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An Op-Amp Integrator, How Does it Work?



Basic Integrator


If you strap a capacitor around an op-amp as a feedback in an inverting configuration, instead of a resistor, you wind up with an interesting basic circuit called an integrator. The name comes from the fact that the circuit basically does a mathematically integrating function, expressed in the output transfer function.

This circuit can be very confusing to many. I think the best way to understand it is from a practical point of view, by way of analogies and equivalent circuits. And once you are able to get a mastery of its functions and behaviors, there’s a lot of interesting opportunities where the basic circuit can be useful.

Current Flow and Output Equation


The output is a derivation applying KCL (Kirchoff’s Current Law) at the amplifier’s summing node, and the op-amp’s ideal assumptions namely that the positive and negative terminal are at the same potential, and no input bias current flows. The voltage input creates a current in the resistor whose other end is at a ground potential. The same current then flows in the capacitor. Using the formula of the capacitor current in the time domain (meaning a function of time, t as the variable) , the output is derived as follows.



Note that the input terminals of the op amp is at ground potential, and that the output is negative assuming the direction of current flow in the figure above. What we have above is the time domain analysis of the integrator, where one gets insights of its behavior as a function of time. It is most often helpful to analyze circuits imagining what plays out in the realm of time as soon as an input or power is applied in the circuit. I often do it in slow motion and as if time is suspended. In this case, when a dc is applied at the input, a constant current, i,  is created and is forced to flow to the capacitor. This will create a constant increase of voltage in the capacitor, a ramp, with a slope proportional to the current, as shown in the first equation of the current above.

Plot of Vin and Vout vs Time
Note that the Vout is shown negative in the figure above for convenience. Because it is in an inverting configuration, the actual Vout is a negative ramp.

Equally important is the analysis in the frequency domain. The capacitor is replaced by its impedance equivalent, Z.What comes next is the usual analysis of the op-amp inverting configuration.

Inverting Op Amp


Where,


The j is an operator signifying a complex number with 90 degrees in phase. In straight calculation without regard for phase, we simply substitute the frequency value, in Hertz, to get the impedance in Ohms. Z2 is equal to R.

This tells us several important points.

a.       The impedance at DC is infinite, and the output of the integrator is infinite as well. In reality, the output will be limited by the op amp output swing.
b.      At very high frequencies, the impedance is very low close to zero, and the integrator output is close to zero as well.
c.       At intermediate frequencies where the impedance is at some specific value based on the frequency, the integrator has an output equivalent to the close loop gain.
Plot of Output in Absolute Value vs Frequency

It is interesting to note how the basic integrator behaves over frequency, as plotted on a graph, called Bode Plot, shown in the figure above with the output in absolute value in the y-axis and the input frequency in the x-axis. At the frequency where the graph crosses the x-axis, the gain is 1 (0dB) where the capacitive reactance (Z) is equal to the resistance (R). Observe that the graph has a constant slope of -20dB/decade or -6dB/octave. -20dB is a gain of 1/10 while -6dB is ½. It is worth emphasizing that at DC (0 Hz), the gain is infinite.

These circuit behaviors open up a lot of opportunities for many applications, examples are low pass filtering, waveform generations such as triangular wave, and dc error corrections in feedback circuits. We will have separate topics for those interesting applications.

Comments and feedback are appreciated.

For related literature and references:
Op Amp Basic and Integrator by Brad Albing
Op Amp Circuit Collections, Texas Instruments



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