Operational Amplifiers

The op-amp is one of the most useful ICs in analog electronics. A single chip gives you a high-gain differential amplifier that, with a few resistors and capacitors, becomes a precision amplifier, active filter, integrator, comparator, signal conditioner, or oscillator. It's the Swiss Army knife of analog design — understanding op-amp circuits means being able to build almost any linear analog function from scratch.

The Ideal Op-Amp

Property	Ideal value	Typical real value	Impact of non-ideality
Open-loop voltage gain (A_OL)	Infinite	10,000–1,000,000	Negligible with negative feedback
Input impedance	Infinite	10 MΩ – 10 TΩ (FET input)	Loads the signal source slightly
Output impedance	Zero	10–100 Ω open-loop; ~mΩ with feedback	Limits drive capability
Bandwidth	Infinite	1 MHz – 1 GHz (gain-dependent)	Gain falls at high frequency
Input offset voltage (Vos)	Zero	0.1–10 mV	DC error at output = Vos × gain
Input bias current	Zero	1 pA – 1 µA	Voltage error across source impedance
CMRR	Infinite	60–120 dB	Common-mode signals leak through

The Golden Rules (for negative feedback circuits)

1. The output does whatever it must to make V+ = V−  (virtual short)
2. No current flows into either input                 (infinite Zin)

Apply these two rules to any op-amp circuit in negative feedback and you can derive the gain without knowing anything else about the part.

Negative feedback makes the op-amp adjust its output until its two inputs are nearly equal. The inverting circuit applies the signal through Rin and reverses its polarity; the non-inverting circuit applies the signal to the high-impedance + input and preserves polarity.

Common Circuit Configurations

Inverting Amplifier

Inverting op-amp amplifier: Vin passes through R_in to the − input, while R_f feeds the output back; the grounded + input makes the output invert the input signal.

Gain = −R_f / R_in    (negative = phase inverted)

R_in=10kΩ, R_f=100kΩ → Gain = −10
1V in → −10V out

Input impedance = R_in

Non-Inverting Amplifier

Non-inverting op-amp amplifier: Vin drives the + input and the R_f–R_g divider feeds the output to the − input, producing an in-phase amplified output.

Gain = 1 + R_f / R_g    (positive = same phase)

R_f=9kΩ, R_g=1kΩ → Gain = 10
1V in → +10V out

Input impedance ≈ infinite (looking into + terminal)

Voltage Follower (Unity-Gain Buffer)

Voltage follower: direct output-to-inverting-input feedback makes Vout follow Vin at unity gain while providing current drive.

Gain = 1

The output follows the input exactly in voltage but provides current drive. The critical use: isolating a high-impedance source from a low-impedance load without loading the source. The sensor output doesn't care what's downstream.

Summing Amplifier (Mixer)

Summing amplifier: three input voltages feed the inverting input through R1, R2, and R3, and R_f produces their weighted, inverted sum at the output.

Vout = −R_f × (V1/R1 + V2/R2 + V3/R3)

If R1=R2=R3=R_f:  Vout = −(V1 + V2 + V3)

A simple audio mixer. Each input is independent (virtual ground at −). Adding more input resistors doesn't change the others' gain.

Difference Amplifier (Subtractor)

Difference amplifier: resistor networks apply V1 and V2 to opposite op-amp inputs so the output amplifies their voltage difference.

If R1=R2=R, R_f=R_g:  Vout = (R_f/R) × (V2 − V1)

Amplifies the difference between two signals, rejecting common-mode voltage (CMRR). The basis of instrumentation amplifiers for sensor signals.

Integrator

Op-amp integrator: an input resistor and feedback capacitor make the output the inverted time integral of Vin.

Vout = −(1/RC) × ∫Vin dt

Output is the running integral of the input. A square wave in produces a triangle wave out. Used in PID controllers, waveform generators, ADCs, and signal processing. Without a reset switch across C or high-value parallel resistor, DC drift causes the output to rail.

Differentiator

Op-amp differentiator: an input capacitor and feedback resistor make the output proportional to the inverted rate of change of Vin.

Vout = −R_f × C × dVin/dt

Output proportional to rate of change. Edge detector — a sudden step produces a spike at the output. Sensitive to noise (noise is fast = high derivative); usually needs an input resistor to limit high-frequency gain.

Comparator (Open-Loop)

Open-loop comparator: Vin at the + input is compared with V_ref at the − input, driving Vout to a supply rail according to which voltage is higher.

No feedback. Output slams to +rail or −rail depending on which input is higher.

Use a dedicated comparator IC (LM393, LM339, TLV1801) instead of an op-amp for this — comparators have faster response, open-collector outputs, and are designed not to latch. An op-amp used as a comparator often has slow response and can oscillate in the transition region.

Common Op-Amp ICs

Part	Supply	GBW	Vos	Notes
LM741	±15V	1 MHz	1–6 mV	Historic, teaching only — poor in modern designs. High offset, limited bandwidth.
LM358	3–32V single	1 MHz	3 mV	Single-supply, two channels. Cheap and everywhere. Slow, limited rail-to-rail.
TL071 / TL081	±15V	3 MHz	3 mV	JFET input — high input impedance, low bias current. Audio, general analog.
NE5532	±5 to ±15V	10 MHz	0.5 mV	Low noise, high drive current. The studio audio standard.
OPA2134	±2.5 to ±18V	8 MHz	0.05 mV	Low distortion, JFET input. Hi-fi audio.
AD8620	±2.3 to ±16.5V	25 MHz	0.025 mV	Precision FET, very low noise and offset.
MCP6002	1.8–6V single	1 MHz	3 mV	Rail-to-rail, microcontroller compatible. Microchip's workhorse for 3.3V/5V systems.
LMV321	2.7–5V single	1 MHz	7 mV	Single-supply, tiny packages. Sensor conditioning near microcontrollers.

Gain-Bandwidth Product

GBW = open-loop gain × frequency  (constant for a given op-amp)

A_v_max at frequency f:  A_v = GBW / f

Example: TL071 GBW = 3 MHz
  At f=30 kHz: max gain = 3,000,000 / 30,000 = 100
  At f=300 kHz: max gain = 10
  At f=3 MHz: max gain = 1

Always check: GBW / desired_gain  >  highest signal frequency + margin

Single-Supply Operation

Many op-amps require dual supplies (±12V, ±15V). For single-supply operation (5V or 3.3V), use a rail-to-rail op-amp and create a virtual ground at V_supply/2 for the signal reference:

Single-supply virtual ground: equal divider resistors R1 and R2 create V_ref at half the supply voltage for biasing analog signals.

Buffer V_ref with a voltage follower to provide low impedance.
AC-couple the input and output with capacitors to block DC bias.

References

Horowitz & Hill — The Art of Electronics, 3rd ed. Chapters 4–5
Franco — Design with Operational Amplifiers and Analog Integrated Circuits, 4th ed.
Texas Instruments — Op Amp Applications (PDF) ↗
Analog Devices — Op Amp Applications Handbook ↗