What is the time constant of an RC series circuit?

Explanation:

Time Constant of an RC Series Circuit

Definition: The time constant of an RC series circuit is a measure of the time it takes for the voltage across the capacitor to charge or discharge to approximately 63.2% of its final value (or decrease to 36.8% of its initial value) when subjected to a step input of voltage. It is an important parameter in analyzing the transient response of RC circuits.

Formula: The time constant (denoted by τ, pronounced "tau") for an RC circuit is given by:

τ = R × C

Where:

• R is the resistance in ohms (Ω).
• C is the capacitance in farads (F).

This formula indicates that the time constant is directly proportional to both the resistance and the capacitance of the circuit. A larger time constant implies a slower charging or discharging process, while a smaller time constant indicates a faster transient response.

Working Principle: In an RC series circuit, when a DC voltage is applied, the capacitor begins to charge through the resistor. The rate of charging is influenced by the resistor's resistance and the capacitor's capacitance. The time constant (τ) quantifies the duration over which the capacitor reaches a significant proportion of its full charge.

The voltage across the capacitor (V_c) at any time t during charging is given by:

V_c(t) = V_max × (1 - e^-t/τ)

Here:

• V_max is the maximum voltage applied to the circuit.
• t is the time elapsed since the application of the voltage.
• e is the base of the natural logarithm (approximately 2.718).

During discharging, the voltage across the capacitor decreases exponentially and is given by:

V_c(t) = V_initial × e^-t/τ

Where V_initial is the initial voltage across the capacitor at the start of the discharge.

Correct Option Analysis:

The correct option is:

Option 4: RC

This option is correct because the time constant τ is the product of the resistance (R) and the capacitance (C) in the circuit. This relationship is fundamental to the analysis of RC circuits and is derived from the governing equations of the circuit.

Additional Information

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

Option 1: R/C

This option is incorrect. The ratio R/C does not represent the time constant of an RC circuit. Instead, the time constant is the product of resistance and capacitance, not their ratio.

Option 2: C/R

This option is also incorrect. The ratio C/R does not have any direct relevance to the time constant of an RC circuit. The time constant is determined by multiplying R and C, not dividing them.

Option 3: 1/RC

This option is incorrect as well. While 1/RC is related to the frequency of an RC circuit in certain contexts (such as the cutoff frequency in filters), it does not represent the time constant. The time constant is given by RC, not its reciprocal.

Conclusion:

The time constant (τ) of an RC series circuit is a fundamental parameter that determines the transient response of the circuit. It is defined as the product of the resistance (R) and capacitance (C) in the circuit. This relationship is crucial for understanding how quickly a capacitor charges or discharges in an RC circuit. The correct option, RC, accurately represents the time constant, while the other options are incorrect representations of this concept.