The lamp that has to be slow

The gain a tolerance demands found that a Wien oscillator from 1 per cent parts must have its gain set to 3.072 for 999 units in a thousand to start, and the gain an amplifier asks for, squared added the amplifier’s share to the same margin. Both essays priced a margin that every unit carries for life, and both ended on the same alternative: put something in the gain network that adjusts the gain to whatever each unit needs, so that the margin is used at start-up and then stops being spent.

The classic thing to put there is a lamp. A small tungsten filament is a resistor whose value rises as it warms, and in the lower arm of the gain network it makes a loop: more amplitude, more heat, more resistance, less gain. The amplitude settles at the level where the filament has exactly the resistance the network asks for, and a limiter’s diodes are not needed at all. What the limiter charges for measured what those diodes cost in distortion, and it is a large number, because they are the only nonlinear thing in the loop and every harmonic is theirs. A filament is not nonlinear on the time scale of one cycle. It is a linear resistor whose value is being changed slowly.

The word slowly is the whole of what this essay measures. The arrangement replaces one limit, a waveform being bent, with a control loop that has dynamics of its own. Those dynamics turn out to be a resonance, and its two numbers are tied to the distortion by a law with no free parameter in it.

Three equations and no envelope #

The network is the one the earlier measurements used: a series arm of ten kilohms and ten nanofarads, a shunt arm of the same, running at 1.59 kHz. The amplifier here is ideal, because the question is the lamp’s and because the gain an amplifier asks for, squared has already said how small the amplifier’s share of the gain is. The upper gain resistor is 500 Ω, and the lower arm is a filament modelled by the simplest law that is still a filament: its resistance is 50 Ω cold and rises in proportion to its power,

$$R = R_0\left(1 + \frac{\hat P}{P_0}\right), \qquad \tau\,\frac{d\hat P}{dt} = \frac{v^2}{R} - \hat P,$$

with $\hat P$ the power after the filament’s thermal time constant $\tau$ has low-passed it. A real filament’s resistance is a power law in its temperature, and its time constant moves as it heats; a straight line is what the argument needs, and every result below is stated through the one number the law reduces to at the operating point.

The virtual earth makes the circuit simple in a useful way. The inverting input follows the non-inverting one, so the voltage across the filament is the network’s own node voltage. Two capacitors and one filament are three states, and the march steps those three states through every cycle of the carrier with fourth-order Runge–Kutta, thousands of cycles at a time. Nothing about an envelope, an average or a slowly varying amplitude goes into it. The envelope is read from the peaks afterwards, and an analysis then has to agree with it.

Where it settles #

The operating point needs no march. The network asks for a gain of three, the upper arm is fixed, and so the filament must settle at $R_f/(k - 1) = 250$ Ω. The filament’s law then gives its power, 2 mW, and the power gives the voltage: a peak of exactly 1.000 V at the node, 3.00 V at the output. The march arrives at that number to four figures, and no diode, rail or device parameter is in it. Those were what set the amplitude in the amplitude nothing linear predicts, where a linear model’s poles could say how fast the envelope grew and never where it stopped.

Getting there is less tidy. Cold, the filament is 50 Ω and the stage asks for a gain of eleven, so the poles are nowhere near the axis, the output hits the 12 V rail within a few cycles and stays there. For as long as it stays there the filament is being heated by far more than its settled power, and with a time constant of 100 ms the output comes off the rail after about 2 ms. By then the filament is on its way to more heat than the operating point wants, and the amplitude falls through its settled value and keeps going, reaching within 1 per cent of it only after 639 ms.

So a lamp oscillator’s switch-on is violent. The waveform that should be 3 V begins as a clipped 12 V wave, and the node reaches 5.55 times its settled peak. The filament that was meant to be gentle has a thermal mass, and a cold thermal mass is an open loop.

An integrator and a lag #

Why the amplitude rings at all takes two lines, and they are the essay’s main result. An oscillator’s envelope does not respond to a gain, it integrates one. With the poles near the axis, the logarithm of the amplitude changes at a rate set by the excess gain,

$$\frac{d\ln A}{dt} = \frac{\omega_0}{2}\,(k - 3),$$

which is the growth rate two exponentials, and where they meet measured, read in the small-excess limit. An excess of one part in a thousand does not move the amplitude by one part in a thousand and stop. It keeps moving it for as long as it is there. The filament then lags that integrator by its own time constant. Write $a$ for the fractional change in amplitude and $p$ for the fractional change in filament power, and let $m = 1 - R_0/R_{hot}$, which is how strongly the resistance follows the power at the operating point. Here it is 0.8. Linearising the two laws about the operating point gives

$$\tau s^2 + (1 + m)\,s + 2\omega_0 m = 0.$$

That is a second-order loop, an integrator and a lag, and it is the shape of every control loop whose plant is an integrator. It has a natural frequency and a damping ratio, and neither was designed.

