1st order circuits/ op amp

we started class with learning about Integrator:  an op amp circuit whose output is proportional to the integral of the input signal.

If the feedback resistor Rf in the familiar inverting ampliflier of a is replaced by a capacitor and we obtain an ideal integrator.

We draw following graphs of vin vs t and vout vs t. It’s graph of integrator op amp. They saturate really quickly. If Vin is sinusoid then Vout would look like something in the picture.

Differentiator: is an op amp circuit whose output is proportional to the rate of change of the input signal. If the input resistor is replaced by a capacitor , the resulting circuit is a differentiator. iR=iC

Vo = -RC (dvi/dt)

Differentiator circuits are electronically unstable because any noise within the circuit is exaggerated by the differentiator.

above picture also shows graph of op amp differentiator. 

LAB: Inverting Differentiator: In this lab we study the forced response of a circuit which performs a differentiation. The circuit output is the derivative with respect to time of the input to the circuit.

PreLAB: we did following equation to solve for Vout as a function of the circuit input.  W=2f The frequency f has units of hz 

We constructed the circuit shown below using R=470 ohm C= 470 nF . we used oscilloscope to measure both the input and output voltages .

Then we applied sinusoidal input voltage with frequency = 1kHz A= 1V and offset=0V.

Below is image of oscilloscope window showing the wave forms and their measured amplitudes.

Measured A= 1.12 and calculated was  1.39. % Error was 19.4%

Following is the output signal at 1Khz

Following is the input signal graph

Then we applied sinusoidal input voltage with frequency = 500Hz A= 1V and offset=0V.

Below is image of oscilloscope window showing the wave forms and their measured amplitudes.

% Error was 15.9% when compared with calculated values of 0.69V. measured values were 0.58               

 output sine wave signal of 1V at 2KHz

Then we applied sinusoidal input voltage with frequency = 2kHz A= 1V and offset=0V.

Below is image of oscilloscope window showing the wave forms and their measured amplitudes.

Measured A= 2.21 and calculated was A= 2.78. % Error was 20.5%

Output signal graph 

Input signal graph

Error source: op amp has really big saturation which caused big % error.

 Below is our calculations and measured values showing % error.. 

Step Response of an RC Circuit:  its behavior when the excitation is the step function, which may be a voltage or current source.

v = v _f + v _n

v _f = V and  v_n = (V₀ − V_s )e^{− t/T}

Vn is the natural response of the circuit.

The natural response or transient response is the circuit’s temporary response that will die out with time. The forced response or steady state response is the behavior of the circuit a long time after an external excitation is applied. 

Below is problem we did in class to get step response circuit

More problems done in class..

Functions are either discontinuous or have discontinuous derivatives. Types include unit step, unit impulse, and unit ramp functions:

Summary: we learned about op amps and how they integrate in RC circuits. we did lab to see how  inverting differentiator and signularity functions work.