Part II: looking for plausible E-I curves
At the end of the last installment I asked the rhetorical question of whether you'd ever tried to balance a light switch midway between "on" and "off." We've all tried that; few have ever succeeded. It's difficult because most light switches exhibit mechanical hysteresis, pretty much the same phenomenon that I said was the lies-to-children explanation for the behaviour of the neon tube relaxation oscillator. The light switches in particular are designed that way because the manufacturers don't want you operating them half-on; doing so would create an arc between the switch contacts (a plasma discharge actually very much like the ones we're studying inside the glow lamp), and the heat of the arc would vaporize the contacts and possibly start a fire. So the switches are built to be hysteretic because hysteretic systems abhor being caught in the middle.
Here's the incorrect E-I curve, one last time:
Pretty much every relevant law of physics is pushing the operating point of the glow lamp toward the corners of the hysteresis loop - away from the negative resistance point X. This isn't just because we drew the diagram inaccurately; in fact, the correct E-I diagram (to be discussed soon, I promise) is even worse. Instead it's a fundamental feature of negative resistance. Negative resistance is inherently unstable.
Let's say we wanted to operate the lamp at the point marked X. Figuring that point A is at 70V, and point B is at 50V (which are nominally correct values for a standard NE-2), we can guess that X should be 60V. So let's plug the bulb into a 60V supply, in the simulator. Don't do this in real life.
If you just run a basic DC analysis, Qucs will produce reasonable-looking DC bias values, because both the simulator and my model are designed in such a way as to try to force it to behave well in that kind of analysis. But as soon as you attempt a transient analysis, describing the circuit's behaviour over time instead of in a timeless DC steady state, here's what you get:
Oddly enough, it says the problem is in the DC analysis, even though we could run the DC analysis alone and that would work. I think that's because the automatic DC analysis baked into the transient analysis algorithm has higher standards for what constitutes a valid solution. It doesn't converge and can't even start the transient part of the analysis.
The 60V supply corresponds to a vertical line on the E-I chart. It says, okay, I don't care what the current will be, but the voltage has to be 60V. That vertical line passes through the neon tube's E-I curve in three places, one of them being X. Then the simulator is trying to solve the corresponding system of partial differential equations numerically (that's how such simulators work) and that means trying to construct a proof that the voltage and current have specific values and no others, to within some tolerance. Trouble is, there are three different points on the chart that would be consistent with the partial differential equations governing the simulation. No unique solution is possible. It can't decide which one should be correct - the "off" point below X, the negative-resistance "neither off nor on" point X itself, or the "on" point above X. The simulator tries for a while and then throws an error message.
If you try it in real life, the laws of physics aren't allowed to throw an error message. So the lamp and the power supply must choose one of the three possible operating points consistent with their respective principles of operation. It won't be X for reasons I'll get to in a moment. If you're lucky, it'll be the "off" point below X. (That's where the simulator's DC analysis goes, when it works.) If you're not lucky, it'll be the "on" point above X, which corresponds to a current of something like a million amperes because of additional kinks in the real-life E-I curve not shown on the simplified diagram. Then the glow lamp explodes, spraying you with superheated microscopic glass shrapnel. Or, maybe it just cracks open with a pathetic little popping noise. It's hard to predict.
Let's talk about why it can't choose point X. Suppose at some point in time the lamp did happen to be at X: voltage 60V, current whatever it is at that point. In fact, with my NE-2 model, the current at 60V in the negative-resistance region is exactly 1 microamp. What happens next?
Voltage and current randomly fluctuate, always. One reason for that is that electricity isn't really a continuous, infinitely divisible, Newtonian fluid. It's actually made up of electrons - little badly behaved hard shiny bullets of energy-stuff. On the very precise scale you can't say that the current is exactly a particular value all the time. Sometimes, just as one electron is passing your reference point, it looks like a whole lot of current just for a moment. In between the electrons it looks like no current just for a moment. Inside the lamp you've also got rather larger and heavier neon ions bouncing around. A "current" is only a current when you average it over time and then you have to care how long a time scale you're looking at. So let's say that just for a moment, on the time scale the lamp cares about, the current appears to be slightly more than a microamp.
