Transformer design

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This page deals primarily with the design of electrical transformers. For the definition, function, and history, see Transformer.

Transformer design requires knowledge in several different areas of electricity, and in the materials they are manufactured with. Transformer designers use both pencil and software to design them. Either way, they both have to adhere to the standard formulas that have been around since transformers were first introduced. These equations are deduced from the Maxwell equations which in turn resulted from Faraday's law. Today, this is known as field theory. One would be surprised in the fact that not much has changed since the time of Maxwell, Faraday, and others. The only improvements have been in the materials as the math is still the same. These improvements are the core material, wire, and insulation used.

Contents 1 Power transformer design 1.1 Initial calculations 1.2 Type of iron (steel) 1.3 Secondary turns calculation 1.4 Thickness of windings 2 Wire selection 3 The core stack 4 Watts versus volt-amperes 5 Rectifier transformers 6 Equations 6.1 Imperial measurement system 6.2 Simpler formulae 6.3 Equation notes 7 Electrical steel types 7.1 Silicon steel 7.2 Other alloys 8 References 9 Footnotes

[edit] Power transformer design

The designer first needs several known factors to design a transformer. For a transformer using a sine or square wave, one needs to know the incoming line voltage, the operating frequency, the secondary voltage(s), the secondary current(s), the permissable temperature rise, the target efficiency, the physical size one can use, and the cost limitations. Once these factors are known, one can begin the design.

[edit] Initial calculations

One first starts with the primary voltage and frequency. Since they are a known factor, they are the first numbers to be plugged into the equations. One then will find the power in watts (or volt-amperes) of each secondary winding by multiplying the voltage by the current of each coil. These are added together to get the total power the transformer must provide to the load(s).

The transformer losses in watts are estimated and added to this sum to give a total power the primary coil must supply. The losses are from both the wires resistance (I² R loss), and the loss in the core from hysteresis due to eddy currents. Both losses are dissapated by heat. Here, the permissable temperature rise must be kept in mind. Each type of core material will have a loss chart whereby one can find the loss in watts per pound by looking up the operating flux density and frequency it will be ran at. Next, one selects the type of iron by what efficiency is stated, and the maximum cost the user specified. Once the iron is selected, the flux density is selected for that material.

[edit] Type of iron (steel)

Each type of iron (steel) has a maximum flux density it can be ran at without saturating. The designer refers to B-H curves for each type of steel. They will select a flux density where the knee either starts on the curve, or slightly up on it. This will give a transformer with the lowest weight possible for that material. The curve shows that as saturation begins, the magnetic field strength in Oersteds (H) raises rapidly as compared to any increase in flux density (B). Once the designer knows the primary voltage, frequency, waveform, total power in watts or volt-amperes, and the maximum flux density, he can then begin to use the equations to his benefit. When using the equations, the two most important are the number of turns (N), and the core area (a). One needs to find the core area in square centimeters or inches, and match it to the total power in watts or volt-amperes. The larger the core, the more power it will handle. Once this core size is calculated, one can then find the number of turns for the primary. One then is looking at a transformer whos primary voltage will cause a flux density of a specified amount due to the number of turns in a certain type/size of core.

For sine wave operation, one can now use either the two short formulas, or begin using the long formulas which are more exact, and whereby all the factors can be changed. For square wave operation, refer to the notes at the end of the equations section. Either way, it's time to start using a transformer design sheet. The design sheet has places to write the details such as the flux density, the number of turns, calculate the turns per layer, and thickness of the coil.

[edit] Secondary turns calculation

Once the number of turns of the primary are calculated, the secondary windings numbers can be calculated with the same turns per volt figure. If the primary has 120 turns for 120 volts input, we would have 1 turn per volt. If we needed a 12 volt secondary, then we would require 12 turns on it. This is for a perfect transformer without losses though.

In reality, there are losses that have to be added, or the 12 turn coil will not produce 12 volts but a lower voltage. A rule of thumb is to allow for 5% in losses. In this case, we would multiply the 12 turns by 1.05 to get a new number of turns equalling 12.6 turns. Since half turns are not advisable, 13 turns would be used. It is best to have a slightly higher voltage as to have one too low. Beware, smaller transformers which have a higher turns per volt, have higher losses, and the efficiency drops as the size goes down.

