ALTERNATING CURRENT

ALTERNATING AND DIRECT CURRENT

An alternating current (A.C.) is one which periodically changes in magnitude and direction. I = I0 sin (ωt + ϕ)

If ϕ is positive then current leads voltage and if ϕ is negative then current lags the voltage.

The source of alternating emf may be a dynamo or an electronic oscillator.

The alternating emf E at any instant may be expressed as E = E0 sinωt
where ω is the angular frequency of alternating emf and E0 is the peak value or creast value or amplitude of emf. E is the intantanous value.

Direct current (D.C.) is that current which may or may not change in magnitude but it does not change its direction.
ADVANTAGES OF A.C. OVER D.C.
The generation of A.C. is cheaper than that of D.C.
Alternating voltage can be easily stepped up or stepped down by using a transformer.
A.C. can be easily converted into D.C. by rectifier. D.C. is converted to A.C. by an inverter.
A.C. can be transmitted to a long distance without appreciable loss.
AVERAGE VALUE OF ALTERNATING CURRENT
The average value of AC over one full cycle (sine or cosine) is zero since there are equal positive and negative half cycles.
The average current for half cycle is 2I0 /π where I0 is the peak value of current.

2 2
(Eavg= E =0.637 Eo =63.7 % of E o∨I avg= I o=0.637 I o=63.7 % of I o)
π o π
 T
T
Iavg= ∫ sin kwt ∨cos kwt=
2 2
Average value of square of sine or cosine function for time period T is T/2.
0 2
t2

Average value of function y=f(t) from t1 to t2 is defined as

∫ f ( t ) dt
t1
y avg =
t 1−t 2

RMS VALUE OF ALTERNATING CURRENT

The rms value of alternating current can also be defined as the direct current which produces the same heating effect in a given resistor in a given
time as is produced by the given A.C. flowing through same resistor for the same time. Due to this reason the rms value of current is also known as
effective or virtual value of current.

Io
The rms value of alternating current is called the effective or virtual value of alternating current (or emf). I rms = =0.707 I o
√2
Vo
Similarly the rms value of alternating voltage is called the effective or virtual value of alternating voltage (or emf). V rms = =0.707 V o
√2
yo
Rms value of sine or cosine y = yosinwt or y = yocoswt the yrms =
√2
Rms value for combined sine and cosine function y= yo+ yo1(sinw1t+ϕ1)+ yo2(sinw2t+ϕ2)+…..+ yon(sinwnt+ϕn)

√ √ √
n
1 2 4 9 31
Yrms= y 2o + ∑ y Ex: y= 3 + 2 sinwt+ 3 sinwt then yrms= 9+ + =
i=1 2 oi 2 2 2
2π
Time period: The time taken by A.C. to go through one cycle of changes is called its period. It is given as T =
w
Phase: It is that property of wave motion which tells us the position of the particle at any instant as well as its direction of motion.
Phase angle: Angle associated with the wave motion (sine or cosine) is called phase angle.

Lead: Out of the current and emf the one having greater phase angle will lead the other e.g., in equation i = i 0 sin and e = e0 sin ωt, the

current leads the emf by an angle .

Lag: Out of current and emf the one having smaller phase angle will lag the other. In the above equations, the emf lags the current by .
 Io Io
I rms I o π = = √2
Form factor of AC = = X =1.11 Peak factor of AC= I rms i o
I avg √ 2 2 I o
√2

POWER IN AN A.C. CIRCUIT

The power is defined as the rate at which work is being done in the circuit. In ac circuit, the current and emf are not necessarily in the same phase,
therefore we write E = E0 sin ωt & I = I0 sin (ωt + φ).

Eo I o
Instantaneous power, P = EI = E0 I0 sin ωt sin (ωt + φ) Apparent Power =ErmsIrms= Reactive power (watt less power) = Erms Irms sin φ
2

Average power (true power or watt full power) Pav = Erms Irms cos φ ∴

Here cos φ is known as power factor. ie, cosine of phase angle difference between current and voltage. The value of cos φ depends on the nature
Average po wer
of the circuit. Cosϕ =
Apparent power

RESISTANCE OFFERED BY INDUCTOR, RESISTOR AND CAPACITOR TO A.C.

Alternating current in a circuit may be controlled by resistance, inductance and capacitance, while the direct current is controlled only by
resistance.

