Oscillators & Operational Amplifiers Notes

Last reviewed 16 Sept 2026 · 6 min read

Oscillators

An oscillator generates a periodic waveform (sine, square, triangular) without an external input signal, converting DC power into AC. It is an amplifier with positive feedback.

Barkhausen criterion

For sustained oscillations:

Loop gain magnitude $∣ A β ∣ = 1$ (in practice slightly greater than 1 to start oscillations).
Total phase shift around the loop = 0° or 360°.

Oscillations start from noise and build up until non-linearity limits the amplitude.

RC oscillators (audio frequencies)

FormulaRC oscillators

RC phase shift oscillator (three identical RC sections giving 180°, plus 180° from an inverting amplifier):

f = \frac{1}{2 π R C 6}, required amplifier gain \geq 29

Wien bridge oscillator (lead–lag network with zero phase shift at resonance, non-inverting amplifier):

f = \frac{1}{2 π R C}, required gain = 3

Wien bridge oscillators give low distortion, easily tunable output — widely used in audio signal generators.

LC oscillators (radio frequencies)

FormulaLC oscillators

Tank circuit frequency: $f = \frac{1}{2 π L C _{e q}}$

Hartley oscillator — tapped inductor (or two inductors) with one capacitor: $L_{e q} = L_{1} + L_{2} + 2 M$ Colpitts oscillator — two capacitors with one inductor: $C_{e q} = \frac{C _{1} C _{2}}{C _{1} + C _{2}}$ Clapp oscillator — Colpitts with an extra series capacitor for better stability

Crystal oscillators

A quartz crystal exhibits the piezoelectric effect — mechanical vibration ↔ electrical voltage.
Equivalent to a very high-Q resonant circuit with series and parallel resonant frequencies very close together.
Excellent frequency stability — used in clocks, watches, microprocessors, communication transmitters, frequency standards.

555 timer

A versatile IC (comparators, flip-flop, discharge transistor).

Formula555 timer

Astable multivibrator (free-running square wave):

t_{hi g h} = 0.693 (R_{1} + R_{2}) C, t_{l o w} = 0.693 R_{2} C, f = \frac{1.44}{( R _{1} + 2 R _{2} ) C}

Duty cycle $= \frac{R _{1} + R _{2}}{R _{1} + 2 R _{2}}$

Monostable (one-shot) pulse width: $T = 1.1 R C$

Applications: timers, pulse generation, LED flashers, PWM, alarms.

Operational amplifiers

An op-amp is a high-gain, direct-coupled differential amplifier IC with two inputs — inverting (−) and non-inverting (+) — and one output. The IC 741 is the classic general-purpose op-amp.

Characteristics

Parameter	Ideal op-amp	Typical 741 (order of magnitude)
Open-loop voltage gain $A_{O L}$	Infinite	About $2 \times 1 0^{5}$
Input impedance	Infinite	About 2 MΩ
Output impedance	Zero	About 75 Ω
Bandwidth	Infinite	Unity-gain bandwidth about 1 MHz
CMRR	Infinite	About 90 dB
Slew rate	Infinite	About 0.5 V/µs
Input offset voltage	Zero	A few mV

CMRR (common mode rejection ratio) $= \frac{A _{d}}{A _{c m}}$ — ability to reject signals common to both inputs (noise); usually in dB.
Slew rate — maximum rate of change of output voltage; limits the maximum undistorted frequency: $f_{ma x} = \frac{S R}{2 π V _{m}}$ .
Offset voltage/current — output not zero with zero input; nulled externally.

Virtual short (golden rules for negative feedback)

With negative feedback and very high gain:

Voltage difference between inputs ≈ 0 — $V_{+} = V_{-}$ (virtual short; if + is grounded, − is a virtual ground).
Input currents ≈ 0 (infinite input impedance).

Linear applications

FormulaOp-amp circuits

Inverting amplifier: $V_{o} = - \frac{R _{f}}{R _{1}} V_{in}$ (input impedance ≈ $R_{1}$ )

Non-inverting amplifier: $V_{o} = (1 + \frac{R _{f}}{R _{1}}) V_{in}$ (very high input impedance)

Voltage follower (buffer): $V_{o} = V_{in}$ (gain 1, high input impedance, low output impedance)

Summing (adder) amplifier (inverting): $V_{o} = - R_{f} (\frac{V _{1}}{R _{1}} + \frac{V _{2}}{R _{2}} + \dots)$

Difference (subtractor) amplifier (matched resistors): $V_{o} = \frac{R _{f}}{R _{1}} (V_{2} - V_{1})$

Integrator: $V_{o} = - \frac{1}{R C} \int V_{in} d t$ (square wave → triangular wave)

Differentiator: $V_{o} = - R C \frac{d V _{in}}{d t}$ (triangular wave → square wave; sensitive to noise)

Instrumentation amplifier — three op-amps; high input impedance, high CMRR, gain set by one resistor — used with strain gauges, load cells, thermocouples and bridge sensors in structural and geotechnical monitoring.
Active filters (low-pass, high-pass, band-pass), precision rectifiers, voltage-to-current converters, logarithmic amplifiers.

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