Pacejka ’94 parameters explained – a comprehensive guide

The Pacejka / Magic Formula (MF) equations were conceived to fit the data gathered from experimental tests with real tires. The equations can then predict the behavior of each specific real tire with great precision. Real Pacejka/MF data sets are heavily protected intellectual property of the tire manufacturers. However, when used in video games where gameplay is important, the Pacejka curves can be fine tuned in a variety of ways in order to achieve the desired result.

This guide explains how each parameter of the Pacejka ’94 specification affects the resulting curve, as well as the typical range for the values.

Longitudinal Force

Pacejka ’94 Longitudinal Force parameters:

Parameter Role Units Typical range Sample
b0 Shape factor 1.4 .. 1.8 1.5
b1 Load influence on longitudinal friction coefficient (*1000) 1/kN -80 .. +80 0
b2 Longitudinal friction coefficient (*1000) 900 .. 1700 1100
b3 Curvature factor of stiffness/load N/%/kN^2 -20 .. +20 0
b4 Change of stiffness with slip N/% 100 .. 500 300
b5 Change of progressivity of stiffness/load 1/kN -1 .. +1 0
b6 Curvature change with load^2 -0.1 .. +0.1 0
b7 Curvature change with load -1 .. +1 0
b8 Curvature factor -20 .. +1 -2
b9 Load influence on horizontal shift %/kN -1 .. +1 0
b10 Horizontal shift % -5 .. +5 0
b11 Vertical shift N -100 .. +100 0
b12 Vertical shift at load = 0 N -10 .. +10 0
b13 Curvature shift -1 .. +1 0

Pacejka ’94 longitudinal formula

Coefficient Name Parameters Formula
C Shape factor b0 C = b0
D Peak factor b1, b2 D = Fz · (b1·Fz + b2)
BCD Stiffness b3, b4, b5 BCD = (b3·Fz2 + b4·Fz) · e(-b5·Fz)
B Stiffness factor BCD, C, D B = BCD / (C·D)
E Curvature factor b6, b7, b8, b13 E = (b6·Fz2 + b7·Fz + B8) · (1 – b13·sign(slip+H))
H Horizontal shift b9, b10 H = b9·Fz + b10
V Vertical shift b11, b12 V = b11·Fz + b12
Bx1 (composite) Bx1 = B · (slip + H)

F = D · sin(C · arctan(Bx1 – E · (Bx1 – arctan(Bx1)))) + V

where:

  • F = longitudinal force in N (newtons)
  • Fz = vertical force in kN (kilonewtons)
  • slip = slip ratio in percentage (0..100)

Initial values for setting up the longitudinal curve

Reference load 4000
Maximum load 13000
b0 1.5
b2 1100
b4 300
b8 -2
all other 0

Load-independent parameters

Load don’t have influence on these. If all other parameters are 0 then the curve keeps itself invariant with respect to the vertical load.

The first four (b0, b2, b4, b8) are the most relevant parameters that define the curve’s shape.

Parameter Typical range Description Related
b0 1.4 .. 1.8 General shape of the curve. Defines the amount of falloff after the peak.
The Pacejka model defines b0 = 1.65 for the longitudinal force.
C
b2 900 .. 1700 Friction coefficient at the peak (vertical coordinate) *1000. D
b4 100 .. 500 Peak’s horizontal position specified as “ascent rate”. BCD
b8 -20 .. +1 Curvature at the peak. The more negative = more “sharp”. Has influence on the falloff afterwards. E
b10 -5 .. +5 Curve’s horizontal shift Sh
b11 -100 .. +100 Curve’s vertical shift Sv
b13 -1 .. +1 Adjustment of the curvature at the peak. Similar to b8 E

Load-dependent parameters

The desired curve’s shape must be configured at the reference load. Typically the related parameters (load independent) must also be tweaked in order to conserve the shape.

These load-dependent parameters must be verified along all the load range, from minimum to maximum, in order to ensure their coherency. The coherency limit of each parameter can be located at the maximum load. The exception is b12, which must be verified at the minimum load.

