# How does changing the angular frequency ω of the generator affect the current flowing in the circuit and the power transferred to the electric motor?

When an electric motor is used with an electric generator, the situation can be viewed as an RLC circuit where the generator is an AC power source. The source produces a sinusoidal AC voltage with an value of V and an angular frequency ω. The work done by the motor is reflected in the resistance R of the circuit. How does changing the angular frequency ω of the generator affect the current flowing in the circuit and the power transferred to the electric motor?

Let's define inductive reactance $$\chi_L=\omega L$$ and capacitive reactance $$\chi_c=\frac{1}{\omega C}$$. Then, using resistance $$R$$, the impedance of the circuit is $$Z=\sqrt{R^2+(\chi_L-\chi_C)^2}$$ The rms current is $$I=\frac{V}{Z}$$ where $$V$$ is rms voltage.
The phase between the current and voltage is given by $$\tan\delta=\frac{\chi_L-\chi_C}{R}$$ Then, the power is simply $$P=IV\cos\delta$$ Inserting the initial variables, $$P=\frac{V^2}{\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}}\cdot\sqrt{\frac{\omega L-\frac{1}{\omega C}-R}{R}}$$ The power is the highest when $$\delta=0$$ which happens when $$\chi_L=\chi_C$$.