The self inductance of the motor of an electric fan is 10 H. In order to impart maximum powr of 50 Hz, it should be connected to a capacitance of [closed]

Question

how can i solve this (*)The self inductance of the motor of an electric fan is 10 H. In order to impart maximum powr of 50 Hz, it should be connected to a capacitance of

what is the concept behind the formula

i know this is the formula for c

Answer 1

In the equation

$$X_{C}=\frac{1}{2πfC}$$

$X_C$ is the capacitive reactance of a capacitor in response to a sinusoidal voltage source of frequency $f$ in Hz. The capacitive reactance is the magnitude of the impedance of the capacitor. Basically, it is the magnitude of the capacitor's resistance to current flow.

Since you have stated that you have not yet learned about phasor diagrams, I will attempt to explain the concept behind this equation without phasor diagrams.

The concept behind this equation is rooted in the general form of Ohm's Law and the basic relationship between the current in and voltage across a capacitor.

Per Ohm's law the relationship between the rms voltage across, rms current, and the magnitude of the capacitor's resistance to current flow is

$$I=\frac{V}{X_C}$$

Now according to the first equation above, the capacitive reactance decreases with an increase in the frequency $f$. This behavior of the capacitor comes from the basic relationship between the current and voltage for a capacitor

$$i(t)=C\frac{dv(t)}{dt}$$

Note that the greater the rate of change of voltage (the greater the frequency $f$), the greater the magnitude of the current in the capacitor. This is the equivalent of saying the greater the frequency, the lower the capacitor reactance in the Ohm's law equation.

MAXIMIMIZING POWER ($I^{2}R$)

Going beyond your specific question, for an inductor the inductive reactance (magnitude of the impedance of the inductor) is

$$X_{L}=2πfL$$

Note that for the inductor, frequency has the exact opposite effect as the capacitor. An increase in frequency increases the inductor's resistance to current flow. We take this into account by representing the impedance of the capacitor and inductor on the complex plane where the imaginary component $j$ is the y-axis and real component (resistance R) is on the x-axis. Then, the capacitor impedance is $-jX_C$, the inductor impedance is $+jX_L$, and the resistor impedance is $R$ on the real axis. For the series RLC circuit the equivalent impedance is then

$$Z_{eq}=R+Z_{C}+Z_{L}=R-jX_{C}+jX_{L}$$

The current, and thus power, is maximized in the circuit when the equivalent impedance is minimized. This occurs when the inductive and capacitive impedances cancel, i.e., when they are equal, or

$$X_{C}=X_{L}$$

$$\frac{1}{2πfC}=2πfL$$

or

$$C=\frac{1}{(2πf)^{2}L}$$

Hope this helps.