Basic Electrical & Electronics Engineering — Signal Generator, NEC licence examination syllabus (Nepal Engineering Council).
RC Oscillators: making sine waves without a single coil
Resistors and capacitors only — which is why every audio oscillator uses them.
Inductors are bulky, expensive, and at audio frequencies they'd have to be enormous. So for low frequencies we build the frequency-selecting network from resistors and capacitors instead. Here is the challenge. A single RC stage can shift the phase by at most 90°, but we need a specific total. So these circuits are all about stacking or balancing RC sections cleverly.
RC Phase-Shift Oscillator
A common-emitter amplifier already inverts, giving 180°. To satisfy Barkhausen we need another 180° from the feedback network. One RC section can approach but never reach 90°, so we cascade three sections at 60° each.
Frequency of oscillation (three equal RC sections):
f = 1/(2πRC√6) ≈ 1/(15.39 RC)
Feedback fraction at that frequency: β = 1/29
Minimum amplifier gain: A ≥ 29
Phase budget:
amplifier (CE) 180°
3 × RC sections 3 × 60° = 180°
TOTAL 360° ✔ Barkhausen satisfied
Wien Bridge Oscillator
A cleverer approach. Instead of building up 180°, use a network whose phase shift is zero at exactly one frequency, paired with a non-inverting amplifier (also 0°). Total: 0° — Barkhausen satisfied with far less attenuation.
Frequency of oscillation:
f = 1/(2πRC) ← much simpler than the √6 version
Feedback fraction: β = 1/3
Minimum gain: A ≥ 3
For an op-amp non-inverting stage, A = 1 + R_f/R₁ = 3
→ R_f = 2R₁ (the classic Wien bridge ratio)
Compare the two. The phase-shift oscillator needs a gain of 29; the Wien bridge needs only 3. That's because the Wien network throws away far less signal. Lower required gain means the amplifier operates more linearly, so the Wien bridge produces a much purer sine wave — which is why it's the standard for audio test oscillators.
⚔️ Phase-shift vs Wien bridge
Phase shiftf = 1/(2πRC√6). β = 1/29, needs A ≥ 29. Simple to build with one transistor. Higher distortion. Awkward to tune (three RC pairs must change together).
Wien bridgef = 1/(2πRC). β = 1/3, needs A ≥ 3. Very low distortion, easily tuned with a dual-gang potentiometer. Needs amplitude stabilisation — classically a small incandescent lamp in the feedback path, whose resistance rises as it warms, automatically holding Aβ at 1.
Worked numerical 1 — phase-shift oscillator design
Design an RC phase-shift oscillator for 1 kHz using C = 10 nF. Find R and the minimum gain.
Design a Wien bridge oscillator for 2 kHz using R = 10 kΩ. Find C, and the feedback resistors for an op-amp version.
f = 1/(2πRC)
C = 1/(2πfR) = 1/(2π × 2000 × 10 000)
= 1/(1.2566 × 10⁸)
= 7.96 × 10⁻⁹ F ≈ 7.96 nF → use 8.2 nF standard
With C = 8.2 nF the actual frequency becomes:
f = 1/(2π × 10⁴ × 8.2 × 10⁻⁹) = 1941 Hz
(3% low — acceptable, or trim R to 9.7 kΩ)
Amplifier gain requirement: A = 3
A = 1 + R_f/R₁ = 3 → R_f/R₁ = 2
Choose R₁ = 10 kΩ → R_f = 20 kΩ
Practical note: make R_f slightly larger (say 22 kΩ) so
A = 3.2 and oscillation starts reliably, then use a lamp
or a diode limiter to pull the amplitude back.
Worked numerical 3 — comparing the two at the same frequency
Both oscillators are to run at 5 kHz with C = 4.7 nF. Compare the required R and gain.
PHASE-SHIFT:
R = 1/(2πfC√6) = 1/(2π × 5000 × 4.7n × 2.449)
= 1/(3.617 × 10⁻⁴) = 2765 Ω ≈ 2.77 kΩ
Gain needed: 29
WIEN BRIDGE:
R = 1/(2πfC) = 1/(2π × 5000 × 4.7 × 10⁻⁹)
= 1/(1.4765 × 10⁻⁴) = 6773 Ω ≈ 6.77 kΩ
Gain needed: 3
Summary:
R needed Gain needed
Phase-shift 2.77 kΩ 29
Wien bridge 6.77 kΩ 3
The Wien uses larger resistors (√6 = 2.449× larger) but
needs about a tenth of the gain.
💡 Exam angle: numericals here are almost always "find f given R and C" or the reverse. Memorise both formulas and note the difference is just the √6 factor. The comparison marks come from stating β and the minimum gain (1/29 & 29 versus 1/3 & 3) and explaining that the Wien bridge's lower gain requirement gives lower distortion. Mentioning the lamp for amplitude stabilisation is a classic bonus point.
Syllabus points
RC phase-shift & Wien bridge (frequency numerical)
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