# Differentiator and Integrator

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A non-inverting and inverting amplifier can be adjusted to have an approximate differentiator and integrator. Approximate differentiator and integrator circuits can also be used as high-pass and low-pass filter circuits (Huijsing, 2017). The purpose of the laboratory session is to analyze the operation of the differentiator and the integrator to determine the efficiency of the amplifier filters and to equate them with the theoretical performance requirements.
Filter circuits can be used to exclude unwanted frequency spectrum from the desired signal frequency and to increase the intensity of the desired signal frequency (Lenk, 1999). Following figures depict basic approximate differentiator and low-pass operational amplifier filter circuits.

Figure 1: Approximate Differentiator Circuit

Figure 2: Low-pass op-amp filter

2. Theory

Approximate Differentiator

The theory of approximate differentiator is simple. The input capacitor and resistor forms an input reactance that blocks DC current and allows AC current to pass through based on the frequency of the signal. The hither the frequency i.e. the rate of change of the AC signal the less is the reactance and more is the output voltage whereas the less is the frequency the higher is the reactance and lower is the output voltage. Thus, the output voltage is the approximate replica of the rate of change of the input signal, which is mathematically known as differentiation operation.

The output voltage of the op-amp is given by,

Vout = –Ic1 × R2

The charge through the capacitor is given by, Q = C1 × Vin

­The rate of change of charge, = C1

Or, Ic1 = C1

Or, = C1

Or, Vout = R2C1

This mathematical expression of the output voltage indicates that the output voltage is the time rate of change of the input voltage or a differentiation of the input voltage.

Again, the reactance of the differentiator is given by,

R1 =

Or, fc =

This the expression for the critical frequency.

3. Calculations

Critical Frequency, fc = = 1061 Hz

Output voltage, Vop = = × 2= 1.272 Vp-p

B. Cut-off frequency of the filter is given by,

fc = = = 1061 Hz

Low frequency gain, HLF = = = 10

Log magnitude of H/HLF = 20 log10 ()

= 20 log10 () = -8.65×10­­­­-5

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