Understanding the Inductive Circuit Time Constant

An inductive circuit, typically an RL circuit (Resistor-Inductor circuit), exhibits a transient behavior when a voltage source is applied or removed. During the 'on' transient (when voltage is applied), the current does not instantly jump to its final steady-state value. Instead, it increases gradually over time.

Defining the Time Constant ($\\tau\$)

The time constant, denoted by the Greek letter tau ($\\tau\$), is a crucial parameter in characterizing the transient response of an inductive circuit. For a simple RL circuit consisting of a resistor (R) and an inductor (L), the time constant is defined as:

$$ \\tau \= \frac \{L} \{R} \$$

Where:

• $\\\tau \$ is the time constant in seconds.
• $\L \$ is the inductance in Henries (H).
• $\R \$ is the resistance in Ohms ($\\\Omega \$).

Current Rise in an Inductive Circuit

The current $\I(t) \$ in an inductive circuit at any time $\t \$ after applying the voltage follows an exponential rise towards its final steady-state value $\I_{final} \$. The formula for this current rise is:

$$ \I(t) \= \I_{final} \\\left( \1 - e^{-\frac \{t} \\tau \} \\right) \$$

The time constant ($\\\tau \$) represents the time it takes for the current to reach approximately 63.2% of its final steady-state value. Let's see why:

If we set $\t = \\\tau \$ in the formula:

$$ \I(\\\tau \) \= \I_{final} \\\left( \1 - e^{-\frac \{\\\\tau \} \{\\\tau \} \} \\right) \$$

$$ \I(\\\tau \) \= \I_{final} \\\left( \1 - e^{-1} \\right) \$$

Since $\e^{-1} \$ is approximately $0.3679$, the equation becomes:

$$ \I(\\\tau \) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\$\\approx 0.6321 \\times \I_{final}\

This means that after one time constant, the current reaches approximately 63.2% of its final value. This value is often rounded to 63% for practical purposes.

Evaluating the Options

Let's look at the given options:

• 1. 63%: This is the correct value, representing the current reached after one time constant ($\\\\\tau \$).
• 2. 90%: The current reaches about 90% of its final value after approximately $2.3\\\\\tau \$.
• 3. 37%: At $t = \\\\tau \$, the remaining voltage across the inductor is $\V_{final} \e^{-1} \$, which is about 37% of the initial voltage. This relates to the voltage decay, not the current rise percentage at $t=\\\\\tau \$.
• 4. 100%: The current theoretically reaches 100% only after an infinite amount of time, although it gets very close (over 99%) after about $5\\\\\tau \$.

Therefore, the time constant of an inductive circuit is defined as the time required for the circuit current to rise to approximately 63% of its final steady-state value.