Mathcad 14 File:
Power Fit
Power Approximations are very poor
Torque Shape:
Insane Mode Specifications
3.1 seconds 0-60 mph
Front and Back Motor Spec: Power 691 hp
Conventional Fuse: 1300 amp battery limitation
Battery Peak Power: 1300 A x 398.6 = 695 hp
System Power Efficiency: Best Guess 81%
Inverter and/or Algorithm sets max accel ~ 1g.
Ludicrous Mode Specifications
2.8 seconds 0-60 mph with Ludicrous Speed Upgrade
Front and Back Motor Power Spec: 762 hp
Electronic Fuse: 1500 amp "effective" battery upgrade
Battery @50% SOC Peak Power: 1500 A x 360V = 724 hp
System Power Efficiency: Best Guess ~ 81% for 586 hp
Peak Motor Torque: 713 ft lb.
New Inverter Algorithm to maximize Ludicrous acceleration.
85 kWh battery with all-wheel drive
90 Regenerative Braking upgrade increases range 6%
90 kW important for X
P85 260 miles range (EPA). Estimate 90 kW 272 miles@65 mph
New 253 mile EPA
3.1 seconds 0-60 mph
2.8 seconds 0-60 mph with Ludicrous Speed Upgrade
11.7 seconds ¼ mile
10.9 seconds ¼ mile with Ludicrous Speed Upgrade
259 hp
713 lb-ft Total motor torque
Ludicrous Mode
155 mph top speed
Model S Emergency Response Guide says
We assume that the implementation of the New Algorithm for Rear Motor Torque & Power Split in Ludicrous Mode, will provide the maximum tire grip acceleration, 1.1g, and not optimize Efficiency, that is,
What is the minimum acceleration needed to meet the 0 to 60 mph in 2.8 seconds?
Thus a constant 21.43 mph/s in sufficient to meet our spec.
However, torque or acceleration is not constant. The 713 ft lb is peak torque.
This more in-depth analysis is what follows.
For the acceleration let's units of g, that is, of the earths' gravitation pull =
Model S (Dual Motor 4WD) Ludicrous P85D Specs
1/4 mile: 10.9 seconds
0 – 60mph: 2.8 seconds (3.2s for 10% heavier Model X)
Tech details:
10% acceleration improvement (1.1 gs)
New fuse electronic fuse allows current increased from 1300 to 1500 Amps, but ‘Ludicrous Mode’ will only be available between 30-60mph due to traction limitation at lower speeds.
Macro Model:
Vehicle mass (Mcurb), Coefficient of tire friction
The vehicle also has rotational energy from rotating tires, motor rotor, and gear box.
A factor km, which multiplies the mass,
Mcurb = 4936 lbm,
Then acceleration (a) is given by:
Newton's Second Law:
See pg 3 for Section on Traction Control.
Then the Torque required to get to g = 1.1, requires that Torque be at least:
Torque_max_g
Dual Motor Model S is a categorical improvement on conventional all-wheel drive systems. With two motors, one in the front and one in the rear, Model S digitally and independently controls torque to the front and rear wheels. The result is unparalleled traction control in all conditions.
Effective Mass Coeff*: Rotating Inertia Impact on Propulsion and Regenerative Braking for Electric Motor Driven Vehicles
Motor Inductance, Resistance and Maximum RPM: Estimated vs Actual
It will likely be a long time before anyone can actually measure these parameters to verify, but i'll take a shot at calculating them.
Why is this important or of any interest? Because these electrical characteristics are used in the calculations made by the inverter to drive the motor using a Direct Torque Control (DTC) scheme, described in patents and discussed in another thread.
For a 3-phase, 4-pole wye motor with the poles wired 2s2p, and with 4 Turns per pole using 16 AWG copper wire 12-in-hand:
phase inductance ~ 493 nH
phase resistance ~ 5.3 mR
What is the Estimated Motor Power needed to meet the 0 - 60 mph in 2.8 s performance,
Ludicrous Mode: The current has been increased from 1300 to 1500 amps (15 %), the total dual motor power increased from 682 hp to 762 hp (12%), and the total torque from 707 to 713 ft lbf.
