Some people refer to it as "Torque Multiplication"... we simply call it added acceleration!
The principle behind this is simple. It's easier for the engine to turn the rear wheels if the torque gearing is "low". Low gearing is actually a numerically higher ratio than "high" or "tall" gearing.
Why are there different differential ratios?
Different amounts of torque and horsepower in different engines make the need for different gear ratios necessary. When an engine has a lot of torque, it generally means that it has a low maximum RPM, therefore, to attain reasonable highway speeds, you need a "tall" differential. When RPM levels are higher, diff ratios can be made lower to give faster acceleration. BMW does not chose differential ratios arbitrarily, they are chosen for a specific compromise of acceleration and cruising comfort.
Why chose a different torque ratio?
You and BMW may have a different idea of how your car should perform. Having the right torque gear ratio and lowering your ratio will give you better acceleration, but at a loss of theoretical top speed, and a higher cruising RPM.
What is theoretical top speed?
Theoretical top speed is the mathematical calculation of maximum engine RPM, divided by the top gear ratio of the transmission, divided by the differential ratio, which gives the number of wheel rotations per minute. When you take the circumference of the wheel and divide it into a mile, you can determine the minutes taken to cover a mile, or, the miles per hour the vehicle will travel.
Here's an example using some typical numbers. Assuming a maximum engine speed of 7000 RPM, a top gear ratio 0.83:1, a differential ratio 3.15:1 and a wheel circumference of 80 inches, we find that at your maximum RPM:
If we have...
7,000 / 0.83 = 8,434 driveshaft revolutions per minute
8,434 / 3.15 = 2,677 wheel revolutions per minute
Inches in one mile = 63,360
63,360 / 80 = 792 wheel revolutions per mile
...we can calculate...
2,677 / 792 = 3.38 miles per minute
...or in miles per hour...
3.38 x 60 = 202.8 mph
If you do the calculations, you'll see that most vehicles are incapable of reaching their theoretical top speed. In some cases this is due to factory-installed electronic vehicle speed limiters, but just as often, the engine simply cannot overcome the vehicle's aerodynamics.
How much am I losing by going to a lower differential ratio?
Again, you can do the math. Take your current gear ratio, and subtract it from the new ratio. Divide the difference by the original ratio and multiply the result by 100. This will give you your percentage change. Let's look at another example...
If your current differential ratio is 3.15, and your desired ratio 3.46:
3.46 - 3.15 = 0.31
0.31 / 3.15 = 0.098
0.098 x 100 = 9.80 (or 9.8%)
Your difference is 9.8%. Now, you can take your current RPM at any given speed and add 9.8% to see your new engine speed. If you are currently turning 3000RPM at 70mph with a 3.15:1 differential, you will be turning 3294RPM with a 3.46:1 differential.
The major gain here is getting the engine into it's peak power band at a 10% lower road speed, which translates to faster acceleration.
What Limited Slip Options Do I Have?
There are multiple choices of limited slip available for various BMW applications. Choosing the correct one for your needs can be confusing. The primary choices are as follows;
Salisbury Style: This is a clutch type limited slip differential. BMW has used Salisbury style differentials almost exclusively.
Advantages: Very flexible in design, can be set up for mild street use to prevent wheel spin in snow/gravel, through to various versions of race application. Static Lock is always available and can be set from 10% to 75% of engine torque depending on the customer's need. Dynamic lock application can be custom set with use of different ramp angles on the pressure plates.
Disadvantages: Generates heat in hard use, requiring frequent fluid changes and high quality parts. Cheap clutches will burn out very quickly. If one wheel loses traction, differential reverts back to static lock.
Viscous Style: BMW used this style first in the rear of the E30 325iX, it is also used by Subaru and others. The true viscous unit uses the shearing action of discs splined to the side gears rotating in proximity of discs splined to the housing to generate heat. The unique properties of this fluid are such that heat increases its viscosity ( instead of lessens) and it is this "drag" that hinders relative moment of the rear wheels.
Advantages: No wear parts, and the ability to generate lock with only one wheel getting traction.
Disadvantages: Very limited application styles, virtually impossible to properly rebuild. Very reactionary in that ther has to be relative wheel motion to generate heat, and therefore there is a significant delay in lock activation. The total amount of lock is the least of all types of limited slip.
Motorsport M-Variable Style: The M-variable is really a hydraulic/friction disc mechanism. That a viscous fluid is used offers some sealing and pressure advantages, but the fluid itself is not responsible for lock. The dynamic lock, over and above the static, is a result of fluid being forced into a moveable chamber that transfers pressure onto the disc pack from the "pumping" action of the one spinning rear wheel.
Advantages: Can generate dynamic lock with only one wheel on the ground
Disadvantages: Reactionary, in that when the minimal amount of static lock ( which dissipates from day one, with no mechanism to maintain the percent) is overcome and one wheel spins, there is a delay until the pump moves the fluid from the reservoir to the pressure chamber. And, you can never get 100% lock with this one either.
Torsen/Gleason style: This is a gear driven unit that has no clutches. It uses centrifugal forces to create the torque biasing required to drive both wheels. BMW used a Torsen brand differential in some Z3 models, but not the M versions. Quaife is a commonly known brand that uses the Torsen/Gleason method.
Advantages: No wear parts
Disadvantages: Reactionary, due to no static lock. Reverts to an open differential if one wheel completely looses traction, such as on ice, or if it lifts from the road surface.