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AExE0105 Signal generator

Oscillator principle

An oscillator converts DC supply energy into a periodic output without requiring an external periodic input. An amplifier and frequency-selective positive-feedback network commonly form the loop.

Let loop gain be \(A(j\omega)\beta(j\omega)\). For sustained sinusoidal oscillation at \(\omega_0\), the Barkhausen conditions are

\[ |A\beta|=1, \qquad \angle A\beta=0^\circ\pmod{360^\circ}. \]

Startup requires a small disturbance or noise and usually \(|A\beta|>1\) initially. Nonlinearity or automatic gain control then reduces effective loop gain toward 1 to stabilize amplitude. Barkhausen conditions identify the oscillation condition; they do not by themselves guarantee a practical circuit will start and settle cleanly.

Sinusoidal oscillator families

Family Frequency-selective network Best recognition
RC phase shift Cascaded RC sections provide phase shift Sine generation at relatively low frequencies
Wien bridge Lead-lag RC bridge Low-distortion, conveniently tunable sine wave
Hartley Two inductive sections and one capacitor LC oscillator; inductive divider
Colpitts One inductor and two capacitors LC oscillator; capacitive divider
Crystal Piezoelectric resonator Very high \(Q\) and excellent frequency stability

For equal \(R\) and \(C\) in the common Wien bridge form,

\[ f_0=\frac{1}{2\pi RC}, \]

and the amplifier must provide a gain of 3 at balance. Practical amplitude control is needed to avoid decay or clipping.

For an ideal LC tank,

\[ f_0=\frac{1}{2\pi\sqrt{LC}}. \]

Hartley uses an effective inductance determined by its tapped/coupled inductors. Colpitts uses

\[ C_{eq}=\frac{C_1C_2}{C_1+C_2}. \]

A crystal has a motional RLC branch plus shunt capacitance and has nearby series and parallel resonances. Its high \(Q\) gives a narrow bandwidth and stable frequency, but only limited pulling/tuning range.

\[ Q=\frac{f_0}{\Delta f}. \]

Higher \(Q\) means narrower resonance. Avoid memorizing one universal crystal or LC \(Q\) value; it depends on the actual resonator and loading.

Waveform generators

Sinusoidal feedback oscillators are not the only signal generators.

Generator Operating idea Typical output
Relaxation oscillator Capacitor repeatedly charges and discharges between thresholds Square/pulse plus exponential ramp
Schmitt trigger + integrator Hysteretic switching drives linear integration Square and triangle
Constant-current capacitor ramp Nearly constant \(i=C\,dv/dt\) Triangle or sawtooth
Function generator Shaping and switching stages Sine, square, triangle
555 in astable mode Threshold/trigger comparators charge and discharge a capacitor Rectangular pulses

Oscillator versus waveform generator: an oscillator is any self-sustained periodic source; "waveform generator" often emphasizes selectable nonsinusoidal shapes.

Common traps

  • Positive feedback is required at the oscillation frequency; negative feedback generally stabilizes an amplifier instead.
  • Phase shift can total \(0^\circ\) or any integer multiple of \(360^\circ\).
  • Crystal is the standard answer for highest \(Q\) and frequency stability.
  • LC is naturally tunable by changing \(L\) or \(C\).
  • RC oscillators avoid inductors and suit lower-frequency sine generation.
  • No oscillator creates energy; the DC supply provides output and loss power.

Signal-generator example

For a Wien bridge with \(R=10\ \text{k}\Omega\) and \(C=10\ \text{nF}\),

\[ f_0=\frac{1}{2\pi(10^4)(10^{-8})}\approx1.59\ \text{kHz}. \]

Signal-generator revision box

  • Sustained oscillation: loop magnitude 1 and phase \(0^\circ\) modulo \(360^\circ\).
  • Startup normally needs loop gain above 1; amplitude control brings it back.
  • Hartley = inductive divider; Colpitts = capacitive divider.
  • Crystal = highest \(Q\)/stability; LC = tunable; RC = lower-frequency convenience.
  • Relaxation oscillators switch between thresholds and make nonsinusoidal waves.