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 R_N 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 I_C = -V_i/R, it follows:

The constant V_o0 represents the initial condition: V_o0 = V_o (t = 0 ) = Q₀/C.

11.4.2. Initial Condition

An integrator can often be used only if its output voltage V_o (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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