This concept was invented in 2001|
There appears to be a straightforward way to greatly improve the performance efficiency of automotive engines. The result will certainly be greater gas mileage, higher efficiency, and cleaner burning, and much more available power.
This means that an automobile might have a rather tiny engine, where it could commonly get 50 or more miles per gallon during most driving, but where the same vehicle could accelerate from 0-60 mph in three or four seconds! Or a quarter-mile dragstrip in maybe 8 seconds!
No new or exotic technology is involved!
Anyone who has ever tried to improve the performance of an automotive engine knows to try to improve certain things. Specifically for this discussion, airflow in and out of the cylinders is very important, as is the temperature of the intake air along the way.
ALL the possible improvements are centered on the goal of getting MORE OXYGEN inside the cylinder. With a fuel injection system and computer control, an appropriate amount of gasoline can then be squirted in to make the proper fuel-air mixture.
The more air-fuel mixture that can be put into the cylinder and ignited, the more FUEL is in there, which means the more CHEMICAL ENERGY is inside that cylinder. It is nearly proportional that the amount of gasoline ignited determines the amount of power developed (as long as the mixture is maintained at a ratio that can burn well.)
The more fuel burned inside a single cylinder, then the greater total force there will be on the top of the piston and the more power the engine will produce. In the same way, the LESS air-fuel mixture that can be put into the cylinder and ignited, the better gas mileage there will be, but less power will be generated! These are obviously two diametrically opposed situations!
Giant V-8 engines were developed in the 1950s and 1960s to be able to have more fuel in each cylinder, to produce impressive amounts of power, but they tended to have terrible gas mileage. By the 1980s, the size of engine displacement was made much smaller, so each cylinder would contain LESS fuel-air mixture, to need to use up less gasoline, but the small engines could not produce much power. Detroit and the other vehicle manufacturers have always thought that they are forced to choose one or the other, that either they can make ECONOMY CARS that have good gas mileage but little performance, or make GAS-GUZZLERS that have impressive acceleration.
They are all wrong!
By 2002, I had invented an engine that accomplishes both! I was quite amazed when all three of the Detroit car manufacturers had absolutely NO interest in it at that time (or since)! It appears that they have an arrogance that only THEY can come up with good ideas! They may be wrong about that, especially since all three of them are teetering on the edge of corporate bankruptcy!
I do NOT give any manufacturer permission to use the information in this presentation or this invention of mine, UNLESS they make legal arrangements with me first. I see that modern society really needs this advancement NOW, and so I see cause to disclose many of the details of it now.
Regarding the mixing aspects, Ford has spent massive R&D funds pursuing a cylinder design called Stratified Charge, and Honda created their CVCC cylinder design. One major complication in most engines is that the gasoline and air are mixed a tiny fraction of a second before having to burn. For engines that have individual fuel injectors in each cylinder, at high engine speeds this can involve only 0.001 second or so! For carburetted engines or those that have TBI (throttle body injection), there is more mixing time but there are other complications that occur.
There are several possible ways this concept can be configured. We will discuss the configuration that was first developed by 2002, using an existing V-8 engine as a basis. When we refer to specific numbers here, they will be for a small-block Chevy 350 V-8 engine.
Since we will only use HALF the cylinders for producing power, but each of those cylinders will produce around FOUR TIMES the power, the net TOTAL POWER OUTPUT OF THE ENGINE IS ABOUT DOUBLE of what it was as a standard V-8!
Imagine a standard V-8 engine, but where ONLY four of the cylinders are actually used. In an engine that is built to have a firing sequence of 14286573, we might use cylinders 1, 2, 6 and 7, so that the firing of them will be equally spaced in time, so no additional vibration would develop. This would involve two cylinders on each side and two in front and two in back, so engine stresses would be relatively balanced.
There have been some production automobiles that have had the capabilities of operating as a V-8 or as a V-6 or a V-4. I believe the first was a Cadillac V-8 engine option which had the valve mechanism electronically controllable. When high power was needed all eight cylinders were used, but in highway cruising, only four were used with the (intake) valves of the other four remaining closed to not consume fuel-air mixture. That approach was interesting but it caused some new turbulence problems, too. It was also very expensive and subject to breakdowns that no one could repair!
