Op-Amps, Feedback and Oscillators
Ideal op-amp rules, inverting, non-inverting, summing, difference, integrator and differentiator circuits, CMRR and slew rate, feedback topologies, Barkhausen criterion and RC and LC oscillators.
The operational amplifier is a high-gain differential amplifier. With negative feedback, its circuits behave almost exactly as the external resistors and capacitors dictate.
Ideal op-amp rules (with negative feedback)
- No current into the inputs (infinite input resistance).
- Virtual short: V₊ = V₋.
Standard circuits
| Circuit | Output |
|---|---|
| Inverting | V_o = −(R_f/R₁) V_in |
| Non-inverting | V_o = (1 + R_f/R₁) V_in |
| Voltage follower | V_o = V_in |
| Summing (inverting) | V_o = −R_f Σ(Vᵢ/Rᵢ) |
| Difference (equal ratios) | V_o = (R_f/R₁)(V₂ − V₁) |
| Integrator | V_o = −(1/RC) ∫ V_in dt |
| Differentiator | V_o = −RC dV_in/dt |
Non-ideal parameters
- CMRR = differential gain/common-mode gain (often in dB).
- Slew rate: maximum dV_o/dt. For a sine of amplitude V_m: full-power bandwidth f_max = SR/(2πV_m).
- Input offset voltage, bias current, finite gain–bandwidth product.
- Comparators and the Schmitt trigger (positive feedback, hysteresis) use the op-amp without negative feedback.
Feedback
Negative feedback with loop gain Aβ: A_f = A/(1 + Aβ). It reduces gain and distortion and increases bandwidth by (1 + Aβ).
| Topology | Input resistance | Output resistance |
|---|---|---|
| Voltage-series | Increases | Decreases |
| Voltage-shunt | Decreases | Decreases |
| Current-series | Increases | Increases |
| Current-shunt | Decreases | Increases |
Oscillators
Barkhausen criterion: loop gain Aβ = 1 in magnitude with a total phase shift of 0° (or 360°).
| Oscillator | Frequency | Gain needed |
|---|---|---|
| RC phase shift (3 RC sections) | f = 1/(2πRC√6) | A ≥ 29 |
| Wien bridge | f = 1/(2πRC) | A ≥ 3 |
| Colpitts (LC) | f = 1/(2π√(L C_eq)), C_eq = C₁C₂/(C₁ + C₂) | |
| Hartley (LC) | f = 1/(2π√((L₁ + L₂ + 2M)C)) | |
| Crystal | Very stable, series or parallel resonance |
Worked example
An op-amp with slew rate 0.5 V/µs drives a 10 V peak sine.
- f_max = 0.5 × 10⁶/(2π × 10) = 7.96 kHz.
Practice
An RC phase-shift oscillator uses R = 1 kΩ and C = 1 µF. What is its frequency in Hz? (Give 2 decimal places.)
Show answer and explanation
Answer: 64.97f = 1/(2πRC√6) = 1/(2π × 10⁻³ × 2.449) = 64.97 Hz.
An inverting summer has R_f = 10 kΩ and inputs of 1 V through 10 kΩ and 2 V through 5 kΩ. What is V_o in V?
Show answer and explanation
Answer: -5V_o = −10k(1/10k + 2/5k) = −(1 + 4) = −5 V.
An amplifier has A = 1000 and β = 0.01. What is its closed-loop gain? (Give 2 decimal places.)
Show answer and explanation
Answer: 90.91A_f = A/(1 + Aβ) = 1000/11 = 90.91.
What minimum amplifier gain does a Wien bridge oscillator need?
Show answer and explanation
Answer: 3At f₀ the Wien network attenuates by 1/3 with zero phase shift, so A must be at least 3.