**Transient Current**

An electric current which vary for a small finite time, while growing from zero to maximum or decaying from maximum to zero, is called a transient current.

**Growth of Current in an Inductor**

Growth of current in an inductor at any instant of time t is given by

I = I_{o}(1 – e ^{-Rt / L})

where, I_{o} = maximum current, L = self inductance of the inductor and R = resistance of the circuit.

Here R / L = τ, is called time constant of a L – R circuit.

Time constant of a L – R circuit is the time in which current in the circuit grows to 63.2% of the maximum value of current.

Decay of current in an inductor at any time t is given by

I = I_{o}e ^{-Rt / L}

Time constant of a L – R circuit is the time in which current decays to 36.8% of the maximum value of current.

**Charging and Discharging of a Capacitor**

The instantaneous charge on a capacitor on charging at any instant of time t is given by

q = q_{o}(1 – e ^{– t / RC})

where RC = τ, is called time constant of a R – C circuit.

The instantaneous charge on a capacitor in discharging at any instant of time t is given by q = q_{o}e ^{– t / RC}

Time constant of a R – C circuit is the time in which charge in the capacitor grows to 63.8% or decay to 36.8% of the maximum charge on capacitor.

**Alternating Current**

An electric current whose magnitude changes continuously with time and changes its direction periodically, is called an alternating current.

The instantaneous value of alternating current at any instant of time t is given by

I = I_{o} sin ωt

where, 10 = peak value of alternating current.

The variation of alternating current with time is shown in graph given below

Mean or average value of alternating current for first half cycle

I_{m} = 2I_{o} / π = 0.637 I_{o}

Mean or average value of alternating current for next half cycle

I’_{m} = – 2I_{o} / π = – 0.637 I_{o}

Mean or average value of alternating current for one complete cycle = O.

Root mean square value of alternating current

I_{v} = I_{rms} = I_{o} / √2 = 0.707 I_{o}

Where, I_{o} = peak value of alternating current.

Root mean square value of alternating voltage

V_{rms} = V_{o} / √2 = 0.707 I_{o} = 0.707 V_{o}

Reactance

The opposition offered by an inductor or by a capacitor in the path of flow of alternating current is called reactance.

Reactance is of two types

(i) **Inductive Reactance** (X_{L}) Inductive reactance is the resistance offered by an inductor.

Inductive reactance (X_{L}) = Lω = L2πf = L2π / T

Its unit is ohm. X_{L} ∝ f

For direct current, X_{L} = 0 (f = 0)

(ii) **Capacitive Reactance** (X_{c}) Capacitive reactance is the resistance offered by an inductor

Capacitive reactance,

X_{c} = 1 / Cω = 1 / C2πf = T / C 2π

Its unit is ohm X_{c} ∝ 1 / f

For direct current, X_{c} = ∞ (f = 0)

**Impedance**

The opposition offered by an AC circuit containing more than one out of three components L, C and R, is called impedance (Z) of the circuit.

Impedance of an AC circuit, Z = √R^{2} + (X_{L} – X_{C})^{2}

Its SI unit is ohm.

**Power in an AC Circuit**

The power is defined as the rate at which work is being in the circuit.

The average power in an AC circuit,

P_{av} = V_{rms} i_{rms} cos θ

= V / √2 i / √2 cos θ = Vi / √2 cos θ

where, cos θ = Resistance(R) / Impedance (Z) is called the power factor 0f AC circuit.

**Current and Potential Relations**

Here, we will discuss current and potential relations for different AC circuits.

(i) **Pure Resistive Circuit (R** circuit)

(a) Alternating emf, E = E_{o} sin ωt

(b) Alternating current, I = I_{o} sin ωt

(c) Alternating emf and alternating current both are in the same phase.

(d) Average power decay, (P) = E_{v} . I_{v}

(e) Power factor, cos θ = 1

**(ii) Pure Inductive Circuit (L** Circuit)

(a) Alternating emf, E = E_{o} sin ωt

(b) Alternating current, I = I_{o} sin (ωt – π / 2)

(c) Alternating current lags behind alternating emf by π / 2.

(d) Inductive reactance, X_{L} = Lω = L2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iii)** Pure Capacitive Circuit**

(a) Alternating emf, E = E_{o} sin ωt

(b) Alternating current, I = I_{o} sin (ωt + π / 2)

(c) Alternating current lags behind alternating emf by π / 2.

