# Electricity And Magnetism Of Transformers Technology Essay

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The transformer is essentially just two (or more) inductors, sharing a common magnetic path. Any two inductors placed reasonably close to each other will work as a transformer, and the more closely they are coupled magnetically, they become more effective.

When a changing magnetic field is in the coil of wire (an inductor), a voltage is induced into the coil which is a result of the applied magnetic field. A static magnetic field has no effect, and generates no output. Many of the same principles apply to generators, alternators, electric motors and loudspeakers.

When an electric current is passed through a coil of wire, a magnetic field is created - this works with AC or DC, but with DC, the magnetic field is static. For this reason, transformers cannot be used directly with DC, for although a magnetic field exists, it must be changing to induce a voltage into the other coil.

Try this experiment. Take a coil of wire (a loudspeaker crossover coil will do nicely for this), and a magnet. Connect a multimeter (preferably analogue) to the coil, and set the range to the most sensitive current range on the meter. As you move the magnet towards or away from the coil, you will see a current, shown by the deflection of the meter pointer. As the magnet is swung one way, the current will be positive, the other way it will be negative. The higher the coil's inductance and the stronger the magnet, the greater will be the induced current.

Move the magnet slowly, and the current will be less than if it is moved quickly. Leave it still, and there is no current at all, regardless of how close the magnet may be. This is the principle of magnetic induction, and it applies to all coils.

The permeability of transformer cores varies widely, depending on the material and any treatment that may be used. The permeability of air is 1, and most traditional cores have a much higher (i.e. > 1) permeability.

As well as permeability, magnetic cores (with the exception of air) have a maximum magnetic flux they can handle without saturation. When a magnetic core is saturated, it can carry no more magnetic flux. At this point, the magnetic field is no longer changing, so current is not induced into the winding.

You will be unable to saturate your nails with the magnet, as there is a very large air gap between the two pole pieces. This means that the core will always be able to support the magnetic flux, but the efficiency is also very much lower because the magnetic circuit is open. Nearly all the transformers you will see have a completely closed magnetic circuit, to ensure that as much of the magnetism induced into the core as possible will pass through the winding(s).

## Essential Workings of a Transformer

Figure above shows the basics of all transformers. A coil (the primary) is connected to an AC voltage source - typically the mains for power transformers. The flux induced into the core is coupled through to the secondary, a voltage is induced into the winding, and a current is produced through the load.

The diagram also shows the various parts of a transformer. This is a simple transformer, with two windings. The primary will induce a magnetic field into the core in sympathy with the current produced by the applied AC voltage. The magnetic field is concentrated by the core, and nearly all of it will pass through the windings of the secondary as well, where a voltage is induced. The core in this case is typical of the construction of a "C-Core" transformer, where the primary and secondary are separated. The magnitude of the voltage in the secondary is determined by a very simple formula, which determines the "turns ratio" (N) of the component - this is traditionally calculated by dividing the secondary turns by the primary turns.

## N = Ts / Tp

Tp is simply the number of turns of wire that make up the primary winding, and Ts is the number of turns of the secondary. A transformer with 200 turns on the primary and 20 turns on the secondary has a turn ratio of 1:10 (i.e. 1/10 or 0.1)

## Vs = Vp * N

Mostly, you will never know the number of turns, but of course we can simply reverse the formula so that the turns ratio can be deduced from the primary and secondary voltages.

## N = Vs / Vp

If a voltage of 240V (AC, naturally) is applied to the primary, we would expect 24V on the secondary, and this is indeed what will be measured. The transformer has an additional useful function - not only is the voltage "transformed", but so is the current.

## Is = Ip / N

If a current of 1A were drawn by the primary in the above example, then logically a current of 10A would be available at the secondary - the voltage is reduced, but current is increased. This would be the case if the transformer were 100% efficient, but even this - the most efficient "machine" we have - will sadly never be perfect. With large transformers used for the national supply grid, the efficiency of the transformers will generally exceed 95%, and some will be as high as 98%.

Smaller transformers will always have a lower efficiency, but the units commonly used in power amplifiers can have efficiencies of up to 90% for larger sizes.

## 2.   TERMINOLOGY

Coercivity - is the field strength which must be applied to reduce (or coerce) the remanent flux to zero. Materials with high coercivity (e.g. those used for permanent magnets) are called hard. Materials with low coercivity (those used for transformers) are called soft.

