Mathematics Computation

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Transformers

A transformer is a device that consists of a magnetic core with two independent coils of wire wound around the core.

When an electric current flows through the primary coil it produces a magnetic field in the core which then in turn induces an electrical current in the secondary coil.

The voltage induced depends on the rate at which the magnetic field is generated or decays. A direct current will only produce a secondary current when applied or when cut. Transformers are therefore mainly use with alternating current systems.

The purpose of a transformer is to change the voltage (or current) and this can be done by stepping up or down of the voltage. In principal the ratio of the input to output voltage is directly proportional to the number of the turns of the primary to secondary coils.

Electrical losses are mainly resistive losses better known as I²R losses where I is the current in amperes and R is the resistance in ohm. If the current is reduced then so are the losses and for this reason electricity is distributed at high voltages and low currents. The power distributed is the product of the Voltage and the Current:

P = V x I

High voltages are however dangerous and therefore step down transformers are used to convert the supply to low voltage at the point of distribution. Standard transmission voltages range between 20 to 120 kV whilst the outlet voltage range between 110 to 240 Volts for single phase supply.

The standard frequencies of distribution is either 50 or 60 Hertz (cycles per second)

The normal construction of a transformer consists of an iron core made up of thin steel laminates and forming a rectangular frame.

The Legs have a width D and a thickness C. The window inside has a width W and a length L

To design a transformer there are a number of things which are based on experimental data that allows a preliminary estimate of the size and quantity of material required:

· The current density in the copper windings should lie between 1.5 and 2.1 amps per square millimetre

· The volts per turn = 0.012 √watts

· The maximum flux Φ in the iron core should be between 7000 and 9000 lines per square cm

· The flux density B in lines per sq. cm is proportional to the ampere-turns per cm

· The loss in the core depends on the flux density and the amount of iron

· The core area D x C = Φ / B cm

· The approximate dimensions for the core are:

D = √Φ/B

C = D + 10% D

W = 1.4 D

L = 2 to 3 x W

· The induced E.M.F. is

E = 4 k f T Φ_m 10^-8 volts

E = effective E.M.F.

k = constant 1.11 for a sine wave

f = frequency in cycles per second

T = total turns in series

Φ_m = max flux in lines

· The efficiency η of a transformer is less than 1 due to the losses in the iron and the copper.

P = η V I watt

· The resistance R of the copper can be estimated as

R = l_m / (58 x A)

L_m = length of the wire in metre

A = area of the wire in mm²

· The mass of the core will be the volume multiplied by the density of the material. Use a figure of 6 g per cub cm and a density efficiency of 0.9 to allow for the insulation.

M = 2x 0.9 ((L +2D) + W) x C x D x 6/ 1000 kg.

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