How to build a transformer or inductor

The technique for building a transformer and inductor are the same, however designing them are different.


 * ** If you are interested in a quick design of an inductor or transformer, with out the details or precision follow the red double asterisk, ' ** '

=Variables, Units, and Acronyms= You have to keep track of units during design. Its easy to get mixed up. Wire gauges are generally in non-metric, as well as some other variables. For the most part keep units in metric and distances in centimeters

The following quantities are specified, using the units noted:


 * Universal constants
 * Permittivity of free space $$\mu_o$$ (Wb A-1 m-1)
 * $$\mu_o = 4\pi 10^{-7}$$ (Wb A-1 m-1)


 * Wire variables:
 * $$\rho$$, Wire resistivity (Ω-cm)
 * $$I_{tot}$$, Total RMS winding currents (A)
 * $$I_{m,max}$$, Peak magnetizing current (A)
 * $$I_{RMS}$$, Max RMS current, worst case (A)
 * $$P_{cm}$$, Allowed copper loss (W)
 * $$A_c$$, Cross sectional area of wire (cm2)


 * Transformer/inductor design parameters
 * $$n_1, n_2$$, turns (turns)
 * $$L_m$$, Magnetizing inductance (for an xformer) (H)
 * $$L$$, Inductance (H)
 * $$K_u$$, Winding fill factor (unitless)
 * $$B_{max}$$, Core maximum flux density (T)


 * Core parameters
 * EC35, PQ 20/16, 704, etc, Core type (mm)
 * $$K_g$$, Geometrical constant (cm5)
 * $$K_{gfe}$$, Geometrical constant (cmx)
 * $$A_c$$, Cross-sectional area (cm2)
 * $$W_A$$, Window area (cm2)
 * $$MLT$$, Mean length per turn (cm)
 * $$l_m$$, Magnetic path length (cm)
 * $$l$$, or $$l_g$$, Air gap length (cm)
 * $$\mu$$, Permittivity (Wb A-1 m-1)
 * $$\mu_r$$, Relative Permittivity (unitless)
 * $$\mu = \mu_o \mu_r$$


 * Acronyms
 * RMS: root-mean-squared - $$x_\text{rms} = \sqrt{ \langle x^2 \rangle} \,\!$$ (where $$\langle \ldots \rangle$$ denotes the arithmetic mean)
 * MLT: mean length turn
 * AWG: American wire gauge

=Design=

Selecting the size and type of core
This is a very very basic way to choose a core. The type of core has a lot to do with the frequency, current, and power. The size and type of the core has to do with core loss, power loss in the core.


 * Low frequency such as 60-120Hz should use laminated core **
 * High frequency 1kHz-1MHz should use a ferrite core. **

The size of the core depends on the power of the transformer, and expected power loss in the core (core loss).

Core size
$$ K_g \geq \frac{\rho L^{2}_{M} I^{2}_{tot} I^{2}_{M,max}}{B^{2}_{max} P_{cu} K_{u}} 10^8 (cm^5) $$

Be careful about units
 * Copper (wire) resistivity $$\rho$$
 * $$\rho = 1.724 \times 10^6 \Omega$$ cm @ 20 C
 * $$\rho = 2.3 \times 10^6 \Omega$$ cm @ 100 C
 * Transformers, inductors can get warm, and hot so it may not be room temp
 * $$B_{max}$$, Max core magnatizing flux density (T)
 * $$I_{RMS}$$, Max RMS current, worst case (A)
 * $$K_{u}$$, Winding fill factor (unitless)
 * $$L$$, Inductance (H)

The core $$ K_g = \frac{A^{2}_{c} W_A}{(MLT)} $$
 * When selecting a core you have the following parameters
 * $$MLT$$, Mean length per turn (cm)
 * $$A_c$$, Core cross sectional area (cm2)
 * $$W_A$$, Core window area (cm2)
 * $$K_g$$, Core geometrical constant (cm5)
 * Types of Cores Magnetic core

Selecting the wire gauge
The selection on the wire gauge has to do with the amount of resistance that is acceptable, the current flowing through the inductor, and if all the turns can fit in the area of the transformer. The size of the transformer can always be increased if more area is needed

Resistive Loss
Variables in calculating resistance
 * $$MLT$$, Mean Length per Turn, the average length of wire to complete a full turn (cm)
 * $$A_w$$, Cross sectional area of wire (cm2)
 * $$n$$, number of turns
 * $$\rho$$, resistivity of copper, 1.724 10^–6 (Ω-cm)

Measure Mean Length per Turn (MLT). The easy way to measure MLT is to take a wire and wrap it around the core or bobbin, loosely. If you plan on have multiple turns try to make an average loop, but looser to be safe. Measure the wire and that's your MLT. Some cores will give you the MLT in the specification. Keep in mind that the specification is for a fully filled core, but use it to be safe. Always be conservative, and make the length longer.

