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Larry:
You are correct that if you lower the voltage in proportion to lowering the frequency for the same transformer, then the same saturation level is reached.
The integration of voltage over time scaled by some constant determines the flux density. What is integration? It is the summation of very small areas from one limit to another. The very small areas in this case are the magnitude of the voltage times an infinitely small increment of time. What the integration does is calculate the area under the voltage curve from one point in time to another.
So consider a sine wave. Let zero degrees be the positive zero crossing. The integral from 0 deg to 0 deg is 0 because there was zero area under the voltage curve from 0 to 0. Note as we move from 0 deg to 180 deg the value of the area under the curve increases. The integral of sin u du is -cos u. When you evaluate the integral from one point to another the final value is evaluated first and the initial value second and subtracted from said first.
The integral from 0 deg (0 radians) to 90 deg (Pi/2 radians) is - cos 90 - (-cos 0) = -0 - (-1) = 1 . Next consider the integral over the full half cycle. The result is - cos 180 - (- cos 0) = - (-1) - (-1) = 2 .
If we want to know the average value of a half wave rectified voltage, then we divide the area under the curve by the base of the curve over which the average is being determined. For this calculation the angle is measured in radians. The base for a full cycle is 2*Pi. Thus, the average is 2/2*Pi = 1/Pi = 0.318 . If we want the full wave rectified average value, then double this value and the result is 0.636 . This you might recognize as the constant to convert the peak value of a sine wave to its full wave average value.
In a similar fashion you can derive the constant for conversion of sine wave peak to its RMS value. The integral of sine squared is (u/2) - (sin 2u/4). RMS is the root mean square. For 1/2 cycle this is (Pi/2 - 0) - ( 0 - 0 ) = Pi/2 . To average this divide by the time of 1/2 cycle, (Pi/2)/Pi = 1/2 . Then take the square root (the root part of RMS) and the result is 0.707 . Again a familiar constant.
Back to the magnetic circuit. As you progress from the positive zero crossing of the voltage to the negative zero crossing the flux is increasing according to the function K * ( 1 - cos u ). Therefore maximum flux density in the positive direction occurs at the negative zero crossing. Look at photo P7 on my web site at
www.beta-a2.com/EE-photos.html
Here you see the maximum magnetizing current pulses occurring at the negative zero crossing, about 0.9 CM on the scope scale for the first peak. These pulses result from the core going into saturation.
In the magnetic circuit the flux is a result of the time the voltage is applied and thus a scaling factor has to be applied in front of the integral that is proportional to the period of the sine wave. Period is the inverse of frequency.
The integral alone of the sin gives us the shape of the buildup of the flux. The absolute magnitude of the flux and therefore flux density is determined by the length of time for one cycle and other parameters, such as, number of turns, core material, and cross-sectional area of the core.
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