## Sunday, March 23, 2008

### Concept Review Ch.39 Alternating Current

Alternating Current

The current produced by a battery is always in one direction only.

But the electricity that we use in our house is generally alternating current.

If the direction of the current in a resistor or in any other element changes alternately, the current is called an alternating current.

The equation for current, that varies sinusoidally with time, is given by

i = i0sin(ωt+φ)

Where i = current and t = time.
i0 is the maximum value of the current and it is called peak current or the current amplitude.

When we plot the current on graph, the current repeats its value after each time interval T = 2π/ω

Production or Generation of Alternating Current (AC)

Alternating current or AC is produced using an AC generator also called AC dynamo.
It consists of three important parts:

1. a magnet
2. an armature with slip rings and
3. brushes

Magnet may be a permanent magnet or an electromagnet whose poles face each other producing a strong uniform magnetic field between them. It is attached to the body and is called stator.

Armature: It is a coil wound over a soft iron core which is attached or mounted to a shaft that rotates. On this shaft termed as a rotor, at one end there are two rings. These rings are called slip rings. The coil terminal points are connected to the two slip rings.

Brushes: There are two brushes attached to the stator that maintain contact continuously with slip rings and the brushes are connected to the external circuit.

As the armature is rotated in the magnetic field emf is generated in the coil.

Instantaneous and RMS current

i = i0sin(ωt+φ) ... (39.1) defines the instantaneous current at any instant t.

We can define average current as

ī (i bar) = ∫idt/∫dt = 1/T∫idt (integration limits are 0 to t)

Mean square current is average of square of instantaneous current. The square root of mean square current is called root mean square current or rms current. This is also known as the virtual current.

The rms current or the virtual current corresponding to the sinusoidal current i = i0sin(ωt+φ) is

irms = i0/√2

Alternating voltage

An alternating voltage (potential difference) may be written as

V = V0sin(ωt+φ)
This gives instantaneous voltage. The mean voltage V(bar) is zero over a complete cycle. The mean square voltage over a cycle is V0²/2 and the root mean square voltage (rms voltage or virtual voltage) is V0/√2.

Significance of RMS values

A constant voltage Vrms applied across a resistor produces the same thermal energy as that produced by the voltage V = V0sin(ωt+φ).

Phase factor

Current and emf are in general not in phase in an AC circuit.
If the emf is Є = Є0sin ωt
The current will be i = i0sin(ωt+φ)

The constant term φ is called the phase factor.
For purely resistive circuit, φ = 0;
For a purely capacitive circuit, φ = π/2 and
For a purely inductive circuit φ = - π/2.

Choke Coil

Choke coils are used with fluorescent mercury-tube fittings (tube lights) in houses. Choke coil is a coil having large inductance but a small resistance. The choke coil is used to reduce the voltage across the tube or the resistor.

An additional resistor could be used for the same purpose, but a resistor would consume power. The inductance will not consume power.

Hot-wire instruments

To measure alternating currents or voltages, we have to use a property so that the deflection of the moving part depends on i² and not on i. Hot wire instruments work on this principle.

Hot-wire ammeter

A platinum-iridium wire is used as a hot wire, whose rise in temperature is proportional to i².

Hot-wire voltmeter

The construction is similar to ammeter except that a shunt is connected in parallel to hot wire in ammeter and whereas a high resistance is connected in series with the hot wire in voltmeter.

DC dynamo

DC dynamo supplies current in one direction only in the circuit connected to it. A system split cylinders on the armature is used for this purpose and the split cylinders with brushes is called a slip ring commutator.

In this system, the ends of the armature are connected separately to the split cylinder halves C1 and C2. The carbon brushes (B1 and B2) press against two halves. In the rotation for half the time B1 is connected to C1 and for half the time it is connected to B2. Therefore in the external circuit current goes in the same direction. But its magnitude varies. To reduce variation in the current another coil perpendicular to the first one is added to the system in such a way that emf from this coil is maximum when emf from the first coil is minimum and vice versa. The sum of the two contains less oscillations and the current is more nearly constant. The coils can be increased further to reduce the variation.

DC Motor

A motor converts electrical energy into mechanical energy. The motor construction is similar to the DC dynamo. But power is supplied to the armature from a DC dynamo. Because the armature is in magnetic field, torque acts on it and it rotates. As the armature is on a shaft to which load is attached, work is done. As the coil rotates an induced emf e is produced opposite to applied emf Є. If the resistance of the circuit is R, the current at any instant is given by i = (Є-e)/R.

Transformer

A transformer changes the AC voltage applied to the primary coil to a higher or lower value at the output end of the secondary coil. It consists of two coils wound separately on a laminated soft-iron core. The source of alternating voltage is connected to one of the coils named as primary coil. An induced emf appears across the ends of the secondary.

If there N1 turns in the primary and N2 turns in the secondary, and if an alternating emf Є1 is applied across primary, the emf at out end of secondary will be

Є2 = -N2 Є1/N1

The minus sign shows that Є2 is 180° out of phase with Є1.

Current i2 = -N1*i1/N2

The minus sine shows that i2 is 180° out of phase with i1.

Step-up and step down transformer

If N2>N1, the secondary emf Є2 is larger in magnitude than the primary emf Є1. This type of transformer is called a step-up transformer. But the secondary current is less than the primary current. The primary coil has to sustain the high current and hence it is made with thick wire.

If N2 is less than N1, the secondary emf is smaller and it called step down transformer. In this case current is more in secondary, and this coil is to be made from thick wire.

Efficiency of transformer

There is some loss energy in the transformer due to resistances, hysteresis in the core, eddy currents in the core etc. but efficiencies up to 99% are easily achieved. Efficiency is defined as output/input in terms of power.

Transmission of Power

At the electricity generation plant power is stepped up to 66 kV and fed to the transmission lines. In a town or city, the voltage is stepped down to the required value such as 220V. Why is it done that way? As we saw above in step down transformer current is more in secondary and less in primary. As voltage is more in the primary current is less and we lose less in i²Rt losses if ‘i' is less.