[p0lar]

Almost as touchy a subject as the existence of the God (Higgs-Boson) particle, the true winner of a French civil war, or the effectiveness of nuclear proliferation in the middle east, is that of E46 ///M3 brake bias.

I aim to clear the confusion by posting rationale, equations, diagrams, and sample calculations for consumption en masse with respect only to the braking system itself. I do not aim to explain how brakes work except with relation to the formulas themselves as it's been timelessly reiterated in just as many different ways all over the Internet. I also will not factor tire widths and the coefficient of friction between various tire and surface combinations.

Though I should really start with rationale and theory, I'm going to get to the core of it all -- equations, because that's what everyone wants.

Please do not hesitate to contact me by whatever means necessary with corrections to any part of this post - it is meant to serve as a reference!

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Dimensions required: (default E46 M3 specifications listed in parentheses)

1. Master cylinder piston(s) and Sizes (MK60-based vehicles, >09/2002: 25mm front, 22mm rear) (MK20-based vehicles, <09/2002: 25mm front, 20mm rear)
2. Pedal ratio (3:1 mechanical lever)
3. Coefficient of friction of pad material, front and rear (0.36)
4. Brake pad width, front and rear (Front: 61mm; Rear: 54mm)
5. Caliper types [fixed/floating] (floating)
6. Number and sizes of all pistons in the brake caliper (Front: 1 piston, floating, 60mm; Rear Standard: 1 piston, floating, 42mm; Rear ZCP/CSL: 1 piston, floating, 46mm)
7. Rotor diameter (Front Standard: 325mm; Front ZCP/CSL: 345mm; Rear: 328mm)

Constants:

• pi = 3.141592653589793238462643383279502884197169399375 10582097494459230781640628620899862803482534211706 79821480865132823066470938446095505822317253594081 28481117450284102701938521105559644622948954930381 96442881097566593344612847564823378678316527120190 91456485669...
  

Variables:

• Leg Force = 150 pounds

The first element necessary to get started is the individual line pressures generated for both the front and rear pistons within the master cylinder, obtained by Equation 1:
Equation 1: P = F_pedal / A_piston

• P = Pressure
• F_pedal = Force due to pedal operation
• A_piston = Area of master cylinder piston

The Force exerted by the pedal is obtained by Equation 2:
Equation 2: F_pedal = F_leg · Ratio_pedal

• F_pedal = Force exerted by pedal operation
• F_leg = Force exerted on pedal from leg
• Ratio_pedal = Distance from exertion of leg force to pivot point divided by distance from pivot point to master cylinder piston actuator

The Area of the master cylinder piston is obtained by Equation 3:
Equation 3: A_piston = pi · r_piston²

• A_piston = Area of master cylinder piston
• r_piston = radius of master cylinder piston

Equation 4 is used to determine the radius from a given diameter:
Equation 4: r_piston = D_piston / 2

• r_piston = radius of master cylinder piston
• D_piston = diameter of master cylinder piston

Brake torque generated at the wheel, individually T_f and T_r, is given by Equation 5:
Equation 5: T_{f,r} = P · A_total · R_effective · c_F

• T_{f,r} = Torque at wheel
• P = Pressure
• A_total = Combined area of all pistons within a caliper
• R_effective = Effective radius of pad placement to rotor
• c_F = Coefficient of friction of pad material

The total combined effective area of the caliper pistons is obtained by utilizing Equations 6, 7, 8 and 9 based on their respective caliper configurations (Floating: single-piston, Floating: 2-piston, Fixed: 4-piston, Fixed: 6-piston, Fixed: 8-piston):

For a Floating: single piston caliper:
Equation 6: A_total = 2 · pi · r_piston²

• A_total = Total combined effective area
• r_piston = radius of piston ^{*
  *} - refer to Equation 4 to obtain radius from diameter)

For a Floating: 2-piston and Fixed: 4-piston caliper:
Equation 7: Atotal = 2 · pi · (rpiston_1² + rpiston_2²)

• A_total = Total combined effective area
• r_{piston_1} = radius of piston 1 ^*
• r_{piston_2} = radius of piston 2 ^{*
  *} - refer to Equation 4 to obtain radius from diameter)

