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Subtracting Circuits
11.2.1. Reduction to an Addition
A subtraction operation can be reduced to the problem of an addition by inverting the signal to be subtracted. This requires the circuit shown in Fig. 11.2. 11.4. Integrators
One of the most important applications of the operational amplifier in analog computing circuits is as an integrator. Its output voltage can be expressed in the general form
11.4.1. Inverting Integrator
The inverting integrator shown in Fig. 11.7 differs from the inverting amplifier in that the feedback resistor RN is replaced by the capacitor C. The output voltage is then expressed by
where Q0 is the charge on the capacitor at the beginning of the integration (t =0). As IC = -Vi/R, it follows:
The constant Vo0 represents the initial condition: Vo0 = Vo (t = 0 ) = Q0/C.
11.4.2. Initial Condition
An integrator can often be used only if its output voltage Vo (t=0) can be set independently of the input voltage. Using the circuit shown in Fig. 11.10, it is possible to stop integration and set the initial condition.
If the switch S1 is closed and S2 is open, the circuit operates like that in Fig. 11.7; the voltage V1 is integrated. If switch S1 is now opened, the charging current becomes zero in the case of an ideal integrator, and the output voltage remains at the value that it had at the time of switching. This may be of use if we want to interrupt computation; for example, in order to read the output voltage at leisure. To set the initial condition, S1 is left open and S2 is closed. The integrator becomes an inverting amplifier, with an output voltage of
11.4.3. Summing Integrator
Just as the inverting amplifier can be extended to become a summing amplifier, so an integrator can be developed into a summing integrator as shown in Fig. 11.12. The relationship given for the output voltage can be derived directly by applying KCL to the summing point.
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