The system of Example 7.1 and Fig. 7-3 has two samplers. The system characteristic equation is derived

Question:

The system of Example 7.1 and Fig. 7-3 has two samplers. The system characteristic equation is derived in Example 7.1 as

Show that the same characteristic equation is obtained by opening the system at the second sampler.

Example 7.1

Consider the system of Example 5.2, which is repeated in Fig. 7-3(a). The flow graph of the system is opened at the first sampler as shown in Fig. 7-3(b). The effect of the second sampler is included, by denoting its input as E₁(s) and its output as E^*₁ (s). From this flow graph we write E = GE-GHE E = -GE Starring the first equation and solving for Et, we obtain R(s) + T E = G(s) GE 1 + GH* + Eo(s) Ei(s) Gy(s) TV (b) Ey(s) Ej(s) -H(s) G2(s)

Fig 7-3(a), (b) R(s) + T G(s) + (a) T H(s) G(s) C(s) E(s) E,(s) G(s) (b) E (s) E (s) -H(s) G(s)

Example 7.2

Consider the system shown in Fig. 5-9. The original signal flow graph is shown in Fig. 5-10.
The system equations are Starring these equations, we obtain and The sampled flow graph can then be drawn from these equations as R(s) + T FIGURE 5-9 Sampled-data system for Example 5.2. R 1 E RC G(s) E G FIGURE 5-10 Original flow graph R* E G -G E -GH* G G C 20