Signal generator

An oscillator is a circuit that produces a continuous periodic signal (sine, square, triangular) without any input signal, by feeding part of its output back to its input in phase. Oscillators provide clock signals in computers, carrier signals in radio transmitters, and test signals in labs. This topic covers the Barkhausen criterion, the main RC, LC and crystal oscillator circuits, and non-sinusoidal waveform generators such as multivibrators and the 555 timer.

Key concepts

• Basic principle: An amplifier of gain $A$ with a feedback network of gain $\beta$ will sustain oscillations when the feedback is positive (regenerative). The energy lost in each cycle is made up by the amplifier, which takes power from the DC supply. An oscillator therefore converts DC energy into AC energy.
• Barkhausen criterion: For sustained oscillations, (1) the loop gain magnitude $|A\beta| = 1$, and (2) the total phase shift around the loop is 0° or 360°. In practice $|A\beta|$ is made slightly greater than 1 at start-up so oscillations build up from noise, and non-linearity then limits the amplitude.
• Types of oscillators:

Type	Frequency range	Frequency set by	Examples
RC	audio (Hz to ~1 MHz)	resistors and capacitors	RC phase shift, Wien bridge
LC	radio frequency (100 kHz to hundreds of MHz)	tank circuit L and C	Hartley, Colpitts, Clapp
Crystal	very stable fixed frequency	quartz crystal	Pierce, crystal-controlled Colpitts

• RC phase-shift oscillator: A CE amplifier (180°) plus three RC sections giving 60° each (180° total). The amplifier gain must be at least 29.
• Wien bridge oscillator: Uses a lead-lag (series RC and parallel RC) network that gives 0° phase shift at $f = 1/(2\pi RC)$, with feedback fraction 1/3. A non-inverting amplifier gain of 3 is needed. Widely used for audio-frequency generators because it is easily tuned.
• LC tank circuit: Energy oscillates between the magnetic field of L and the electric field of C at $f = 1/(2\pi\sqrt{LC})$; losses are made up by the amplifier.
• Hartley oscillator: Tank uses a tapped inductor (two inductors $L_1$, $L_2$) and one capacitor.
• Colpitts oscillator: Tank uses a split capacitor ($C_1$, $C_2$) and one inductor. The Clapp oscillator adds a small capacitor in series with L for better frequency stability.
• Crystal oscillator: Quartz shows the piezoelectric effect — mechanical stress produces voltage and vice versa. Its electrical equivalent is a series RLC branch in parallel with a holder capacitance, giving a series resonant frequency $f_s$ and a slightly higher parallel resonant frequency $f_p$. Very high Q (10⁴ to 10⁶) gives excellent frequency stability; used in watches (32.768 kHz) and microcontroller clocks.
• Waveform generators (non-sinusoidal):
  • Astable multivibrator: no stable state, free-running square wave generator.
  • Monostable multivibrator: one stable state; produces one pulse of fixed width when triggered.
  • Bistable multivibrator: two stable states; this is a flip-flop.
  • Schmitt trigger: comparator with hysteresis, converts slowly varying or noisy signals to clean square waves.
  • Triangular wave: produced by integrating a square wave (op-amp integrator); a function generator combines a Schmitt trigger and integrator, then shapes the triangle into a sine.
  • 555 timer: popular IC used as astable or monostable.

High-yield points

• Barkhausen: $|A\beta| = 1$ and loop phase shift 0° (360°).
• RC phase shift: gain ≥ 29, $f = 1/(2\pi RC\sqrt6)$.
• Wien bridge: gain = 3, feedback 1/3, $f = 1/(2\pi RC)$.
• Hartley = tapped inductor; Colpitts = split capacitor.
• Crystal: piezoelectric, very high Q, parallel resonance $f_p$ slightly above series $f_s$.
• 555 astable: $f = 1.44/[(R_A+2R_B)C]$; monostable $T = 1.1RC$.

Formulas

$f_{\text{RC phase shift}} = \frac{1}{2\pi RC\sqrt{6}}$

$f_{\text{Wien}} = \frac{1}{2\pi RC}$

$f_{\text{Hartley}} = \frac{1}{2\pi\sqrt{(L_1+L_2+2M)C}}, \qquad f_{\text{Colpitts}} = \frac{1}{2\pi\sqrt{L C_{eq}}},\ C_{eq} = \frac{C_1C_2}{C_1+C_2}$ $R$ in Ω, $C$ in F, $L$ in H, $M$ = mutual inductance (H).

555 astable: $f = \dfrac{1.44}{(R_A + 2R_B)C}$; monostable pulse width $T = 1.1RC$.

Worked example

A Wien bridge oscillator uses $R = 10$ kΩ and $C = 10$ nF. Find the frequency.

1. $RC = 10^4 \times 10^{-8} = 10^{-4}$ s.
2. $f = 1/(2\pi\times10^{-4}) = 1/(6.283\times10^{-4}) \approx 1592$ Hz ≈ 1.59 kHz.
3. Required amplifier gain = 3, e.g. $1 + R_f/R_1 = 3$ with $R_f = 2R_1$.

Exam tips

• Oscillators use positive feedback; amplifiers use negative feedback for stability.
• Minimum gain: RC phase shift = 29; Wien bridge = 3.
• Hartley = tapped inductor; Colpitts = tapped capacitor (remember "C for Colpitts, C for capacitor").
• Crystal oscillators are chosen for frequency stability, not tunability.
• A bistable multivibrator is a flip-flop; an astable has no stable state.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. A Wien-bridge oscillator uses $R = 10$ kΩ and $C = 10$ nF in both arms of the frequency-selective network. Its frequency of oscillation is about:

• A. 6.5 kHz
• B. 1.59 kHz
• C. 159 Hz
• D. 10 kHz

Answer: B. $f = \frac{1}{2\pi RC} = \frac{1}{2\pi \times 10^4 \times 10^{-8}} \approx 1.59$ kHz.

2. In a transistor RC phase-shift oscillator with three identical RC sections, each section ideally contributes a phase shift of:

• A. 90°
• B. 120°
• C. 180°
• D. 60°

Answer: D. The amplifier gives 180°; the three RC sections must together give 180°, i.e. 60° each (for identical sections).

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