User:Msiddalingaiah/Transformer design

Switching Supply Core

The approximate power a core can support is

$P\approx {\frac {(NaB)^{2}f}{L}}$

or

$P\approx {\frac {5\times 10^{5}B^{2}a^{\frac {3}{2}}f}{\mu _{r}}}$

The derivation comes from these fundamental equations:

$V=N{\frac {d\phi }{dt}}$

$V=L{\frac {di}{dt}}$

$E={\frac {1}{2}}Li^{2}$

$L\approx 3.937\times 10^{-5}{\frac {(rN)^{2}}{9r+10l}}$

where,

• P is the power in Watts
• 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 meters,
• B is the peak magnetic flux density in Teslas,
• L is the inductance in Henries,
• r is the radius of a solenoid in meters,
• l is the length of a solenoid in meters,
• $\mu _{r}$ is the dimensionless relative permeability
• $5\times 10^{5}$ is a constant with units of meters/Henry
• $3.937\times 10^{-5}$ is a constant with units of Henries/meter

Equations

There are 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:

$E={{\frac {2\pi fNaB}{\sqrt {2}}}{10^{-8}}}\!=4.44fNaB{10^{-8}}$

thus,

$E={4.44fNaB{10^{-8}}}\!$

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 or Webers per square meter, 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:

$N={\frac {E10^{8}}{4.44fBa}}\!$

$B={\frac {E10^{8}}{4.44fNa}}\!$

$a={\frac {E10^{8}}{4.44fBN}}\!$

$P={0.707JfWaB}\!$

The derivation of the above formula is actually quite simple. The maximum induced voltage, e, is the result of N times the time-varying flux:

e = N dφ/dt

If using RMS voltage values and E equal the rms value of voltage then:

e = E${\sqrt {2}}$

and

E = dφ/dt${\frac {N}{\sqrt {2}}}$

Since the flux is created by a sinusoidal voltage, it too varies sinusoidally:

φ(t) = Φ$_{max}\sin wt$ = A$B_{max}\sin wt$, where A = area of the core

Taking the derivative we have:

dφ(t)/dt = wA$B_{max}\cos wt$

Substituting into the above equation and using

w = 2 ${\pi }$f and the fact that we are only concerned with the maximum value yields

E = ${\frac {{2\pi }fNAB}{\sqrt {2}}}$

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):

$E={28.638faNB}\!$

$N={\frac {E10^{8}}{28.638fBa}}\!$

$T={\frac {10^{8}}{28.638faB}}\!$

$B={\frac {E10^{8}}{28.638fNa}}\!$

$a={\frac {E10^{8}}{28.638fBN}}\!$

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

$P={\frac {fBWa}{17.26S}}\!$

$Wa={\frac {17.26SP}{fB}}\!$

$D={\frac {17.26PS}{WCfB}}\!$

where,

• P is the power in volt amperes or watts,
• T is the volts per turn,
• E is the RMS voltage,
• 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.

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, rearranging, 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):

$a={0.1725{\sqrt {P}}}\!$

$T={\frac {4.85}{a}}\!$

And for 12 kilogauss at 50 Hz:

$a={0.206{\sqrt {P}}}\!$

$T={\frac {5.82}{a}}\!$

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 (77,400 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