Sinusoidal Response of Parallel RC Circuit MCQ Quiz - Objective Question with Answer for Sinusoidal Response of Parallel RC Circuit - Download Free PDF

Explanation:

Nature of Supply Current in Parallel RC Circuit:

Definition: In a parallel RC (Resistor-Capacitor) circuit, the resistor and capacitor are connected in parallel to each other, and the supply current is distributed between the resistor and capacitor branches. The overall nature of the supply current is determined by the combined behavior of these two components concerning the applied voltage.

Correct Option Analysis:

The correct answer is:

Option 3: Leads by 90 degrees.

In a parallel RC circuit, the capacitor causes a phase shift in the current with respect to the voltage. Capacitors are reactive elements, and their current leads the voltage by 90 degrees. On the other hand, resistors do not cause any phase shift, and their current is in phase with the voltage.

When these two currents combine in a parallel circuit, the supply current leads the voltage due to the dominant effect of the capacitor's current. The resulting phase angle depends on the relative magnitude of the resistive and capacitive currents. However, in general, the supply current leads the voltage, and this behavior is characteristic of parallel RC circuits.

Mathematical Analysis:

The supply current in the parallel RC circuit is the vector sum of the currents through the resistor (I_R) and the capacitor (I_C).

Let:

• V = Applied voltage
• R = Resistance
• X_C = Capacitive reactance

The current through the resistor is given by:

I_R = V / R

The current through the capacitor is given by:

I_C = V / X_C

In a capacitor, the current leads the voltage by 90 degrees. Hence, I_C is out of phase with I_R. The total supply current (I_Supply) is obtained by the phasor addition of I_R and I_C. This results in a phase angle between the supply current and the voltage, where the current leads the voltage.

Advantages of Understanding:

• Helps in circuit design and analysis in AC systems.
• Supports the proper selection of components for desired phase characteristics.
• Aids in understanding power factor correction techniques.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: In phase.

This option is incorrect because, in a parallel RC circuit, the supply current cannot be in phase with the voltage due to the presence of the capacitor. Capacitors inherently cause a phase shift where the current leads the voltage. Therefore, the supply current cannot be entirely in phase with the applied voltage.

Option 2: Lags by 90 degrees.

This option is incorrect because the capacitive current leads the voltage by 90 degrees, not lags. Although inductive elements can cause the current to lag the voltage by 90 degrees, this is not the case for capacitive elements in the RC circuit.

Option 4: Compensates and becomes zero.

This option is incorrect because, in a practical parallel RC circuit, the supply current does not become zero. Both the resistive and capacitive components contribute to the overall current, and their vector sum results in a non-zero supply current.

Option 5: Not applicable.

This option is irrelevant as it does not provide a valid description of the behavior of the supply current in a parallel RC circuit.

Conclusion:

The supply current in a parallel RC circuit leads the voltage due to the capacitive component's influence. Understanding this phase relationship is crucial for analyzing AC circuits, designing systems with specific phase requirements, and implementing power factor correction techniques.

Explanation:

• Since we know that, for the parallel circuit, the voltage and the voltage phase difference is same irrespective of the elements.
• Only the phases of current changes.
• So, In a parallel RC circuit, the phase difference between the applied voltage and the voltage across R and C in parallel will be 0° only.

I = 0.3 A

P = 11 W

\(\Rightarrow \frac{{{V^2}}}{R} = 11\) 

\( \Rightarrow \frac{{{{\left( {110} \right)}^2}}}{R} = 11\) 

⇒ R = 1100 Ω

\({I_R} = \frac{V}{R} = \frac{{110}}{{1100}} = 0.1\;A\) 

\(I = \sqrt {I_R^2 + I_C^2} \) 

\( \Rightarrow {I_C} = \sqrt {{{\left( {0.3} \right)}^2} - {{\left( {0.1} \right)}^2}} = 0.28\;A\) 

\({V_C} = {I_C}{X_C}\) 

\(\Rightarrow {V_C} = {I_C}\frac{1}{{\omega C}}\) 

\( \Rightarrow C = \frac{{{I_C}}}{{\omega {V_C}}} = \frac{{0.282}}{{2\pi \times 60 \times 110}} = 6.82\;\mu F\) 

Explanation:

• Since we know that, for the parallel circuit, the voltage and the voltage phase difference is same irrespective of the elements.
• Only the phases of current changes.
• So, In a parallel RC circuit, the phase difference between the applied voltage and the voltage across R and C in parallel will be 0° only.

Explanation:

• Since we know that, for the parallel circuit, the voltage and the voltage phase difference is same irrespective of the elements.
• Only the phases of current changes.
• So, In a parallel RC circuit, the phase difference between the applied voltage and the voltage across R and C in parallel will be 0° only.

