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ZTRANSFORMS
Equation (25.4) may now be written CA(Z) = Tc(nT  hT)z (25.5) If we let m = 1  A, we may write for the argument of c in Eq. (25.5) nT  AT = nT  (1  m)T = (n  l)T + mT Equation (25.5) may now be written CA(Z) = 2 c[(n  l)T + mT]z (25.6) (25.7) If we let n = n  1, Eq. (25.7) may be written CA(Z) = 5 c[(n + ~)T]z z~ n =O or CA(Z) = zl 2 c[(n + m)T]z n =O (25.9) (25.8) This last expression is the definition of the modified Ztransform. Replacing the index n with n, to avoid an awkward symbol in the definition, we have the expression for the modified Ztransform: CA(Z) = c(z,m) = zl 2c[(n + m)T]z n=O (25.10) The symbol C(z,m) has replaced CA(Z) and m = 1  A. Tables of transform pairs have been developed that relate a function f(t) to its modified Ztransform. Table 22.1 provides the modified Ztransforms for the functions of t listed in the table. PULSE TRANSFER FUNCTION FOR MODIFIED ZTRANSFORM
Returning to Eq. (25.3), we may write c;(s) = G;W *W Writing this in terms of Ztransforms, we have
CA(z) (25.11) GA(Z)F(Z) (25.12) The convention has been established to replace the Ztransform of the delayed function, such as C,+(Z) in Eq. (25.12), with the symbol C(z,m) where m is SAMPLEDDATA CONTROL SYSTEMS
related to A by the relation m = 1  A. Changing the subscripted symbols in Eq. (25.12) according to this convention gives the equivalent expression C(z,m) = .G(z,m)F(z) (25.13) where G(z ,m) is the Ztransform corresponding to GA(S). Remember that GA(S) is simply the transfer function for the process G(s) multiplied by the transfer function for the fictitious delay; thus: GA(s) = G(s)~* ~ (25.14) To find G(z. ,m) one may refer to a table of transforms and find the entry G(z ,m) corresponding to the desired G(s). To find the output of the block diagram in Fig. 25.4 at times other than sampling instants, one uses the modified Ztransform and writes SUMMARY OF USE OF THE MODIFIED ZTRANSFORM.
C(z,m) = G(z,m)F(z) (25.15) It should be realized that C (z , m) is simply a Ztransform and that it can be inverted by the same procedures used for inverting ordinary Ztransforms. Furthermore, the inversion gives information about the response only at sampling instants. The result from the inversion of the modified Ztransform gives the values of c(t) between sampling instants. By choosing m between 0 and 1, one can obtain the values of c(t) at any desired time within the sampling intervals. For convenience, one may apply the following rule to determine the effect of the size of m on the time into the sampling intervals. Rule. Inversion of C (z , m) gives a response that is equivalent to stepping back one sampling period in c(t) and advancing mT units of time. Thus, for nz = 1, there is no delay and the result is the same as that obtained from the ordinary Ztransform. This can be seen from the fact that A = 1  m = 1  1 = 0. On the other extreme, for m = 0, one has a delay that approaches a full sampling period. The system presented earlier in Fig. 25.2 for the purpose of explaining the timing of the various signals will now be worked as an example using the method just discussed. Example 25.1. Obtain by means of ordinary and modified Ztransforms the response for the system shown in Fig. 25.5. Determine the response c(t) at sampling instants and at times positioned 0.3T from the beginning of sampling instants. c*(r) I A , T c*(s)

