Question 9.5.1: Series Blocks and Loop Reduction Determine the transfer func...
Series Blocks and Loop Reduction
Determine the transfer function X(s)/F(s) for the system whose diagram is shown in Figure 9.5.8a
When two blocks are connected by an arrow, they can be combined into a single block that contains the product of their transfer functions. The result is shown in part (b) of the figure. This property, which is called the series or cascade property, is easily demonstrated. In terms of the variables X(s), Y (s), and Z(s) shown in the diagram, we can write
X(s) = \frac{1}{s + 12} Y (s) Y (s) = \frac{1}{s + 6} Z(s)
Eliminating Y (s) we obtain
X(s) = \frac{1}{s + 6} \frac{1}{s + 12} Z(s)
This gives the diagram in part (b) of the figure. So we see that combining two blocks in series is equivalent to eliminating the intermediate variable Y (s) algebraically.
To find the transfer function X(s)/F(s), we can write the following equations based on the diagram in part (b) of the figure:
X(s) = \frac{1}{(s + 6)(s + 12)} Z(s) Z(s) = F(s) − 8X(s)
Eliminating Z(s) from these equations gives the transfer function
\frac{X(s)}{F(s)} = \frac{1}{s^{2} + 18s + 80}