Into how many time constants is an exponential curve divided?
The correct answer and explanation is :
An exponential curve is divided into five time constants (denoted as τ) to reach approximately 99.3% of its final value.
Explanation:
When analyzing exponential growth or decay in systems such as electrical circuits (RC and RL circuits), thermal systems, and population dynamics, the time constant (τ) plays a crucial role in determining how fast the system approaches its steady-state value.
Definition of Time Constant (τ)
The time constant is the time required for a system to change a significant portion of its final value. In exponential decay, the value decreases to approximately 36.8% (1/e) of its initial value in one time constant.
Dividing the Exponential Curve
The response of an exponential system follows the equation:
[
y(t) = y_{\infty} + (y_0 – y_{\infty}) e^{-t/\tau}
]
where:
- ( y(t) ) is the value at time ( t ),
- ( y_{\infty} ) is the final steady-state value,
- ( y_0 ) is the initial value,
- ( \tau ) is the time constant.
The system progresses as follows:
- 1τ → 63.2% of the final value.
- 2τ → 86.5% of the final value.
- 3τ → 95.0% of the final value.
- 4τ → 98.2% of the final value.
- 5τ → 99.3% of the final value.
After five time constants, the system is considered to have effectively reached its steady-state, as the remaining change is negligible.
Conclusion
An exponential curve is effectively divided into five time constants, beyond which the difference from the final value is too small to be significant. This principle is widely applied in engineering and physics to analyze transient behaviors in dynamic systems.