## Why the Simpson’s Sum Does Not Get Used In Circuit Simulations

In recent postings I have been sharing my experiences, learning to use the software ‘NG-SPICE’, which uses numerical methods to simulate circuit-diagrams. Well, to simulate ‘Netlists’ anyway, that represent circuits. And the GUI which I have has as drawback, not being as fancy as some commercial GUIs, and only allowing me to perform certain types of simulations, that include DC Sweeps, AC Sweeps, and Pulses. I think that if I was to delve deeper, and edit my Netlists using a text-editor, I might be able to expand the range of possibilities…

But then I do think that a premise of how ‘SPICE’ works in general, is to state the Voltage as a Primary phenomenon, to which Current is Secondary. By that I mean, pure capacitors are simulated as having current, that is the derivative of voltage, while in pure inductors, the current is merely the integral of voltage. ( :2 ) And so, SPICE uses numerical approximations of both derivatives and integrals. ( :1 ) And in the many settings my GUI does offer me, I get to choose which method of integration out of two I prefer: ‘Trap’ or ‘Gear’.

The question could just pop into somebody’s head: ‘Methods of numerical integration were taught to me, which are more accurate than Trap, such as The 3-point Simpson’s Sum. (Actually, I was taught to compute 2/3 times the Midpoint, plus 1/3 times the Trap Sum, not their average.) Why can’t I select that?’ And the answer I would suggest is as follows: I already wrote a posting about the simplest method of numerical differentiation, and about how, if the step-size is too long, it can generate differentials which are too high in amplitude. If this was combined with an unsuitable method of integration, one of two paradoxical results could follow:

1. An LC tank circuit, aka a pure inductor connected to a pure capacitor, could end up unstable, gaining amplitude, or
2. The same, simulated circuit could lose momentum, apparently to nowhere, and stop ringing.

Either result is counter to what happens in Physics. And so it would seem that the medium-range errors in the Trap method, happen to complement the errors exactly, in the simplest method of differentiation. If the differentiation came into being because consecutive samples were subtracted, then simply to add them again, will reproduce what we started with. And so our pure, lossless resonant circuit, would resonate forever, as it should… The engine has no place for ‘dampened integrals’ here.

The other method available, ‘Gear’, is also known as ‘The Backward Differentiation Formula’, or the ‘BDF’. It’s mainly suited for trying to simulate systems which are ‘stiff’ i.e., where the step-interval is assumed to be too long, and where heavy bodies interact with great force, approximated with coarse time-steps. It’s like The Simpson’s Sum on steroids. I’ve heard bad things about it. One main reason not to use it, is the History by which it will stabilize a simulated circuit, while the same circuit, when actually etched into silicon, became unstable. There might be cases where only the Gear Method can be used, but it should be used as a last resort.

The (simpler) ‘Riemann Sum’ has as a problem, that it must either be conceptualized as being ‘left-handed’ or ‘right-handed’, which means, that each input sample must either represent an abstract rectangle that follows it, or that preceded it. With critically-sampled – i.e., long stepped – signals, doing so would introduce a phase-shift. The Trap Sum alleviates such a phase-shift.

(Updated 06/23/2018, 19h35 … )

## Sometimes, we misunderstand things.

One of the facts which I’ve been blogging about, concerns the software called ‘NG-SPICE’, which stands for ‘Simulation Program with Integrated Circuit Emphasis’. When using it, I can sometimes seem to recognize the parameters with which it defines components, those parameters will correspond to concepts which I already know about – such as capacitance, or ‘transconductance’ – but in the modern context, those parameters can stand for the real-world properties of a MOSFET, that do not match old-world properties, by the same name.

So as an example of this, I can demonstrate the following ‘SPICE’ definition, of an old-fashioned, discrete MOSFET, which was named the ‘2N7000′:

http://dirkmittler.homeip.net/text/2N7000.mod.txt

In this specification there is a parameter named ‘CGSO’ , of which I might say, ‘It stands for Gate-Source capacitance’. Its value is close to 1.79·10-7 . If this value was in Farads, it would actually mean that the transistor to be modeled has 180nF input-capacitance, when not active. This would be a very high capacitance-value, which in turn, would lead me to think that the transistor-type was of very low quality. But in reality, this parameter is in Farads /Meter. And so what I would now think, after noticing this detail, is that because the transistor in question only has a width of 100μM, its passive input-capacitance is really only 18pF.

The same goes for ‘CGDO’.

This will make a huge difference, in terms of how fast circuits can become, that use this transistor. And I am back to having faith, in how NG-SPICE simulates its circuits.

At the same time, when such a transistor is functioning in an active circuit, it will exhibit the property of voltage-gain indirectly. This voltage-gain will reduce the apparent capacitance, between the Gate and Common, because it reduces the voltage-changes between the Gate and the Source of the transistor, which determine how much current will flow due to its real Gate-Source capacitance, with respect to how large the changes are, in Gate-to-Common voltages. However, this phenomenon needs to be held in suspicion, if we are expecting it to reduce input capacitance by more than a factor of 1000, because:

• The real voltage-gain of the transistor-circuit is only rarely that high, and
• There are additional factors which will increase input-capacitance, so that such an ideal will not be reached.

