Hi all,

I am working through the physics of transformer design, and on the attached slide, hit an inquiry of where the divide by 2 factor came from?

Any insight? I understand the rest.

Regards,

 

Most calculations involve the angular velocity of the AC waveform - ω - radians/second.

ω=2 π FHz

An easier to understand explanation is below, from This site, towards bottom of page

The rms voltage induced per turn is determined from Faraday's Law: √2 e = (2πf)BmaxA x 10-8

....
Two basic equations are used in transformer design. The first is essentially Faraday's law, E = dΦ/dt x 108 V, where Φ is maxwells (gauss times square centimeters). For transformers, this is written Bmax = √2 E 108 / 2πfNAK, where E = rms voltage in a winding, N = number of turns, f = frequency (Hz), A = area of core (cm2), K = stacking factor (the proportion of A occupied by iron). A sinusoidal variation in flux is assumed, which is a reasonable assumption when the core does not saturate, but by no means exact. Bmax is generally assumed as a design parameter, as well as a value for E/N in volts per turn, and the necessary A results. A typical value for a small transformer is E/N = 0.1 V/turn.


The second is Ampere's law, H = 0.4πNI/l, where l is the length of the magnetic circuit. From the value of Bmax, the value of Hmax can be found from the magnetization curve of the core material, and this equation used to determine N, when a reasonable value of I is assumed (say, 5% of the full-load current).

 

The direct answer to your question is that your extract specifically states "the positive peak". You need to double this to get the negative peak as well.

You should observe that the frequency of the power waveform is twice that of the voltage, current or flux waveform.

I will use sinusoids.

If \(\Phi\) is the maximum value of the flux and f the frequency then the flux changes from +\(\Phi\) to -\(\Phi\) in half a cycle i.e. 1/2f seconds.

So the average rate of change of flux is 2\(\Phi\) /0.5f = 4f\(\Phi\)

Note the 2\(\Phi\) is the distance between positive and negative peaks.

But the average rate of change of flux = average emf induced.

Thus induced emf = 4f\(\Phi\) volts per turn

Multiply this by the form factor (1.11 for sinusoids) to get the RMS value and we have the well known equation

EMF induced per turn = 4.44Nf\(\Phi\) where N is number of turns.

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A calculus method is as follows; If \(\phi\) is the instantaneous flux then

\(\phi\) = \(\Phi\)sin2π ft

Instantaneous induced voltage per turn is

-d\(\phi\)/dt volts

= -\(\Phi\) x cos2πft volts (note the minus sign)

= 2πf\(\Phi\) x sin(2πft -π/2)

Thus the max value occurs when the sin term is 1 and equals 2πf\(\Phi\) volts per turn.

Thus the RMS value = √2 x 2πf\(\Phi\) volts per turn = 4.44f\(\Phi\) as before.

This is a shortened version of thatoneguys's link.

 

Thanks guys