Transformer Voltage vs. Frequency

[sus4nx]

Homework Statement

Using two solenoids on a ferromagnetic core, I made a primitive transformer and graphed this graph of voltage against frequency which is attached.

Homework Equations

[What I'm looking for.]

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.

 

[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 because 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

 

[kerimek]

The linear pattern in the earlier part of the curve can be described by the equation V = N * dΦ/dt, where V is the voltage, N is the number of turns on the primary coil, and dΦ/dt is the rate of change of magnetic flux in the core. This equation is derived from Faraday's law of induction, which states that the induced voltage is proportional to the rate of change of magnetic flux.

The equation V = 4.44f is known as the transformer equation and it relates the voltage and frequency on the secondary side to the voltage and frequency on the primary side. This equation is valid for ideal transformers, where there is no energy loss due to hysteresis or other factors. In your experiment, since you are using a primitive transformer with solenoids and a ferromagnetic core, the voltage drop due to hysteresis is significant and cannot be ignored.

To model the behavior of the voltage drop due to hysteresis, you can use the equation V = V0 * e^(-k*f), where V0 is the initial voltage, k is a constant that depends on the material and geometry of the core, and f is the frequency. This equation takes into account the inverse relationship between hysteresis and voltage, and the increase in hysteresis as frequency increases.

In addition, you can also consider the skin effect, which is the tendency of alternating current to flow on the surface of a conductor rather than through its entire cross-section. This can be modeled by the equation V = V0 * e^(-k*f^2), where k is a constant that depends on the properties of the conductor and f is the frequency.

I hope this helps in understanding the behavior of your transformer and in modeling it accurately. Good luck with your lab!