Here's where cut-and-try still comes into transformer design. Since the primary coil has to be wound with a wire that is large enough to handle the total power the transformer will handle at a certain flux density, it must fit within the cores window(s) once the overall size is calculated after adding the bobbin and paper thickness of each layer. Most of the time, the design has to be played with over this because the coil is too big for the windows. If the coil doesn't fit, there is a few options. One can use a larger core with larger window openings having the same core area, or one can raise the flux density by reducing the turns on the primary. Once these turns are reduced, the turns in the secondary will be reduced. This since the number of turns in the primary equal the number of turns in the secondary minus losses. However, this is at the expense of raising the flux density, the magnetizing current, the temperature, and lowering the efficiency. It's much better to select a larger core which has larger windows to accept the coil. The depth or thickness of the new core can be adjusted to equal the old core area in square centimeters or square inches. This measurement is the cores tounge width multiplied by its depth or thickness. As the core size goes up, so does the tounge width.

[edit] Thickness of windings

When calculating the coil thickness, several things needs to be considered. The voltage that each layer of coil sees will determine the wires insulation thickness. Once this voltage is known, the diameter of the insulated wire can be used. By knowing the wire diameter, one then calculates the number of turns per layer, and the number of layers by using the window height. Next, one needs to adjust the thickness of the insulation paper for the layers of each winding due to the voltage between the coils. This thickness is added to the total coil thickness by multiplying the paper thickness by the number of layers. The paper that seperates two different windings is always thicker than the layer paper to match the voltage difference between the windings. Last, the bobbin thickness is added. All is then added to the design sheet and the total calculated. This total thickness is compared to the window dimensions for a fit.

A smaller coil with less layers is always recommended. A coil with a large number of layers will run much hotter than one with a few. Each winding has a "hot spot" which is always located mid-way at its center. If the winding has a large amount of layers, the heat will increase at this hot spot. The hot spot is almost always where the winding will fail due to heat. The heat from each winding has to travel through each layer and is dissapated from the outside of the coil. This means that the winding closer to the core will be hotter than the outer ones. Since this is the case, and most of the time the winding closest to the core is the primary, the largest wire that will fit should be used. The exception to having the primary here is using a winding with very small diameter wire. Since the coil will expand due to heat, a small wire coil on the outside could break because of the expansion. Being at the core, it would expand less and not break the wire. Most small bias windings, rated at a few miliamperes, used in vacuum tube circuits are wound in this manner.

For using a two section bobbin (for a two winding transformer), the above is not necessary. These are used by jumble-winding the wire on each section of the bobbin. Jumble-winding by definition means that the wire is wound on the bobbin in a random way without layers seperated by paper. However, the amount of wire used for each winding has to fit withing the bobbin so it too will fit inside the cores windows. A huge amount of transformers are manufactured this way to save cost.

[edit] Wire selection

[edit] The core stack

[edit] Watts versus volt-amperes

[edit] Rectifier transformers

[edit] Equations

There is two approaches used in designing transformers. One uses the long formulas, and the other uses the Wa product. The Wa product is simply the cores window area multiplied by the cores area. Some say it simplifies the design, especially in C-core (cut core) construction. Most manufacturers of C-cores have the Wa product added into the tables used in their selection. The designer takes the area used by a coil and finds a C-core with a similar window area. The Wa product is then divided by the window area to find the area of the core. Either way will bring the same result.

For a transformer designed for use with a sine wave, the universal voltage formula is:

thus,

where,

• E is the sinusoidal rms or root mean square voltage of the winding,
• f is the frequency in hertz,
• N is the number of turns of wire on the winding,
• a is the cross-sectional area of the core in square centimeters or inches,
• B is the peak magnetic flux density in teslas per square centimeter, gausses per square centimeter, or lines (maxwells) per square inch.
• P is the power in volt amperes or watts,
• W is the window area in square centimeters or inches and,
• J is the current density.
• Note: 10 kilogauss = 1 tesla.

This gives way to the following other transformer equations for cores in square centimeters:

[edit] Imperial measurement system

The formulas for the imperial (inch) system are still being used in the United States by many transformer manufacturers. Most steel EI laminations used in the US are measured in inches. The flux is still measured in gauss or teslas, but the core area is measured in square inches. 28.638 is the conversion factor from 6.45 x 4.44 (see note 1). The formulas for sine wave operation are below. For square wave operation, see Note (3):

To determine the power (P) capability of the core, the core stack in inches (D), and the window-area (Wa) product, the formulas are:

where,

• P is the power in volt amperes or watts,
• T is the turns per volt,
• S is the current density in circular mils per ampere (Generally 750 to 1500 cir mils),
• W is the window area in square inches,
• C is the core width in square inches,
• D is the depth of the stack in inches and,
• Wa is the product of the window area in square inches multiplied by the core area in square inches. This is especially useful for determining C-cores but can also be used with EI types.