The neon lamp sees the reduced current and increases its voltage drop, moving down and to the right on the curve. Then the power supply sees the voltage has increased to more than 60V, and reduces the current to compensate. Then the neon lamp sees the further-reduced current and tries again to increase its voltage drop. The lamp and power supply fight their way down to the "off" state on the lower limb of the curve. If the initial random fluctuation had been in the opposite direction, they could just as easily shoot up to the "on" state on the upper limb. It could go either way, but it's not going to remain at X for long. The other two states are stable, at least until the lamp explodes. The negative-resistance state is unstable.
I am anthropomorphizing the power supply by describing what it sees and what it tries to do, and that sort of suggests I may be talking about some kind of regulated power supply, with at least as much intelligence as let's say a 78xx-series regulator chip. But it does not have to be so. A perfectly ordinary battery with a reasonably low, linear, output impedance, would still exhibit this behaviour. To blow up the glow lamp the power supply only needs to be a thing that obeys Ohm's Law, and it only needs to even do that in a rough qualitative way.
There is a way out of this mess. Most of us are probably accustomed to thinking of applying a voltage to a component and seeing what happens. We determine the voltage; we impose it on the component from outside; and the component decides the current. That is really the wrong way to think about glow lamps. As described above, if we try to get to point X by applying the correct voltage, Bad Things happen. But we don't have to use a constant-voltage supply! We could use a constant-current supply instead. Those aren't easy to come by in real life, but they're a basic primitive of the simulator.
One way of describing the problem with the glow lamp is that its current is not a function of its voltage drop. To be a mathematical function there has to be exactly one output value for every input value; and between the maintaining and starting voltages, the glow lamp instead has three different current values for each voltage. But if we switch E and I, then it becomes a function. For any given current, there is only one possible voltage. A constant-current supply forces the system onto some horizontal line on the E-I chart. A horizontal line can cut the glow lamp's curve in only one place, so it seems we should be able to get stable behaviour even in the vicinity of point X. Here's the circuit in the simulator. I set up a "parameter sweep" to try a bunch of different values for the constant current and plot the resulting curve.
Look at that E-I curve. It's transposed relative to the other ones I've been displaying up to this point, with the voltage along the vertical axis and the current along the horizontal, but that's hardly the worst problem with it. In fact, this curve bears pretty much no resemblance to the earlier ones. I told you the earlier curves were lies. Have fun finding the hysteresis loop on that curve, let alone biasing the device into the middle of it.
It's easy to guess that maybe my curve looks unexpected because I'm simply using a bad model in the simulator. This model I'm using is one I cooked up myself, so maybe I just built it wrong. If you've been paying attention and you're experienced with graphing things, maybe you've also thought of an adjustment to try that might make the graph better-behaved. But of course I'm presenting this backwards anyway: I didn't start out with this model and then wonder why it didn't match the curves. Instead I started out with curves like the ones I already showed you, built a model to imitate them, found it didn't work, deduced that those curves were incorrect, and then I went and found some correct (or at least, more plausible) curves from better sources and built a model to imitate them.
The first place I found a halfway reasonable diagram of the actual neon glow lamp E-I function was in an article by the irreplaceable Forrest M. Mims III, in the December 1976 issue of Popular Electronics. I think I once owned a physical copy of that issue, but I don't seem to anymore. I hope it went to a good home. Here is a scan of the article and some related notes, courtesy of Nuts and Volts, and below is a clip of the important diagram.
The voltages shown there, which look like 94V on and 82V off, certainly don't match the NE-2, but it's plausible they could match some real-life neon glow lamp. I think some of the proportions are wrong, too. But the general shape of the curve - which you may note still looks nothing like the earlier "lies to children" curves nor the simulation curve I plotted - is a lot better. If you transpose this to put the voltage on the horizontal axis, you do get a couple of negative-resistance switchbacks. One corresponds to the start of glow discharge at about 10 attoamperes (what do you mean your multimeter isn't calibrated in attoamperes?), and that's the one we care about; the other corresponds to the start of arc discharge at about 100mA, which is lamp-destroying territory. The most important thing about this diagram, however, is that it's semi-logarithmic. That's really the issue with the simulation diagram I showed earlier: the interesting things are happening at the microamp level or below, and so of course they're not going to show up when I plot the function on a linear scale that goes up to 10mA. The cranks with their not-to-scale hand-drawn plots don't label the quantities on their current axes at all, and it's misleading and detrimental.
