# Transformer Voltage vs. Frequency

• sus4nx
In summary: TheoryIn summary, the conversation discusses the use of two solenoids on a ferromagnetic core to create a primitive transformer and the resulting graph of voltage against frequency. The conversation also explores equations for the transformer, with a focus on the turns ratio and Lenz's Law. The conversation also mentions the effects of hysteresis on the transformer's behavior and suggests finding the equation for the hysteresis curve to determine the relationship between frequency and voltage.
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.

#### Attachments

• voltvsfreq.jpg
29.3 KB · Views: 995
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

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!

## 1. What is the relationship between transformer voltage and frequency?

The relationship between transformer voltage and frequency is known as the transformer voltage-frequency ratio. This ratio is determined by the transformer's turns ratio, which is the ratio between the number of turns on the primary winding and the number of turns on the secondary winding. As frequency increases, the transformer voltage decreases, and vice versa.

## 2. How does frequency affect the performance of a transformer?

The frequency of the input power can significantly impact the performance of a transformer. A higher frequency can cause the transformer to heat up and experience losses, while a lower frequency can cause voltage fluctuations and impact the transformer's efficiency. Properly matching the transformer's frequency to the input power is crucial for optimal performance.

## 3. Can a transformer operate at varying frequencies?

The ability of a transformer to operate at varying frequencies depends on its design and construction. Some transformers, such as audio transformers, are designed to operate at a specific frequency range. However, others, such as power transformers, can operate at a wide range of frequencies within their specified limits. It is essential to consult the transformer's specifications to determine its frequency range.

## 4. How do I calculate the output voltage of a transformer at a specific frequency?

To calculate the output voltage of a transformer at a specific frequency, you can use the transformer's turns ratio and the input voltage. Simply multiply the input voltage by the turns ratio to get the output voltage. For example, if the transformer has a turns ratio of 1:10 and the input voltage is 120V, the output voltage would be 1200V.

## 5. What happens if I apply a frequency that is outside of the transformer's frequency range?

If a frequency is applied that is outside of the transformer's frequency range, the transformer may not operate properly or may become damaged. Applying a higher frequency can cause the transformer to overheat and potentially fail, while a lower frequency can cause the transformer to experience voltage fluctuations and reduced efficiency. It is crucial to operate a transformer within its specified frequency range to ensure safe and efficient operation.

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