Effective Area - of a core is the cross sectional area of the centre limb for E-I laminations, or the total area for a toroid. Usually this corresponds to the physical dimensions of the core but because flux may not be distributed evenly the manufacturer may specify a value which reflects this.

Effective length - of a core is the distance which the magnetic flux travels in making a complete circuit. Usually this corresponds closely to the average of the physical dimensions of the core, but because flux has a tendency to concentrate on the inside corners of the path the manufacturer may specify a value for the effective length.

Flux Density - (symbol; B, unit; Teslas (T)) is simply the total flux divided by the effective area of the magnetic circuit through which it flows.

Flux leakage - in an ideal inductor the flux generated by one turn would be contained within all the other turns.

Magnetomotive Force - MMF can be thought of as the magnetic equivalent of electromotive force. It is the product of the current flowing in a coil and the number of turns that make up the coil.

Magnetic Field Strength - (symbol: H, unit; ampere metres (A m-1)) when current flows in a conductor, it is always accompanied by a magnetic field. The strength, or intensity, of this field is proportional to the amount of current and inversely proportional to the distance from the conductor (hence the -1 superscript).

Magnetic Flux - (symbol: ; unit: Webers (Wb)) we refer to magnetism in terms of lines of force or flux, which is a measure of the total amount of magnetism.

Permeability - (symbol; µ, units: henrys per metre (Hm-1) is defined as the ratio of flux density to field strength, and is determined by the type of material within the magnetic field - i.e. the core material itself. Most references to permeability are actually to "relative permeability", as the permeability of nearly all materials changes depending upon field strength (and in most cases with temperature as well).

Remanence - (or remnance) is the flux density which remains in a magnetic material when the externally applied field is removed. Transformers require the lowest possible remanence, while permanent magnets need a high value of remanence.

## Cores

Laminated core transformer showing edge of laminations at top of photo

## Laminated steel cores

Transformers for use at power or audio frequencies typically have cores made of high permeability silicon steel. The steel has a permeability many times that of free space, and the core thus serves to greatly reduce the magnetizing current, and confine the flux to a path which closely couples the windings. Later designers constructed the core by stacking layers of thin steel laminations, a principle that has remained in use. Each lamination is insulated from its neighbors by a thin non-conducting layer of insulation.

The effect of laminations is to confine eddy currents to highly elliptical paths that enclose little flux, and so reduce their magnitude. Thinner laminations reduce losses, but are more laborious and expensive to construct. Thin laminations are generally used on high frequency transformers, with some types of very thin steel laminations able to operate up to 10 kHz.

One common design of laminated core is made from interleaved stacks of E-shaped steel sheets capped with I-shaped pieces, leading to its name of "E-I transformer". Such a design tends to exhibit more losses, but is very economical to manufacture. The cut-core or C-core type is made by winding a steel strip around a rectangular form and then bonding the layers together. It is then cut in two, forming two C shapes, and the core assembled by binding the two C halves together with a steel strap

## Solid cores

Powdered iron cores are used in circuits (such as switch-mode power supplies) that operate above main frequencies and up to a few tens of kilohertz. These materials combine high magnetic permeability with high bulk electrical resistivity. For frequencies extending beyond the VHF band, cores made from non-conductive magnetic ceramic materials called ferrites are common. Some radio-frequency transformers also have movable cores (sometimes called 'slugs') which allow adjustment of the coupling coefficient (and bandwidth) of tuned radio-frequency circuits.

## Toroidal cores

Small toroidal core transformer

Toroidal transformers are built around a ring-shaped core, which, depending on operating frequency, is made from a long strip of silicon steel wound into a coil, powdered iron, or ferrite. A strip construction ensures that the grain boundaries are optimally aligned, improving the transformer's efficiency by reducing the core's reluctance. The closed ring shape eliminates air gaps inherent in the construction of an E-I core. The cross-section of the ring is usually square or rectangular, but more expensive cores with circular cross-sections are also available. The primary and secondary coils are often wound concentrically to cover the entire surface of the core. This minimizes the length of wire needed, and also provides screening to minimize the core's magnetic field from generating electromagnetic interference.

Toroidal transformers are more efficient than the cheaper laminated E-I types for a similar power level. Other advantages compared to E-I types, include smaller size (about half), lower weight (about half), less mechanical hum (making them superior in audio amplifiers), lower exterior magnetic field (about one tenth), low off-load losses (making them more efficient in standby circuits), single-bolt mounting, and greater choice of shapes. The main disadvantages are higher cost and limited power capacity (see "Classification" above). Because of the lack of a residual gap in the magnetic path, toroidal transformers also tend to exhibit higher inrush current, compared to laminated E-I types.