In the write-up that goes with this diagram, Mims goes into some detail about what the marked current ranges ("normal glow," "abnormal glow," and "arc") mean. One thing he says which doesn't really make much sense is that "most commercial devices are rated at a continuous current of 0.1 to 10mA," which places the normal operating point right in the middle of the range labelled "abnormal glow" (a positive-resistance section of the plot, mind you); but also that "Prolonged operation in the abnormal glow region [...] will destroy the lamp." I suspect that may have been an error introduced during editing of the article rather than an error by Mims, because another source I'm about to describe says something more sensible that an inattentive editor could reasonably distort into what was printed.
I don't believe that the breakdown current of a real glow lamp is actually 1E-17 amps (10 attoamps). With a 94V breakdown voltage as shown, that would mean the lamp when turned off has an equivalent resistance of (positive) 9.4E+18 ohms: 9.4 exaohms or 9.4 trillion megaohms. I don't believe the glass envelope of the bulb, appearing across the leads in parallel with the gas, is such an amazing insulator as that implies, even if it's possible to imagine that the gas itself could have so high a resistance in some theoretical sense (which I also doubt, but don't really know for sure). Nonetheless, this diagram shows behaviour that makes basic theoretical sense given the physics of how the lamp works, and that (when translated into simulation terms, with some important adjustments) makes my simulated circuits behave as real ones should.
While I'm talking about realistic E-I curves, I should mention that there exists a machine one can buy or build called a "curve tracer" (usually realized as an attachment to a general-purpose oscilloscope) and with some care to make sure the tracer is operating in a constant-current rather than constant-voltage mode, one can actually plug in a glow lamp and see how it behaves instead of trusting simulations. Jack Smith, at Clifton Laboratories, has a page about the relaxation oscillator where he does just that - and the resulting curve looks very much like the one from my simulation earlier. The machine produces a linear scale, just like the one in my simulation. The linear scale isn't really very informative but that's a fact of how it operates, and the voltages and shapes match up to the necessary transposition. Ronald Dekker has a lengthy page about neon ring counters, which goes into a lot of topics beyond the scope of the current article, but it includes some real-life curve tracer photos.
My favourite neon tube E-I curve is the one from the 1966 General Electric Glow Lamp Manual, scan courtesy of SV3ORA. Clip below; notice that this one is a lot more detailed than any of the others seen to date. I would suggest ignoring the handwritten notes; the copy that got scanned no doubt went through any number of hands between 1966 and whenever it made it onto the Net, and there's no guarantee that those earlier readers knew what they were looking at.
In the associated text, it gives what I'm willing to believe is an accurate account of how glow lamps really operate. At very small currents, we are just looking at the high resistance of the insulator. That is the section of the curve from A to B (note there are three points labelled B). At some level of current, the voltage suddenly increases as the gas ionizes. That is a phase transition, something like melting ice. Current is held constant as the voltage rises until all of the first round of ionization is complete. The actual current level where this happens is variable; it depends mostly on the background ionization of the gas, which in turn depends on how much light and other radiation is floating around.
More radiation increases this fixed current level (shifting point B to the right) as well as decreasing the voltage at the top (the voltage of points D and E). Many tubes were made with built-in radioactivity to ensure a minimal level of ionization, primarily so that the top voltage wouldn't be too high when the tube was trying to start in pitch darkness, the dreaded "dark effect." As Ronald Dekker discussed in his article, these radioactive additives only had a half-life of about ten years. Any glow lamps actually dating from 1966 are pretty much stone cold by now, and radioactivity is a lot less politically correct nowadays so I imagine that recent-manufacture lamps tend not to be compensated for dark effect even when brand new. People in the 1960s probably got to do all KINDS of stuff we're not supposed to do anymore.
As mentioned, I have my doubts about whether anybody in the 1960s actually had access to test equipment capable of measuring this kind of thing accurately in the zeptoamp range, as the diagram implies they did, nor that the insulation is good enough to keep the leakage currents from swamping this part of the curve completely, but never mind; it's a reasonable extrapolation from measurements they could really have made, and not the important part of the diagram anyway.
