Operational amplifiers
The operational amplifier is a versatile building block: with a few resistors and capacitors, it amplifies, sums, filters, and compares signals.
What an op-amp is
An operational amplifier is an integrated differential amplifier with two inputs, inverting (-) and non-inverting (+), and one output. It amplifies the difference between the inputs by a very large open-loop gain, often 100,000 or more. The name comes from analogue computers, where op-amps performed mathematical operations. The classic 741 appeared in 1968.
The ideal op-amp
Analysis usually assumes an ideal op-amp: infinite open-loop gain, infinite input impedance (so no current flows into the inputs), and zero output impedance. With negative feedback, the output adjusts until the two input voltages are equal, the virtual short. These two rules, no input current and equal input voltages, make op-amp circuits easy to analyse.
Inverting amplifier
In the inverting amplifier, the input signal connects through \(R_{in}\) to the inverting input, a feedback resistor \(R_f\) runs from output to that input, and the non-inverting input is grounded. The gain is \(\frac{V_{out}}{V_{in}} = -\frac{R_f}{R_{in}}\). With \(R_f = 100\) k\(\Omega\) and \(R_{in} = 10\) k\(\Omega\), the gain is -10, inverting and amplifying the signal.
Non-inverting amplifier and buffer
In the non-inverting amplifier, the signal drives the + input, and a divider from output to ground feeds the - input: gain \(1 + \frac{R_f}{R_g}\). Its input impedance is very high. Connecting the output directly to the - input gives a voltage follower with gain 1, a buffer that lets a weak, high-impedance source drive a heavy load.
Summing, differentiating, and integrating
An inverting amplifier with several input resistors sums signals, the basis of audio mixers: \(V_{out} = -R_f(\frac{V_1}{R_1} + \frac{V_2}{R_2})\). Replacing the feedback resistor with a capacitor creates an integrator, whose output is proportional to the integral of the input; a capacitor at the input creates a differentiator. Active filters combine op-amps with resistors and capacitors for sharp, gain-adjustable frequency responses.
Comparators and positive feedback
Without negative feedback, an op-amp's huge gain drives the output to one supply limit or the other depending on which input is higher, acting as a comparator, useful for detecting thresholds such as low battery voltage. Positive feedback adds hysteresis (a Schmitt trigger), preventing chattering when noisy signals hover near the threshold.
Real op-amp limits
Real op-amps have limited output swing (rail-to-rail types get close to the supplies), finite gain-bandwidth product (for example 1 MHz, meaning a gain of 10 is available up to about 100 kHz), slew-rate limits on how fast the output can change, input offset voltage and bias currents, and noise. Choosing an op-amp means matching these specifications to the application.
Worked example
A sensor produces 0 to 50 mV and must span 0 to 5 V for an analogue-to-digital converter, a gain of 100. A non-inverting amplifier with \(R_g = 1\) k\(\Omega\) and \(R_f = 99\) k\(\Omega\) gives \(1 + 99 = 100\). With an op-amp whose gain-bandwidth product is 1 MHz, a gain of 100 is available only up to about 10 kHz, which is fine for a slowly changing temperature sensor.
Key takeaways
Op-amps are differential amplifiers with very high open-loop gain; the 741 appeared in 1968. Ideal op-amp rules with negative feedback: no input current and equal input voltages, the virtual short. The inverting amplifier has gain -R_f/R_in; the non-inverting amplifier has gain 1 + R_f/R_g and high input impedance; the follower buffers with gain 1. Summing amplifiers mix signals; capacitors create integrators, differentiators, and active filters. Without feedback, op-amps act as comparators, and positive feedback gives Schmitt-trigger hysteresis. Real op-amps have limited swing, gain-bandwidth product, slew rate, offsets, and noise.
Flashcard
High-gain differential amplifier
Check your understanding
An inverting amplifier has R_in = 4.7 kilohms and R_f = 47 kilohms. What is its gain?