IMPEDANCE (Z): Total opposition offered by all circuit elements to the flow of a.c is called the impedance and is denoted by Z,

i.e., SI unit is ohm. Impedance triangle

REACTANCE (X): The opposition offered by inductance or capacitance or both to the flow of ac in an ac circuit is called reactance and is
denoted by X. Thus when there is no ohmic resistance in the circuit, the reactance is equal to impedance. The reactance due to inductance alone is
called inductive reactance and is denoted by X L, while the reactance due to capacitance alone is called the capacitive reactance and is denoted by
XC. Its unit is also ohm.

1
ADMITTANCE (Y): The inverse of impedance is called the admittance and is denoted by Y, i.e., y= Its SI unit is ohm –1 or mho or Siemen(S)
Z

If R = 0, cos φ = 0 and P av = 0 i.e., in a circuit with no resistance, the power loss is zero. Such a LC circuit is called the wattless circuit and the
current flowing is called the wattless current. We can resolve I rms into two components I rms cos ϕ and I rms sin ϕ. Here, the component I rms cos ϕ
contributes towards power dissipation and the component I rms sin ϕ does not contribute towards power dissipation. Therefore, it is called wattles
current.
Pure Resistor Pure Inductor Pure capacitor
Circuit
diagram

Phasor
diagram
And phase
difference
ϕ= 0 cos ϕ= cos 90o=0
cos ϕ= cos 90o=0
Graph emf, I
vs wt

Equation Emf E = E0 sinωt E = E0 sin ωt. E = E0 sin ωt

Equation for
current

Iavg= 0 for full cycle Iavg= 0 for full cycle Iavg= 0 for full cycle
2 Eo E 2 Eo E 2 Eo E
Half cycle Iavg= ,Irms = rms Half cycle Iavg= ,Irms = rms Half cycle Iavg= ,Irms = rms
πR R πR R πR R

Impedance Z=R Z = ω L = XL =2πfL 1 1

Z= Xc = = ML2T-3A-2
Unit of XL ohm Dimension ML2T-3A-2 wC 2 πfc
Power factor cos ϕ=1 Phase difference φ = -π/2 φ = + π/2
Power Instantaneous Power=EoIosin2wt Average power = 0. ie., no power is Average power = 0. ie., no power is dissipated
Average power dissipated in pure inductor circuit. in pure capacitor circuit.
=ErmsIrmscosϕ=ErmsIrms=Erms2/R=I2rms.R But Apparent power (rate at which But Apparent power (rate at which energy stored
energy stored in inductance) = Erms.Irms
in capacitance) = Erms.Irms
Note Impedance of circuit is equal to the For high frequency of ac XL becomes For high frequencies Xc becomes 0 and
ohmic resistance ꝏ and the inductor behaves as open capacitor acts as a high pass filter. Therefore it
circuit I=0. So inductor offers high allows ac current easily.
resistive path to a.c.

RL series circuit RC series circuit LC series circuit

Circuit
diagram

Phasor
diagram
And phase
differnce

Graph emf, I
vs wt

Equation for E = E0 sin ωt Eo=√ V R 2+V L2 E = E0 sin ωt Eo=√ V R 2+V C2 E = E0 sin ωt E = VC – VL

Emf
Equation for E o E
o Eo Eo E
current I=Iosin(wt-ϕ) Io = Z = 2 I=Iosin(wt+ϕ) Io = = 2 I =I o sin ( wt ± ϕ )I=
X C− X L
√ R + X L2 Z √ R + X c2
 Impedance Eo Eo E 1
=√ R + X L where XL = ωL =√ R + X c Z = 0 , Reactance X= =X c − X L =( −wL)
2 2 2 2
Z= Z= I wC
Io Io
Power factor R R π
cos ϕ = p.f = cos ϕ = Φ=± P.f cos ϕ=0.
√ R + X L2
2
√R + X
2
c
2 2

Power Eo I o
2
Eo ErmsR
2
Eo I o
2
ErmsR Pavg = Erms I rms cosϕ=0 and Papp=Irms . Erms
Pavg= cosϕ= cos ∅ = 2 cosϕ=
√ R + wL2
√
2 2 Pavg = 2 2 1 and
R+
and Papp=Irms . Erms wC
2

Papp=Irms . Erms

SERIES LCR CIRCUIT

Consider a circuit containing a resistance R, inductance L and capacitance C in series having an alternating emf. E = E 0 sin ωt.
Phasor diagram

Let I be the current flowing in circuit. VR, VL and VC are respective potential differences across resistance R, inductance L and capacitance C.

