Engine Management

The magic formula....

It all starts with something called charged air mass. The steady state fueling is calculated based on mass air flow combined with a targeted air:fuel ratio. The mass of fuel is then applied to the system by injectors of fixed flow rate measured in pounds per hour.

The first variable is air charge, which is the quantity of air that is moved through the system in a single period of time. In a four cycle engine, this is a single cam cycle, or two revolutions of the crank shaft  Volume is the second variable, and is relatively constant, but differs in each instance of an engine. Volume is the total displacement of the engine and is measured as the sum of the difference of each cylinder volume from lowest point to highest point in the cycle. Third is a more complex concept for a variable; Volumetric Efficiency (VE) is the effectiveness of the engine to move the total volume in a single cam cycle, and is measured as a ratio or percentage. The three variables relate as:

AirCharge = Volume * VE

Next is the conversion from air charge to air mass. the ratio between air charge and air mass is rho. The value is comprised of the volume of air and the mass of air. The mass of air is considered as dry air and because air density is relevant, the barometric pressure will be factored into the equation:

            AirMass

rho = --------------

          AirCharge

Now it is necessary to add in the ideal gas law to provide a base for calculations:

pV = nRT

But the value of rho is a dynamic value based on the volume flow of air mass. The static value of air mass has been measured extensively and is provided in the Handbook of Chemistry and Physics.
(weight of dry air at sea level 28.966 g/mol::F-201 Handbook of Chemistry and Physics 56th ed.) This is the value of n, and V is the total displacement of the engine. Because the system is dynamic, the equivalent component for rho is:

            n
rho := ---
            V

Last is the controllable element of the operation. The equation's goal is calculating fuel mass. The mass of fuel is combined with the volume of air based on a target ratio of fuel to air, or the Air-Fuel Ratio(A:F)

            AirMass
A:F = --------------  and  AirMass = AirCharge * rho
           FuelMass

                     AirCharge * rho
FuelMass = ----------------------
                               A:F

Air charge can be replaced with Volume * VE, and because, from the ideal gas law, rho equates to:

             p
rho = --------
           R * T

The equation for fuel mass becomes:

                     Volume * VE * p
FuelMass = ----------------------
                          A:F * R * T

In the total of the equation, there is one constant R which is the ideal gas constant and has the value: Calculated from base information Boltzmann's Constant k = 1.3807e-23 J/K combined with Avagadro's Number N(a) = 6.0220e23 /mol provides a value:

k * N(a) = 8.3145754 J/mol-K

- The value of pressure is absolute and is measured in Newtons per square meter (Pascals).

- Temperature is absolute and measured in degrees Kelvin, VE and A:F are both ratios and have no units, and because 1 Joule => 1 m³-Pa, the measurement for volume is in cubic meters.

The last part of the process is to eliminate units based on fuel mass:

                               m³ * Pa

FuelMass => --------------------------  => mol

                        m³-Pa/mol-K * K

Then multiplying the value by the mass of dry air, in g/mol, yields the fuel mass in grams, and the final equation resolves to:

                      Volume * VE * p * 28.966
FuelMass = -----------------------------------
                          A:F * T * 8.3145754

The units for each of the items is:

FuelMass - grams

Volume - cubic meter

p - Pascals

T - degrees Kelvin

To provide workable sized variables, so the intermediate values don't get too big or too small to manage with integers only, some base 10 elements are implemented. Vehicle engines have their displacement rated in cubic inches or liters, and often in cubic centimeter. Conversions to all these units are readily available, and since the equation started with cubic meters, cubic centimeter (cc) is chosen to provide sufficient resolution. Pressure sensors are rated in kilo-Pascals or often BAR (base atmosphere), so the selection for pressure is kilo-Pascal (kPa). Because temperature is in degrees kelvin, there is no need to change to another unit, and leave conversion to Centigrade (C) or Fahrenheit (F) to the user interface. Last, because grams is a very large value in fueling, the fuel mass will be in miligrams.

Conversions:

grams = 10³ miligrams
cubic meter = 10⁶ cubic centimeter
Pascal = 10^-3 kilo-Pascals

                           V * VE * P * 28.966 * 10^3
FuelMass = ------------------------------------------------
                      A:F * T * 8.3145754 * 10^6 * 10^-3

Leaving a single constant 3.483761781

Now for my stump speech about not using floating point...

Don't use floating point for embedded processes. The conversions eat up all sorts of valuable time, and it is really quite unnecessary if the operations are well planned.  For the fueling equation, VE and A:F are both ratios, so any value used for the fixed point method will divide out. In other words, if you multiply a ratio by 1000 so that 99.9% becomes 999 integer value, if the same multiplier is used in the numerator and denominator, they will cancel each other out. If temperature is stored as an absolute value K * 100 and pressure is stored as kilo-pascals * 100, the same elimination will occur. This leaves only the constant 3.483761781.

Planning this out, it is possible to use a numerator portion and a denominator portion. When I do this type of operation, I like to use either the numerator or the denominator as a power of 2. This way, the multiplication, or division operation is resolved to a simple logical shift right or left.  After some playing with values, I came up with

16384

--------- =  3.483733787 (I think that's close enough)

  4703

So now the total fueling calculation only has the penalty of multiplication and division. If the calculations are done in a proper sequence, the error loss is minimized. The result is fuel mass in mg per CAM cycle.