## Air cores

A physical core is not an absolute requisite and a functioning transformer can be produced simply by placing the windings near each other, an arrangement termed an "air-core" transformer. The air which comprises the magnetic circuit is essentially lossless, and so an air-core transformer eliminates loss due to hysteresis in the core material. The leakage inductance is inevitably high, resulting in very poor regulation, and so such designs are unsuitable for use in power distribution. They have however very high bandwidth, and are frequently employed in radio-frequency applications, for which a satisfactory coupling coefficient is maintained by carefully overlapping the primary and secondary windings. They're also used for resonant transformers such as Tesla coils where they can achieve reasonably low loss in spite of the high leakage inductance.

## Windings

Windings are usually arranged concentrically to minimize flux leakage.

The conducting material used for the windings depends upon the application, but in all cases the individual turns must be electrically insulated from each other to ensure that the current travels through every turn. For small power and signal transformers, in which currents are low and the potential difference between adjacent turns is small, the coils are often wound from enamelled magnet wire, such as Formvar wire. Larger power transformers operating at high voltages may be wound with copper rectangular strip conductors insulated by oil-impregnated paper and blocks of pressboard.

High-frequency transformers operating in the tens to hundreds of KHz often have windings made of braided Litz wire to minimize the skin-effect and proximity effect losses. Large power transformers use multiple-stranded conductors as well, since even at low power frequencies non-uniform distribution of current would otherwise exist in high-current windings. Each strand is individually insulated, and the strands are arranged so that at certain points in the winding, or throughout the whole winding, each portion occupies different relative positions in the complete conductor. The transposition equalizes the current flowing in each strand of the conductor, and reduces eddy current losses in the winding itself. The stranded conductor is also more flexible than a solid conductor of similar size, aiding manufacture.

For signal transformers, the windings may be arranged in a way to minimize leakage inductance and stray capacitance to improve high-frequency response. This can be done by splitting up each coil into sections, and those sections placed in layers between the sections of the other winding. This is known as a stacked type or interleaved winding.

Both the primary and secondary windings on power transformers may have external connections, called taps, to intermediate points on the winding to allow selection of the voltage ratio. In power distribution transformers the taps may be connected to an automatic on-load tap changer for voltage regulation of distribution circuits.

## 4. WORKING

The whole principle of operation is based on induced magnetic flux, which creates a voltage and current in the secondary and primary. It is this characteristic that allows any inductor to function as expected, and the voltage generated in the primary is called a "back EMF" (electromotive force). The magnitude of this voltage almost equals (and is in the same phase as) the applied EMF.

For a sinusoidal waveform, the current through an inductor lags the voltage by 90 degrees. Since the induced current is lagging by 90 degrees, the internally generated voltage is shifted back again by 90° so is in phase with the input voltage. Imagine an inductor or transformer (no load) with an applied voltage of 230V. For the effective back EMF to resist the full applied AC voltage (as it must), the actual magnitude of the induced voltage (back EMF) is just under 230V. The output voltage of a transformer is always in phase with the applied voltage (within a few thousandths of a degree).

For example, a transformer primary operating at 230V input draws 150mA from the mains at idle and has a DC resistance of 2 ohms. The back EMF must be sufficient to limit the current through the 2 ohm resistance to 150mA, so will be close enough to 229.7V (0.3V at 2 ohms is 150mA).

When you apply a load to the output (secondary) winding, a current is drawn by the load, and this is reflected through the transformer to the primary. As a result, the primary must now draw more current from the mains. Somewhat intriguingly perhaps, the more current that is drawn from the secondary, the original 90 degree phase shift becomes less and less as the transformer approaches full power. The power factor of an unloaded transformer is very low, meaning that although there are volts and amps, there is relatively little power. The power factor improves as loading increases, and at full load will be close to unity (the ideal).

The impedance ratio of a transformer is equal to the square of the turns ratio

## Z = N²

Transformers are usually designed based on the power required, and this determines the core size for a given core material. From this, the required "turns per volt" figure can be determined, based on the maximum flux density that the core material can support. Again, this varies widely with core materials.

A rule of thumb can be applied, that states that the core area for "standard" (if indeed there is such a thing) steel laminations (in square centimeters) is equal to the square root of the power. Thus a 625VA transformer would need a core of (at least) 25 sq cm, assuming that the permeability of the core were about 500, which is fairly typical of standard transformer laminations. This also assumes that the core material will not saturate with the flux density required to obtain this power.

