Operational amplifier circuits

Why it matters

Strain gauges, thermocouples, load cells and current-sense resistors produce millivolt signals that must be amplified, offset and scaled before an ADC can read them. Op-amp circuits do that job, and they also build the error amplifiers, integrators and buffers inside every analog controller and sensor interface in a mechatronic system. A handful of standard circuits, analysed with two simple rules, covers most of what you will meet.

Key ideas

What an op-amp is. A high-gain differential voltage amplifier: v_out = A_OL·(v₊ − v₋), with open-loop gain A_OL typically 10⁵–10⁶. It runs from a supply (for example ±15 V or a single 5 V rail) and its output cannot go beyond that supply; it saturates a little short of the rails unless it is a rail-to-rail type.

Ideal op-amp model. Infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth, zero offset. Two "golden rules" follow when there is negative feedback and the output is not saturated:

1. No current flows into either input.
2. The output adjusts so that v₊ = v₋ (virtual short). If v₊ is grounded, v₋ is a virtual ground.

These rules are invalid without negative feedback (open loop or positive feedback): the op-amp then acts as a comparator and the output sits at a rail.

Why negative feedback. It trades excess gain for a closed-loop gain fixed by resistor ratios, independent of the uncertain A_OL. It also widens bandwidth, reduces distortion, and moves input and output impedances in the useful direction.

Standard linear circuits.

• Inverting amplifier: input through R1 to v₋, R_f from output to v₋, v₊ grounded. Gain −R_f/R1; input impedance equals R1.
• Non-inverting amplifier: input to v₊, divider R_f and R1 from output to ground feeds v₋. Gain 1 + R_f/R1 (never below 1); input impedance very high.
• Voltage follower (buffer): output tied to v₋, input to v₊. Gain +1, used to stop a high-impedance source being loaded.
• Summing amplifier: several inputs, each through its own resistor, into the virtual ground of an inverting stage; output is a weighted, inverted sum. Used for mixing, offsetting and simple DACs.
• Difference amplifier: amplifies v₂ − v₁ and rejects the common-mode voltage, provided the resistor ratios are matched. The three-op-amp instrumentation amplifier adds two input buffers for very high input impedance and sets gain with one resistor; it is the standard bridge/strain-gauge front end.
• Integrator: capacitor in the feedback path of an inverting stage; output is the negative time integral of the input. A large resistor across C prevents drift into saturation from offset. Used in PI controllers and ramp generators.
• Differentiator: capacitor at the input, resistor in feedback; output proportional to −dv_in/dt. It amplifies high-frequency noise, so a series input resistor is added in practice.

Practical limits.

• Gain-bandwidth product (GBW): for a voltage-feedback op-amp with a single dominant pole, closed-loop bandwidth ≈ GBW / (noise gain), where noise gain = 1 + R_f/R1 for both inverting and non-inverting stages.
• Slew rate (SR): maximum rate of change of output (V/µs). A sine of peak V_p needs SR ≥ 2π·f·V_p, otherwise it distorts into a triangle.
• Output saturation: the output cannot exceed the supply rails, whatever the formula says.
• Input offset voltage and bias currents: produce DC errors, multiplied by the noise gain; integrators accumulate them.
• CMRR: a real op-amp responds slightly to common-mode input; resistor mismatch in a difference amplifier usually limits CMRR more than the op-amp does.

Formulas

v_out = A_OL · (v₊ − v₋) (open loop) A_v = −R_f / R1 (inverting) A_v = 1 + R_f / R1 (non-inverting); A_v = 1 (follower) v_out = −R_f · (v₁/R₁ + v₂/R₂ + … + v_n/R_n) (inverting summer) v_out = (R₂/R₁) · (v₂ − v₁) (difference amplifier with R₃ = R₁, R₄ = R₂) v_out = −(1/(R·C)) · ∫ v_in dt + v_out(0) (integrator) v_out = −R·C · dv_in/dt (differentiator) f_CL = GBW / (1 + R_f/R1) (closed-loop −3 dB bandwidth) f_max = SR / (2π · V_p) (full-power bandwidth)

• v: voltages (V); R: resistances (Ω); C: capacitance (F); t: time (s); A_v: closed-loop voltage gain (V/V); A_OL: open-loop gain (V/V); GBW: gain-bandwidth product (Hz); SR: slew rate (V/s); V_p: output peak (V); f: frequency (Hz).
• All closed-loop formulas assume an ideal op-amp with negative feedback and an output inside the supply rails.

