Integrators and Differentiators are basic building blocks of analog computers. They enable summation and subtraction operations (hence, the core component of them is called “operational” amplifier). For multiplication, multiple integrators are used in parallel, along with exponential and logarithmic elements (non-linear op-amp circuits), to achieve the effect via the following transformation:
ln(ab) = ln(a) + ln(b) = c
e^(c) = ab
This is why analog multiplicators are expensive, costing up to $20 a pop.
All the circuits below require very specific values for their components. Changing the value of the components will do more than changing the properties of the output, it can cause instability and disrupt the circuit’s operation. As an oscilloscope is required to diagnose these circuits, some preset values that have been tested in the lab are also included in this memo for DIY projects at home.
Monostable multivibrator (one-shot timer)
Monostable multivibrator has one stable state and it will change to the unstable state for a period of time when a trigger pulse (negative edge) is introduced as input.
The circuit can be derived from astable multivibrator circuit; the only new addition is the grounded diode in parallel to the output capacitor. If all values are appropriate, connecting the diode will dampen all output and (negative) feedback generated by the powered astable multivibrator without any input. After this dampening characteristic has been achieved, negative edge trigger input can be introduced to complete the circuit.
The basic circuit is as seen below:
The timing period T is the amount of time it takes for the circuit to return switch from unstable back to the stable state. The timing period is given by
T = RC ln[1 / (1-B)]
where B is the regenerative feedback as described in Memo: Schmitt trigger. The units of the remaining variables are as follow:
T: seconds (s)
R: ohms (Ω)
C: farad (F).
Cheat sheet: when R1 = R2, the timing period T = 0.693 RC
Similarly, the charging period is the amount of time the circuit must wait before it can be triggered again. This is given by
T(charging) = RC ln[(1+B) / B]
In some circuits, an additional RC differentiator circuit can be connected to the input (sometimes, only a single 0.01uF capacitator is sufficient). The purpose of this extra circuit is to transform rectangular signal into trigger pulse signal as seen below
The complete monostable circuit is as follow:
Integrator functions like an average filter, it’s often used as a low-pass filter.
The output of the integrator is given by
or in Laplace domain, it is
Vout = -Vin/(sRC)
If the integrator starts from zero (no charge in the capacitor), the output is simply given by
where Vout(t0) represents the output voltage of the circuit at time t = t0.
Op-amp integrator suffers from the same frequency response limitation as other closed-loop op-amp circuits. It has a cut-off frequency at -3 dB and a decreased output at high frequencies. In addition to this, the integrator also has run-away output issue where it can drift to either power rail due to constant noises and it must be reset periodically to prevent this problem.
The drift is caused by any of the three conditions:
The input Vin has a non-zero DC component,
Input bias current is non-zero,
Input offset voltage is non-zero.
A more complex, grounded integrator circuit prevents this drift
A simple switch in parallel to the negative feedback capacitor allows resetting the integrator to zero.
For the grounded integrator circuit, the output is given by
The differentiator, in contrast, is a high-pass filter. It has poor high-frequency response and any sudden disturbance at the input will cause it to ring at natural frequency
The transfer function of the above circuit is as follow:
or in Laplace domain:
Vout = -sVinRC
Bonus: Non-inverting integrator
Like closed-loop amplifiers, non-inverting integrators and differentiators circuits are easily achievable by switching GND and Vin. A possible circuit for non-inverting integrator is as shown below and it makes use of an RC passive low pass filter circuit at the non-inverting input.