(d) Inductive reactance, X_{L} = Cω = C2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iv) **R – C Circuit**

E = E_{o} sin ωt

I = E_{o} / 2 sin (ωt – φ)

Z = √R^{2} + (1 / ωC)^{2}

tan φ = – 1 / ωC / R

Current leading the voltage by φ

V^{2} = V^{2}_{R} = V^{2}_{C}

(v) **L – C Circuit**

(vi) **L – C – R Circuit**

(a) Alternating emf, E = E_{o} sin Ωt

(b) Alternating current, I = I_{o} sin (Ωt ± θ)

(c) Alternating current lags leads behind alternating emf by ω.

(d) Resultant voltage, V = √V^{2}_{R} + (V_{L} – V_{C})^{2}

(e) Impedance, Z = √R^{2} + (X_{L} – X_{C})^{2}

(f) Power factor, cos θ = R / Z = R / √√R^{2} + (X_{L} – X_{C})^{2}

(g) Average power decay, (P)= E_{V}I_{V} cos θ

**Resonance in AC Circuit**

The condition in which current is maximum or impedance is minimum in an AC circuit, is called resonance.

(i) **Series Resonance Circuit**

In this circuit components L, C and R are connected in series.

At resonance = X_{L} = X_{C}

Resonance frequency f = 1 / 2π√LC

A series resonance circuit is also known as acception circuit.

(ii)** Parallel Resonance Circuit**

In this circuit L and C are connected in parallel with each other.

At resonance, X_{L} = X_{C}

Impedance (Z) of the circuit is maximum.

Current in the circuit is minimum.

**Wattless Current**

Average power is given by

P_{av} = E_{rms} = I_{rms} cos θ

Here the I_{rms} cos φ contributes for power dissipation. Therefore, it is called wattless current.

**AC Generator or Dynamo**

It is a device which converts mechanical energy into alternating current energy.

Its working is based on electromagnetic induction.

The induced emf produced by the AC generator is given by

e = NBAω sin ωt = e_{o} = sin ωt

There are four main parts of an AC generator

(i) **Armature** It is rectangular coil of insulated copper wire having a large number of turns.

(ii) **Field Magnet**s These are two pole pieces of a strong electromagnet.

(iii) **Slip Rings** These are two hollow metallic rings.

(iv) **Brushes** These are two flexible metals or carbon rods, which remains slightly in contact with slip rings .

**Note** An DC generator or dynamo contains split rings or commutator inspite of slip rings.

**DC Motor**

It is a device which converts electrical energy into mechanical energy.

Its working is based on the fact that when a current carrying coil is placed in uniform magnetic field a torque acts on it.

Torque acting on a current carrying coil placed in uniform magnetic field

τ = NBIA sin θ

When armature coil rotates a back emf is produced in the coil.

Efficiency of a motor,

η = Back emf / Applied emf = E / V

**Transformer**

It is a device which can change a low voltage of high current into a high voltage of low current and vice-versa.

Its working is based on mutual induction.

There are two types of transformers.

(i) **Step-up Transformers** It converts a low voltage of high current into a high voltage of low current.

In this transformer,

N_{s} > N_{P}, E_{s} > E_{P}

and I_{P} > I_{S}

**(ii) Step-down Transformer** It converts a high voltage of low current into a low voltage of high current.

In this transformer,

N_{P} > N_{S}, E_{P} > E_{S} and I_{P} < I_{S}

**Transformation Ratio**

Transformation ratio,

K = N_{S} / N_{P} = E_{S} / E_{P} = I_{P} / I_{S}

For step-up transformer, K > 1

For step-down transformer, K < 1

**Energy Losses in a Transformer**

The main energy losses in a transformer are given below

- Iron loss
- Copper loss
- Flux loss
- Hysteresis loss
- Humming loss

**Important Points**

- Transformer does not operate on direct current. It operates only on alternating voltages at input as well as at output.
- Transformer does not amplify power as vacuum tube.
- Transformer, a device based on mutual induction converts magnetic energy into electrical energy.
- Efficiency, η = Output power / Input power

Generally efficiency ranges from 70% to 90%.

- A choke coil is a pure inductor. Average power consumed per cycle is zero in a choke coil.
- A DC motor connects DC energy from a battery into mechanical energy of rotation.
- An AC dynamo/generator produces are energy from mechanical energy of rotation of a coil.
- An induction coil generates high voltages of the order of 1OS V from a battery.

It is based on the phenomenon of mutual induction.

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