av ) = The average of all the instantaneous values of an alternating currents over one complete cycle is called average value.
It is zero for one full cycle since there are equal positive
and negative half cycles. The average current for half cycle is 2I 0 /p.
Iav = 0 0
I
= I
p
Average value of a function over a period of T is given
by (^) ( ) ( )
0
T F t F t dt T
< > = (^) ∫
Eg:-
sin 2
< w t > =
cos 2
< w t > =
< sin 2 w t > = (^0) ; < cos 2 w t > = 0
R.m.s. Value of A.C. (Irms) Or Effective Value Of A.C. (Ieff):
It is equal to that direct current which produces same heating in a resistance as is produced by the A.C. in same resistance during same time.
\ I
eff
= I
rms
0
2
I
Form factor =
rms value
average value over half cycle
rms 0
av 0
I I
I I
p
A.C. Through A Resistor
(i) The instantaneous value of alternating emf is E = E 0 sinwt R
~
(ii) The instantaneous value of alternating current is
I = I 0 sinwt
Peak value of current, 0 0
E
I =
R
Phasor diagrams:-
I, E
Current
E.m.f
O wt
2p p 3p (^22)
p
1 2 2
0
0
∫
∫
T
rms (^) T
I dt
I
dt
ALTERNATING CURRENT
Chapter
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2 CUET 2023-24 P
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I R
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The value of impedance (Z)
V
I
= is equal to R and reactance (X) is zero.
A.C. Through an Inductor
L
~
E = E 0 sinwt 0 sin^ 2
I I t
p w
\ Peak value of current
0 0
E
I
w L
Inductive Reactance (XL)
Its value is XL = wL = 2pfL.
It by passes D.C. but offers finite impedance to the flow of A.C.
Phasor diagram
I, E E.m.f
O p
2 p wt
Current
3 2
p 2
p
V (^) L
p/2 I (^) L
Hence we conclude that in a purely inductive circuit (zero ohmic resistance the current lags behind the applied voltage by an angle p/2.
A.C. Through a Capacitor
C
~
E = E 0 sinwt
0 sin^ 2
I I t
p = (^) w +
Peak value of current, 0 0 1
E
I
C
w
Capacitive Reactance (Xc)
X C
C fC
w p
Phasor diagram
I (^) C
V (^) C
p/
In a purely capacitive circuit the current leads the applied emf by an angle of p/2.
A.C. Through LR Series Circuit
L
V (^) L V (^) R
R
E
I
~
E = E
0 sinwt
I = I (^) 0 sin ( w − f t )
where
1 XL
tan
R
− ^
f =
p < f <
Peak value of current, 0 0 0 2 2 2 2 2 L
E E
I
R X R L
E
R f L
where X L = wL = 2pf L is inductive reactance
Phasor Diagram VL
V
f V^ R I
2 2 rms L rms
E
Z R X
I
A.C. Through CR Series Circuit
E = E 0 sinwt
I = I (^) 0 sin ( w + f t )
where
1 tan
XC
R
− ^
f = (^)
p − < f <
4 CUET 2023-24 P
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The average power dissipated in an a.c. circuit is also called
the true power and Irms Erms is called apparent power or virtual
power.
In a pure resistive circuit true power = appearent power.
In a pure inductive circuit
P = E
rms
I
rms cos^0 2
p =
Thus, no power loss takes place in a circuit having inductor only.
In a pure capacitive circuit
P = ErmsIrms cos 0 2
p =
Thus, no power loss takes place in a circuit having capacitor only.
In any circuit power factor cos
R
Z
f =
For RC circuit, 2 2 2
cos 1
R
R
C
f =
w
For LR circuit, 2 2 2
cos
R
R L
f =
For LCR circuit, 2 2
cos 1
R R
Z R L C
f = =
In series LCR circuit power is dissipated only in resistor which becomes maximum at resonance.
Transformer
A transformer works on the principle of mutual induction. It is a static device that is used to increase or decrease the voltage in an AC circuit. On a laminated iron core two insulated copper coils called primary and secondary are wound. If the primary is connected to an alternating source of emf, by mutual induction, an emf is induced in the secondary.
Flux loss
(Magnetic flux loss can be minimized by winding the primary and secondary coils one over the other.)
When efficiency of transformer is 100%
(Transformer ratio or turn ratio) s s^ p
p p s
E N I
K
E N I
where E s
, I
s
, N
s , are the potential, current and number of turn of secondary coil. E p
, I
p
, N
p , are the potential, current and number of turn of primary coil.
Efficiency of Transformer
η =^
output mechanical power
Input electric power
η =^ =
output power
Input power
E I
E I
s s
P P