Number of turns (n) is gotten when you calculate the inductance for an inductor or turns ratio for a transformer.

Cross sectional area of wire (Aw) is based on the size wire you choose, obviously. The size of wire goes on the American wire gauge (AWG)). Wikipedia has a chart of wire gauges with there Area. See American wire gauge.

The equation for resistance is:

$$ R=\rho \frac{n MLT}{A_w} $$

Remember to keep you units correct.

Fill-factor
** Another factor you need to be aware of is will all the turns of wire fit in your core. This is called the fill-factor. If the amount of turns you need with the wire size you need does not fit you can always use a bigger core.

The variables are
 * $$W_A$$, Window area (cm2)
 * $$A_W$$, Wire area (cm2)
 * See American wire gauge for wire areas
 * $$n$$, Number of turns
 * $$K_u$$, Window utilization factor, the fill-factor
 * $$K_u$$, must be less than 1
 * Realistic fill-factors
 * 0.5 for simple low-voltage inductor
 * 0.25 to 0.3 for off-line transformer
 * 0.05 to 0.2 for high-voltage transformer (multiple kV)
 * 0.65 for low-voltage foil-winding inductor

So you must follow this equation

$$ K_u W_A \geq n A_W $$

For multiple wire types the equation would be

$$ K_u W_A \geq n_1 A_W1 + n_2 A_W2 + ... $$

Inductor
In general inductors use a ferrite core.

Inductance for a coiled bobbin, with a magnetic core

$$ L=\frac{\mu A_c n^2}{l} $$

In general you control the inductance by making l an air-gap, which will be very small and μ the permittivity of free space $$ l=l_g $$, $$ \mu = \mu_o = 4 \pi x 10^-7 M/m $$

where
 * $$n$$, number of turns
 * $$A_c$$, cross section area of the core
 * $$\mu$$, permittivity of the free space or if no air gap the permittivity of the ferrous material
 * $$l$$, length of the air gap or if no air gap the length of the ferrous materials loop

Inductance for a toroid

$$ L=\frac{\mu A_c n^2}{2 \pi r} $$

where
 * $$n$$, number of turns
 * $$A_c$$, cross section area of the core
 * $$\mu$$, permittivity of the free space or if no air gap the permittivity of the ferrous material
 * $$r$$, radius of the toroid (to the center/middle of the ferrous material)

Inductance for a short air core coil

$$ L=\frac{r^2 n^2}{9 r + 10 l} $$

where
 * $$n$$, number of turns
 * $$r$$, radius of the coil
 * $$l$$, length of the coil

Advanced Inductor design
Nearly full description of Inductor design. Very good, but very technical. Recommended for building inductors in optimizing power, size, losses, and precise inductance.

Chapters from a power electronics course
 * Chapter 14 Inductor Design
 * Chapter 13. Filter Inductor Design

Transformer
The base equations for a transformer.

$$ \frac{V_1}{V_2}=\frac{n_1}{n_2} $$

$$ \frac{I_1}{I_2}=\frac{n_2}{n_1} $$

In most cases you don't want inductances in a transformer, unless you are using it in a switching converter, or filter. Inductance only has to be a modeled on one side, as. $$L_m$$ If your transformer has no air gap the inductance will be low, and can be ignored.....
 * Transformers have inductance.

$$ L_m=\frac{\mu A_c n^2_1}{l} $$

=Construction=

Winding
There are easy ways to wind a core and there are hard ways. Well semi-easy.

Ferrite core, bobbin
Winding a ferrite core is very easy. You just need to wrap the wire around the bobbin.

Toroid
If you only need a few windings the solution is simple. Just wind it.

When there are many windings, the easiest way to wind a Toroid is to make a needle like show in the laminated core image. The needle needs to be thinner, and the length of the needle dictates the length of the wire you can wrap without splicing two wires.

Making the needle: **
 * Get a soft semi-flexible piece of plastic, or what ever u can find
 * Cut it in the shape shown in the image
 * poke a hole in the needle to have the wire start at
 * Wind your needle
 * Don't make it thicker that the Toroid (obviously)

To wind it: **
 * hold one end of the wire
 * thread the needle trough the Toroid.
 * Wrap it around the torrid
 * Make sure the loops are tight, and close together. Well wrapped loops increase the number of windings you can make.
 * Repeat

Laminated iron core

 * to be: **

Air core

 * Get a plastic screw
 * Width of the screw being twice the radius of the coil.
 * Thread size to match the number of turns with the length of the coil. It wont be perfect, but you can compress or stretch the coil to the correct length
 * Bring the wire and a ruler to the hardware store

=References=
 * Chapter 14 Inductor Design: very good.
 * Chapter 13. Filter Inductor Design: very good
 * Electronic Components: very good
 * Transformer_design
 * http://www.geofex.com/Article_Folders/xformer_des/xformer.htm
 * https://nicoreindia.com/transformer-core/

From HowTo Wiki, a Wikia wiki.
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