Key Points - Alternating Current

- Alternating Current:
The current whose magnitude changes with time and direction reverses periodically is called alternating current. a) Alternating emf E and current I at any time am given by:
E = E0 sin ωt
Where E0 = NBAω
I = I 0 sin(ωt − φ )
Where I 0 =
NBAω
R

2. T
Where T is the time period.
Values of Alternating Current and Voltage a) Instantaneous value:
It is the value of alternating current and voltage at an instant t. b) Peak value:
Maximum values of voltage E0 and current I0 in a cycle are called peak values. c) Mean value:
For complete cycle,
ω = 2π n =

- T
< E >=
1
Edt = 0
T ∫0
T
1
< I >= ∫ Idt = 0
T 0
Mean value for half cycle: Emean =

2.E0
π d) Root – mean- square (rms) value:
E
Erms = (< E 2 >)1/2 = 0 = 0.707 E0 = 70.7% E0
2
I
I rms = (< I 2 >)1/2 = 0 = 0.707 I 0 = 70.7% I 0
2
RMS values are also called apparent or effective values.

- Phase difference Between the EMF (Voltage) and the Current in an AC Circuit a) For pure resistance:
The voltage and the current are in same phase i.e. phase difference φ = 0 b) For pure inductance:
The voltage is ahead of current by
π

2.i.e. phase differenceφ = +
π
2
. c) For pure capacitance:
The voltage lags behind the current by

- 

2.i.e. phase difference φ = −
π

Reactance:
Reactance a)
E
E E
X = = 0 = rms ± π / 2
I
I0
I rms b)

- π
Inductive reactance
X L = ω L=2π nL
Capacitive reactance c)
1
1
XC =
=
ωC 2π nC

Impedance:
Impedance is defined as,
Z=
E E0 Erms
=
=
φ
I
I0
I rms
Where φ is the phase difference of the voltage E relative to the current I. a) For L – R series circuit:
Z RL = R 2 + X L2 = R 2 + ω L2
? ωL ?
−1 ? ω L ? tan φ = ?
? orφ = tan ?
?
? R ?
? R ? b) For R – C series circuit:
? 1 ?
Z RC = R 2 + X C2 = R 2 + ?
?
? ωC ? tan φ =
2
1
? 1 ?
Or φ = tan −1 ?
?
ωCR
? ωCR ? c) For L – C series circuit:
Z LCR = R 2 + ( X L − X C ) 2

1.?
?
= R2 + ?ω L −
ωC
?

- 2

1.?
?
1
?
ωL −
?ωL −
?
?
ωC ?
ωC tan φ = ?
Or φ = tan −1 ?
R
R
?
?

Conductance:
Reciprocal of resistance is called conductance.
?
?
?
?
?
2

1.mho
R

- Power in and AC Circuit: a) Electric power = (current in circuit) x (voltage in circuit)

P = IE b) Instantaneous power:
Pinst = Einst x Iinst c) Average power:
1
Pav = E0 I 0 cos φ = Erms I rms cos φ

2.d) Virtual power (apparent power):
1
= E0 I 0 = Erms I rms
2

- Power Factor: a) Power factor
P
R cos φ = av =
Pv Z
G= b) For pure inductance
Power factor, cosφ =1 c) For pure capacitance
Power factor, cosφ =0 d) For LCR circuit
Power factor, cosφ =
R

1.?
?
R + ?ωL −
ωC
?
2
2

- 

1.?
?
X = ?ωL −
ωC
?
Wattless Current:
The component of current differing in phase by

- π

2.relative to the voltage, is called wattles current.
The rms value of wattless current:
I
= 0 sin φ
2
= I rms sin φ =
I0 ? X ?
? ?

2.Z ?
Choke Coil: a) An inductive coil used for controlling alternating current whose self- inductance is high and resistance in negligible, is called choke coil. b) The power factor of this coil is approximately zero.
Series Resonant Circuit a) When the inductive reactance (XL) becomes equal to the capacitive reactance (XC) in the circuit, the total impedance becomes purely resistive (Z=R). b) In this state, the voltage and current are in same phase ( φ = 0), the current and power are maximum and impedance is minimum. This state is called resonance. c) At resonance,
1
ωr L =
ωr C

- Hence, resonance frequency is,

1.fr =

2. LC d) In resonance, the power factor of the circuit is one.

Half – Power Frequencies:
Those frequencies f1 and f2 at which the power is half of the maximum power (power at resonance), i.e., f1 and f2 are called half – power frequencies.
1
Pmax
2
I
I = max
2
P
∴ P = max
2
Band – Width: a) The frequency interval between half – power frequencies is called band – width.
∴ Bandwidth ?f = f 2 − f1
P=
b) For a series LCR resonant circuit,

1.R
?f =

2. L

Quality Factor (Q):
Maximum energy stored
Q = 2π ×
Energy dissipated per cycle

2. Maximum energy stored
=
×
T
Mean power dissipated
Or
ωL fr f
1
Q= r =
=
= r
R
ωr CR ( f 2 − f1 ) ?f