The march is kicked off its operating point by about one per cent and the ring and decay are read from the peaks. At 100 ms the decay is 9.0 per second against the pair’s 9.0, and the ring is 400 rad/s against 400. At 3 ms, where the ring has only about four carrier cycles in each of its own and an envelope is barely a meaningful idea, they still agree to one per cent: 297 against 300, and 2,283 against 2,290. The kick has to be small for this. A kick of a tenth of the filament’s power puts the envelope twenty per cent out at 100 ms, and the ring comes back 6.6 per cent low, as a softening spring’s does. The linear loop is a small-signal statement and the march knows it.

The decay has no frequency in it #

Read the pair’s two roots apart and they say something odd. The decay rate is

$$\sigma = \frac{1 + m}{2\tau},$$

which contains the filament and not the oscillator. An amplitude disturbance dies at the same rate in seconds on a 20 Hz oscillator and a 20 kHz one that share a lamp. The ring frequency, $\sqrt{2\omega_0 m/\tau}$, does contain the carrier, so the faster oscillator rings faster and for more swings over the same decay. For this lamp at 100 ms the damping ratio is 0.0225: a Q of about 22 in a loop nobody meant to be a resonator.

There is a practical reason for that to matter, and it is the disturbance itself. Any change in anything the oscillator cares about, a step in the supply or in the load, or a frequency range switched, moves the gain the loop needs by a small amount, and the amplitude answers with a decaying wobble at sixty-odd hertz that lasts half a second. On an audio test oscillator that is the bounce a user sees on a level meter after changing frequency, and it is not a defect of any part. The best damping is not the one to build found settling exquisitely sensitive to damping near an optimum; this loop is nowhere near an optimum and cannot be taken there by choosing $\tau$, since slowing the filament reduces the damping as $1/\sqrt{\tau}$.

The distortion is a ripple #

A filament is not perfectly slow. Its power is $v^2/R$, and for a sine wave at the node that power has a large component at twice the carrier. The thermal mass low-passes it by a first-order factor, $1/(2\omega_0\tau)$ when that is small, and what gets through moves the resistance by a fraction $m$ of itself and the gain by twice that, since $k - 1 = R_f/R$ and the operating point sits at $k - 1 = 2$. A gain moving at $2\omega_0$ multiplying a carrier at $\omega_0$ produces a term at $3\omega_0$, and the loop then closes around it, so the Wien network’s response at three times its centre decides how much of it stands at the output. That response is exact: $|1 - 3\beta(3j\omega_0)| = (8/3)/|3 + j8/3|$, or 0.664.

Put together, the third harmonic over the fundamental is $0.2509\,m/\omega_0\tau$, which for this lamp is $0.201/\omega_0\tau$. Measured on the marched output with a window at the oscillator’s own measured period, it is 0.671 per cent at $\omega_0\tau = 30$ against 0.669, 0.0669 at 300 against 0.0669 and 0.00669 at 3,000 against 0.00669. Three significant figures over three decades is a good deal better than an analysis this short has a right to expect, and the reason is that the filament really is doing what the analysis says.

The contrast with the limiter is in kind and not only in size. Two diodes across the feedback resistor make harmonics by bending the wave, and their level depends on how hard they are working, which is the excess gain. The filament bends nothing. Over one cycle the circuit is linear with a time-varying coefficient, and the harmonic is a modulation product. That is also why there is no fifth harmonic to speak of: it would need the ripple at $4\omega_0$, which is a product of a product.

The time constant cancels #

Now the two results can be multiplied. The envelope settles within one per cent in $\ln(100)/\sigma = 9.21\tau/(1 + m)$. The distortion is $0.2509\,m/\omega_0\tau$. In the product $\tau$ disappears, and if settling is counted in carrier cycles rather than seconds $\omega_0$ disappears too:

$$D_3 \times N_{settle} = \frac{0.2509 \times 9.21}{2\pi}\,\frac{m}{1 + m} = 0.368\,\frac{m}{1 + m}.$$

For this lamp that is 0.163 of a cycle. The figure is a designer’s whole choice with a lamp in it, drawn on one line.

A designer can have 0.01 per cent of distortion, and then the amplitude takes 1,634 cycles to settle. At 1.59 kHz that is 1.03 seconds, which is tolerable. At 20 Hz it is 82 seconds, which is what the slow-settling low ranges of real audio generators are, and now there is a reason to expect it rather than a complaint to make. The figure’s dots lie on the product within a few per cent across the whole range where the ripple is small, 0.163 at $\omega_0\tau$ of 100 and of 3,000 alike. Choosing the lamp barely helps: running the filament cooler, $m = 0.5$, brings the product down only to 0.123, and it weakens the control the filament has over the amplitude to do it.

So a thermally slow element in a gain network is not a free lunch with a long delay in it. It is a fixed exchange rate. The rate is low enough that the bargain is excellent at audio and above, and this is the reason a lamp in a Wien bridge became the standard circuit for a low-distortion source, which two diodes never could.