However, as we saw in section I, the System Power to the motor @81% System Efficiency is limited to 649 hp
There is a basic relationship between Torque, Motor RPM, and Motor Power:
Assume that there is No Traction Control, this is tires can slip.
Thus initially, full torque is applied to the wheels, until max motor power limits the torque.
Refer to the below plot and examine the Power versus time profile.
The Power is given by:
This is shown in the graph below. Tire velocity, vPmax, to get to max motor power and Torque =
The time to get to max motor power is tPmax.
There are 2 paths to get to the max power:
Elon Musk said that the peak acceleration is 1.1 g.
http://www.bbc.com/autos/story/20150720-teslas-28-second-model-s-the-p85d-ludicrous
Velocity at Max Power
What is more important T or hp? They are related: Both must be sufficient for goal.
They are not independent of one another.
Given one, that specifies the other.
Horsepower = (Torque x RPMs) / 5252.
Also,
Most important: a = F/a.
At 5252 RPMs the horsepower and torque will be exactly the same torque supplies the force that gets you accelerating, but must have enough Power to overcome Power losses, e.g. windage, that increases with velocity.
This gives an L/R time constant of 93 usec. If we use 3 tc to reach full current, then the motor max speed ~ 26,858 rpm.
With a 9.7:1 gearbox the theoretical max speed would be 228 mph.
For a P85D to reach 155 mph the motor(s) will need to spin at 18,200 rpm and make at least 50 kW each. The inverter would need to be generating a current waveform of 606 Hz, and the per phase L/R time constant would have to be less than 138 usec.
Send data when you get some...
Pm( ω) = T(ω)⋅k⋅2⋅ π⋅ ω RPM
The average power from the start at 0 power to the peak of Powermax is 1/2 Powermax.
The relationship; is shown at the right.
Because of limits on tire traction, the Initial Peak Acceleration in Ludicrous mode is at 1.1 g.
a = Peak Motor Torque x GR/m.
Past a certain velocity (v),
Torque-Power Limited = Max Power*GR/v_limit.
The 2 motor power curves belong to
two different Traction Models:
#1: No Traction Control, Red Curve
#2: With Traction Control, Blue Curve
The most critical part
of the model is the
1.11 seconds of Peak
Power from the time
the acceleration falls
from 1.1 g to the
2.8 second spec limit.
tPmt is the time to Peak Power = 1.46s
tgfall is the time for acceleration(g)
to fall below 1.1g = 1.69 s.
These times are in seconds.
We need sufficient power to get to get an additional 5.7 mph in an additional 0.55 s.
This requires an "average" acceleration of a = 5.7 mph/0.55 s = 0.47 g.
Estimate of EPA Rating
for 90 kW hr battery,
Given 260 miles for 85.
For a vehicle velocity, v,
The vehicle also has rotational energy from rotating tires, motor rotor, transmission
The factor km times the mass,
By setting these two equal and using a little algebra we find tmax.
What Max Power do we need to meet the 0 - 60 mph in 2.8 s with No Traction Control?
For a vehicle velocity, v,
AR gives the height, height =
High Performance Tires, AR ~ 60
Ultra High Performance Tires, AR <= 55
Let's say we want a Power sufficient to maintain 1.1 g for
v = 1.1 g x 2.25s = 54.3 mph
Assume a 5% Power Loss
System Efficiency:
100% & 50% SOC Voltage:
"width mm" / "AR" Z "R = radius(in)"
265 /35ZR21
High-performance tires are usually wider than other types of tires. This provides a greater surface area that clings to paved surfaces. It's important to note that most high-performance tires are intended for driving in warm, dry conditions. There are also high-performance all-season and winter tires available. They will help your vehicle grip the road in bad weather.
