Characteristics and response of zero-order system

A zero-order system has the model $a_0 q_o = b_0 q_i$, so $q_o = K q_i$ with static sensitivity $K = b_0/a_0$.

Characteristics

• The output is proportional to the input at every instant. There is no lag, delay, phase shift or distortion.
• Amplitude ratio $M = 1$ (relative to $K$) and phase angle $\phi = 0$ at all frequencies.
• The only parameter is $K$, found by static calibration.
• No energy storage (no mass, spring or capacitance) in the model.

Response

For any input, a step, ramp or sine, the output has the same shape scaled by $K$.

 q_i |  ____        q_o |  ____   (scaled by K)
     | |    |           | |    |
     +-+----+--> t      +-+----+--> t

Potentiometer as example

A linear potentiometer of length $L$ and supply $E$ gives $e_o = (E/L)\,x$, so $K = E/L$ V/m. The output follows the wiper position with no dynamic lag.

Limits

A zero-order model is an idealization valid only when the input changes slowly compared with the instrument's natural frequency. In a real potentiometer, wiper mass and inertia, wire-wound resolution, friction, stray capacitance and inductance, and the loading error of a finite load give time-dependent behaviour at high frequency, so the true response is of higher order.