For a Fixed: 6-piston caliper:
Equation 8: Atotal = 2 · pi · (r_{piston_1}² + r_{piston_2}² + r_{piston_3}²)

• A_total = Total combined effective area
• r_{piston_1} = radius of piston 1 ^*
• r_{piston_2} = radius of piston 2 ^*
• r_{piston_3} = radius of piston 3 ^{*
  *} - refer to Equation 4 to obtain radius from diameter

For a Fixed: 8-piston caliper:
Equation 9: Atotal = 2 · pi · (r_{piston_1}² + r_{piston_2}² + r_{piston_3}² + r_{piston_4}²)

• A_total = Total combined effective area
• r_{piston_1} = radius of piston 1 ^*
• r_{piston_2} = radius of piston 2 ^*
• r_{piston_3} = radius of piston 3 ^*
• r_{piston_4} = radius of piston 4 ^{*
  *} - refer to Equation 4 to obtain radius from diameter

The effective radius is calculated using Equation 10:
Equation 10: R_effective = r_rotor - (w_pad / 2)

• R_effective = effective radius of rotor
• r_rotor = radius of rotor ^*
• w_pad = width of brake pad ^{**
  *} - refer to Equation 4 to obtain radius from diameter
  ^** - width of brake pad is an estimate of the distance the center of the piston(s) from the rotor's diameter and may not be the most precise way to gauge application of force (Noted by IntelOne)

Finally, Equation 11 gives the ratio of the front brake torque to that of the rear and is an index reflecting overall front:rear bias:
Equation 11: Bias = T_f / T_r

• Bias = index representing front to rear wheel brake torque ratio
• T_f = Brake torque generated at front wheel
• T_r = Brake torque generated at rear wheel

Equation 12 is used to convert from a ratio to a percent front:rear bias:
Equation 12:

% Bias_front = (T_f · 100%) / (T_f + T_r)

% Bias_rear = (T_r · 100%) / (T_f + T_r)^*

• % Bias_front = Bias percentage at front wheel
• % Bias_rear = Bias percentage at rear wheel
• T_f = Torque generated at front wheel
• T_r = Torque generated at rear wheel
  ^*alternately; one bias percentage can be calculated by subtracting the other from 100%

[p0lar]

This is a quick chart I've constructed using Excel to illustrate the bias generated by various combinations of brake calipers/rotors. Please eMail me any additional caliper/rotor/pad combinations you would like to see crossed, and dimensions of the pad swept area, rotor diameter and piston size and count, if possible.

Edit | 08.02.2012: Added WilWood SL6R to front calipers with 325mm rotor

Edit | 08.09.2012: Added C6 Z06 Corvette to front calipers with 355mm rotor

Edit | 09.23.2012: Added a multitude of calipers. Included FULL bias, inclusive of master cylinder as well as % bias front/rear. Added deviation from standard/ZCP E46 M3 biases in terms of %F (front) or %R (rear) movement - this could give more data to the layman. This was a massive update, please review to let me know if there are any issues with my calculations or data presentation.

Edit | 09.24.2012: Added 4-Piston RacingBrake caliper.

Edit | 03.27.2013: Removed differentiation from OEM bias charts - they were causing way too much confusion. Added charts for differences in DSC (MK60 (>09/2002) vs DSC3 (<09/2002). Also added a small table to show calculated bias results for OEM E46 M3s and a warning for those who wish to downgrade to ZCP/CSL brakes on a DSC3-based system from <09/2002.

FULL BIAS INCLUDING MASTER CYLINDER PISTON DIFFERENTIATION

E46 M3 with MK60-based Master Cylinder (Front: 25mm, Rear: 22mm) (post-09/2002)


E46 M3 with DSC3-based Master Cylinder (Front: 25mm, Rear: 20mm) (pre-09/2002)


Warning: If you replace your pre-09/2002 DSC3-based OEM E46 M3 brakes with ZCP/CSL brakes, including the rear caliper, you are doing yourself a grand disservice. This causes an almost 5% shift in rear bias from the actual ZCP configuration, which was never designed for the DSC3-based master cylinder. Consider upgrading to a MK60-based master cylinder at the same time - the MK60-based power-booster may also be required.

