GATE 2026 EE Q.26: solution and answer

Official answer: BSource: Official answer key, GATE 2026 EE (IIT Guwahati)

1. RMS values. The source is a pure 50 Hz sinusoid, so $V_{rms}=\frac{325}{\sqrt{2}}\text{V}$. The current has a 50 Hz fundamental and a 150 Hz (3rd-harmonic) component: $I_1=\frac{12}{\sqrt{2}}\text{A}$ and $I_3=\frac{9}{\sqrt{2}}\text{A}$.

2. Total rms current. Components of different frequency are orthogonal over a period, so their rms values add in quadrature: $$I_{rms}=\sqrt{I_1^2+I_3^2}=\frac{\sqrt{12^2+9^2}}{\sqrt{2}}=\frac{15}{\sqrt{2}}\text{A}$$

3. Real power. Average power comes only from voltage and current components of the same frequency. The voltage has no 150 Hz component, so the 3rd-harmonic current contributes no real power. The fundamental current, $12\sin (2\pi 50t)$, is in phase with $v_s$, so $\cos {\varphi }_1=1$: $$P=V_{rms}{I}_1\cos {\varphi }_1=\frac{325}{\sqrt{2}}\cdot \frac{12}{\sqrt{2}}\cdot 1=1950\text{W}$$

4. Apparent power: $$S=V_{rms}{I}_{rms}=\frac{325}{\sqrt{2}}\cdot \frac{15}{\sqrt{2}}=2437.5\text{VA}$$

5. Power factor: $$\text{PF}=\frac{P}{S}=\frac{1950}{2437.5}=0.80$$

6. Check with the factorised form: PF = (displacement factor) × (distortion factor) $=\cos {\varphi }_1\times \frac{I_1}{I_{rms}}=1\times \frac{12}{15}=0.80$. Answer: (B).