Circuit Response
every model has an edge, and the figure carries it
Concept

Barkhausen criterion — where it appears

The condition for a loop to sustain an oscillation: a loop gain of exactly one with zero phase at one frequency. Since no network holds a gain of exactly one, a real oscillator starts with more and relies on a nonlinearity to pull the gain back.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The closed-loop poles at a gain of 3.05. The locus of the two poles as the amplifier's gain runs from 2.7 to 3.3. It crosses the imaginary axis at a gain of 3.000000 — bisected on the netlist, not quoted — and at 3.05 the real part is 2.500e+2 radians a second, which is an envelope multiplying by 1.17015 every cycle. The crosses are the closed form ω₀(k−3)/2 and they sit on the measured circles.

The gain that is exactly one

An oscillator is designed by making the loop gain one at the frequency where the phase is zero. The gain at which this circuit's poles reach the imaginary axis is 3.000000000000, bisected on the netlist — an equality, not a range. A gain three per cent high multiplies the envelope by 1.0987 every cycle and reaches the rails in 61 milliseconds; three per cent low divides it by the same factor. A one per cent resistor cannot hold the condition, and neither can any other component.

applied · Oscillator The gain a Wien oscillator needs, when one of its four parts is off its value. computed by solving, not by drawing: for each part moved alone from −5% to +5%, the gain at which the poles reach the axis, bisected on the netlist, against 1 + Rs/Rp + Cp/Cs (to ten digits at every point). A series resistor or shunt capacitor 1% high raises the gain needed to 3.0100; a shunt resistor or series capacitor 1% high lowers it to 2.9901. Every part moves the frequency the same way, by half its error: 1% high on any one of them, 1.58 kHz against 1.59 kHz. The gain needed moves by the error itself, a third of it relative to three.

The gain a tolerance demands

A Wien oscillator needs a gain of exactly three only when its four network parts are exactly equal. In general it needs 1 + Rs/Rp + Cp/Cs — bisected on the netlist to ten digits — so a series resistor 1% high asks for 3.0100 and a shunt resistor 1% high for 2.9901. Four thousand seeded oscillators from 1% parts set to the textbook's comfortable 3.02 leave a quarter of them silent; 99.9% start only at 3.072, and at that setting the median unit distorts 2.6% — the limiter has to absorb a margin the tolerance forced. The frequency spreads by one part's own spread, 0.58% for 1% parts, as large as the limiter's own 0.771% shift. Parts from one array, tracking to 0.05%, need only 3.0035, because the gain depends on ratios and the frequency on products.

applied · Oscillator The gain a Wien oscillator needs above three, against its amplifier's gain–bandwidth product. computed by solving, not by drawing: the closed-loop gain at which the oscillating pair reaches the axis, bisected on the netlist with a one-pole amplifier of 100 dB direct-current gain, for equal 10 kΩ and 10 nF parts. At 10 times the frequency, 3.15339; at 30 times the frequency, 3.02349; at 100 times the frequency, 3.00257; at 1000 times the frequency, 3.00012. Every point agrees with 3(1 + δ) + 3ε²/(1 + δ + 3ε), where ε = kf₀/GBW and δ = k/a0, to two parts in ten million. The bandwidth's share falls as the square of the ratio; below it sits a floor of 9.00e-5 from the finite direct-current gain, and the two are equal near a ratio of √(3a0) = 548.

The gain an amplifier asks for, squared

A Wien oscillator built round a real amplifier needs more than a gain of three, and the excess is easy to overrate: the amplifier's lag moves the frequency in proportion to the lag and the required gain only in proportion to its square. At a gain–bandwidth product a hundred times the oscillation frequency the frequency is 4.22 per cent low and the gain needed is 3.00257 — fifty-one times smaller in proportion. The square is also what makes the bias spread: an amplifier 30 per cent slow asks for 1.94 times the typical excess. Against 1 per cent parts the amplifier matches the network's share at a ratio of 16; against a network matched to 0.05 per cent, not until 86.

applied · Oscillator A Wien oscillator with a lamp for its lower gain arm, τ = 100 ms: the envelope rings 63.6 times in ten time constants at 63.6 Hz and settles at 1.000 V. computed by solving, not by drawing: three state equations marched through every cycle of the 1.59 kHz carrier. The peak at the network's node, over its settled value (solid), from a millivolt with a cold filament, against time in units of the filament's time constant τ = 100 ms. Cold, the 50 Ω filament asks for a gain of 11 and the output sits on the 12 V rail until the filament has warmed. It leaves the rail after 1.99 ms, falls through the operating point, overshoots and rings at 63.6 Hz with a damping ratio of 0.0225. The dashed bound is the linear loop's decay, exp(−(1 + m)t/2τ) with m = 0.8, and it contains neither the carrier's frequency nor the amplitude. The settled amplitude, 1.000 V at the node and 3.00 V at the output, is the one at which the filament is at 250 Ω.

The lamp that has to be slow

A small lamp in a Wien oscillator's gain network adjusts the gain to whatever each unit needs, so the margin a limiter spends is no longer spent, and the price is an amplitude loop of its own. The envelope is an integrator and the filament is a lag, which makes a second-order loop that rings at the square root of 2ω₀m/τ and decays at (1 + m)/2τ. The filament's ripple at twice the carrier is the only distortion, 0.201/ω₀τ. So the slowness that lowers the distortion lengthens the settling in exact proportion: distortion times settling is 0.163 cycles at any time constant, and 0.01 per cent takes a second at 1.59 kHz and 82 seconds at 20 Hz.

applied · Oscillator

Named alongside it

The objects these essays reach for when they reach for this one.

Wien bridgeComponent toleranceDesign tradeoffLoop gainAmplitude stabilisationMonte carloTotal harmonic distortionDamping ratioGain–bandwidth productLimit cycleModel rangePoles

All concepts