Just in case everyone is now running off to get themselves a high powered panel after having a look at that last top speed file, here is another examining the power to weight ratios of cars with the 2011 ballasting formulas in play (the second worksheet in the file looks at the 2011 formulas in comparison to those used over the past decade).
As can be seen, lower powered cars have superior power to weight ratios (providing that the car chassis weight is not too heavy) which means more drive force per kg and this will ultimately result in better car acceleration.
This is however only considering panel powers and overall car weights. On top of this, the motor torque at stall and low rpm (ie at the start of a model solar car race) for a higher powered car is also somewhat less compared to that of a lower powered car in relation to panel power (due to the higher motor current and therefore greater resistive losses).
As a quick example, consider one of the DC motors that the challenge cars use at stall. Here, the motor has no mechanical output power (defined as motor torque multiplied by shaft speed) and no frictional/windage losses (dependent on motor speed). All the power is therefore basically going into heating up the windings of the motor (resistive losses) and this is given by P = I^2 x R. Now, for the electronics units that most AIMSCC cars are using these days these are perhaps around 70 to 80% efficient at stall. So if the above formula is rearranged we get:
I(motor current) = SQRT[(Panel power x elect. eff.)/motor resistance]
Which is then the current running through the motor. As you may know the motor torque is directly proportional to the motor current and so using the motor's torque constant from the data sheet the motor torque can then be calculated (motor torque = motor current x torque constant).
Say we wanted to then calculate the stall torque for a 5 and 10W panel in 100% sun where the starting elec. eff. is maybe 80% (for argument's sake let's say that it's the same for both powers) and the motor is the 2232 6V with a torque constant of 8.03 mNm/A and resistance of 0.9 Ohms (motor is 0.81 plus a little extra for the wiring).
For a 5W panel this gives:
8.03 x SQRT(5 x 0.8 / 0.9) = 16.9 mNm
For a 10W panel this gives:
8.03 x SQRT(10 x 0.8 / 0.9) = 23.9 mNm
So in other words the 5W car has roughly 16.9/23.9 x 100 = 71% of the stall torque yet the panel power is at 50%. For a high and low powered car with the same gear ratio and wheel size (let's assume) this then means that the drive force of the 5W car at stall will also be around 71% of the 10W car.
At the same time a 5W car with a 300g chassis will only weigh 0.65 kg (using the 2011 ballasting formula) compared to the 1.9 kg (34%) of a 10W car with a same-weight chassis.
For those of you that are familiar with Newton's second law of motion (F = ma or force equals mass times acceleration) you should then see what this all means straight away (assuming that the drive force dominates). For those of you that aren't, consider the following:
F = ma can be rearranged and car acceleration (m/s/s) then be seen to be equal to a = F/m (note that this force refers to the sum of the forces acting on the car however the slope and rolling resistance forces will be neglected in this example).
If this is then applied to the 5W car, where the drive force is 71% and the mass 34% of the 10W car, this will lead to an initial take off acceleration of a = (0.71*F)/(0.34*m) = 2.1F/m (ie more than twice the initial car acceleration).
Now while this all looks very impressive, it must be remembered that this is only at the very start of a model solar car race (initial take off). As soon as the car starts to move and speed up, the motor current will quickly drop and this will then also quickly reduce the drive force of the car. Because of this drop in drive force it will then no longer dominate and instead begin to balance out with the forces opposing the car's motion (ie rolling resistance/cornering and air drag).
In other words, the sum of the forces on a model solar car will eventually fall to zero and forward car acceleration cease to occur (ie a top speed will be reached).
Due to the drop in motor current the higher resistive losses in the motor of the 10W car will also be reduced considerably once in motion (motor current and torque are just about halved after 1 or 2m of racing meaning that resistive losses are quartered).
By the time a car has moved just 5m in a race in good sunlight, motor speed will typically be up at around 10000rpm and at these sorts of speeds the 2232 motor will not be all that far away from its max efficiency (on either a high or low powered car).
This being the case, the motor efficiency of the 5W car will no longer be superior and, providing that the elect. eff. is the same for both cars, the mechanical output power of the motor on the 5W car will drop to basically half of that of the 10W car (ie the motor torque and car drive force of the 5W car will be half of that of the 10W car at the same car speed if both cars have the same gear ratio).
Having twice the drive force at the same car speed therefore means that the 10W car has more drive force to overcome rolling resistance, cornering and air drag. Although rolling resistance is dependent on the mass of the car and will therefore be greater for a heavier and higher powered car (nearly 3 times as much if rolling resistance is taken to be directly proportional with car weight and independent of speed), air drag is however related to car aerodynamics and velocity (squared) which means that it will be similar for high and low powered cars alike. Since air drag is the greatest retarding force at top speed in good sun conditions (around twice the rolling resistance drag of a higher powered car and 4 or 5 times for a lower powered car), the extra drive force to overcome it is therefore the reason why higher powered cars ought to have a higher top speed.
So if a notably superior car acceleration for a low powered car only exists for a relatively short distance at the start of a model solar car race, is it important? Sure it is. In this period it gives the lighter and lower powered cars the biggest opportunity to get ahead in a race before slowly being reeled back in by a car with a higher power.
In order to make use of this acceleration to the full extent wheel slip must be minimised on a lower powered car. If not reduced, too much wheel spin will occur and this can quite easily cost half a second in race time.
Since wheel slip depends on the load that the drive wheel is carrying, the drive force (dependent on motor torque, gearing and wheel size) and the wheel slip coefficient, having such a high drive wheel force to car weight ratio makes lower powered cars far more prone to suffering from wheel slip at the very start of a race. Keep this in mind when designing your car.
Before finishing up it should be mentioned that while the gear ratios of the 5 and 10W cars in the above example were taken to be the same, the ratio of a higher powered car is in general a little higher.
One reason for this is because higher powers tend to have a higher voltage panel and so can reach higher motor rpm without the motor voltage exceeding the panel's max power voltage.
Another reason is to get the motor current to drop more quickly in an attempt to improve motor and electronics efficiencies during the earlier stages of a race. This may then reduce the power loss over the course of a 1 lap race slightly and lead to a small improvement in race time, even though it will result in the end top speed of the car suffering somewhat (due to exceeding the panel's Vmp and therefore no longer receiving max panel power).
Finally, remember to concentrate on car quality and minimising your losses rather than focusing on panel power. The 2011 ballasting formula is there for a reason and this is to try and make things as fair as possible for all powers.
A well built and designed low powered car (particularly one with very good aerodynamics) will very readily have a higher top speed than a not so well built and designed high powered car. Similarly, a good quality high powered car can quite easily accelerate away from the start gate more quickly than a not so good lower powered car.
Just a few more things to consider,
Marc
Approximating Model Solar Car Top Speeds (2011 ballasting).xls
