First-Order OP AMP Circuit(Integrator)

Purpose
    The purpose of this lab is to explore the OP Amp integrator, and make sketches of output waveforms for 1kHz sine wave, triangle wave, and square wave inputs.

Analysis    
    To illustrate how these circuits perform integration, consider the circuit in Figure 1.  

Figure 1. A Practical Integrator.

Since we can treat the voltages at the op amp inputs are equal and practically zero. Plus, the currents going into the op amp inputs are ideally zero, then the current through R1 is equal to the sum of the currents through C and Rf. From Figure 1, making Rf>>R1 will make the R1 current practically the same as the C current, ic.  The current through R1 is Vin/R1, so ic is very close to Vin/R1 if Rf>>R1.
By KCL,
Vin/R1 = ic + (0-Vout)/Rf (this term can be ignored since Rf is greater than R1 by a factor of 100 times)
==>Vin/R1 = -C*(dVout)/dt
==>Vout = -1/C * integral (Vin/R1*dt) + Vinitial


Assemble

Figure 2. Build-up Circuit on Breadboard

Figure 3. Build-up Circuit connected to MAC

Results

• Sine wave input

Figure 4. Output Obtained from Sine Wave Input

• Triangle wave input

Figure 5. Output Obtained from Triangle Wave Input

• Square wave input

Figure 6. Output Obtained from Square Wave Input

Questions

    What is that 10M resistor (Rf) for? What happened when it is removed?

If a small DC component was present in the input waveforms, the capacitor (C) offers infinite resistance and so the integrator circuit will be like an inverting opamp amplifier with infinite feedback resistance (Rf = ∞). The equation for the voltage gain (A) of an  opamp amplifier in inverting mode is  A = -(Rf/R1). Substituting Rf=∞ in the present scenario we get A=∞. Therefore the small input offset voltage will get amplified by this factor and there will be an error voltage at the output. In addition, Vinitial would be affected by a small DC component of input. Adding a feedback Rf in parallel with the capacitor will fix the low frequency gain (A) of the circuit to a fixed small value and so the input offset voltage will have practically no effect on the output offset voltage and variations in the output voltage is prevented. The gain goes from negative infinity to some finite value -Rf/R1.