The next step is to calculate the number of turns per volt for the primary winding. This varies with frequency, but for a 50Hz transformer, the turns per volt is (approximately) 45 divided by the core area (in square centimetres). Higher performance core materials may permit higher flux densities, so fewer turns per volt might be possible, thus increasing the overall efficiency and regulation. These calculations must be made with care, or the transformer will overheat at no load.

You can determine the turns per volt of any transformer (for reasons that will become clearer as we progress) by adding exactly 10 turns of thin "bell wire" or similar insulated wire to an existing transformer, wound over the existing windings. When powered from the correct nominal supply voltage, measure the voltage on the extra winding you created, and divide by 10 to obtain the turns per volt rating for that transformer.

Assume for a moment that you have a transformer for a fair sized power amplifier. The secondary voltage is 35-0-35V which is much too high to power the preamp circuit or even its power supply - but you will be able to do that with a single 16V winding. Another transformer would normally be used, but you can also add the extra winding yourself. This is almost too easy with toroidal transformers, but with others it may not be possible at all. If the transformer uses (say) 2 turns per volt, a mere 32 extra turns of bell wire (or similar) will provide 16V at the typical 100mA or so you will need.

# A very interesting phenomenon exists when we draw current from the secondary. Since the primary current increases to supply the load, we would expect that the magnetic flux in the core would also increase (more amps, same number of turns, more flux). In fact, the flux density decreases! In a perfect transformer with no copper loss, the flux would remain the same - the extra current supplies the secondary only. In a real transformer, as the current is increased, the losses increase proportionally, and there is slightly less flux at full power than at no load.

## 5. ENERGY LOSES

An ideal transformer would have no energy losses, and would be 100% efficient. In practical transformers energy is dissipated in the windings, core, and surrounding structures. Larger transformers are generally more efficient, and those rated for electricity distribution usually perform better than 98%.

Experimental transformers using superconducting windings achieve efficiencies of 99.85%. The increase in efficiency from about 98 to 99.85% can save considerable energy, and hence money, in a large heavily-loaded transformer; the trade-off is in the additional initial and running cost of the superconducting design.

Losses in transformers (excluding associated circuitry) vary with load current, and may be expressed as "no-load" or "full-load" loss. Winding resistance dominates load losses, whereas hysteresis and eddy currents losses contribute to over 99% of the no-load loss. The no-load loss can be significant, so that even an idle transformer constitutes a drain on the electrical supply and a running cost; designing transformers for lower loss requires a larger core, good-quality silicon steel, or even amorphous steel, for the core, and thicker wire, increasing initial cost, so that there is a trade-off between initial cost and running cost.

Transformer losses are divided into losses in the windings, termed copper loss, and those in the magnetic circuit, termed iron loss. Losses in the transformer arise from:

## Winding resistance

Current flowing through the windings causes resistive heating of the conductors. At higher frequencies, skin effect and proximity effect create additional winding resistance and losses.

## Hysteresis losses

Each time the magnetic field is reversed, a small amount of energy is lost due to hysteresis within the core. For a given core material, the loss is proportional to the frequency, and is a function of the peak flux density to which it is subjected.

## Eddy currents

Ferromagnetic materials are also good conductors, and a core made from such a material also constitutes a single short-circuited turn throughout its entire length. Eddy currents therefore circulate within the core in a plane normal to the flux, and are responsible for resistive heating of the core material. The eddy current loss is a complex function of the square of supply frequency and inverse square of the material thickness. Eddy current losses can be reduced by making the core of a stack of plates electrically insulated from each other, rather than a solid block; all transformers operating at low frequencies use laminated or similar cores.

## Magnetostriction

Magnetic flux in a ferromagnetic material, such as the core, causes it to physically expand and contract slightly with each cycle of the magnetic field, an effect known as magnetostriction. This produces the buzzing sound commonly associated with transformers, and can cause losses due to frictional heating.

## Mechanical losses

In addition to magnetostriction, the alternating magnetic field causes fluctuating forces between the primary and secondary windings. These incite vibrations within nearby metalwork, adding to the buzzing noise, and consuming a small amount of power.

## Stray losses

Leakage inductance is by itself largely lossless, since energy supplied to its magnetic fields is returned to the supply with the next half-cycle. However, any leakage flux that intercepts nearby conductive materials such as the transformer's support structure will give rise to eddy currents and be converted to heat. There are also radiative losses due to the oscillating magnetic field, but these are usually small.