Inverting Differentiator

Today we went over differentiators and integrators. They are based on the inverting amplifier. A differentiator circuit can be obtained by replacing the feedback resistor with an inductor or replacing the input resistor with a capacitor. An integrator circuit can be obtained by replacing the feedback resistor with a capacitor or replacing the input resistor with an inductor. Since inductors are expensive and rarely used, we only use capacitor to implement differentiator or integrator.

Then the concept of singularity is introduced. There are three common singularity functions. They are unit step function, unit impulse function and unit ramp function. They can be used to represent step response.

We did a lab on inverting differentiator. Here is a schematic of the circuit

Here is the setup of the circuit

We measured the experimental resistance of the resistor

We applied three different voltage inputs to the circuit. They are 100 Hz sinusoid, 250 Hz sinusoid, and 500 Hz sinusoid. We observed the output voltage on the oscilloscope

100 Hz

250 Hz

500 Hz

The phase shift on the graph is exactly π/2, which corresponds to the phase difference between a sinusoidal function and its derivative. We also compared the amplitude of the output voltage curve and the theoretical output.

100 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	100.00	100.00	0.00%
ω (rad/s)	628.32	628.32	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	0.43	0.43	0.64%

250 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	250.00	250.00	0.00%
ω (rad/s)	1570.80	1570.80	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	1.07	1.07	0.17%

500 Hz

	Experimental	Theoretical	Percent Error
A (V)	1.00	1.00	0.00%
f (Hz)	500.00	500.00	0.00%
ω (rad/s)	3141.59	3141.59	0.00%
R (Ω)	681.00	680.00	0.15%
C (μF)	0.96	1.00	4.00%
τ (ms)	0.65	0.68	3.86%
Vout (V)	2.12	2.14	0.76%

Summary: Differentiators and integrators can be implemented by using capacitors and resistors. It can be used to shift the phase of a sinusoidal input. Additionally, differentiators and integrators are also operational amplifiers circuits. It shows that operational amplifiers can not only be used to implement basic operations such as addition, subtraction, multiplication and division, but also calculus operations such as differentiation and integration.