Zero-Order-Hold Transfer Function Derivation

The Zero-Order-Hold (ZOH) is a fundamental component in digital control systems. Its primary function is to reconstruct a continuous-time signal from a sequence of discrete samples, holding each sample's value constant over the sampling interval.

The impulse response, h(t), of an ideal ZOH circuit, assuming the sampling interval is T, is a rectangular pulse defined as:

$ h(t) = \begin{cases} 1 & \text{for } 0 \le t < T \\ 0 & \text{otherwise} \end{cases} $

The transfer function, H(s), in the Laplace domain is obtained by taking the Laplace transform of the impulse response h(t):

$ H(s) = \mathcal{L}\{h(t)\} = \int_0^\infty h(t) e^{-st} dt $

By substituting the definition of h(t) and the integration limits (from 0 to T):

$ H(s) = \int_0^T 1 \cdot e^{-st} dt $

Evaluating this definite integral:

$ H(s) = \left[ -\frac{1}{s} e^{-st} \right]_0^T $

Applying the limits of integration:

$ H(s) = \left( -\frac{1}{s} e^{-sT} \right) - \left( -\frac{1}{s} e^{-s \cdot 0} \right) $

$ H(s) = -\frac{1}{s} e^{-sT} + \frac{1}{s} $

Simplifying the expression by factoring out 1/s:

$ H(s) = \frac{1}{s} (1 - e^{-sT}) $

This result represents the transfer function of the Zero-Order-Hold system for a sampling interval T.