First-Order Unity Feedback System

Theory

A first-order system is one in which the highest power of \( s \) in the denominator of the transfer function defines the order of the system. For a first-order system:

\[ \frac{C(s)}{R(s)} = \frac{1}{sT + 1} \]

Where \( C(s) \) is the output and \( R(s) \) is the input.

Unit-Step Input

For a unit-step input:

\[ R(t) = 1 \quad \text{and} \quad R(s) = \frac{1}{s} \]

Substituting \( R(s) = \frac{1}{s} \) into the transfer function:

\[ C(s) = \frac{1}{sT + 1} \cdot \frac{1}{s} \]

Expanding \( C(s) \) into partial fractions:

\[ C(s) = \frac{1}{s} - \frac{T}{sT + 1} \]

Taking the inverse Laplace transform:

\[ C(t) = 1 - e^{-t/T} \quad \text{for} \quad t \geq 0 \]

At \( t = T \), the output reaches 63.2% of its final value:

\[ C(t) = 1 - e^{-1} = 0.632 \]

The time constant \( T \) is the time required for the output to reach 63.2% of its final value.

Unit-Impulse Input

For a unit-impulse input:

\[ R(s) = 1 \]

Substituting \( R(s) = 1 \) into the transfer function:

\[ C(s) = \frac{1}{sT + 1} \]

Taking the inverse Laplace transform:

\[ C(t) = \frac{1}{T} e^{-t/T} \quad \text{for} \quad t \geq 0 \]