FuelMass = Volume * VE;

FuelMass /= AF;

FuelMass *= PortPressure;

FuelMass /= PortTemp;

FuelMass *= 16384;

FuelMass /= 4703;

Because of the values involved, it is a good idea to use a 32bit long for all the intermediate values (casting when necessary).

Because injectors are rated in pounds per hour, a conversion to mg is needed. The value in calibration is all that is necessary so the conversion can be performed at the UI level.

453.592 grams per pound

3600 seconds per hour

1 lb/hr = 125.998 mg/s

Build this into a spreadsheet and play around with the numbers and let me know what you find.

Jack

A little more math calisthenics. The injectors are rated in lb/hr, but are stored in calibration in mg/s so a 40# injector is stored as 5040 (5039.92 rounded integer). Remembering the scheduler resolves to system tics (2MHz). The pulse width should be calculated as number of tics and still retain significant digits using integer math.  The total possible volume is a summation of all the injectors.

The total fuel mass per cam cycle is in mg. The total pulse width per cam cycle then is:

     fuel mass * TICS_PER_SEC

     -------------------------------------

         total injector volume

For batch and staggered, the pulse is done twice so the pulse width is divided in two.  For TBI the pulse is done four times so the pulse width is divided by 4.

When to do calculations is now what is necessary. The formula provided is for steady state fueling, where there are no changes to demand or RPM, and the A:F ratio remains generally constant.  The equations can be performed generally at any time, meaning they do not need to be done in the tooth interrupt. However, the volume of fuel required does change and needs to be updated regularly and at a high enough rate to maintain smooth operation.  Examining the information, it is possible to not some items that do not change during runtime, and can be performed during engine stop, summation of injectors as an example.

For the timing of events, the scheduler is employed in exactly the same manner as before, for TBI and staggered batch, four evenly spaced events, and for standard batch two evenly spaced events. The last case of sequential fueling, needs to activate the injector during the intake stroke of the cylinder. This begins at the top of the exhaust stroke, which is 360 degrees from TDC, which is the value of the tach event from ignition timing.  Using another array for the injector start point, it is a simple case to add provision for injector scheduling the same as with ignition.

I kind of glossed over something from before, the value of advance was determined in units of degrees, and I kind of fudged the value for dwell. In actuality, dwell is a time value for how long it takes to bring the coil to a full charge. Since the operation of the scheduler is based on angle, it is necessary to convert the dwell time into a dwell angle.  This is done using the magic number "base" described before. The value of base was used in the scheduling operation to convert the value of angle to time, but it can also be used in the reverse operation.  Because the dwell time value doesn't change significantly, dwell angle only needs to be calculated as speed changes, usually in the mS routine.  It is also possible at the same time to do the same operation with pulse width. Dividing by 2 (batch) or 4 (TBI) can be done during the scheduling event, because it is only a right shift operation. If at this point it starts to look like the 1mS operation is getting too fat with processes, it is always possible to run the calculations interleaved.  Doing the ignition calculations on one pass, then the fueling calculation on the second pass. This is something that would require some physical testing to see if it is a concern.  Time to run the experiment.

Jack

A little more math calisthenics. The injectors are rated in lb/hr, but are stored in calibration in mg/s so a 40# injector is stored as 5040 (5039.92 rounded integer). Remembering the scheduler resolves to system tics (2MHz). The pulse width should be calculated as number of tics and still retain significant digits using integer math.  The total possible volume is a summation of all the injectors.

The total fuel mass per cam cycle is in mg. The total pulse width per cam cycle then is:

     fuel mass * TICS_PER_SEC

     -------------------------------------

         total injector volume

For batch and staggered, the pulse is done twice so the pulse width is divided in two.  For TBI the pulse is done four times so the pulse width is divided by 4.

When to do calculations is now what is necessary. The formula provided is for steady state fueling, where there are no changes to demand or RPM, and the A:F ratio remains generally constant.  The equations can be performed generally at any time, meaning they do not need to be done in the tooth interrupt. However, the volume of fuel required does change and needs to be updated regularly and at a high enough rate to maintain smooth operation.  Examining the information, it is possible to not some items that do not change during runtime, and can be performed during engine stop, summation of injectors as an example.

For the timing of events, the scheduler is employed in exactly the same manner as before, for TBI and staggered batch, four evenly spaced events, and for standard batch two evenly spaced events. The last case of sequential fueling, needs to activate the injector during the intake stroke of the cylinder. This begins at the top of the exhaust stroke, which is 360 degrees from TDC, which is the value of the tach event from ignition timing.  Using another array for the injector start point, it is a simple case to add provision for injector scheduling the same as with ignition.

I kind of glossed over something from before, the value of advance was determined in units of degrees, and I kind of fudged the value for dwell. In actuality, dwell is a time value for how long it takes to bring the coil to a full charge. Since the operation of the scheduler is based on angle, it is necessary to convert the dwell time into a dwell angle.  This is done using the magic number "base" described before. The value of base was used in the scheduling operation to convert the value of angle to time, but it can also be used in the reverse operation.  Because the dwell time value doesn't change significantly, dwell angle only needs to be calculated as speed changes, usually in the mS routine.  It is also possible at the same time to do the same operation with pulse width. Dividing by 2 (batch) or 4 (TBI) can be done during the scheduling event, because it is only a right shift operation. If at this point it starts to look like the 1mS operation is getting too fat with processes, it is always possible to run the calculations interleaved.  Doing the ignition calculations on one pass, then the fueling calculation on the second pass. This is something that would require some physical testing to see if it is a concern.  Time to run the experiment.

Jack