Derive expression for impedance
Dear Student,
If we have the following circuit,
Then we have current and voltage as
....(i)
the voltage drops across the capacitor and inductor can be written as cosines since displacing a sine function by 90 degrees transforms the function into a cosine. With calculus, one would realize the voltage drops across the inductor and capacitor would have cosine dependence because they are related by derivatives to the current.
We can now sum the voltages to get:
Using double-angle formulas, one can rewrite the expression on the right:
The impedance is Z, and the phase difference between the current and the voltage of the source is f
Regards.
If we have the following circuit,
Then we have current and voltage as
....(i)the voltage drops across the capacitor and inductor can be written as cosines since displacing a sine function by 90 degrees transforms the function into a cosine. With calculus, one would realize the voltage drops across the inductor and capacitor would have cosine dependence because they are related by derivatives to the current.
We can now sum the voltages to get:
Using double-angle formulas, one can rewrite the expression on the right:
The impedance is Z, and the phase difference between the current and the voltage of the source is f
Regards.