In a parallel resistance (R) and capacitor (C) circuit, if resistance R increases, the phase angle ϕ __________. 

Explanation:

Parallel Resistance (R) and Capacitor (C) Circuit

Definition: A parallel resistance and capacitor circuit is an electrical circuit configuration where a resistor (R) and a capacitor (C) are connected in parallel to a common voltage source. Such circuits are widely used in AC (alternating current) applications to analyze impedance, phase angles, and time constants.

Working Principle: In an AC circuit, a capacitor introduces a phase shift because the current leads the voltage by 90 degrees, while the resistor does not introduce any phase shift (current and voltage are in phase). The combination of these two components results in a net phase angle (ϕ) between the total current and the source voltage, which is dependent on the relative values of resistance (R) and capacitive reactance (X_c).

Phase Angle (ϕ): The phase angle (ϕ) in a parallel R-C circuit is determined by the relationship between the current through the capacitor and the current through the resistor. Mathematically, the phase angle is given by:

tan(ϕ) = I_C / I_R

Where:

• I_C = Current through the capacitor
• I_R = Current through the resistor

Since the current through the capacitor is given by I_C = V / X_C and the current through the resistor is I_R = V / R, the phase angle can also be expressed as:

tan(ϕ) = X_C / R

Where:

• X_C = 1 / (2πfC), the capacitive reactance
• R = Resistance

From the above equation, it is evident that the phase angle ϕ depends on the ratio of X_C (capacitive reactance) to R (resistance).

Effect of Increasing Resistance (R):

When the resistance (R) in the parallel R-C circuit is increased:

• The denominator of the term X_C / R increases, resulting in a decrease in the value of tan(ϕ).
• As tan(ϕ) decreases, the phase angle ϕ also decreases since the tangent function is directly related to the angle.

Hence, when resistance (R) increases, the phase angle (ϕ) decreases.

Correct Option:

The correct answer is:

Option 4: Decreases

As explained above, increasing the resistance reduces the phase angle in a parallel R-C circuit due to the inverse relationship between resistance and tan(ϕ).

Additional Information

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

Option 1: Increases

This option is incorrect because increasing resistance (R) in the parallel R-C circuit reduces the phase angle (ϕ). The relationship between tan(ϕ) and resistance is inversely proportional, so an increase in R leads to a decrease in tan(ϕ) and consequently a decrease in ϕ.

Option 2: Becomes zero

This option is incorrect because the phase angle (ϕ) does not become zero simply by increasing the resistance. The phase angle becomes zero only in a purely resistive circuit (when the capacitor is removed or its effect is negligible). In a parallel R-C circuit, the phase angle will decrease but will not reach zero as long as the capacitor is present.

Option 3: Remains the same

This option is incorrect because the phase angle (ϕ) is dependent on the resistance (R) and capacitive reactance (X_C). When R changes, the phase angle also changes. Therefore, the phase angle cannot remain the same when resistance is increased.

Conclusion:

In a parallel R-C circuit, the phase angle (ϕ) decreases when the resistance (R) increases. This is due to the inverse relationship between resistance and tan(ϕ), as shown in the mathematical derivation. Understanding the behavior of phase angles in R-C circuits is essential for analyzing AC circuits and their impedance characteristics.