Two exponentials, and where they meet

Four essays on this circuit have measured four properties of its settled state. The poles either side of the crossing, the amplitude no linear model contains, the distortion the limiter charges for and the frequency that is not the design equation’s are all answers about an oscillator that has already been running for some time.

Every one of those answers came out of a march that started at a millivolt on one capacitor and took between forty-three and a hundred and forty-five cycles to arrive. The march recorded the peak of every one of those cycles, and nothing has ever read them.

That record is the only part of this circuit a linear model gets exactly right, which makes it the part worth checking a closed form against rather than an expectation. And the amplitude the record ends at belongs to a device parameter nothing has ever varied: the saturation current of the diodes that stop the growth. Both halves are exponentials — one in time, one in voltage — and the whole behaviour of the circuit is where they meet.

What the growth rate is, exactly #

The characteristic equation of the ideal loop is $s^2 + s(3-k)/RC + 1/(RC)^2 = 0$, so the pole pair sits at

σ = ω₀(k − 3)/2 and ω_d = ω₀√(1 − ((k − 3)/2)²), with ω₀ = 1/RC.

The netlist returns those numbers rather than being told them. At a gain of 3.20 the measured real part is 999.9999999999985 against a closed form of 1000.0000000000009 s⁻¹, and the damped frequency is 9949.874371066198 against 9949.8743710662. Both are read off a pole extraction on the assembled matrix, in the way where the behaviour is written down sets out, and neither is used to produce the other.

The envelope is therefore multiplied each cycle by $e^{2\pi\sigma/\omega_d}$, which is

exp( π(k − 3) / √(1 − ((k − 3)/2)²) ).

The denominator is the part that is usually dropped, and dropping it is exact in the limit of a small excess gain. It is not exact anywhere else.

The reproduction, which is the fit #

A growth rate taken off a march is not a measurement of anything until the stretch it is fitted over is one where the linear model applies. The diodes are open only while the output is small, so the fit is made on the cycles below eight per cent of the settled amplitude and on no others.

Where there are enough of those, the agreement is close and it is one-sided in a way worth noting. At a gain of 3.01 the fitted growth is low by 0.001 per cent against the poles; at 3.05 by 0.003; at 3.1 by 0.005; at 3.2 by 0.011. The march is always slightly slower than the linear model, and it is slightly slower by more as the gain rises, because a larger excess gain means the diodes begin to take a bite out of the top of the envelope earlier in the growth. That is not error; it is the thing being measured arriving before the measurement is finished.

The bisection on the netlist that the first essay in this ladder reported still returns 3.00000000000000 for the gain at which the real part changes sign, so the two ends of the range are both anchored on the same machinery.

The range that closes as the error opens #

The number of small cycles available to fit is not a detail of the method. It is the reciprocal of the quantity being measured, and it collapses:

The approximation is worst exactly where the circuit can no longer be asked whether it is right. At 3.8 the envelope goes from four and a half millivolts to a volt in two cycles; there is no small-signal stretch to fit, so the marched envelope has nothing to say about a form that is a fifth low there. The claim has to be carried by the closed form and the pole extraction, and the fit is what entitles those to be trusted at the other end of the range.

That is the same shape as the calibration in the assumption that is a geometry, arriving from the other direction: there the exact case was the geometry nobody builds, and here it is the gain nobody designs at, because a third of a per cent above three is not a gain any pair of resistors holds.

What the denominator is #

The correction that is usually dropped has a physical object behind it, and the object is the oscillation frequency rather than the growth.

A cycle is $2\pi/\omega_d$ long, and $\omega_d$ is not $\omega_0$. It falls as the gain rises, because the pole pair moves out along a locus of constant radius: the real part grows and the imaginary part must shrink to keep the product of the roots equal to $1/(RC)^2$. Measured on the netlist against an $\omega_0$ of exactly 10,000 radians a second, the damped frequency is 9999.87 at a gain of 3.01, 9996.87 at 3.05, 9949.87 at 3.20, 9797.96 at 3.40 and 9165.15 at 3.80.

So at a gain of 3.8 each cycle lasts 9.11 per cent longer than a cycle at the natural frequency, and the envelope grows at its own rate for the whole of that time. The growth per cycle is therefore larger than $e^{\pi(k-3)}$ by exactly the ratio $\omega_0/\omega_d$ in the exponent, which is where the square root comes from and why the error is one-sided: the dropped form is never high.

That is also the reason the growth law and the frequency that is not the design equation’s are the same measurement read two ways. The frequency essay found the oscillation running below $1/2\pi RC$ by amounts of a per cent or less, and attributed them to the amplifier and the limiter. The pole pair’s own contribution is separate from both and is much larger — half a per cent at a gain of 3.2 and 8.35 per cent at 3.8 — and it does not appear in the settled frequency at all, because by the time the circuit has settled the diodes have brought the effective gain back to three and the poles back to the axis. It is visible only during the growth, which is the stretch nothing was reading.