Since the entire displacement of the eight cylinders in this engine total 350 cubic inches, one cylinder therefore has a displacement (internal volume) of 350/8 or 43.75 cubic inches. If the original engine was designed to have a 9:1 compression ratio, then this volume gets squeezed into 1/9 of that volume, or just under 5 cubic inches. This volume is usually all in the irregular space which is carved into the underside of the heads of the engine.
This engine needs to operate in several VERY DIFFERENT modes of operation! We will discuss the POWER mode in this presentation. The other modes are extremely different, which allow the economy operation for normal driving.
We need to note that these POWER cylinders require air at around three or four times the volume of what it used to be. The compressing cylinders can only provide TWO times the original amount of air (by a modification discussed below). This causes the POWER mode to ONLY be possible for a designed number of seconds, before the engine has to revert to a lower-power output mode of operation due to air starvation.
This works out to be excellent, as it provides truly massive power for a few seconds at a time, while necessarily causing much more fuel-conserving operation for the other 99% of the time of driving. I sometimes note to people that it eventually dawned on me that my high-performance-capable Corvette generally does only maybe thirty "power blasts" from stop lights in an entire year, each only involving it being floored for maybe two seconds. For all the time I drive the Corvette each year (which I truly enjoy), to realize that there probably is only ONE TOTAL MINUTE EACH YEAR where I actually USE everything that is designed into that vehicle, seemed both amazing and rather wasteful!
THIS engine design is centered on ALWAYS operating as a four-cylinder engine, and other features (not described in this presentation) to ensure that 99% of driving occurs using a configuration that is somewhat of a sissy regarding power, but which therefore has great fuel economy. A mid-sized car only needs around 60 horsepower to cruise at 60 mph, and the Economy mode of this engine should provide around 120 horsepower, which is plenty for all normal driving. But it ALSO has this POWER mode always available where it provides around 480 horsepower, double what it had been able to provide as a V-8 engine! The best of both worlds. Great fuel economy and awesome power and acceleration!
So now consider if the connecting rods of those four pistons are replaced by SHORTER con rods which are 0.8" shorter. What is the effect of this? There are three major effects:
Even though we are using only FOUR cylinders as power cylinders, we already have at least a 50% increase in available engine total power! It will actually be greater than this, for two separate reasons discussed below.
In the original engine, since exhaust gases have not all left the cylinder from the previous firing, that means that we could NOT bring in a new 43.75 cubic inches of fuel-air mixture, but instead a lesser amount. This depends a lot of how the valve timing is designed, but it is common to only be able to bring in a new 35 cubic inches of fresh fuel-air mixture. But in this new design, the exhaust gases will RUSH out of the cylinder as soon as the exhaust valve opens, due to there being much higher pressure remaining in the cylinder (as seen below). Even better, after that pressure subsides, and the intake valve opens, we have yet another advantage!
So we see that there are advantages in having the power cylinders still producing the desired 9:1 compression ratio, but in doing it using a 3:1 cylinder compression! This means that we would need to have a very large supply of "compressed air" available at about 30 PSIG (45 PSIA).
As soon as an intake valve opened, instead of the downward moving piston having to SUCK the fresh air in, this compressed air would FORCE ITS WAY quickly into the combustion chamber. For a brief time, the exhaust valve is still open, so this pressurized intake air actually also FORCES the remaining exhaust gases out! We can therefore get much closer to the 43.75 cubic inches of intake air in.
But note that after the exhaust valve has closed, the cylinder gets filled with compressed air, actually about THREE TIMES as much actual air as normal (Ideal Gas Law). When an appropriate amount of gasoline is injected into the cylinder (due to computer control of fuel injection), we then have three times the amount of fuel-air mixture inside the cylinder to burn, including around 135 "standard" cubic inches of air. The computer would ensure that the appropriate amount of gasoline had been injected to provide the best fuel-air ratio for best burning.
Therefore, we have THREE TIMES AS MUCH chemical energy in the gasoline inside the cylinder, which will all burn very efficiently due to proper fuel-air mixture. Actually, due to the effects of a variant of an inter-cooler, we will actually have around FOUR times as much air and gasoline and energy inside the cylinder. (discussed below)
When this fuel-air mixture burns well, it HAS TO create about four times the amount of power as normal. So, even though only half the cylinders are used for power, each one is able to create about four times as much torque and horsepower, and so the engine would now create about DOUBLE the torque and horsepower as the original engine did as a V-8!
