Question

The working frequency of an RLC circuit should be its resonant frequency. The maximum current in the circuit, however, is not enough for this to occur. By what factor should the resistance of the circuit be changed so that the maximum current reaches the value in order for the circuit to work?

Answer 1

The described circuit has a current of

$$I_1=\frac{U}{\sqrt{R^2+[\omega L-1/(\omega C)]^2}}.$$

But it must have a current with magnitude of

$$I_2=\frac{U}{R},$$

since it must work at the resonant frequency. At the resonant frequency the capacitive and inductive reactances are equal to each other and thus the total impedance decreases.

To make $I_1$ equal to $I_2$, we can decrease the resistance in the first equation for the current (now label it $r$). Determine it from the condition of equal currents:

$$R=\sqrt{r^2+[\omega L-1/(\omega C)]^2},\\ \\ r=\sqrt{R^2-[\omega L-1/(\omega C)]^2}.$$

So, the initial resistance $R$ should be decreased by $k$ times:

$$k=\frac{r}{R}=\frac{\sqrt{R^2-[\omega L-1/(\omega C)]^2}}{R}.$$