TABLE OF CONTENTS Show that the systems depicted in Figure 2.9, and in Figure 2.10, are equivalent (we have left out the argument s in the transfer functions of the various blocks).
Recall that two systems are equivalent if their transfer functions are equal. The transfer functions of the systems depicted in Figure 2.9 are:
These three systems are equivalent.
For the first system depicted in Figure 2.10, we have:
For the second system, we obtain:
which leads to the same expression as before. Consequently, these two systems are equivalent.
Calculate the transfer function G ( s) of the RC system depicted at the top of Figure 2.11.
Figure 2.11: RC systems
Explain why the transfer function of the two RC systems depicted at the bottom of Figure 2.11 is not equal to the product G ( s) G ( s).
Show that the RC system is equivalent to a closed-loop system with:
From Ohm's law, we derive:
where Ri( t) and 1/ C ? i( t) dt represent the voltage drops across the resistor and capacitor, respectively. Equation (2.24) expresses the fact that the input voltage v ( t) is the sum of the voltage drops across the resistor and capacitor. Taking the Laplace transform,
we derive:
This simple system is commonly used as a...
