**Subject :**Advance control system

## The z-Transform

The effect of sampling within a system is pronounced. Whereas the stability and transient response of analog systems depend upon gain and component values, sampled-data system stability and transient response also depend upon sampling rate. Our goal is to develop a transform that contains the information of sampling from which sampled-data systems can be modeled with transfer functions, analyzed, and designed with the ease and insight we enjoyed with the Laplace transform. We now develop such a transform and use the information from the last section to obtain sampled-data transfer functions for physical systems. The above Equation is the ideal sampled waveform. Taking the Laplace transform of this sampled time waveform, we obtain

(1)

Now, letting z = e^{TS}, Eq. (1) can be written as

(2)

Equation (2) defines the z-transform. That is, an F{z) can be transformed to f(kT), or an f(kT) can be transformed to F(z). Alternately, we can write

(3)

Paralleling the development of the Laplace transform, we can form a table relating f(kT), the value of the sampled time function at the sampling instants, to F(z). A partial table of z-transforms is shown in Table 1, and a partial table of z-transform theorems is

**Table 1**

**TABLE 2**

Shown in Table 2. For functions not in the table, we must perform an inverse Z-transform calculation similar to the inverse Laplace transform by partial-fraction expansion.