I = 0.3 A

P = 11 W

\(\Rightarrow \frac{{{V^2}}}{R} = 11\) 

\( \Rightarrow \frac{{{{\left( {110} \right)}^2}}}{R} = 11\) 

⇒ R = 1100 Ω

\({I_R} = \frac{V}{R} = \frac{{110}}{{1100}} = 0.1\;A\) 

\(I = \sqrt {I_R^2 + I_C^2} \) 

\( \Rightarrow {I_C} = \sqrt {{{\left( {0.3} \right)}^2} - {{\left( {0.1} \right)}^2}} = 0.28\;A\) 

\({V_C} = {I_C}{X_C}\) 

\(\Rightarrow {V_C} = {I_C}\frac{1}{{\omega C}}\) 

\( \Rightarrow C = \frac{{{I_C}}}{{\omega {V_C}}} = \frac{{0.282}}{{2\pi \times 60 \times 110}} = 6.82\;\mu F\) 

A series RC circuit is drawn as:

The voltage across the capacitor using the voltage division rule is:

\({V_C}\left( \omega \right) = \frac{{{V_S}}}{{j\omega C}}\left( {\frac{1}{{R + \frac{1}{{j\omega C}}}}} \right)\)

\({V_C}\left( \omega \right) = \frac{{{V_s}}}{{j\omega C}}\left( {\frac{{j\omega C}}{{1 + Rj\omega C}}} \right)\)

\({V_C}\left( \omega \right) = \frac{{{V_s}}}{{1 + j\omega RC}}\)

\({V_C}\left( \omega \right) = \frac{{{V_s}}}{{\sqrt {1 + {\omega ^2}{R^2}{C^2}}(\tan ^{ - 1}}\left( { \omega RC} \right) {}}\)

\({V_C}\left( \omega \right) = \frac{{{-V_s}}}{{\sqrt {1 + {\omega ^2}{R^2}{C^2}} }}{\tan ^{ - 1}}\left( { \omega RC} \right)\)

\(\phi =-{\tan ^{ - 1}}\left( { \omega RC} \right)\)

Explanation:

Nature of Supply Current in Parallel RC Circuit:

Definition: In a parallel RC (Resistor-Capacitor) circuit, the resistor and capacitor are connected in parallel to each other, and the supply current is distributed between the resistor and capacitor branches. The overall nature of the supply current is determined by the combined behavior of these two components concerning the applied voltage.

Correct Option Analysis:

The correct answer is:

Option 3: Leads by 90 degrees.

In a parallel RC circuit, the capacitor causes a phase shift in the current with respect to the voltage. Capacitors are reactive elements, and their current leads the voltage by 90 degrees. On the other hand, resistors do not cause any phase shift, and their current is in phase with the voltage.

When these two currents combine in a parallel circuit, the supply current leads the voltage due to the dominant effect of the capacitor's current. The resulting phase angle depends on the relative magnitude of the resistive and capacitive currents. However, in general, the supply current leads the voltage, and this behavior is characteristic of parallel RC circuits.

Mathematical Analysis:

The supply current in the parallel RC circuit is the vector sum of the currents through the resistor (I_R) and the capacitor (I_C).

Let:

• V = Applied voltage
• R = Resistance
• X_C = Capacitive reactance

The current through the resistor is given by:

I_R = V / R

The current through the capacitor is given by:

I_C = V / X_C

In a capacitor, the current leads the voltage by 90 degrees. Hence, I_C is out of phase with I_R. The total supply current (I_Supply) is obtained by the phasor addition of I_R and I_C. This results in a phase angle between the supply current and the voltage, where the current leads the voltage.

Advantages of Understanding:

• Helps in circuit design and analysis in AC systems.
• Supports the proper selection of components for desired phase characteristics.
• Aids in understanding power factor correction techniques.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: In phase.

This option is incorrect because, in a parallel RC circuit, the supply current cannot be in phase with the voltage due to the presence of the capacitor. Capacitors inherently cause a phase shift where the current leads the voltage. Therefore, the supply current cannot be entirely in phase with the applied voltage.

Option 2: Lags by 90 degrees.

This option is incorrect because the capacitive current leads the voltage by 90 degrees, not lags. Although inductive elements can cause the current to lag the voltage by 90 degrees, this is not the case for capacitive elements in the RC circuit.

Option 4: Compensates and becomes zero.

This option is incorrect because, in a practical parallel RC circuit, the supply current does not become zero. Both the resistive and capacitive components contribute to the overall current, and their vector sum results in a non-zero supply current.

Option 5: Not applicable.

This option is irrelevant as it does not provide a valid description of the behavior of the supply current in a parallel RC circuit.

Conclusion:

The supply current in a parallel RC circuit leads the voltage due to the capacitive component's influence. Understanding this phase relationship is crucial for analyzing AC circuits, designing systems with specific phase requirements, and implementing power factor correction techniques.