Operational Amplifier Circuits

Overview

This demonstration shows several op-amp circuits, including a multivibrator, integrator and differentiator.

Model

Multivibrator with comparator

../../_images/multivibrator_circuit_DIN.svg — Fig. 1 Multivibrator

The multivibrator amplifier is an astable oscillator which generates rectangular waveforms at the output using an RC network at the inverting input and a voltage divider at the non-inverting input of the amplifier.

This configuration switches between its two unstable states, with the time spent in each state controlled by the charging or discharging of the capacitor through a resistor.

Initially, the capacitor starts charging up to \(V_{\mathrm{out}}\). Once the voltage at the inverting terminal becomes equal to or greater than the voltage at the non-inverting terminal, \(\lambda V_{\mathrm{out}}\), the op-amp output clamps to the negative supply rail, thus causing the capacitor charge go to zero and start charging to the new value of \(V_{\mathrm{out}}\). This goes on until the negative supply rail reaches the \(-\lambda V_{\mathrm{out}}\) threshold, and the output changes state again, reinitiating the cycle. This produces a steady, continuous square wave pulse train at the output.

The RC time constant determines the rate of capacitor charge/discharge, or the period of the output waveform, and the voltage divider network sets the reference voltage level, \(\lambda V_{\mathrm{out}}\).

A small capacitance \(C_{\mathrm{in}}\) is required for decoupling of negative feedback.

Formula derivation

From voltage divider,

\[\begin{aligned} \lambda =\frac{R_{1}}{R_{1}+R_{2}} \end{aligned}\]

General charging equation for a capacitor with an original charge:

\[\begin{aligned} q =CV\cdot(1-e^{\frac{-t}{RC}})+q_{0}\cdot e^{\frac{-t}{RC}} \end{aligned}\]

For \(V=V_{\mathrm{out}}\) and \(q_{\mathrm{0}}=\lambda CV_{\mathrm{out}}\),

\[\begin{aligned} q =-CV_{\mathrm{out}}\cdot(1-e^{\frac{-t}{RC}})+\lambda CV_{\mathrm{out}}\cdot e^{\frac{-t}{RC}} \end{aligned}\]

\[\begin{aligned} T=2RC\cdot \ln \left(\frac{1+\lambda}{1-\lambda}\right) \end{aligned}\]

This result is also obtained for the discharging period of the operation, assuming the magnitudes of the rails are equal, meaning \(t_{\mathrm{charge}}=t_{\mathrm{discharge}}\).

Multivibrators are used in a variety of applications where square waves or timed intervals are required, such as a flashing light.

Inverting integrator

An integrator produces an output voltage, which is proportional to the integral of the input voltage.

\[\begin{aligned} V_{\mathrm{out}}=-\frac{1}{RC}\cdot \int_{0}^{t}v_{\mathrm{in}}\,dt \end{aligned}\]

Formula derivation

Because of virtual ground and infinite impedance of the input terminals of the op-amp, all of the input current flows through R and C:

\[\begin{aligned} i_{\mathrm{in}}=\frac{v_{\mathrm{in}}-V_{-}}{R}=\frac{v_{\mathrm{in}}}{R}=i_{R}=i_{C} \end{aligned}\]

\[\begin{aligned} v_{C}=V_{-}-V_{\mathrm{out}}=-V_{\mathrm{out}} \end{aligned}\]

\[\begin{aligned} i_{C}=C\cdot \frac{\mathrm{d}v_{C}}{\mathrm{d} t}=-C\cdot \frac{\mathrm{d}V_{\mathrm{out}}}{\mathrm{d} t}=\frac{v_{\mathrm{in}}}{R} \end{aligned}\]

\[\begin{aligned} \frac{\mathrm{d}V_{\mathrm{out}}}{\mathrm{d} t}=-\frac{1}{RC}\cdot v_{\mathrm{in}} \end{aligned}\]

\[\begin{aligned} V_{\mathrm{out}}=-\frac{1}{RC}\cdot \int_{0}^{t}v_{\mathrm{in}}\,dt \end{aligned}\]

An application for this circuit could be integrating water flow and measuring the total quantity of water that has passed by the flowmeter.

Inverting differentiator

The differentiator op-amp configuration produces an output voltage that is proportional to the rate of change of the input voltage by measuring the current through a capacitor:

\[\begin{aligned} V_{\mathrm{out}}=-RC\cdot \frac{\mathrm{d}v_{\mathrm{in}}}{\mathrm{d} t} \end{aligned}\]

The right-hand side of the capacitor is held at \(0\) volts due to the virtual ground effect. Therefore, current through the capacitor is solely due to change in the input voltage. A steady input voltage will not cause a current through C, but a changing input voltage will. The faster the voltage changes, the larger the magnitude of the output voltage.

Formula derivation

Because of virtual ground and the infinite impedance of an op-amp, all current flowing through the capacitor also flows through R1:

\[\begin{aligned} i_{C}=i_{R_{1}}=\frac{V_{-}-V_{\mathrm{out}}}{R_{1}}=\frac{-V_{\mathrm{out}}}{R_{1}} \end{aligned}\]

\[\begin{aligned} v_{C}=v_{AC}-i_{C}R_{2}\approx v_{AC} \end{aligned}\]

(the very small resistance R2 is needed for convergence purposes)

\[\begin{aligned} i_{C}=C\cdot \frac{\mathrm{d}v_{C}}{\mathrm{d} t}=C\cdot \frac{\mathrm{d}v_{AC}}{\mathrm{d} t}=-\frac{V_{\mathrm{out}}}{R_{1}} \end{aligned}\]

\[\begin{aligned} V_{\mathrm{out}}=-R_{1}C\cdot \frac{\mathrm{d}v_{AC}}{\mathrm{d} t} \end{aligned}\]

An application for this circuit could be monitoring the rate of change of temperature in an environment where too high or too low of a temperature rise is detrimental and would, thus, trigger an alarm or a notification using additional circuitry on the output.

Simulation

Multivibrator with comparator

Increase the capacitance value to \(100\,\mu\mathrm{F}\) and observe the increase in the amount of time it takes to charge up the capacitor, as shown in Fig. 4. Increase the left-most resistor to \(25\,\mathrm{k}\Omega\) and observe the change in the output voltage switching frequency.

../../_images/multivibrator_scope.svg — Fig. 4 Multivibrator circuit simulations comparison

Inverting integrator

Set the amplitude and frequency of the AC voltage source to \(0\) and the DC voltage source to \(1\,\mathrm{V}\). The output should look like the one in Fig. 5. If a fixed voltage is applied to the input of an integrator, the output voltage will be a ramp with a constant slope of the negative input voltage multiplied by a factor of 1/RC.

Inverting differentiator

Set the amplitude and frequency of the AC voltage source to \(0\) and the initial capacitor voltage to \(1\,\mathrm{V}\). Observe the output after there is no change in the input voltage, as shown in Fig. 6. After the capacitor C discharges, do you see what you expected? If the input voltage is constant, dv/dt is zero and the output voltage is zero.

Conclusion

Operational amplifiers are a core part of analog electronics and can perform many different operations depending on the passive component configurations around them. For more op-amp examples, visit the Analog Electronics Academy page on Plexim’s website.