The p.d VR is in phase with current I. The p.d VC lags behind the current by angle π/2. The p.d. VL leads the current by angle π/2.

∴ Resultant applied emf, E=√ V R +¿ ¿ ¿ ¿ = √(RI )2 +¿ ¿ ¿ ¿

2

i.e., ∴ Impedance,
E
Z= =√ (R) + ¿ ¿ ¿ ¿
2
I

The phase leads of current over applied emf is given by

It is concluded that :

If XC > XL, the value of φ is positive, i.e., current leads the applied emf.

If XC < XL, the value of φ is negative, i.e., current lags behind the applied emf.

If XC = XL, the value of φ is zero, i.e., current and emf are in same phase. ie, ϕ=0.

Therefore, cosϕ=1. LCR circuit behaves as pure resistive circuit. Power factor is unity. Current is maximum. Impedance is minimum. This is
called the case of resonance and resonant frequency for condition XC = XL, is given by :

i.e., ∴

Thus the resonant frequency depends on the product of L and C and is independent of R.

At resonance, impedance is minimum, Zmin = R and current is maximum

Note: Before resonance the current leads the applied emf, at resonance it is in phase, and after resonance it lags behind the emf. LCR series circuit
is also called as acceptor circuit and parallel LCR circuit is called rejecter circuit.

Note:
Wrong: Adding impedances / reactances /resistors algebraically. Correct. For these physical quantities, vector addition must be done
Wrong: Kirchhoff’s laws are applicable in D.C. circuit only
Correct. Kirchhoff’s laws are applicable in A.C. circuit also (which may include inductor and capacitor).

Q - Factor

The sharpness of tuning at resonance is measured by Q-factor or quality factor of the circuit and is given by

Higher the value of Q-factor, sharper is the resonance i.e. more rapid is the fall of current from maximum value (I 0) with slight change in
frequency from the resonance value.

Bandwidth: It is the band of allowed frequencies and is defined as the difference between upper and lower cut-off frequencies, the frequency at

which power becomes half of maximum value and current becomes .

Summary:

Unless mentioned otherwise, all a.c. currents and voltages are r.m.s. values.

For resonance to occur, the presence of both L and C elements in the circuit is a must.
In series resonant circuit, current is maximum at resonance. In a parallel resonant circuit, current is minimum (or zero) at resonance but p.d across
the combination is maximum.

While adding voltage across different elements in an a.c. circuit we should take care of their phases.

The average current over a complete cycle in an a.c circuit is zero but the average power is not zero.

An inductor offers negligibly low resistance path to d.c. and a resistive path for a.c.

A capacitor acts as a block for d.c and a low resistance path to a.c.

TRANSFORMER

A transformer is a device for converting high voltage into low voltage and vice versa, without change in power and frequency.
There are two types of transformers.
Step up transformer: It converts low voltage into high voltage. Step down transformer: It converts high voltage into low voltage.
The principle of a transformer is based on mutual induction and a transformer always works on AC. The input is applied across primary
terminals and output is obtained across secondary terminals. In a transformer, the input emf and the output emf differ in phase by π radians.

ns
The ratio of number of turns in secondary and primary is called the turn ratio. i.e., Turn ratio K= .
np
E s ns IP
If EP and ES are alternating voltages, IP and IS the alternating currents across primary and secondary terminals resply. then, = =K=
Ep np IS

Output Power Pout E s I s

Efficiency of transformer, Ƞ= = =
Input Power P¿ EP I P
The large scale transmission and distribution of electrical energy over long distances is done with the use of transformers. The voltage output of
the generator is stepped-up (so that current is reduced and consequently, the I2R loss is cut down) and it is transmitted to long distances then it is
stepped down at area substation and again stepped down at distribution substation by 240V reaches our home.

POWER LOSSES IN A TRANSFORMER

Copper loss: This is due to resistance of the winding of primary and secondary coil (I2 R)
Iron loss or Eddy current loss: Loss due to leakage of magnetic flux. To reduce this transformer core is laminated.
Magnetic loss or Flux leakage: Loss occurred due to flux leakage. It can be reduced by winding the primary and secondary coils one over the
other.
Hysteresis Loss: Due to repeated magnetization and demagnetization of iron core. To minimize these losses, the transformer core is made up of a
laminated soft iron strips.
Humming loss: Due to vibration.
Inspite of all these losses, we have transformers with efficiency of more than 90%. For an ideal transformer efficiency is 100%.