Worked examples

Example 1 (standard: bridge signal with a difference amplifier). A strain-gauge bridge gives v₁ = 2.00 V and v₂ = 2.05 V. A difference amplifier uses R₁ = R₃ = 10 kΩ and R₂ = R₄ = 100 kΩ, on ±12 V supplies. Find v_out.

1. v_out = (R₂/R₁)·(v₂ − v₁).
2. R₂/R₁ = 100 kΩ/10 kΩ = 10.
3. v_out = 10 × (2.05 − 2.00) V = 10 × 0.05 V = 0.5 V.
4. The 2 V common-mode voltage is rejected (ideal, matched resistors); 0.5 V is well inside the rails.

v_out = 0.50 V.

Example 2 (GATE level: bandwidth and slew rate). A non-inverting amplifier uses R1 = 1 kΩ and R_f = 99 kΩ with an op-amp of GBW = 1 MHz and SR = 0.5 V/µs. (a) Find the gain and the closed-loop bandwidth. (b) What is the highest frequency at which it can deliver an undistorted 10 V peak sine?

1. A_v = 1 + R_f/R1 = 1 + 99/1 = 100.
2. f_CL = GBW/A_v = 1 × 10⁶ Hz / 100 = 10 kHz.
3. Slew limit: f_max = SR/(2π·V_p) = (0.5 × 10⁶ V/s)/(2π × 10 V) = 7.96 kHz.
4. The slew limit (7.96 kHz) is lower than the small-signal bandwidth (10 kHz), so it governs at full output.

A_v = 100, f_CL = 10 kHz, f_max ≈ 7.96 kHz for 10 V peak.

Example 3 (integrator). An integrator has R = 10 kΩ and C = 0.1 µF, starts at v_out(0) = 0, and receives a +1 V step for 2 ms (supplies ±12 V).

1. RC = 10⁴ Ω × 10⁻⁷ F = 1 ms.
2. v_out = −(1/RC)·v_in·t = −(1/10⁻³ s) × 1 V × 2 × 10⁻³ s = −2 V, falling linearly.

v_out = −2.0 V after 2 ms (a ramp of −1 V/ms).

Common mistakes

• Using the virtual-short rule in a circuit with no negative feedback (a comparator).
• Forgetting the minus sign of the inverting, summing and integrator stages.
• Writing the non-inverting gain as R_f/R1 instead of 1 + R_f/R1.
• Reporting an output beyond the supply rails; the real output saturates.
• Dividing GBW by the signal gain of an inverting stage (R_f/R1) instead of the noise gain (1 + R_f/R1).
• Confusing bandwidth limits with slew-rate limits; large outputs are usually slew limited.
• Assuming a difference amplifier rejects common mode when its resistor ratios are not matched.

For GATE ME

Questions are mostly direct applications of the ideal model: output of an inverting, non-inverting, summing or difference circuit for given resistors and inputs; integrator output for a step or square input; occasionally closed-loop bandwidth from GBW or a saturation check against the supply. Practise writing KCL at the inverting node with the virtual-short rule; it solves almost every circuit.

Quick check

1. An inverting amplifier has R1 = 2 kΩ and R_f = 10 kΩ. What is its gain?
2. A non-inverting amplifier needs a gain of 11 with R1 = 1 kΩ. What R_f is required?
3. Why is the inverting input called a virtual ground in an inverting amplifier?
4. An op-amp with GBW = 2 MHz is used at a noise gain of 20. What is the closed-loop bandwidth?
5. An inverting summer has R_f = 10 kΩ, inputs 0.5 V through 10 kΩ and −0.2 V through 5 kΩ. What is v_out?

Answers: 1. −5. 2. 10 kΩ. 3. Negative feedback forces v₋ = v₊ = 0 V, yet no current flows into the input. 4. 100 kHz. 5. −0.1 V.