Basic Electrical & Electronics Engineering — Signal Generator, NEC licence examination syllabus (Nepal Engineering Council).
Oscillators: an amplifier that feeds itself
No input signal at all, yet a clean sine wave comes out. Where does it come from?
This looks like it shouldn't work. An amplifier needs an input to amplify. But take an amplifier and route part of its output back to its own input, in the right phase. It will then produce a continuous signal from nothing. The energy comes from the DC supply. The starting kick comes from the tiny random noise present in every circuit. The condition that decides whether it oscillates or just sits there is one of the most quoted results in electronics.
The feedback loop
A_f = A / (1 − Aβ) With beta = 0.01, A = 100, Af = 200, Abeta = 0.5.
Raise β until the loop gain Aβ approaches 1 and watch the closed-loop gain run away. That is the Barkhausen condition: at Aβ = 1 the amplifier sustains an output with no input, which is an oscillator rather than a broken amplifier.
With feedback, the closed-loop gain is
A_f = A / (1 − Aβ)
A = open-loop amplifier gain
β = feedback fraction (how much output returns)
Aβ = LOOP GAIN
Now look at the denominator. If Aβ → 1, then A_f → ∞:
finite output from zero input. That is oscillation.
The Barkhausen criterion
Sustained oscillation requires BOTH conditions:
1. MAGNITUDE: |Aβ| = 1
(loop gain exactly unity)
2. PHASE: ∠Aβ = 0° or 360°
(total phase shift around the loop is a whole cycle,
i.e. the feedback is POSITIVE / regenerative)
Frequency of oscillation = the one frequency at which the
phase condition is satisfied.
Both conditions are needed, and exams love testing whether you know that. Satisfy the phase condition alone and you get nothing. Satisfy the magnitude condition alone and you get nothing. There is a practical subtlety worth knowing. Designers deliberately set |Aβ| slightly above 1, so that oscillation starts reliably every time. The amplifier's own non-linearity (or an automatic gain control) then pulls the gain back to exactly 1. Without that, the amplitude would keep growing until the wave clipped into a square.
🎯 Where the first signal comes from
Thermal noiseEvery resistor generates tiny random voltage noise containing all frequencies. There is always something to amplify.
Selective growthThe feedback network satisfies the phase condition at only one frequency. That component gets amplified round the loop again and again; every other frequency is attenuated and dies.
SettlingAmplitude grows until the amplifier begins to saturate, which reduces its effective gain until |Aβ| = 1 exactly. The oscillation then holds steady.
Classifying oscillators
By waveform:
Sinusoidal — RC, LC, crystal
Relaxation — square/triangular/sawtooth (multivibrators)
By frequency range (choose the right technology):
RC oscillators few Hz → ~1 MHz (audio)
LC oscillators ~100 kHz → ~500 MHz (radio)
Crystal oscillators 10 kHz → ~200 MHz (precision)
Worked numerical 1 — checking the Barkhausen condition
An amplifier has a gain of 60. The feedback network returns 1/50 of the output, in phase. Will it oscillate?
Loop gain: Aβ = 60 × (1/50) = 1.2
Magnitude: |Aβ| = 1.2 > 1 ✔ (will start)
Phase: in phase = 0° ✔ (satisfied)
→ It WILL oscillate. Amplitude grows until the amplifier
saturates and effective gain drops to 50, giving Aβ = 1.
Minimum gain needed for oscillation:
A_min = 1/β = 1/(1/50) = 50
With A = 60 there is 20% excess gain — enough to guarantee
reliable starting without severe distortion.
Worked numerical 2 — the gain requirement
A feedback network attenuates the signal to 1/29 of its input and provides the correct phase. What amplifier gain is needed, and what happens with a gain of 25?
β = 1/29
For oscillation: |Aβ| ≥ 1
A ≥ 1/β = 29
With A = 25:
Aβ = 25/29 = 0.862 < 1
→ oscillation DIES OUT. The circuit produces nothing.
With A = 29: Aβ = 1.0 → marginal, may not start reliably
With A = 35: Aβ = 1.21 → starts reliably ✔
Note: β = 1/29 is exactly the Wien bridge oscillator's
feedback fraction, which is why a Wien bridge needs a
minimum gain of 3 in its amplifier stage (see next topic —
the 1/29 here is for the RC phase-shift type).
Worked numerical 3 — positive vs negative feedback
An amplifier with A = 100 has β = 0.01 feedback. Compare the closed-loop gain for positive and negative feedback.
NEGATIVE feedback (signal subtracted, phase 180°):
A_f = A/(1 + Aβ) = 100/(1 + 100 × 0.01)
= 100/2 = 50
→ gain HALVED, but bandwidth doubles, distortion drops,
stability improves. Used in amplifiers.
POSITIVE feedback (signal added, phase 0°):
A_f = A/(1 − Aβ) = 100/(1 − 1.0)
= 100/0 = ∞
→ OSCILLATION. Used in oscillators.
Same circuit topology, opposite phase, opposite purpose.
An amplifier designer fears Aβ = 1; an oscillator designer
aims for it.
💡 Exam angle: the Barkhausen criterion is a guaranteed 4–5 mark question. Write both conditions explicitly (|Aβ| = 1 AND phase = 0°/360°), draw the feedback block diagram, and explain that noise provides the starting signal while the phase condition selects the frequency. Adding the practical point about |Aβ| slightly exceeding 1 for reliable starting often earns the final mark.
Syllabus points
Barkhausen criterion
Feedback oscillator concept
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