[edit] Simpler formulae

A shorter formula for the core area (a) and the turns per volt (T) can be derived from the long voltage formula by multiplying, rearanging, and dividing out. This is used if one wants to design a transformer using a sine wave, at a fixed flux density, and frequency. Below is the short formulas for core areas in square inches having a flux density of 12 kilogauss at 60 Hz (see note 2):

And for 12 kilogauss at 50 Hz:

[edit] Equation notes

• Note 1: The factor of 4.44 is derived from the first part of the voltage formula. It is from 4 multiplied by the form factor (F) which is 1.11, thus 4 multiplied by 1.11 = 4.44. The number 1.11 is derived from dividing the rms value of a sine wave by the its average value, where F = rms / average = 1.11.

• Note 2: A value of 12 kilogauss per square inch (77,412 lines per sq. in.) is used for the short formulas above as it will work with most steel types used (M-2 to M-27), including unknown steel from scrap transformer laminations in TV sets, radios, and power supplies. The very lowest classes of steel (M-50) would probably not work as it should be ran at or around 10 kilogauss or under.

• Note 3: All formulas shown are for sine wave operation only. Square wave operation does not use the form factor (F) of 1.11. For using square waves, substitute 4 for 4.44, and 25.8 for 28.638.

• Note 4: None of the above equations show the stacking factor (Sf). Each core or lamination will have its own stacking factor. It is selected by the size of the core or lamination, and the material it is made from. At design time, this is simply added to the string to be multiplied. Example; E = 4.44 f N a B Sf

[edit] Electrical steel types

[edit] Silicon steel

Ref: ^[1][2][3][4]

Material	Type See Note (1)	Nominal Silicon %	Max. Permeability μ	Max. Flux Density B	Useage
M-4, M-5, M-6	CRGO	2.8-3.5	15,000	17 kilogauss but magnetizing current raises rapidly over 15 kg	Highest efficiency power transformers
M-7, M-8	CRGO	2.8-3.5	10,000	17 kilogauss but magnetizing current raises rapidly over 15 kg	Large generators and power transformers
M-14	CRNO	4.0-5.0	8,500	14 kilogauss	Power and distribution transformers, high eff. rotating machines
M-15	CRNO	2.8-5.0	8,000	13 to 14 kilogauss	Transformers requiring low core loss and excellent permeability
M-19	CRNO	2.5-3.8	7,500	12 to 13 kilogauss	Communication transformers and reactors
M-22	CRNO	2.5-3.5	7,500	12 kilogauss	Cores of high reactance, intermittent duty transformers
M-27	CRNO	1.7-3.0	7,000	10 to 11 kilogauss	Small transformers operating at moderate induction
M-36	CRNO	1.4-2.2	<7,000	10 kilogauss	Used extensively for rotating machines
M-43	CRNO	0.6-1.3	<7,000	10 kilogauss	Fractional HP motors and relays
M-45	CRNO	0.0-0.6	<7,000	10 kilogauss	Fractional HP motors and relays
M-50	CRNO	0.0-0.6	<7,000	10 kilogauss	Intermittent operating apparatus and pole pieces

• Note 1: CRGO = Cold rolled, grain oriented, and CRNO = cold rolled, non oriented.

• Note 2: In the "M" numberining system set by the ASTM, the smaller number yields the highest efficiency, and lowest core losses. M-43 has a core loss at 12 kilogauss of approx. 2 watts per pound. M-15 at 12 kilogauss is approx. 0.75 watts per pound, and M-6 material has a loss of 0.64 watts per pound at 15 kilogauss^[5][6].

[edit] Other alloys

[edit] References

• Practical Transformer Design Handbook, Lowdon, Eric (1981), ISBN 0-672-21657-4
• Standard Handbook For Electrical Engineers, Fink, Donald (1969), ISBN 07-020973-1
• Reference Data For Radio Engineers, McPherson, W (1981), ISBN 0-672-21218-8
• Magnetic Circuits and Transformers, Engineering Staff, MIT (1949), John Wiley & Sons. ISBN N/A
• ATC-Frost Magnetics

[edit] Footnotes

1. ^ Lowdon,Eric (1981). Practical Transformer Design Handbook. ISBN 0-672-21657-4
2. ^ Fink,Donald (1969). Standard Handbook For Electrical Engineers. ISBN 07-020973-1
3. ^ McPherson,W (1981). Reference Data For Radio Engineers. ISBN 0-672-21218-8
4. ^ Eng. Staff of Massachusetts Institute of Technology (1949). Magnetic Circuits and Transformers. John Wiley & Sons. ISBN N/A
5. ^ Fink, Donald (1969). Standard Handbook For Electrical Engineers. ISBN 07-020973-1
6. ^ Eng. Staff of Massachusetts Institute of Technology (1949). Magnetic Circuits and Transformers. John Wiley & Sons. ISBN N/A