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I was reading about the transformer, when I came across a note by the author that stated that the Frequency is not changed as the flux linked to the secondary coil in a transformer changes. Is there any mathematical proof, why the frequency remains unchanged as the flux changes?

Thank You.

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"The frequency doesn't change" is only true when the core is perfectly linear. For a real transformer, there will be some nonlinear effects (saturation) meaning that the sinusoidal input waveform will create harmonics in the output - second harmonics and higher frequencies will appear.

But if you ignore those, then the flux change will vary sinusoidally at a particular frequency and the induced current in the secondary must follow these changes - it cannot either "run ahead" (follow a change that hasn't happened yet) or "fall behind" (not change when the input is changing).

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if you imagine correctly, the magnetization and demagnetization is happening at the frequency of the AC supply.

Therefore when primary side changes flux in one direction there is the same change in the secondary side and the same time interval.

The same way the opposite direction in another same interval of time. So the frequency the Primary AC had will be transmitted to the secondary AC via core.

No need of a mathematical explanation.

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Let $\color{blue}{i=i_0\sin(\omega t)}$ be the input alternating current to the primary coil with a frequency $\color{red}{\omega}$ then the voltage induced in the secondary coil of transformer is given as $$V_{in}=-M\frac{di}{dt}$$ Where, $M$ is the mutual inductance setting the value of $i$, $$V_{in}=-M\frac{d}{dt}(i_0\sin(\omega t))$$

$$=-Mi_0\omega \cos(\omega t)$$$$=\color{red}{-Mi_0\omega\sin \left(\omega t+\pi/2\right)}$$

It is clear that the frequency of output/induced voltage $V_{in}$ is $\color{red}{\omega}$ which is same as that of the input current.

Hence, in a transformer, the frequency remains unchanged (constant) as flux changes.

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Transformer work on mutual induction. Due to this frequency always be constant. Read mutual induction carefully you will get your answer.

V=L(di/dt)

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