Feedback and Oscillators
One circuit's bug is another's specification. The Barkhausen criterion is a failure in an amplifier and a design target in an oscillator.
Skip to the animationNegative feedback divides the gain by (1 + Aβ) and improves bandwidth, linearity and consistency by the same factor — but if accumulated phase shift turns the subtraction into an addition while the loop gain still exceeds one, the same circuit oscillates, which is the Barkhausen criterion read as a fault or as a specification.
What negative feedback buys
| Quantity | Effect of feedback |
|---|---|
| Gain | Divided by (1 + Aβ) |
| Bandwidth | Multiplied by (1 + Aβ) |
| Distortion | Divided by (1 + Aβ) |
| Sensitivity to device spread | Divided by (1 + Aβ) |
| Output impedance (voltage feedback) | Divided by (1 + Aβ) |
| Input impedance (series feedback) | Multiplied by (1 + Aβ) |
Transistor gain is cheap and stability is not, so this is almost always a good trade. The gain-bandwidth product is conserved, which makes it a choice about where to spend it.
How it turns into oscillation
- 1Negative feedback works because the returned signal arrives in antiphase and subtracts.
- 2Every amplifier stage adds phase shift, and it grows with frequency.
- 3At 180° of extra shift, the subtraction has become an addition — nothing in the circuit was changed to cause it.
- 4If the loop gain is still at least one at that frequency, the circuit sustains a signal with no input.
That is the Barkhausen criterion: |Aβ| = 1 with a total phase shift of 0° or a multiple of 360°. In an amplifier it is a failure; in an oscillator it is the design target.
Compensation
Stability requires the loop gain to fall below one before the phase reaches 180°. The margin between the two is the phase margin, and 45° or more is the usual target.
Internal compensation adds a dominant pole to guarantee it, which is precisely why an op-amp's bandwidth is far below its transistors' capability. The analysis is Bode's, unchanged from control systems, because it is the same problem.
Oscillator topologies
| Type | Frequency-setting network | Typical range | Notes |
|---|---|---|---|
| RC phase-shift | Three RC sections, 60° each | Audio | Simple, moderate stability |
| Wien bridge | Series and parallel RC | Audio | Very low distortion with AGC |
| Colpitts | LC tank, capacitive divider | RF | The common RF choice |
| Hartley | LC tank, tapped inductor | RF | Tapped coil is harder to make |
| Crystal | Quartz mechanical resonance | Any | Q of tens of thousands; ppm stability |
The network decides the frequency and its stability; the amplifier merely supplies enough gain to sustain it. So oscillator design is mostly about the selectivity of that network.
Amplitude control
Loop gain must exceed one for oscillation to start from noise. If it stays above one, the amplitude grows until the rails clip it — and a clipped sine is rich in harmonics.
Practical oscillators reduce the gain to exactly one once running. The classic Wien bridge used a small filament lamp whose resistance rises as it warms; modern designs use a thermistor, a JFET as a variable resistor, or an explicit AGC loop.
Two faults, one method
An amplifier that oscillates and an oscillator that will not start are the same problem read from opposite ends. Both are diagnosed by finding where loop gain and phase cross their critical values.
Most unintended oscillation arrives through the supply rail, which closes a feedback path nobody drew on the schematic. That is what decoupling capacitors are for, and why they belong physically close to the device.
The numbers you will be asked for
- Closed-loop gain
A_f = A / (1 + Aβ)
- Feedback factor
1 + Aβ
the improvement factor for everything else
- Barkhausen criterion
|Aβ| = 1 and ∠Aβ = 0° (mod 360°)
- Wien bridge frequency
f = 1 / (2πRC)
and gain must be exactly 3
- RC phase-shift frequency
f = 1 / (2πRC√6)
and gain must exceed 29
- LC tank frequency
f = 1 / (2π√(LC))
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.