Parameter Typical range Description Related
b1 -80 .. +80 Change of the friction coefficient at the peak.
Positive = more friction with more load. Negative = less friction with more load.
D, b2
b3 -20 .. +20 Change of the peak’s horizontal position.
Positive = increases ascent rate with load (moves to the left). Negative = decreases ascent rate with load (moves to the right).
BCD, b4
b5 -1 .. +1 Lineal change of the peak’s horizontal position. Similar to b3 but more lineal and with reverse effect positive-negative.
Positive = decreases ascent rate with load. Negative = increases ascent rate with load.
BCD, b4
b6 -0.1 .. +0.1 Quadratic change of the curvature at the peak.
Positive = more flat with load. Negative = sharper with load.
E, b8
b7 -1 .. +1 Change of the curvature at the peak. Same as b6 but more lineal.
Positive = more flat with load. Negative = sharper with load.
E, b8
b9 -1 .. +1 Change of the horizontal shift.
Positive = shifts to the left with more load. Negative = shifts to the right with more load.
Sh, b10
b12 -10 .. +10 Vertical shift when approaching zero load.
Must be verified for coherency at the configured minimum load.
Sv, b11

Lateral Force

Pacejka ’94 Lateral Force parameters:

Parameter Role Units Typical range Sample
a0 Shape factor 1.2 .. 18 1.4
a1 Load influence on lateral friction coefficient (*1000) 1/kN -80 .. +80 0
a2 Lateral friction coefficient (*1000) 900 .. 1700 1100
a3 Change of stiffness with slip N/deg 500 .. 2000 1100
a4 Change of progressivity of stiffness / load 1/kN 0 .. 50 10
a5 Camber influence on stiffness %/deg/100 -0.1 .. +0.1 0
a6 Curvature change with load -2 .. +2 0
a7 Curvature factor -20 .. +1 -2
a8 Load influence on horizontal shift deg/kN -1 .. +1 0
a9 Horizontal shift at load = 0 and camber = 0 deg -1 .. +1 0
a10 Camber influence on horizontal shift deg/deg -0.1 .. +0.1 0
a11 Vertical shift N -200 .. +200 0
a12 Vertical shift at load = 0 N -10 .. +10 0
a13 Camber influence on vertical shift, load dependent N/deg/kN -10 .. +10 0
a14 Camber influence on vertical shift N/deg -15 .. +15 0
a15 Camber influence on lateral friction coefficient 1/deg -0.01 .. +0.01 0
a16 Curvature change with camber -0.1 .. +0.1 0
a17 Curvature shift -1 .. +1 0

Pacejka ’94 lateral formula

Coefficient Name Parameters Formula
C Shape factor a0 C = a0
D Peak factor a1, a2, a15 D = Fz · (a1·Fz + a2) · (1 – a15·γ2)
BCD Stiffness a3, a4, a5 BCD = a3 · sin(atan(Fz / a4) · 2) · (1 – a5·|γ|)
B Stiffness factor BCD, C, D B = BCD / (C·D)
E Curvature factor a6, a7, a16, a17 E = (a6·Fz + a7) · (1 – (a16·γ + a17)·sign(slip+H))
H Horizontal shift a8, a9, a10 H = a8·Fz + a9 + a10·γ
V Vertical shift a11, a12, a13, a14 V = a11·Fz + a12 + (a13·Fz + a14)·γ·Fz
Bx1 (composite) Bx1 = B · (slip + H)

F = D · sin(C · arctan(Bx1 – E · (Bx1 – arctan(Bx1)))) + V

where:

  • F = lateral force in N (newtons)
  • Fz = vertical force in kN (kilonewtons)
  • slip = slip angle in degrees
  • γ = camber angle in degrees

Initial values for setting up the lateral curve

Reference load 4000
Maximum load 13000
Camber 0
a0 1.4
a2 1100
a3 1100
a4 10
a7 -2
all other 0

Curve’s shape and horizontal behavior with load

The curve’s shape itself always depend on load at the horizontal axis according to the parameters a3 and a4.