The margin that stops being spent #

The argument for the lamp was the margin, and that has to be checked as well. A Wien network whose series resistor is off nominal asks for a gain of $3 + e$, and the gain a tolerance demands showed what a fixed gain then does: each unit runs at its own excess, and the limiter’s working point differs in every one.

With the lamp, each unit simply finds a different filament resistance, $R_f/(k - 1)$, and the amplitude follows from it. Across ±0.1 of required gain, well outside anything 1 per cent parts produce, the marched output moves 4.69 per cent end to end, matching the operating point to half a per cent at every setting. The same range with the gain fixed at 3.15 and two diodes doing the limiting moves the amplitude 17.3 per cent and the distortion from 1.93 to 6.76 per cent, because the diodes are being asked to absorb an excess that ranges from 0.05 to 0.25.

The lamp’s own distortion barely notices. It depends on the unit only through $m$, which moves from 0.800 by a little at either end, so its third harmonic changes by 2.5 per cent over the whole range. So the distortion stops depending on the tolerance at all, which is the thing no choice of fixed margin could achieve: the gain a tolerance demands could only set the margin’s size and take the result.

What a smaller excess would have cost #

A fair comparison asks what the limiter could do if it were given the same spread with less excess, and what the limiter charges for already answers it: the distortion falls as the 0.74 power of the excess gain, so a fixed gain close to three is quiet in a typical unit and fails to start in an unlucky one. The lamp breaks that coupling, since its start-up margin is the cold filament’s gain of eleven, which no tolerance threatens, while its running distortion is set elsewhere. Start-up certainty and running purity were one number with diodes. With the lamp they are two, and the second has moved to a different axis, which is time.

What the filament’s law left out #

Three simplifications are worth stating, because each one moves a number here and none moves the shape. The filament’s resistance is taken as linear in its power, where a real tungsten filament runs closer to a low power law, and the time constant is taken as fixed where a real filament’s falls as it gets hot and radiates more. Both change the value of $m$ and of $\tau$ at the operating point without changing the structure: an integrator, a lag and a ripple. Since every law above is stated in those two numbers, a measured lamp can be put straight into them.

The amplifier is ideal. The frequency that is not the formula showed that a real one moves the frequency by several per cent and the required gain by a much smaller amount, and here the required gain is simply what the filament settles at, so its share appears as a small change of operating point rather than a spent margin. And the rail is a hard clip, which decides how the switch-on looks and how long it spends on the rail but not the loop’s behaviour once it is off it. Real switch-on transients are not gentler; an amplifier recovering from saturation adds its own delay.

The thermal model has one state where a real filament has several: the filament and its supports cool with different time constants, and a thermistor bead has its own distribution. A second thermal state would add a second lag to a loop already short of damping, which only makes the ring worse. How much worse is not computed here, and it is the one result that could change the qualitative picture: with enough lag in the filament’s thermal path, the amplitude loop itself would go unstable, and the oscillator would squeg.

The number worth carrying #

A lamp in a Wien oscillator’s gain network turns the margin problem into a control loop: the envelope integrates, the filament lags, and they ring at $\sqrt{2\omega_0 m/\tau}$ while decaying at $(1 + m)/2\tau$. The filament’s ripple at twice the carrier is the distortion, $0.2509\,m/\omega_0\tau$. Multiply the two and the time constant cancels: distortion times settling is 0.163 of a cycle for a filament running at 250 Ω, and 0.01 per cent takes 1,634 cycles to settle. That is one second at 1.59 kHz and eighty-two at 20 Hz. The amplitude moves under 5 per cent across any network the parts can make. All of it agrees with a march of every carrier cycle, and no number in the law was fitted.

Still open: the second thermal state, a detector instead of a filament, and the loop that can be compensated #

A filament with two thermal paths. A second, slower thermal state adds lag inside a loop whose damping is already 0.02. A real lamp’s supports and envelope supply one, and the point at which the extra lag makes the amplitude ring forever, so that the oscillator squegs, is a stability boundary this model cannot reach. It is the margin question reappearing in the loop that was meant to remove it.

A peak detector and a field-effect transistor. Replace the filament with a rectifier, a capacitor and a channel resistance, and the lag and the ripple are separately chosen for the first time. The ripple is then at twice the carrier through the detector’s capacitor, and a second time constant can filter it without slowing the loop’s main lag so much. Whether that breaks the fixed exchange rate above, or only moves it, is the measurement that would say what the electronic versions of this circuit actually buy.

A zero in the amplitude loop. A control loop with an integrator and a lag is ordinarily cured with a lead. A small proportional path around the filament, a resistor in parallel with it, adds a zero and changes $m$ at the same time. How much damping a parallel resistor can buy, and what it costs in the amplitude’s regulation against the required gain, is a one-parameter march on this circuit.