# 1st Order System Response

This page summarizes step and frequency responses for first order
system of the form:

- Response is within 2% of steady state value at 4
*τ*.
- Bandwidth = 1/
*τ*. High bandwidth means system responds fast.
*τ*>0: stable.
*τ*<0: unstable.

## Step Response

Input: step *A*

Output:

## Frequency Response

Input: *Asin(ωt)*

Output (steady state):

Output (steady state amplitude trend):

## Change Response Speed

*σ* times faster:

- Gain (
*K*): unchanged
- New time constant:
*τ/σ*

## Examples

- Example 1: Sine wave is input to a known system. Determine output.
- Example 2: Make a given system respond 10 times faster.
- Example 3: Make a system reach target value at specific time.
- Example 4: Estimate model parameters from output measurement.