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For an 'ideal' transformer operating at full load, the answer is yes. But, 'real' Transformers are a little less than 100% efficient so, in practice, the input power will slightly exceed the output power. In most circumstances, for the purpose of calculating primary and secondary currents, we can assume 100% efficiency.

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Yes. Transformers do not change the frequency of the input signal.

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Q: Does the input and output frequency of the transformer is the same?
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What is the basic difference between transistor as an amplifier and a step up transformer?

The output power of an amplifier is greater than its input power, whereas the output power of a transformer is almost the same as its input power. In other words, an amplifier increases acts to increase power, whereas a transformer only increases voltage.


Why PLL better than VCO?

A PLL is different than a VCO. Each has its own use. Actually a PLL (Phase Locked Loop) contains a VCO (Voltage controlled oscillator). A VCO is an oscillator whose frequency is related to an input voltage. You can use it when you need a varying frequency that is controlled by a varying voltage. But it is not great at outputting a consistant exact voltage because it is very sensitive to its environment (e.g. temperature). A PLL will "lock" its output frequency to some input frequency. So it can oscillate at a frequency that is controlled by an input oscillator. Not too useful if the output frequency is the same as the input. But the output frequency can be divided before it is compared to the input. This allows the output frequency to be higher (some multiple of) the input frequency. Once a PLL is "locked on" to an input frequency it can be very stable.


The ratio of coils on the input side of a transformer to coils on the output side is the ratio of the input voltage to the output voltage?

I think you mean 'turns' rather than 'coils' (a coil is made up of a number of turns). The answer is that, yes, the turns ratio is the same as the voltage ratio, for an ideal transformer.


What is the lowest ripple frequency of half wave rectifier?

The ripple frequency of a half-wave rectifier is the same as the input frequency.


Why positive feedback unstable?

A: To feedback more of the same signal from the output to increase the input to farther increase the output and farther increase the input will ultimately saturate the system