# Transformer Voltage vs. Frequency

1. Dec 19, 2009

### sus4nx

1. The problem statement, all variables and given/known data
Using two solenoids on a ferromagnetic core, I made a primitive transformer and graphed this graph of voltage against frequency which is attached.

2. Relevant equations

[What I'm looking for.]

3. The attempt at a solution

What I am having trouble is, what equation describes the linear pattern in the earlier part of the curve? There's one on wikipedia, V=4.44f, but why doesn't this relate to V1/N1=V2/N2, eqn for transformer in an alternating current?

And I know that the voltage drops off because of hysteresis which is inversely proportional to voltage, and that the delaying effects of hysteresis become more and more significant to the shortening period as frequency increases. What function do I need to model this behaviour?

I would be really glad if anyone would help, because the lab's due Monday.

#### Attached Files:

• ###### voltvsfreq.jpg
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2. Dec 19, 2009

### chaoseverlasting

For the transformer equation, youre right, the turns ratio equation does hold. The 4.44f... equation is an approximation.

If you assume the voltage across the primary is purely sinusoidal and no harmonics are present,

$$V_p=V_{max}sin\omega t$$

This causes a current

$$I_p=I_{max}sin(\omega t+\theta)$$ where the phase difference exists beacause of the inductance of the primary coil.

This current creates a magnetic flux

$$\phi _p=\phi _{max} sin(\omega t+ \theta)$$ which is in phase with the current.

Now, the voltage induced in the secondary is given by Lenz's Law:

$$E_s=-n\frac{d\phi}{dt}$$ where n is the number of turns of the secondary

This gives $$E_s=n\phi _{max}\omega sin(\omega t+\theta)$$

$$\omega =2\pi f$$

This secondary induced voltage is approximated by $$E_s=4.44nf\phi$$

As for the hysteresis problem, I'm sure you could find the equation of the hysteresis curve online. From there you know the limiting value of magnetic flux. Using that and the above discussion you could find the relation between frequency and voltage.

Hope that helps.

Chaos