Question 9.6.2: Two Uncoupled RC Loops The circuit shown in Figure 9.6.10a c...
Two Uncoupled RC Loops
The circuit shown in Figure 9.6.10a consists of two series RC circuits wired so that the output voltage of the first circuit is the input voltage to an isolation amplifier. The output voltage of the amplifier is the input voltage to the second RC circuit. The amplifier has a voltage gain G; that is, v_{2}(t) = Gv_{1}(t) . Derive the transfer function V_{o}(s)/V_{s}(s) for this circuit, and for the case G = 1 compare it with the transfer function of the circuit shown in Figure 9.6.7.
The amplifier isolates the first RC loop from the effects of the second loop; that is, the amplifier prevents the voltage v_{1} from being affected by the second RC loop. This in effect creates two separate loops with an intermediate voltage source v_{2} = GV_{1}, as shown in Figure 9.6.10b. Thus, using the results from Example 6.2.8 for the lefthand RC loop, we obtain
\frac{V_{1}(s)}{V_{s}(s)} = \frac{1}{RCs + 1}
For the righthand RC loop,
\frac{V_{o}(s)}{V_{2}(s)} = \frac{1}{RCs + 1}
For the amplifier with gain G,
V_{2}(s) = GV_{1}(s)
To obtain the transfer function V_{o}(s)/V_{s}(s), eliminate the variables V_{1}(s) and V_{2}(s) from these
equations as follows:
\frac{V_{o}(s)}{V_{s}(s)} = \frac{V_{o}(s)}{V_{2}(s)} \frac{V_{2}(s)}{V_{1}(s)} \frac{V_{1}(s)}{V_{s}(s)} = \frac{1}{RCs + 1}G \frac{1}{RCs + 1} = \frac{G}{R^{2} C^{2} s^{2} + 2RCs + 1} (1)
This procedure is described graphically by the block diagram shown in Figure 9.6.11a. The three blocks can be combined into one block as shown in part (b) of the figure.
The transfer function of the circuit shown in Figure 9.6.7 was derived in Example 9.6.2.
It is
\frac{V_{o}(s)}{V_{s}(s)} = \frac{1}{R^{2} C^{2} s^{2} + 3RCs + 1} (2)
Note that it is not the same as the transfer function given by equation (1) with G = 1.
A common mistake is to obtain the transfer function of physical elements connected endto-end by multiplying their transfer functions. This is equivalent to treating them as independent devices, which they may not be if each device “loads” the adjacent device. Here the “device” is an RC loop, and we have seen that in Figure 9.6.7 the righthand loop “loads” the lefthand loop,
thus changing the current and voltage in that loop. An isolation amplifier prevents a loop from loading an adjacent loop, and when such amplifiers are used, we may multiply the loop transfer functions to obtain the overall transfer function.
This mistake is sometimes made when drawing block diagrams. The circuit of Figure 9.6.10 can be represented by the block diagram of Figure 9.6.11, where the transfer function of each block can be multiplied to obtain the overall transfer function V_{o}(s)/V_{i}(s). However, the circuit of Figure 9.6.7 cannot be represented by a simple series of blocks because the output voltage v_o affects the voltage v_1. To show this effect requires a feedback loop, as shown previously in Figure 9.6.8.
In general, even though elements are physically connected end-to-end, we cannot represent them by a series of blocks if the output of one element affects its input or the behavior of any preceding elements
