1. **State the problem:** We are given voltage $v(t) = 100 \sin(\omega t - 10) + 200 \sin(2\omega t - 20)$ and current $i(t) = 10 \cos(\omega t - 50) + 5 \cos(3\omega t - 30)$. We need to find the reactive power of this system. 2. **Recall the formula for reactive power:** Reactive power $Q$ is related to the voltage and current components that are out of phase. For sinusoidal signals, reactive power for each frequency component is given by $$Q = V I \sin(\phi_v - \phi_i)$$ where $V$ and $I$ are the RMS values of voltage and current at the same frequency, and $\phi_v$, $\phi_i$ are their phase angles. 3. **Important rules:** - Only components at the same frequency contribute to power. - Voltage and current components at different frequencies do not interact for power calculation. - Convert peak amplitudes to RMS by dividing by $\sqrt{2}$. - Convert all angles from degrees to radians for sine calculations or use degrees consistently. 4. **Identify frequency components:** - Voltage has components at $\omega$ and $2\omega$. - Current has components at $\omega$ and $3\omega$. 5. **Match frequency components:** - At frequency $\omega$: - Voltage amplitude $V_1 = 100$ (peak), phase $\phi_{v1} = -10^\circ$ - Current amplitude $I_1 = 10$ (peak), phase $\phi_{i1} = -50^\circ$ - At frequency $2\omega$: - Voltage amplitude $V_2 = 200$, phase $\phi_{v2} = -20^\circ$ - No current component at $2\omega$, so no power contribution. - At frequency $3\omega$: - Current amplitude $I_3 = 5$, phase $\phi_{i3} = -30^\circ$ - No voltage component at $3\omega$, so no power contribution. 6. **Calculate reactive power at $\omega$:** - Convert peak to RMS: $$V_{1,rms} = \frac{100}{\sqrt{2}} \approx 70.71$$ $$I_{1,rms} = \frac{10}{\sqrt{2}} \approx 7.07$$ - Calculate phase difference: $$\Delta \phi_1 = \phi_{v1} - \phi_{i1} = -10^\circ - (-50^\circ) = 40^\circ$$ - Calculate reactive power: $$Q_1 = V_{1,rms} \times I_{1,rms} \times \sin(40^\circ)$$ $$Q_1 = 70.71 \times 7.07 \times 0.6428 \approx 321.3$$ 7. **Calculate reactive power at $2\omega$ and $3\omega$:** - No matching current at $2\omega$, no reactive power. - No matching voltage at $3\omega$, no reactive power. 8. **Total reactive power:** $$Q = Q_1 = 321.3$$ **Final answer:** The reactive power of the system is approximately $321.3$ (units depend on voltage and current units, typically VAR).