How many cycles it takes, and when that stops being predictable #

The march starts with a millivolt on one capacitor, and the first cycle’s envelope is not a millivolt: it is 4.35 millivolts at a gain of 3.01, rising to 46.2 at 3.8, because the amplifier’s own gain stands between the capacitor and the output and because a larger excess gain has already done some growing inside the first cycle.

Taken from that first recorded peak, the number of cycles needed to reach nine tenths of the settled amplitude should be the logarithm of the ratio divided by the logarithm of the growth. It is, and then it stops being:

The excess is the diodes, and it is the same excess the fit above measures from the other side. A purely exponential envelope would arrive early; the real one spends its last stretch being held back by a limiter that has already started conducting, so it arrives late, and by more as the two exponentials overlap more. At a gain of 3.8 the linear description of the start-up is out by four fifths — which is not a small correction to a growth law, it is the statement that there is no growth law there at all.

The third thing the gain decides #

The reciprocal of the growth rate is a time, and it is a specification nobody writes down for an oscillator that is switched on and off.

From a millivolt on one capacitor, the output reaches nine tenths of its settled amplitude in 99.3 milliseconds at a gain of 3.01 and in 1.31 milliseconds at 3.8 — a factor of 76.1 across a slider over which the distortion changes by 29.9 and the amplitude by 1.58. So there are three quantities on the one knob, and the two that a designer usually weighs against each other are the two that move least.

An oscillator specified for a tenth of a per cent of distortion is being specified, without anybody saying so, for something close to a tenth of a second of settling. That is a long time in an instrument that switches its source between two frequencies, and it is the reason a low-distortion generator has a limiter that is not two diodes — a thermistor, a lamp, or a control loop with its own bandwidth, none of which this circuit has and all of which trade the same account differently.

The other exponential #

Everything above is a property of three resistors and two capacitors. The amplitude is not; it is a property of the diodes, and until now nothing has passed either of the two numbers that describe them.

A junction’s law is $I = I_S(e^{V/\eta V_T} - 1)$, and the constant everybody is taught is not a constant: the forward voltage moves 59.5 millivolts for every decade of current at an ideality of one, which is $\eta V_T \ln 10$ at 25.852 millivolts of thermal voltage.

The oscillator’s amplitude follows the same logarithm and not at the same rate.

The slope is 1.822 thermal voltages a decade rather than one, and the factor is not a fudge. The amplitude settles where the gain averaged over a cycle is exactly three, so lowering the diode’s turn-on by one $\eta V_T \ln 10$ does not simply lower the peak by that much: the conduction has to be re-established over a fraction of a cycle rather than at a point, and the peak has to fall further to do it. The 1.822 is what that costs and it is a measurement rather than a derivation. Its constancy across the ideality is the evidence that the mechanism is the junction’s exponential and nothing else, because a factor that survived a doubling of $\eta$ unchanged is a property of the shape rather than of the scale.

Two knobs that are very nearly separate #

The essay that priced the limiter measured amplitude and distortion against one another along the gain axis and found them tightly coupled: amplitude as the 0.11 power of the excess gain, distortion as the 0.74 power. What it could not do is ask the same question of the diode, because the parameter did not reach the march.

That is the design result and it is worth stating plainly. Over six decades of saturation current the amplitude runs from 963.75 millivolts to 421.4 — a factor of 2.29 — while the distortion runs from 5.9064 per cent to 4.8635, a factor of 1.21. Over the gain slider the amplitude runs from 703 millivolts to 1.061 volts, a factor of 1.51, while the distortion runs from 0.837 per cent to 12.842, a factor of 15.3.

So the two parameters are close to orthogonal in the plane that matters. A larger output at the same purity is bought by choosing a diode with a smaller saturation current, and it is nearly free. A purer output at the same amplitude is bought by lowering the gain and raising the limiter resistance together, and it is paid for in start-up time. Nothing in the standard account of this circuit separates them, because the standard account has one knob in it.

Where the limiter stops it starting #

There is a limit on the diode axis and it is not a soft one.

A junction has a zero-bias slope of $\eta V_T/I_S$, so a leaky enough diode is a resistor across the feedback network before there is any signal to limit. The two antiparallel branches each carry that slope in series with the limiter resistance, so they shunt the feedback resistor with $(R_\text{lim} + \eta V_T/I_S)/2$, and the oscillation ceases where that shunt brings the closed-loop gain down to exactly three — where the shunted feedback resistance is twice the ground-leg resistor.

That condition is arithmetic on a small-signal model and it contains no march: at a gain of 3.20 with ten kilohms of limiter resistance it puts the boundary at a saturation current of 60.121 nanoamperes. Bisecting the marched envelope’s growth over its first twenty cycles puts it at 60.041 nanoamperes. The two routes share the netlist and nothing else — one of them never integrates anything and the other never linearises anything — and they agree to 0.13 per cent.