Parameter Typical range Description Related
a0 1.2 .. 1.8 General shape of the curve. Defines the amount of falloff after the peak.
The Pacejka model defines a0 = 1.3 for the lateral force.
C
a2 900 .. 1700 Friction coefficient at the peak (vertical coordinate) *1000. D
a3* 500 .. 2000 Peak’s horizontal position at the reference load, specified as “ascent rate”. BCD
a4* 0 .. 50 Change of the peak’s horizontal position with load. Smaller value = bigger change with load. BCD
a7 -20 .. +1 Curvature at the peak. The more negative = more “sharp”. Has influence on the falloff afterwards. E
a9 -1 .. +1 Curve’s horizontal shift Sh
a11 -200 .. +200 Curve’s vertical shift Sv
a17 -1 .. +1 Adjustment of the curvature at the peak. Similar to a7. E

* Configure the horizontal behavior with load

Load-dependent parameters

The desired curve’s shape must be configured at the reference load. Typically the related parameters (load independent) must also be tweaked in order to keep the shape.

These load-dependent parameters must be verified along all the load range, from minimum to maximum, in order to ensure their coherency. The coherency limit of each parameter can be located at the maximum load. The exception is a12, which must be verified at the minimum load.

Parameter Typical range Description Related
a1 -80 .. +80 Change of the friction coefficient at the peak.
Positive = more friction with more load. Negative = less friction with more load.
D, a2
a6  -2 .. +2 Change of the curvature at the peak.
Positive = more flat with load. Negative = sharper with load.
E, a7
a8  -1 .. +1 Change of the horizontal shift.
Positive = shifts to the left with more load. Negative = shifts to the right with more load.
Sh, a9
a12  -10 .. +10 Vertical shift when approaching zero load.
Must be verified for coherency at the configured minimum load.
Sv, a11

Camber-dependent parameters

They must ve verified along all the camber range in order to ensure their coherency. The coherency limit of each parameter can be located at the limits of the camber range. These parameters don’t have effect if camber is 0.

Parameter Typical range Description Related
a5 -0.1 .. +0.1 Change of the peak’s horizontal position.
Positive = decreases ascent rate with camber (moves to the right).
Negative = increases ascent rate with load (moves to the left).
BCD, a3
a10 -0.1 .. +0.1 Change of the horizontal shift.
Same sign as camber = shifts to the left. Opposite sign as camber = shifts to the right.
Sh, a9
a13 -10 .. +10 Change of the vertical shift according to camber and load.
Same sign as camber = shifts upwards. Opposite sign as camber = shifts downwards.
The more load the more camber effect.
Sv, a11
a14 -15 .. +15 Change of the vertical shift.
Same sign as camber = shifts upwards. Opposite sign as camber = shifts downwards.
Sv, a11
a15 -0.01 .. +0.01 Change of the friction coefficient at the peak.
Positive = less friction with camber. Negative = more friction with camber.
D, a2
a16 -0.1 .. +0.1 Change of the curvature at the peak.
Same same sign as camber = more flat. Opposite sign as camber = sharper.
E, a7

Pacejka Simplified FormulaSimplified Magic Formula with constant coefficients

This simplification uses the four dimensionless B, C, D, E coefficients only. Numerical values are based on empirical tire data. The coefficients do not depend on load, camber, or further parameters. Source

Coefficient Name Typical range Typical values for longitudinal forces
Dry tarmac Wet tarmac Snow Ice
B Stiffness 4 .. 12 10 12 5 4
C* Shape 1 .. 2 1.9 2.3 2 2
D Peak 0.1 .. 1.9 1 0.82 0.3 0.1
E Curvature -10 .. 1 0.97 1 1 1

F = Fz · D · sin(C · arctan(B·slip – E · (B·slip – arctan(B·slip))))

where:

  • F = tire force in N (newtons)
  • Fz = vertical force in N (newtons)
  • slip = slip ratio in percentage (0..100, for longitudinal force) or slip angle in degrees (for lateral force)

* The Pacekja model specifies the shape as C=1.65 for the longitudinal force and C=1.3 for the lateral force.