The transistor needs to be a fast one, with low Gate-Source capacitance, if that’s what the circuit-designer is looking for.

Dirk

## Another Simple Output-Amplifier, Using Discrete MOSFET Transistors

One of the facts which I’ve been writing about, is that I possess the open-source version of ‘SPICE’, that is named ‘NG-SPICE’, and that this acronym stands for ‘Simulation Program, with Integrated Circuit Emphasis’. The full, associated suite of programs allows me to edit schematic diagrams graphically, but to export ‘Netlists’, so that I can then simulate the circuit – and see if it works.

And one of the facts which I have also been contemplating, is that by default, SPICE will put transistors, which correspond to micron-sized transistors, which will therefore never be able to drive output-loads, from a hypothetical IC, unless an explicit attempt is made, to design output-buffers, which can. These output-amplifiers have as function, that they should merely follow their input voltage, but draw as little current from their respective inputs as possible – that are outputs of other, more interesting ICs – while allowing low load-resistances to be connected to their own outputs, which correspond to plausible external components, such as 100Ω load-resistors.

I had posted an earlier, conceivable design, of such an output-buffer, which had a major flaw, that I also pointed out in the preceding posting: That amplifier could only produce a range of voltages, which was a direct function of what the Gate-Source threshold voltages would be, of the component transistors used. Hence, because I had also specified low-quality, outdated MOSFET transistors with high threshold-voltages, the output-voltage-range, was also modest but reasonable. But, newer transistors will have lower threshold voltages by design, which would, oddly enough, reduce the voltage-range of that amplifier. This would be an important consideration if the transistors were not in fact discrete, but needed to be incorporated onto the IC, where low-threshold-voltage transistors are already standard. Which means, that I needed to design a better output-buffer.

So below is a better output-buffer, schematic:

And these are the SPICE definitions, of the discrete transistors which I decided to base my design on again, both enhancement-mode MOSFETs:

http://dirkmittler.homeip.net/text/2N7000.mod.txt

http://dirkmittler.homeip.net/text/BS250P.mod.txt

The main disadvantage of this latest design would be, that the transistors which I labeled ‘X2′ and ‘X3′, do in fact conduct current to their combined inputs, which makes the additional transistor ‘X1′ necessary, since this amount of current would already be excessive, to connect to an output, of any pre-existing IC circuits. But then, the advantage goes so far, that ‘X2′ now models a level-shift, which exactly mirrors the level-shift of ‘X4′, and the voltage-level-shift of ‘X3′ now mirrors ‘X5′. There is design beauty in this. But one disadvantage now is, that the Gate-Source threshold-voltage of (1) n-Channel MOSFET (2.2V) plus (1) p-Channel MOSFET (3.2V) gets subtracted from the input-voltage, so that the available voltage-range still suffers, with respect to both the supply, and the input-voltage. Input-voltage now ranges from 5.4V to approximately 12.5V, which is closer to the range of supply-voltages than what the previous circuit allowed, and the resulting output-voltages are graphed below:

(Update 06/20/2018, 0h20 : )

There is another observation which I should add:

In the days of vacuum tubes, ‘transconductance’ was measured in Amperes / Volt, and was therefore given in ‘Mhos’, which were the reciprocal of Ohms. Apparently, in modern days, the transconductance of a MOSFET, also given as its ‘KP’, is in Amperes / Volt2 . This conscious design-decision must follow the real-world behavior of MOSFETs, but makes my earlier Math, of multiplying such a component-property by the series-resistance, to arrive at gain, incorrect. Gate-Source voltage-changes lead to current-changes, but greater Drain-Source voltages, lead to greater current-gain. This is good, because the actual gain of a MOSFET, reduces the apparent capacitance at its Gate.

The low-end output-voltage came into being as follows:

## A Pertinent Question, about Micron-Sized Transistors

If we position two electrodes in free air, 1Centimeter apart, and if we then apply 10000Volts across them, the air’s ability to resist electric current will break down, and an electric arc will appear across it.

Because of this simple observation, the question could (and probably, should) be asked, ‘Can a MOSFET transistor the size of a micron, on an Integrated Circuit, withstand 15Volts of Source-Drain voltage, at all?’

A suggestion to the contrary would be, that 10000Volts /Centimeter, is equal to 1 Volt /Micron. Thus, if the two electrodes were 1Micron apart, and standing free in air, it would take only 1 Volt to cause the air to break down, and for a microscopic arc to appear. Yet, Integrated Circuits are known to exist, which operate at 2 Volts, and which use ‘nanometer technologies’. And so in an effort to answer my own question, I would take two further observations into consideration:

1. I already recall reading elsewhere, that the breakdown voltage of high-quality, semiconductor silicon, is considerably greater than that of air !
2. I possess a suite of programs named “SPICE”, which, when performing a Level-8 simulation of MOSFET transistors, only needs to be given the width and the length of a transistor-instance, and which will, on that basis, compute all the other properties of the resulting transistor, making certain assumptions about its design.

This use of SPICE has been commented on, on the following Bulletin-Board: