Since the current and voltage of the capacitor are integral / differential, the capacitance can be used to do integral and differential operations. Although inductance can also be used to do calculus, but because of its large volume, and the actual value of inductance device is not easy to accurately control, so the actual circuit is generally only realized by capacitance.
1. Integral circuit
The feedback branch element of the integrator is capacitor, as shown in the figure below:
Because integral formula is used, in order to express it more clearly. Instead of phasor symbols, real-time AC symbols V (T) and I (T) denote AC voltage and current.
The voltage current relationship of capacitor is as follows:
And the input voltage and Vi(t) The formula of the equation is as follows:
By combining the above two formulas, we can get the following results
In this way, the integral operation of input signal is realized.
2. Differential circuit
Differential circuit is not as common as integral circuit, because differential signal is easily affected by interference. For example, if due to external reasons, the input signal V of the integrator circuiti(t) There is an instantaneous pulse interference (the duration of the interference is very short). Because the output is integral operation, the value generated by the integration of this instantaneous interference signal is very small, which almost has no impact on the output result.
But the differential is not the same. For the instantaneous pulse input interference, even if the amplitude of the interference itself is not large, but because of its rise and fall speed is very fast, the differential operation (i.e. derivative) will produce a large value at the output end. Therefore, differential circuits are not often used in practical applications.
The figure below shows the basic type of a differential circuit
The output expression of the circuit is as follows:
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