It is actually better than that! Where the exhaust in a normal engine does not entirely leave (often around 10% of it remaining in the cylinder after the exhaust process has ended), this modification greatly improves the breathing of the engine (depending on how the valve timing is arranged, regarding overlap of exhaust and intake valves), in at least two different ways. As soon as the exhaust valve opens, the remaining (fairly high) combustion pressure in the cylinder starts driving the exhaust out, as well as the upward movement of the piston squeezing it out. Actually, the opening of the intake valve should be delayed until AFTER nearly all the exhaust has occurred, so any remaining high pressure in the hot exhaust does not force its way backwards into the (lower pressure, 30 PSIG) intake manifold. In actual current engines, it actually does to some extent, which restricts the inward flow of new fuel-air mixture. However, consider the added advantage with highly pressurized intake air, where the opening of the intake valve allows that compressed air to push the rest of the exhaust gases out! And this is after the even higher pressure exhaust gases rushed out on their own! It can be arranged that essentially NO exhaust gas remains, and so the entire displacement volume of the cylinder is available for new gas-air mixture. Also, the pressurized intake air is not slowed significantly by a little remaining exhaust gas, which could allow the intake valve to be opened a number of degrees earlier, even when there was still 30 PSIG of exhaust pressure inside the cylinder! This provides better purging of the exhaust gases as well as several percent of extra increase in power, and also a lot of new flexibility regarding valve timing. It may not actually be necessary to overlap valve timing at all, if that is desired!
Next, anyone familiar with high speed engine operation knows that the major limitation regarding RPMs has to do with engine breathing. At some point, the suction of the downward moving pistons just cannot overcome the frictional drag of the intake air passing through the intake manifold passageways. Essentially all automotive engines "starve for air" before any other limitations regarding high speed operation. (One popular modification gearheads make to V-8 engines is to replace the intake manifold with one with larger ports and passages, and to mill out the intake ports in the heads to handle that better airflow.)
So now consider the advantage of providing a large supply of COMPRESSED air as the intake. Note that this is not just talking about a FEW INCHES of pressure boost due to a supercharger, but about TWO FULL ATMOSPHERES of added pressure! As soon as the intake valve opens, that pressurized intake air RUSHES into the cylinder, far more efficiently and far faster than in any conventional engine. There is a scientific reason for this. When the pistons need to create suction (which always shows up as intake vacuum), at high air speeds there are effects called cavitation and other special turbulences. When a lot of air needs to be moved through narrow passageways, it is nearly always a huge advantage to PUSH the air instead of SUCKING it. This is why all home furnaces have BLOWERS rather than SUCKERS!
In any case, with a highly pressurized source of intake air, a high engine speed does not run out of air. In theory, an engine should be able to run at least three times as fast. Instead of a 6,000 rpm red line for an engine, maybe it could be 18,000 rpm. (It might blow itself apart due to vibrations before that speed though!) Since delivered torque is relatively constant (for that situation) with engine speed, that suggests that the torque output would not drop off as in normal engines. Also, since horsepower is the product of that torque and engine rpm, the net horsepower produced at 18,000 rpm should be around triple that at 6,000 rpm. Assuming the engine does not explode from the very high speeds!
The engine would NOT have any obvious way to produce the usual vacuum that is used to drive power brakes or the dampers for the heater and A/C system! Maybe that could be drawn off the intake ports of the compressor cylinders.
For existing engines, this suggests that a retro-fit kit could massively increase engine performance, while essentially using the same engine. For future vehicles, much smaller engines might therefore be able to create the power and acceleration and performance desired, while retaining the tiny engine displacement that can be used for extremely high fuel economy.
The air provided to this (POWER ONLY) tank would come from the remaining four cylinders of the engine, each of which would be used as two-cycle air compressors. Those four cylinders (3,4,5,8) would use the normal intake manifold air path, of drawing air through the air filter from the front of the vehicle. These cylinders would be pretty standard, but the pistons' connecting rods would be around 0.8" shorter. (OR the heads would have the larger milled out volume, around 15 ci instead of 5 ci) The result would be that when they reach the top of their motion, the resulting compression ratio would not be 8:1 or 9:1, but only 3:1. The "exhaust manifolds" from these four cylinders would not actually be carrying exhaust, but compressed air at about 30 PSIG (45 PSIA). Those four cylinders would send their compressed air to that "air tank chamber with heat exchanger" out to the side of the engine, maybe in front of a wheel.