Battery and System Power
85 kW-hr Battery
New Algorithm for Rear to Front Motor Torque & Power Split to maintain peak acceleration: = TSplit = 503/259 ~ 1.94
Gear Ratio:
Aspect Ratio, AR = /35
Ludicrous Mode Max Power:
Tires Load Index 98V Michelin PS2 Conti
Limit:
Front 245/35R21
Rear 285/30R21
Max Dual Motor Torque:
From TeslaMotors.com/models
Ultra High Performance Tire
Max 1.1 g
Curb Weight:
Aerodynamic Drag Coeff (TM):
Average Wind Velocity:
100%
85% SOC
60kWh
Cross Wind Drag Coff:
Effective Cross Wind V:
1 RPM =
Shape Correction Factor:
Vehicle Frontal Dimensions:
Air Density, tire resistance:
Drag Frontal Area
Tire Rolling Resist, Hys:
Road Rolling Resistance:
Effective Mass Coefficient:
EPA Range Spec for P85 is 253 miles.
See page 6 for
Some simple motor design parameters that might help explain this include:
the physical geometry (diameter and length) of the motor; the number of magnetic poles which sets the operating speed range; the electrical characteristics of the motor (inductance and resistance) determined by the number of turns in the windings and the wire gage used, which in turn set the voltage and current limits.
For example,
A high pole count motor with large gage wire will have higher low-end torque (motor B) but fall off in speed sooner than a motor A. lower pole count high-speed motor wound with smaller gage wire .
Motor B is wound to use higher currents, and motor A to use higher voltages.
Simple Step Model of Tire Traction (
Depending on road conditions, Tires do not have perfect grip, they may slip. Vehicle acceleration, aveh is limited to the maximum tire traction (tiremax_g) = 1.1g.
Case #1
No Traction Control, but no tire slip.
Max motor power and torque are applied to tires. Perfect Tires that do not slip.
Acceleration can exceed 1.1 g.
Battery Mass:
Angular Velocity Symbol,
RPM/1000 Symbol, ωk
The Motor's Max Torque results in a
peak acceleration, amax = Max Force/mass:
Angular Vel Ω @Max Power:
Convert velocity to RPM
The Max Power can sustain this Torque
until the velocity, vfall, reaches
Tire Velocity at Torque Fall:
Tire Velocity to kRPM:
What?
Then at time, tmax, the a falls
Road Resistance, Ft:
RPMpmax for Max Power:
Note:
Air Drag Force, Fa:
Total Opposing Force, Fo:
Torque/Force Falloff Curve:
The Force applied to the road, F is
Tm is Torque of motor
Fmot, Tractive Force from motor, not from slipping tires:
Estimated battery cost
Agrees with curve Right-->
Total = $24,000.
Need this for Definition of a.veh before a(v)
Newton's Third Law of Motion:
Can only unse a1 for Case of Perfect Tires. If not, then tires spin and v of tires > v of road or vehicle. We can not use Fmotor(v)/m to find a.
This method does not work
Velocity g fall, a
Discrete method works: If accel is slowly varying
a2 acceleration is allowed by high performance tires on dry road.
a as function of time, not velocity
RPM at g fall:
Tesla Controller Patent
Case 3: We end up getting the same effective peak torque, we just don't waste the power put into spinning wheels.
Dot Function:
a
T
Shape
(v)
Drag Times Torque/speed Curve - See Graph Below
Blind Alley - Going around in circles. Just use T@1.1 g:
Both Power and Torque fall at RPM for Pmax
Does not meet Specs
The ¼ mile performance simulation is
approximate, because it depends
<---
W use a crude model for Regeneration and we have no data on aerodynamic parameters.