Approaching that boundary the circuit gets quieter and purer at the same time, which is the one place in this ladder where amplitude and distortion move together downwards. At ninety-five per cent of the critical current the output is 26.4 millivolts with 0.149 per cent of distortion — the purest this oscillator ever is, and a hundred millivolts from not working.

It is also a boundary a real part can cross, and the distance to it can be stated in the units the part is chosen in. Every march elsewhere in this ladder uses $10^{-14}$ amperes, which is what this collection’s diode model defaults to for a small-signal junction; the boundary is 6.78 decades above that. Heating cannot reach it — at 1.26 decades per twenty kelvin it would take a hundred and one kelvin — so the only way across is a different part, and a diode selected for speed rather than for low leakage is chosen from exactly the wrong end of that range. Substituting one into this limiter produces a circuit that does not oscillate, and no direct-current analysis of it says why: the operating point is unremarkable, every node sits where it should, and the failure is in a small-signal quantity that no solve for a bias point reports. That is the class of failure a bias point is a solution is about, met from the far side.

What this opens #

Two things follow that this ladder has not asked.

The first is that the limiter’s own dynamics are still absent. Two diodes respond within a picosecond of the voltage across them, so the amplitude control in this circuit has no bandwidth of its own and the settling is entirely the loop’s. Every quieter limiter — a thermistor, a lamp, a peak detector driving a field-effect transistor — is a second loop with a time constant, and the product of that time constant and the growth rate above decides whether the amplitude settles or rings. The machinery for it is a nonlinearity marched in time, which is what the amplitude nothing linear predicts built, and the question is one this collection can answer.

The second is that the start-up measured here begins from a deliberate millivolt on one capacitor. A real oscillator starts from whatever the resistors’ own thermal noise leaves there, which is smaller by several decades, and the growth law prices a decade exactly: at a gain of 3.01 a decade of amplitude costs 73.29 cycles, which at the measured 1591.35 hertz is 46.06 milliseconds; at 3.20 it is 3.65 cycles and 2.31 milliseconds; at 3.80, 0.84 cycles and 0.55. Starting four decades lower is therefore another 184 milliseconds at the quiet end of the slider and another two at the loud end — a factor of 84.0 in what a decade costs, against the 76.1 the settling figure above measures. The two differ for a reason worth having: the louder setting climbs fewer decades, because its first recorded peak is already ten times larger, and it is held back over more of them by diodes that have started conducting. The growth law is what converts a noise floor into a switch-on delay, and it is the only thing that can.

What it does not say #

It does not say the growth law is unreliable. Below about a five per cent excess gain the dropped denominator is worth under a hundredth of a per cent and the simple form is simply right, which is the range every low-distortion design sits in anyway. The correction matters where the circuit is being started quickly on purpose, and there it matters by a fifth.

It does not say the saturation current is a design variable in the sense the gain is, and the reason is worth a number rather than a caveat. A saturation current is chosen by picking a part, and then it moves on its own: the site’s own temperature model for a junction — a $T^3$ prefactor times a Boltzmann factor in the band gap — puts $I_S$ up by 4.46 for every ten kelvin, which is 1.26 decades over twenty.

Marched at those three saturation currents with the thermal voltage held at its 300 K value, so that the one term is isolated, the settled amplitude is 857.66 millivolts, 788.33 and 722.83. A twenty-kelvin room costs 134.8 millivolts, which is 15.72 per cent of the output, and it is the 108.5 millivolts a decade of the figure above with 1.26 decades put into it. That is not a second-order term beside anything: the gain slider’s entire range moves the amplitude by 58 per cent, and a warm afternoon moves it by a sixth. It is a temperature dependence the resistors do not have, it is the reason a serious amplitude reference is a loop rather than a junction, and the general shape of it — two rising quantities differenced — is two millivolts a kelvin, and the wrong sign.

And it does not say the boundary at 60 nanoamperes generalises. It is computed at one gain and one limiter resistance, and it moves with both: a larger excess gain needs a leakier diode to hold it down, and a larger limiter resistance moves the shunt further from the feedback resistor. What generalises is the mechanism — a limiter whose zero-signal conductance is not negligible is part of the loop gain and not merely part of what happens at the peaks — and the edge is a region rather than a number, in the way the edge that is a region collects.

The number worth carrying #

The envelope grows by exp(π(k − 3)/√(1 − ((k − 3)/2)²)) a cycle, which is 20.462 per cent more than the form usually quoted at the top of a practical gain range and four parts in ten million more at the bottom. The amplitude it stops at falls by 108.5 millivolts for every decade of saturation current and is proportional to the diodes’ ideality to four parts in a thousand. Above 60.121 nanoamperes, which is 6.78 decades above the junction every other march here uses, there is no oscillation at all.

The habit that goes with it is about which parameter a trade is being made in. Every model has an edge, and the edge of the usual account of this circuit is not a frequency or an amplitude but a missing axis: the design is presented as one resistor ratio balanced against purity, and the second axis was in the machinery all along, three decibels of headroom wide and nearly free. Before optimising along an axis, it is worth asking how many the object has.