In other words, these four cylinders would have only one function, that of compressing the intake air by a factor of 3:1, and continuously supplying that air tank. The air in the tank would then be at about 45 PSIA (30 PSIG), since atmospheric air is at 15 PSIA.
Because of the 3:1 compression configuration, these cylinders would never be able to over-pressurize that tank, and it would tend to self-regulate at around 30 PSIG.
The camshaft should be made in a unique way with doubled lobes on opposite sides for these cylinders, so they act as two-cycle air compressors rather than the original standard four-cycle operation of the engine (which is maintained for the remaining four power cylinders). This allows these cylinders to be twice as productive in creating compressing air, rather than including the original compression and power strokes. Other than that, the crankshaft, pistons, cylinders, heads, and valves should be able to be the standard original parts. No special machining is necessary for any of those components. (Again, for production vehicles, the heads would be significantly modified.)
The Ideal Gas Law tells us that compressed air becomes hotter. In fact, since this compression occurs in an isentropic mode, there are simple formulas to show the resulting change in temperature and in pressure due to a change in VOLUME by a factor of three inside the compressing cylinders.
For isentropic (or reversible adiabatic) compression, we have
T1/T2 = (V1/V2)(0.4/1.0)
P1/P2 = (V1/V2)(1.4/1.0)
Given these Ideal Gas equations, if we arrange to have the four compressor cylinders each change the VOLUME by a factor of 3:1, we see that the (absolute, Rankine) temperature must increase the gas temperature by a factor of 1.552, which results in the compressed gas then being about 362°F. The Pressure in the gas rises by a factor of 4.656, up to about 70 PSIA or 55 PSIG. This is the situation of the air as it leaves the compressing cylinders. (the ratio of PRESSURE [4.656] times the ratio of VOLUME [1/3] is equal to the ratio of the TEMPERATURE [1.552] in accordance with the Ideal Gas Laws).
We then use our Intercooler heat exchanger to remove heat from this gas to get the temperature back down to about the original ambient temperature. The Ideal Gas Laws still apply, and the result is that we wind up with the air at around the original ambient air temperature, and at about 26% of the original VOLUME of air and at about 58 PSIA (43 PSIG) pressure in the intercooler tank.
Therefore, a water-filled cooling tank surrounds that compressed air chamber to cool it back down to near ambient temperature. The end result is a substantial continuous supply of cool (near ambient), compressed (extremely super-charged, at around 58 PSIA) air that will be now used as intake by the power generating cylinders. That cooling of the air back down to near ambient causes the volume of that air to shrink (again per the Ideal Gas Laws) by about 1/4. That cooling therefore causes each cubic foot of that contain even more oxygen in it, now around four times the amount of oxygen as is in standard ambient air!
The effect is essentially to provide an incredibly effective super-charger for the engine. Instead of just providing a few "inches" or PSI boost as in a supercharger, this arrangement provides a 43 PSI boost!
The water-filled cooling chamber has a similar goal as the inter-coolers on many high performance engines. However, in those engines, the cooling only has a tiny fraction of a second to cool the air from a turbocharger, where this concept enables excellent (inter-)cooling due to the much longer time air is in the tank, and the much cooler water in its tank surrounding it and the cooler ambient air blowing by it.
The four power cylinders are therefore rather normal cylinders. Again, the primary difference is that the pistons are again very low, due to shortened connecting rods (or the heads are modified), so they also only accomplish a 3:1 compression ratio. But, by accepting cooled air that is already compressed 3:1, the end result is a standard 9:1 compression ratio in the cylinders (or actually slightly higher than that due to the intercooler working so well) (at the moment of ignition), with a VERY high supply (four times as much) of air and fuel.
This concept could NOT work with a carburetted engine and it needs cylinder fuel injection and computer control for best results and greatest safety. The resultant power would be VERY high, around DOUBLE the total engine power (at any rpm) as compared to the original V-8 engine it started as. Due to the improvements in exhaust purging, the net results might be even greater.