Dyno Data by Drag Times for 2015 Tesla S P85D
Dyno Torque (Red)
DynoTorque Shape x 1000 (Blue), Extracted from Dyno Torque
http://www.dragtimes.com/2015-Tesla-
Model-S-Videos-27143.html
Calculated P90 EPA Range: 230 Miles
VI. Graphs Compare with Torque & Power Curves at: http://www.teslamotors.com/performance/acceleration_and_torque.php
Model S
http://www.dragtimes.com/2015-Tesla-Model-S-Videos-27143.html
2015 Tesla Model S P85D
Max 864 ft lbf
Max 413 hp
This speeds up plotting
SOC
f
is 10% at recharge. 400V HV battery idle power is Po.
Change in State of Charge = 1 - SOCf
What do we know?
Is this sufficient to get car to 60 mph
distance per charge = cruise velocity x time of cruise, where time of cruise = Battery Energy/Power Diss
If decelerating, charge battery with Regen fraction of energy.
Refer to:
The US06 cycle represents an 8.01 mile (12.8 km) route with an average speed of 48.4 miles/h (77.9 km/h), maximum speed 80.3 miles/h (129.2 km/h), and a duration of 596 seconds. Sampling can be either 1 Hz or 10Hz
The Federal Test Procedure (FTP) is composed of the UDDS followed by the first 505 seconds of the UDDS. It is often called the EPA75.
EPA 2008 Cycle MPG Fuel Economy Least Squares Fit Regression for Range
Published EPA
Plots of EPA Dynamometer Vehicle Testing Profiles
Petroleum-equivalent efficiency
See also: Electric car: Comparison with internal combustion vehicles (ICEVs)
The Roadster does not actually use gasoline; therefore, petroleum-equivalent efficiency (mpg, l/100 km) cannot be measured directly but instead is calculated using one of several different methods:
A number comparable to the typical Monroney sticker's "pump-to-wheel" fuel efficiency can be calculated based on regulations from the DOE and its energy content for a U.S. gallon of gasoline of 33,705 W·h?gal (also called the Lower Heating Value (LHV) of gasoline):[92][93]
Coefficient of Static Friction (μ) is the ratio of Tire Road Force to Vehicle Weight.
Top Fuel drag car tires are getting a coefficient of friction well over
Part of this discussion came from:
Rubber in contact with a
smooth surface
(glass is often used in testing) generates friction forces mainly by
adhesion
.
Tearing and Wear
As deformation forces and sliding speeds go up, local stress can exceed the tensile strength of the rubber, especially at an increase in local stress near the point of a sharp irregularity. High local stress can deform the internal structure of the rubber past the point of elastic recovery. When polymer bonds and crosslinks are stressed to failure the material can't recover completely , and this can cause tearing . Tearing absorbs energy, resulting in additional friction forces in the contact surface.
Wear is the ultimate result of tearing.
Ftotal = Fadhesive + Fdefformation + Fwear
Deformation Friction and Viscoelasticity
Rubber is elastic and conforms to surface irregularities. But rubber is also viscoelastic; it doesn't rebound fully after deformation.
Hysteresis
Hysteresis, or energy loss, in rubber.
where there is some sliding between the rubber and an irregular surface. If the rubber recovers slowly from the passing irregularity as in the high-hysteresis rubber, it can't push on the downstream surfaces of the irregularities as hard as it pushes on the upstream surfaces. This pressure difference between the upstream and downstream faces of the irregularity results in friction forces even when the surfaces are lubricated.
Wide Tires:
Friction is proportional to the normal force of the asphalt acting upon the car tires. This force is simply equal to the weight which is distributed to each tire when the car is on level ground. Force can be stated as Pressure X Area. For a wide tire, the area is large but the force per unit area is small and vice versa. The force of friction is therefore the same whether the tire is wide or not. However, asphalt is not a uniform surface. Even with steamrollers to flatten the asphalt, the surface is still somewhat irregular, especially over the with of a tire. Drag racers can therefore increase the probability or likelihood of making contact with the road by using a wider tire. In addition a secondary benefit is that the wider tire increased the support base a
Friction force is independent of the apparent area of contact.