Operational Amplifiers

The op-amp is the universal analog building block. Learn the ideal op-amp rules (virtual short), inverting, non-inverting, voltage follower, summing, difference, instrumentation, integrator and comparator circuits, plus real limitations: gain–bandwidth product, slew rate, offset, input bias current and rail-to-rail behaviour.

An op-amp is a very high-gain differential amplifier in an IC. With negative feedback, its behaviour is set almost entirely by external resistors and capacitors — making precise amplifiers, filters, and signal conditioners easy to design.

An analogy

An op-amp is like an extremely strict balancing scale: it adjusts its output until both inputs weigh exactly the same.

Ideal op-amp

Infinite gain, infinite input impedance, zero output impedance, infinite bandwidth. With negative feedback, the output does whatever makes V₋ = V₊; no current flows into the inputs.

Standard circuits

Inverting (gain set by ratio, input impedance = R_in), non-inverting (high input impedance), follower (buffer), summing (audio mixers, DACs), difference and instrumentation amplifiers (bridge sensors), integrators (PID, filters), comparators with hysteresis (Schmitt triggers).

Real limitations

GBW: at gain 11 a 1 MHz op-amp gives ≈ 91 kHz bandwidth. Slew rate limits large, fast signals. Offset voltage and bias currents cause DC errors, multiplied by noise gain. Output swing and input common-mode range depend on supply (rail-to-rail parts help at 3.3 V).

Stability and noise

Phase margin, capacitive loads and compensation; voltage and current noise densities; choosing CMOS vs bipolar vs JFET input op-amps; chopper-stabilised zero-drift amplifiers for precision.

Formulas

• Golden rules (negative feedback): V₊ = V₋ (virtual short), I_in = 0
• Inverting: V_out = −(R_f/R_in) V_in
• Non-inverting: V_out = (1 + R_f/R_g) V_in
• Summing / difference: V_out = −R_f Σ(V_i/R_i); V_out = (R₂/R₁)(V₂ − V₁)
• Integrator: V_out = −(1/RC) ∫ V_in dt
• Bandwidth & slew: BW = GBW / noise gain; SR ≥ 2π f V_peak

Worked example: Inverting and non-inverting

Given: R_f = 100 kΩ, R_in = R_g = 10 kΩ

Find: Gains.

• Inverting: −100/10 = −10
• Non-inverting: 1 + 10 = 11

Answer: −10 and +11

Worked example: Slew-rate check

Given: 20 kHz sine, 10 V peak

Find: Minimum slew rate.

• SR = 2π × 20 000 × 10 = 1.26 V/µs

Answer: ≥ 1.3 V/µs (a 0.5 V/µs part would distort)

Op-amp circuit quick-reference

Inverting amplifier:      Vout = -(Rf/Rin) * Vin          Zin = Rin
Non-inverting amplifier:  Vout = (1 + Rf/Rg) * Vin         Zin ≈ ∞
Voltage follower:         Vout = Vin                       buffer
Summing (inverting):      Vout = -Rf * (V1/R1 + V2/R2 + …)
Difference amplifier:     Vout = (R2/R1) * (V2 - V1)       (matched ratios)
Instrumentation amp:      Vout = (1 + 2R/Rgain) * (V2 - V1)
Integrator:               Vout = -(1/RC) ∫ Vin dt
Differentiator:           Vout = -RC dVin/dt
Comparator (open loop):   Vout = +Vsat if V+ > V-, else -Vsat

With negative feedback, apply the virtual-short rule and KCL at the inverting node to derive any of these. Without feedback (comparator) the output saturates.

Common mistakes

• Wrong: Expecting full gain at high frequency. — Right: BW = GBW/gain — check the datasheet.
• Wrong: Running inputs outside the common-mode range. — Right: Use rail-to-rail parts at low supply voltages.

Applications

• Sensor conditioning: Scaling and filtering before ADCs.
• Active filters: Sallen–Key and multiple-feedback designs.
• Control loops: Analog PID and error amplifiers.

Key takeaways

• V₊ = V₋, no input current (with feedback).
• Inverting −R_f/R_in; non-inverting 1 + R_f/R_g.
• BW = GBW/gain; SR ≥ 2πfV_p.
• Watch offset, rails, stability.

Going further

• Phase margin: Stability with capacitive loads.
• Instrumentation amps: High CMRR bridge amplification.
• Noise analysis: e_n, i_n and resistor noise.