With all the cylinders only creating a 3:1 compression, a number of mechanical advantages accrue, including a number of aspects that should minimize wear and future maintenance. Roughly half of the engine would not be creating massive heat, since it would only be used as an air compressor, and so heat stresses on components and lubricants might be less. The engine would be acting as a four-cylinder engine, with the capability of very high gas mileage that is associated with only supplying fuel for four cylinders rather than eight. But the power generated by this ultra-charged engine would be far GREATER than that of the original eight-cylinder engine's performance. Low-end torque appears to be increased, and torque in general should be far greater than the original V-8 engine had. Top end power is wildly increased due to an overall greater torque at all engine speeds.
There would not be significantly greater stresses on the engine components, as the combustion flame is still at the same flame temperature and so the pressure inside the cylinders should not be materially higher than in the original engine. However, where the original engine only has a very BRIEF period of when high pressure exists inside the cylinder to push on the head of the piston, this concept causes that high pressure to endure far longer inside the cylinder. The result should NOT be much higher PRESSURE pushing the piston down, but rather that pressure is applied to the head of the piston for many times longer duration. THIS allows a high pressure to be pushing the piston down around the time when the piston is halfway down the cylinder, at the point where the crankshaft throw is in the most advantageous position to convert that pressure into engine TORQUE OUTPUT. The four times the engine output is therefore a result of higher torque being applied to the crankshaft for a longer period of time than in conventional engines.
In the original V-8 engine, the compression ratio was around 9:1 which created around 120 PSIA (105 PSIG) cylinder compression pressure (confirming what a compression gauge shows). The process of combustion of gasoline is well-known to increase the maximum pressure by around a factor of 5, or therefore around 600 PSI pressure in the burned gases in the cylinder. This pressure acts on every square inch of the surrounding surfaces. The 12.5 square inches of the piston's top is therefore subject to around 7500 total pounds of force (downward). Unfortunately, this occurs at a time when the crankshaft throw is in a terrible angle, essentially straight up. What happens when you put your bicycle pedal STRAIGHT UP and put your weight on it? Not much! YOU know to put the pedal at an angle around 45 degrees AFTER the TDC of the bicycle pedal! It allows your weight to be (geometrically) transferred to the crankshaft of the pedals, and therefore to scoot you along.
Consider the situation after our piston has traveled ONE INCH DOWNWARD in the cylinder of the standard engine. From previous discussions, we know that one inch of travel accounts for about 10 cubic inches of cylinder volume, and we also know that the head had an available volume of around 5 cubic inches. Therefore, after traveling one inch down, our cylinder volume has increased from 5 ci to 15 ci, three times as large. Due to the Ideal Gas Laws, this has the effect (roughly) of DECREASING the pressure in the cylinder by that same proportion. (We are simplifying by assuming that there is NO cooling system taking energy away from the cylinder, a factor that greatly complicates the math and the logic!)
That means that our 7,500 initial pounds force on the piston has dropped to 2,500 pounds after just that one inch of piston travel! It is easy to learn that this situation occurs at the point that our crankshaft has only rotated about 35 degrees! But the POWER CYCLE of the four-cycle engine is 180 degrees long! We are already down to 1/3 force on the piston by just 35 degrees rotation! By the time we get to BDC (the lowest point for the piston), the cylinder volume has increased by a factor of nine, and the maximum available force on the piston is only around 800 pounds.
This suggests the very low thermal efficiency of standard engines, which used to be around 15% in the 1970s and has risen to about 21% today. ACTUAL POWER (force on the piston head to push it down) really only occurs for a disappointingly short period of time! In fact, because of this, ALL existing engines are designed to have their exhaust valves open DURING this power stroke. They are conceding that there is so little useful power available in the bottom half of the power stroke, that they simply allow it to escape and help blow the exhaust gases out!
IF we also would consider the effects of the cooling system, the pressure inside the cylinder drops even faster!
Now consider the same sequence with our unusual engine! Remember that it has a 3:1 compression ratio in the power cylinders. So, AFTER THE ENTIRE 180 degrees crankshaft rotation of the ENTIRE POWER STROKE, what is the remaining pressure acting on the piston? Since the compression ratio is 3:1, that means that we would have 2500 pounds of force still acting on the piston head (three times as much force on the piston as before, all producing usable power output).
