Regenerative Braking — Force-Speed Envelope & Brake-Pad Thermal Fade (2D)

Braking Event

Initial speed 90 km/h

Brake pedal (target decel) 0.30 g

Motor/inverter power rating 75 kW

Starting battery SOC 55%

Pads cold — full friction capacity

Live Readout

Speed

0 km/h

Deceleration (achieved / demand)

0.00 / 0.00 g

Regen force

0 kN

Friction force

0 kN

Energy recovered

0 Wh

Heat into pads

0 Wh

Brake & Battery State

Pad temperature

25 °C

Friction capacity

100%

Battery SOC

55.00%

Charge accept.

100%

Stop #

0 / 0

Cumulative heat

0 Wh

How it works (2D model)

This 2D companion computes the same regen/friction split as the 3D simulator — F_regen is torque-limited at low speed and power-limited at high speed, tapered by a low-speed cutoff and a battery charge-acceptance limit near full SOC — but plots it directly as a force-vs-speed envelope instead of animating a 3D car, and adds a real physics the 3D scene never models: brake-pad thermal fade.

F_demand = m·a_target
F_regen  = min( m·a_max_regen , P_avail/v , F_demand )   (envelope curve, top chart)
F_need   = F_demand − F_regen                              (asked of the friction brakes)

Pad thermal balance (lumped capacitance):
dT/dt = ( F_friction·v − hA(v)·(T − T_amb) ) / C_pad
hA(v) = hA₀ + hA₁·v            (airflow cools the rotor faster at speed)

Brake fade (friction capacity vs temperature):
k(T) = k₀                                   T ≤ T_fade0
k(T) = k₀·[1 − (1−k_min)·(T−T_fade0)/(T_fade1−T_fade0)]   T_fade0 < T < T_fade1
F_friction = min( F_need , k(T)·m·g )        (capped — can fall short of F_need!)

Every stop dumps friction-brake heat into the pads faster than it can convect away, so pad temperature climbs over the course of a stop and only partly recovers between stops. Cross a fade threshold (here ≈380 °C) and the brakes' own force ceiling starts dropping — so on a hard, repeated stop the achieved deceleration can measurably fall short of what the driver's pedal is asking for, exactly the failure mode that makes engine/regenerative braking on long descents (and disc ventilation on track cars) a genuine safety feature, not just an efficiency one.

Brake pedal (target decel) — a harder demand shifts more load onto the friction brakes whenever the motor is already at its regen ceiling, generating heat faster.
Motor/inverter power rating — a bigger motor pushes the power-limited crossover speed higher, so more of a hard stop stays torque-limited (handled by regen) before friction has to help at all.
Rush-Hour Commute — runs six consecutive stops with a short re-acceleration between them, so pad heat and battery SOC carry over from one stop to the next, the way real stop-and-go traffic does.

The top chart's operating-point dot always sits on the true instantaneous (v, F) pair; the friction-capacity line you see bending downward mid-run is the fade — watch it sag below the demand line on a hot repeated stop.

F demand F regen (envelope) F friction ceiling (fades) speed(t) / T_pad(t)