Operational amplifiers: linear and nonlinear applications
A comparator receiving a sine wave with 50 mV of noise changes state 30 times where it should change 3. With 0.1 V of hysteresis, it changes 3 times again.
01The op amp as a feedback amplifier
An operational amplifier is a differential amplifier with enormous gain, from \( 10^5 \) to \( 10^7 \), with a high-impedance input and a low-impedance output. It only makes sense inside a feedback loop: the loop gain is so large that the gain of the circuit is set by the external resistors. With negative feedback and the output not saturated, the voltage between the inputs is the output divided by the gain, practically zero: this is the virtual short circuit.
An introduction to the component, with the basic circuits, is in the interactive course on operational amplifiers on this site.
02Linear applications
| Circuit | Transfer function | Remarks |
|---|---|---|
| Inverting summer | \( -R_f \sum v_i/R_i \) | The inputs do not interact: the junction is a virtual ground |
| Integrator | \( -\frac{1}{RC}\int v_i\,dt \) | Needs a resistor in parallel with C, or it saturates with the offset |
| Differentiator | \( -RC\, dv_i/dt \) | Amplifies noise; limited with a series resistor |
| Current-to-voltage converter | \( -R\, i_i \) | For photodiodes: the virtual ground keeps the diode at 0 V |
| Instrumentation amplifier | \( (1 + 2R/R_G)(v_+ - v_-) \) | Three op amps; the gain is set with a single resistor |
The instrumentation amplifier is the most important linear application for sensors: very high input impedance, differential gain adjustable with \( R_G \) and a common-mode rejection that does not depend on the matching of the resistors of the first stage. Active filters, another central linear application, were covered in analog filters.
03What the ideal model does not tell
At DC, the real op amp has an input offset voltage \( V_{OS} \) between its inputs, from microvolts to millivolts, and bias currents \( I_B \) that flow through the external resistors. Both appear at the output multiplied by the noise gain \( 1/\beta \):
At AC, the gain falls with frequency, the output cannot change faster than the slew rate and it cannot reach the supply voltages except in rail-to-rail op amps. These limits are studied in the units on frequency response and stability.
04Nonlinear applications
Without negative feedback, the op amp is a comparator: its output saturates to one side or the other depending on which input is larger. With a noisy input that crosses the threshold slowly, the noise makes it cross several times and the output bounces. The solution is positive feedback: the Schmitt trigger has two thresholds, and once the output changes, the threshold moves away from the input:
Two periods of a 1 V sine wave with Gaussian noise, applied to a simple comparator (red) and to a Schmitt trigger with thresholds at ±Vh (blue). The ideal output changes three times. The noise is always the same pseudorandom sequence, scaled.
Other nonlinear applications combine the op amp with diodes. The precision rectifier puts the diode inside the loop, which divides its threshold voltage by the gain: it rectifies signals of millivolts. The peak detector adds a capacitor that holds the maximum, and the logarithmic amplifier uses the exponential of a transistor in the loop to compress a dynamic range of several decades.
05In the lab
Measure the output of a gain-of-1000 inverting amplifier with the input grounded, with and without the compensating resistor, and deduce \( V_{OS} \) and \( I_{OS} \). Compare with the datasheet.
Build an instrumentation amplifier with three op amps for a strain-gauge bridge and measure its common-mode rejection by applying the same signal to both inputs.
Add noise from a generator to a slow sine wave and observe the output of a comparator and of a Schmitt trigger. Adjust the hysteresis until the bouncing disappears.
06Common mistakes
- Applying the virtual short circuit to a comparator. It only holds with negative feedback and the output not saturated.
- Using an op amp as a fast comparator. It recovers from saturation slowly; comparators are designed for that.
- Integrating without a parallel resistor. The offset gets integrated and drives the output into saturation within seconds.
- Giving too much hysteresis. It eliminates the bouncing, but delays the switching and can keep small signals from triggering it.
07Self-assessment
What gain error does a gain-of-1000 noninverting amplifier have with an op amp of A = 10⁵?
\( A\beta = 100 \): the gain comes out 1 % low.
A gain-of-100 inverting amplifier uses an op amp with VOS = 2 mV. What output voltage does it have with the input grounded?
\( 101 \cdot 2 \) mV ≈ 0.2 V.
An instrumentation amplifier with R = 25 kΩ needs a gain of 101. What is the value of RG?
\( 1 + 50\,\mathrm{k}/R_G = 101 \): RG = 500 Ω.
A Schmitt trigger with a ±12 V output needs thresholds of ±0.6 V. What R2/R1 ratio does it take?
20.
08Further reading
- Sergio Franco. Design with Operational Amplifiers and Analog Integrated Circuits. 4th ed., McGraw-Hill, 2015. The practical reference: linear and nonlinear applications with their errors.
- Paul Horowitz and Winfield Hill. The Art of Electronics. 3rd ed., Cambridge University Press, 2015. Op amps from the workbench, with hundreds of annotated circuits.
- Robert F. Coughlin and Frederick F. Driscoll. Operational Amplifiers and Linear Integrated Circuits. 5th ed. (Spanish edition, “Amplificadores operacionales y circuitos integrados lineales”), Prentice Hall, 1999. A text in Spanish with many worked applications.