How about HALFWAY down? The original engine would have dropped by 1.75 inches, and we would be at the 2:1 compression point, which is 4.5 times the final volume. This results in a pressure on the piston of around 1650 pounds. And in the new engine design? We are also at the 2:1 compression point, but that is only out of 3:1 total! We therefore have 7500 * 2/3 or 5,000 pounds of force pushing the piston down! This is again THREE TIMES as much force on the piston as in the original engine, again, all producing usable power).
A numerical Integration of the entire 180 degrees of the Power stroke has been done for both the original V-8 350 engine and this modification of it. The results show that the standard 350 V-8 engine creates maximum torque at 37° after TDC, of 364.4 lb-ft (for that instant). When the created torque is Integrated over the entire 180° of the power stroke, the AVERAGE torque is 190.1 lb-ft.
The new engine shows that the maximum torque is created at a LATER angle (60°) which allows it to have much better leverage due to the improved angle of the crankshaft throw. At that instant, the cylinder would produce 631.2 lb-ft of torque. When the effects are all Integrated over the complete power stroke, the AVERAGE torque is 570.3 lb-ft.
The new engine is therefore applying more total force on top of the piston and for a far longer time, throughout the entire power cycle. This, and the improved geometric leverage angle of the crankshaft throw, therefore has the effect of applying three times the torque on the crankshaft at nearly all times. Essentially, three times the engine power, since horsepower is just the product of torque times RPM.
If the original engine produced (an average of) 190 lb-ft of torque at 4000 rpm, then we can calculate the horsepower generated: 190 lb-ft * (4000/60) rps * 6.28 (number of radians in a full revolution) / 550 (lb-ft/sec)/horsepower, which gives about 145 horsepower produced. This is in agreement with some published GM figures regarding a specific V-8 350 engine they made for their Monte Carlo model some years back.
If the new engine produced 570 lb-ft of torque at 4000 rpm, we get: 570 * (4000/60) * 6.28 / 550 which gives about 435 horsepower.
All these effects for the new engine should actually use the number four instead of three, because of the effect of the intercooler increasing the oxygen density of the air. So it is actually around FOUR TIMES the total power from that piston! Given that there are only half as many power pistons, the total engine might therefore produce only around DOUBLE the horsepower (290 horsepower at 4000 rpm) and torque (380 lb-ft at 4000 rpm rather than 190) that it did as a V-8! However, the computer simulations show so much LONGER production of torque and force, that it might actually be closer to that FOUR TIMES (or 580 horsepower and 760 lb-ft of torque at 4000 rpm). In either case, this is a tremendous increase in the total power and torque developed by an existing standard V-8 engine!
Conventional engines only produce large amounts of that force and torque for around 90° of crankshaft revolution, which is why eight cylinders were commonly used to provide a uniform production of that power and torque. (The computer simulations indicate that such a V-8 engine as has been discussed only produces more than 100 lb-ft of torque between crank angles of around 6° and 134° and only more than 150 lb-ft between 9° and 114°, which is around the needed 90° of productive use.) This new engine concept produces more than 100 lb-ft of torque between crank angles of around 6° and 164° and more than 150 lb-ft between 8° and 156°, which is relatively close to the 180° of the entire stroke. This much greater duration of production of a lot of torque and power MIGHT allow this engine to perform even better as just mentioned.
The chilling effects of the cooling system affects both the original and this modified engine about equally. So even though the math to prove that is incredibly difficult, it is still true that the engine will produce around double the horsepower and torque as it did as a V-8 engine.
The exhaust manifolds for the power cylinders would remain like before, but the exhaust manifolds for the compressor cylinders would have to go to the air tank chamber. Similarly, the intake manifold for the super-charger cylinders could remain as before, but the intake ports of the power cylinders would be connected directly to the air tank chamber.
It is NOT "fuel-efficient" during this full-power mode of operation, but the point is that this type of operation would only occur less than 1% of the time of driving. At all other times, the engine will operate similarly to a moderately normal four-cylinder engine, with the implication that the functional fuel efficiency is BETTER than what it was before as a V-8 engine.
Essentially an existing V-8 engine could use less gasoline during more than 99% of driving, while also having the CAPABILITY of producing truly awesome power for a brief period. Instead of just bolting on an existing GM 6-71 truck supercharger on an engine, this much more effective approach to the supercharging concept gives even better performance, while also providing the excellent fuel economy that all modern drivers now expect.
It also has a peculiar problem regarding starting! Again, a simple solution has already been found for this issue.
C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago