A variable inductor with negligible resistance is connected to an ac voltage source. By what factor does the current in the inductor decrease if the inductance is increased by a factor of 8.0 and the driving frequency is increased by a factor of 4.0?
It's easier than using all that math. Increase the L by 8 and the reactance goes up by 8. Reactance is proportional to inductance. Increase the f by 4 and the reactance goes down by 4. Reactance is inversely proportional to frequency. net, reactance goes up by 2, so the current goes down by 2
For an inductor, Z j*2*pi*f*L If the inductance is increased by 8x, and the frequency is increased by 4x, the net increase in impedance is 32x. So the current drops to1/32 or 3.125% of the original magnitude.
Assuming that the voltage source is ideal, i.e., has zero series impedance, and there are no other elelements in the circuit, the current through the inductor will be V / Z? where V is the voltage of the source, which is also the voltage across the load inductor, and Z? is the impedance of the inductor, with Z? Rs + jωL, where Rs is the equivalent series resistance of the inductor, which in this example is negligible, ω is the angular frequency of the excitation, which is 2π times the driving frequency f, and L is the inductance. The j is of course the principal root of -1. Neglecting the resistive component, the magnitude of the current will therefore be V / (2πfL) If fo is the original frequency and Lo the original inductance, the current after the changes in frequency and inductance will be V / [2π(4fo)(Lo/8)] V / [2πfoLo*(4/8)] 2 V / (2πfoLo) so the current increases by a factor of 2.
XL 2 pi x Fo x L L XL / (2 pi x Fo) It seems to me that in an AC circuit where the resistance is zero and the circuit is pure inductance, the current lags the voltage by a full 90 degrees. Set an initial condition of, say: XL 1000 ohms pi 6.28 Fo 1000 Hz L 1000 / (6.28 x 1000) 0.16 H Let L increase by 8 (0.16 H x 8) 1.28 H XL 6.28 x 1000 Hz x 1.28 8,038 ohms Let Fo increase by 4 XL 6.28 x 1000 x 4 x 1.28 32,154 ohms -------------------------------------- Initial conditon: XL 1000 ohms at circuit current A When L increased by 8, resistance increased by 8, circuit current decreased by A / 8 When Fo was allowed to increase by a factor of 4, resistance increased by 32 times and circuit current dropped by A / 32 ( graph of inductive reactance versus frequency for . Notice the response is linear---increasing the frequency increases the reactance in proportion. However, in real inductors, the reactive response is a bit more complicated, since real inductors have parasitic resistance and capacitance built in.) --------------------
