Advanced Electromagnetism and Electromagnetic Induction

It is a known fact that the nonlinearity of a ferromagnetic material’s B-H curve will cause an inductor’s current to be non-sinusoidal, even when the voltage impressed across the inductor is perfectly sinusoidal:

Unless coil resistance is substantial, the core flux waveform (φ) over time will be just as sinusoidal as the voltage waveform, because without resistance to drop voltage, the relationship between voltage and flux is e = N[(dφ)/dt], the rate-of-change of a perfect sine wave being a perfect cosine wave.

Knowing that the core flux waveform will be sinusoidal allows us to derive the inductor current waveform from the B-H curve using a graphical “trick”: using the B-H curve to correlate instantaneous values of flux over time with instantaneous values of coil current over time. When used in this manner, the B-H curve is called a transfer characteristic, because it is used as a map to “transfer” points on one waveform to points on another waveform. We know that φ is directly proportional to B because B = [(Φ)/A], and the core area is constant. We also know that i is directly proportional to H, because

)/l], and both the core length and the number of turns of wire are constant:

Notice that the flux waveform is nice and sinusoidal, while the current waveform is not.

Based on what you see here, describe how an inductor designer can minimize the current distortion in an inductor. What conditions make this distortion better, and what conditions make it worse?

Faraday’s Law of electromagnetic induction states that the induced voltage across a coil of wire is equal to the number of “turns” in the coil multiplied by the rate of change of magnetic flux over time:

Often you will see a negative sign preceding the right-hand side of the equation, to properly denote polarity of the induced voltage. This is the mathematical expression of Lenz’s Law. In this equation, though, the negative sign is omitted and we pay attention only to the absolute value of induced voltage.

Use calculus techniques to express φ as a function of v, so that we may have an equation useful for predicting the amount of magnetic flux accumulated in an inductor or transformer given the voltage across it (v) and the time of the accumulation (T). Hint: you may treat this as a differential equation with separable variables.

For those who are unfamiliar with calculus, you may still answer this question, albeit in a simpler form: write an equation describing the change in magnetic flux within a coil (∆Φ) given a constant DC voltage across the coil (V) and a certain amount of time (t).