A circuit with an electrical load of impedance 

$Z$ is connected to an AC source with voltage $V(t)=300\sin (400t)$ V. The problem provides two lists:

• List-I describes different load configurations.

• List-II describes possible current waveforms $i(t)$ in the circuit.

The task is to match the correct current waveform from List-II to each load configuration in List-I.

Given:

• Source voltage: $V(t)=300\sin (400t)$ V (angular frequency $\omega =400$ rad/s).

• Load configurations (List-I) include resistors, capacitors, and inductors.

• Current waveforms (List-II) are sinusoidal functions with varying amplitudes and phases.

Answer: A

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Solution:

1. Key Concepts

• Impedance ($Z$): For an AC circuit, the impedance depends on the load:

  • Resistor ($R$): $Z=R$.

  • Inductor ($L$): $Z=j\omega L$.

  • Capacitor ($C$): $Z=\frac{1}{j\omega C}$.

• Current Calculation: The current $i(t)$ is given by:

  $$\mathrm{<span\; style="background-color:\; white;">i</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span>}{\mathrm{<span\; style="background-color:\; white;">Z</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">300</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span>}{\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">Z</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">,</span>$$

  where $\varphi$ is the phase difference between voltage and current.

2. Analyze Each Load Configuration

Assume the following mappings (since the original diagram is unclear, we infer based on standard configurations):

Load P: Pure Resistor ($R=30\mathrm{\Omega }$)

• Impedance: $Z=30\mathrm{\Omega }$.

• Current:

  $$\mathrm{<span\; style="background-color:\; white;">i</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">300</span>}}{\mathrm{<span\; style="background-color:\; white;">30</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Waveform: In-phase with voltage, amplitude 10 A.

• Matches List-II (3).

Load Q: Resistor-Inductor Series ($R=30\mathrm{\Omega }$, $L=25$ mH)

• Impedance:

  $$\mathrm{<span\; style="background-color:\; white;">Z</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">R</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">\omega </span>}\mathrm{<span\; style="background-color:\; white;">L</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">30</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">j</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.025</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">30</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">10</span>}\mathrm{<span\; style="background-color:\; white;">\Omega </span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Magnitude and Phase:

  $$\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">Z</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">=</span>\sqrt{{\mathrm{<span\; style="background-color:\; white;">30</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">1000</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">31.62</span>}\mathrm{<span\; style="background-color:\; white;">\Omega </span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">tan</span>}<span\; style="background-color:\; white;"></span>}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">10</span>}}{\mathrm{<span\; style="background-color:\; white;">30</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\approx </span>{\mathrm{<span\; style="background-color:\; white;">18.43</span>}}^{<span\; style="background-color:\; white;">\circ </span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Current:

  $$\mathrm{<span\; style="background-color:\; white;">i</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">300</span>}}{\mathrm{<span\; style="background-color:\; white;">31.62</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">18.43</span>}}^{<span\; style="background-color:\; white;">\circ </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">9.49</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0.322</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Waveform: Lagging phase, amplitude ≈ 9.49 A.

• Matches List-II (5).

Load R: Resistor-Capacitor Series ($R=60\mathrm{\Omega }$, $C=50\mu F$)

• Impedance:

  $$\mathrm{<span\; style="background-color:\; white;">Z</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">R</span>}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">\omega </span>}\mathrm{<span\; style="background-color:\; white;">C</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">60</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">400</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">60</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">50</span>}\mathrm{<span\; style="background-color:\; white;">\Omega </span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Magnitude and Phase:

  $$\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">Z</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">=</span>\sqrt{{\mathrm{<span\; style="background-color:\; white;">60</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">50</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">6100</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">78.1</span>}\mathrm{<span\; style="background-color:\; white;">\Omega </span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">tan</span>}<span\; style="background-color:\; white;"></span>}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">(</span>\frac{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">50</span>}}{\mathrm{<span\; style="background-color:\; white;">60</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\approx </span><span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">39.8</span>}}^{<span\; style="background-color:\; white;">\circ </span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Current:

  $$\mathrm{<span\; style="background-color:\; white;">i</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">300</span>}}{\mathrm{<span\; style="background-color:\; white;">78.1</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">39.8</span>}}^{<span\; style="background-color:\; white;">\circ </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">3.84</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0.695</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Waveform: Leading phase, amplitude ≈ 3.84 A.

• Matches List-II (2).

Load S: Pure Capacitor ($C=125\mu F$)

• Impedance:

  $$\mathrm{<span\; style="background-color:\; white;">Z</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">\omega </span>}\mathrm{<span\; style="background-color:\; white;">C</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">400</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">125</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">j</span>}\mathrm{<span\; style="background-color:\; white;">20</span>}\mathrm{<span\; style="background-color:\; white;">\Omega </span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Current:

  $$\mathrm{<span\; style="background-color:\; white;">i</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">300</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">90</span>}}^{<span\; style="background-color:\; white;">\circ </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">15</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">400</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1.57</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Waveform: Leads voltage by 90°, amplitude 15 A.

• Matches List-II (1).

3. Correct Matching

Based on the analysis:

• P → (3), Q → (5), R → (2), S → (1).

• This corresponds to Option A.

4. Final Answer

The correct match is:

\boxed{A}