From the knee of the constant-current region, wherever it happens, the voltage remains basically constant. Note that unlike what's shown in the Mims diagram, here the voltage is sitting constant at the starting voltage, the high one. At point E, which I read as roughly 0.1 uA on the diagram, it starts to break down. This is called the "normal glow" region, and it corresponds to an active glow discharge covering only part of the electrode. As the current increases through two more orders of magnitude to 10 uA, point F, the voltage falls to 50V: the maintaining voltage. Note that throughout this zone we have a negative resistance. If we want negative-resistance behaviour, we're going to want to aim for a point midway between E and F: something like 60 volts, 1 uA. (Recall that that's where I earlier said X was located.)
At the maintaining voltage and corresponding current, the curve turns around and starts increasing again. The book describes this as the "abnormal glow" region. We now have a glow discharge covering the whole electrode, but it's no longer behaving as a classic negative-resistance glow discharge because we're forcing extra current through it. Here is something worth emphasizing: the normal operating point of the lamp, specified in the case of the NE-2 to be 0.6 mA (not uA; that's 600 uA) is squarely within the abnormal glow region. It's easy to assume that something called "abnormal" must be perverse and dangerous, but the text makes quite clear when describing the abnormal glow region that "Most glow discharge devices are operated in the lower potion of this region." It stands to reason that that's where you would want to bias it when you're using it as a lamp: in the abnormal glow region, the device has positive resistance, and it will behave itself when connected to a sensible driver circuit. So it's understandable that the devices would be designed to operate in that region as a normal thing.
Once the current gets high enough (circa 1 amp) there's a second breakdown as the device enters the "arc" region. This is apparently a different kind of plasma discharge phenomenon, not the same thing as glow discharge but it looks very similar on the E-I chart. There's another negative resistance region in which the voltage decreases down to about 20V as the current goes to 100A, and the chart seems to suggest that then voltage starts increasing again ("abnormal arc"?) but by that point the lamp has already exploded. In the case of neon glow lamps (fluorescent light tubes are another story) arc discharge is extremely destructive. In order to actually have the lamp survive, we can't safely run it past the middle of the abnormal glow region.
The book goes on to describe in some detail the time-dependent behaviour of the ionization levels in the lamp. The current version of my model does not obey the properties described. Instead, it uses some ideas I got from a research paper I read about arc discharges in fluorescent lamps (remember I started out wanting to talk about fluorescent lamps). I don't know whether those principles are valid for neon glow lamps nor even for fluorescent lamps, and I'm still tinkering with it. One warning sign is that a time constant I'm using inside the lamp model (1.2 us, you may have seen it on the diagrams) is much smaller than the time constants described in the glow lamp book; but that's the size it has to be to get the oscillators' behaviour to match reality using my current model. So the bottom line is that I'm not sure my current model is really right with regard to the time-dependent stuff. Nonetheless, it seems to work pretty well.
I am pretty well satisfied, at least, with the E-I curve in my current model. Here is how it looks - same constant-current curve tracer circuit as before, but with an appropriate logarithmic scale.
I included a parasitic resistor, representing insulation leakage, parallel to the whole works. That's somewhat necessary for modelling, to ensure linear behaviour at zero current, and I think it's realistic. I used a value of 70 Gohm, that is 70000 Mohm, which I think is already unreasonably high; but even at 70 Gohm the insulation leakage is enough to swamp the book's claimed constant-current behaviour at downwards of 100 pA. I think that whole business can be relegated to pure theory, which is nice because that means I don't have to model it. Similarly, I have not attempted to model the arc breakdown nor the hypothesized "abnormal arc" beyond 100 amperes. The intention is never to drive the lamp anywhere near those phenomena. The parts that really matter are the normal and abnormal glow discharges, covering the range 0.1 uA to 1 A.
I'll break here. This installment has turned out longer than I'd planned. Remember that you can download the package of Qucs files to play with. Next installment, whenever it appears, will probably talk about how my model works, or maybe my attempts to build an amplifier out of these. I will close with a classic glow-lamp multivibrator. This is another circuit where the model seems to produce very plausible behaviour. Note I've given the two lamps slightly different parameters - that's realistic (real lamps never match perfectly) and also necessary to get the simulator to work, since it would happily simulate two exactly identical lamps if we let it, with nothing to break the symmetry and start the multivibrator operating.
Nobody in electronics thinks there is anything suggestive about the term "multivibrator."