[p0lar]

Here is a sample calculation for an E46 M3 with the competition package. I apologize for converting to SAE, but most people here will comprehend ft-lbs of torque much easier than metric units.

Front Master Cylinder Pressure Calculation:
Equation 4:
25mm / 2 = 12.5 mm = r_piston

Equation 3:
(12.5 mm)² * 3.141592 = 490.87375 mm² = A_piston

Equation 2:
150 lbs x 3 = 450 lbs = F_pedal

Equation 1:
450 lbs / 490.87375 mm² = 0.917 lbs/mm² = 591 PSI = Pressure (front lines)

Rear Master Cylinder Pressure Calculation:
Equation 4:
22 mm / 2 = 11.0 mm = r_piston

Equation 3:
(11.0 mm)² * 3.141592 = 380.1326 mm² = A_piston

Equation 2:
150 lbs x 3 = 450 lbs = F_pedal

Equation 1:
450 lbs / 380.1326 mm²= 1.183 lbs/mm². = 764 PSI = Pressure (rear lines)
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Front Caliper Brake Torque Calculation:
Equation 4:
60 mm / 2 = 30 mm = r_piston

Equation 6:
2 * 3.141592 * 30 mm² = 5655 mm² = 56.55 cm² = 8.77 in² = A_total

Equation 4:
345 mm / 2 = 172.5 mm = r_rotor

Equation 10:
172.5 mm - (61 mm / 2) = 142 mm = 5.59 in = R_effective

Equation 5:
591 PSI * 8.77 in² * 5.59 in * 0.36 = 10430 in•lb / 12 in = 869 ft•lbs = T_{f}

Rear Caliper Brake Torque Calculation:
Equation 4:
46 mm / 2 = 23 mm = r_piston

Equation 6:
2 * 3.141592 * 23 mm² = 3324 mm² = 33.24 cm² = 5.15 in² = A_total

Equation 4:
328 mm / 2 = 164 mm = r_rotor

Equation 10:
164 mm - (54 mm / 2) = 137 mm = 5.39 in = R_effective

Equation 5:
764 PSI * 5.15 in² * 5.39 in * 0.36 = 7633 in•lb / 12 in = 636 ft•lbs = T_{r}

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Ratio of Front to Rear Torque Calculation:
Equation 11:
869 ft.lbs. / 636 ft.lbs. = 1.37:1 F:R Ratio

If you neglect line pressure, since that's usually constant unless someone changes their master cylinder, and simply look at the variables that are likely to change (rotor diameter, pad width, # & size of pistons, pad compound, etc), you would arrive at

Equation 11 (modified):
(8.77 in² * 5.59 in * 0.36)/(5.15 in² * 5.39 in * 0.36) = 1.77:1 F:R Ratio

Theoretically, if the c_F of the pads is also not changing, or is equivalent, rather, then that could also be dropped from the equation to give a relevant base point for comparison.

[Tarmac_E36_M3]

Very cool.

[lucad2]

This is all Chinese to me, pretty cool though


Sent from my iPhone using Tapatalk

[aznbo187]

Sweet! I wasn't aware that the CSL rear calipers had a larger diameter piston. Do you know how they compensate for that with the rest of the setup? Larger front rotor diameter?

[jcreme]

Hear! Hear! Thanks for putting this up.

[p0lar]

Originally Posted by jcreme

Hear! Hear! Thanks for putting this up.

Well, someone had to do it. There's more coming -- much, much more.

I wish this forum code would let you add sub/superscripts...

[macming]

Wow, thanks for maintaining this chart! The calculations shown here is great for figuring out your brake bias.

I found out the Wilwood setup I'm thinking of running will put me into 1.57:1 ratio, which is VERY rear biased Thankfully, if needed, I have a set of non-zcp rear calipers sitting around, and that will put me back into the 1.8:1 range.

[bmwbob]

welcome to 2002. this isnt the first brake bias chart! great work though!