# e-skate calculator

I don't have my parts and I want to select...

I have already my parts and I want to check...

kg
kph
Ohm
A
0
mAh

0.0 km
0.0 W
0.0 W
0.0 %

Teeth
Teeth
mm
rpm/V
0

1 : 1
0.0 kph
0.0 rpm

kg
kph
s
km
km
0
0
0

0.0 A
0.0 W
0.0 mAh
0.0 Wh/km

kph
mm
Teeth
rpm/V
0

1 : 1
0 T
0.0 rpm

Teeth
Teeth
mm

1 : 1
0 mm
0 T

mAh
A
0
0

0 V
0 mAh
0.0 W

# calculator physics

## Power and range

This calculator is based on physical models to estimate the relationship between power, capacity, and desired performances.

The power is what the motor provides to forward movement, by overcoming forces. Those forces are different whether you are accelerating or staying at a constant speed and riding on flat or uphill. In total, four majors forces are present: inertia, gravity, rolling resistance, aerodynamic drags.

### Inertia

Inertia appears during acceleration. It is the resistance of a mass, like a rider or a board, to changes in its velocity. The heavier this mass is, the hardest it is to start a movement.

This force also depends on the acceleration, that is to say, the change in your velocity over a given time, expressed:

$a = \large { \frac{V}{t_{start}} }$

where:

• $V$ is your desired speed in $m/s$
• $t_{start}$ is the time it takes to reach your speed in $s$

Finally, the formula for inertia is:

$F_{ie} = (m + M).a$

where:

• $m$ is your mass in $kg$
• $M$ is your board mass in $kg$
• $a$ is acceleration in $m/s^2$

#### Gravity

The gravity becomes an opposing force when you go uphill. The formula for gravity force in $N$ is:

$F_g = (m+M).g.\text{sin}\left(\text{arctan}(G)\right)$

where:

• $m$ is your mass in $kg$
• $M$ is your board mass in $kg$
• $g$ is the gravitational acceleration, equals to $9.81 m/s²$
• $G$ is the hill slope, in percentage

### Rolling resistance

This force is due to the friction between wheels and road. This friction is lower with urethane wheels on good quality roads than with pneumatics on grass. Also, the heavier the rider is, the higher this force is.

The formula for rolling resistance in $N$ is:

$F_r = C_{rr}.(m+M).g.\text{cos}\left(\text{arctan}(G)\right)$

where:

• $C_{rr}$ is the coefficient rolling resistance that is dimensionless
• $m$ is your mass in $kg$
• $M$ is your board mass in $kg$
• $g$ is the gravitational acceleration, equals to $9.81 m/s²$

The coefficient rolling resistance $C_{rr}$ translates the quality of the road and the type of wheels. We have estimated the $C_{rr}$ as below

• $C_{rr}=0.015$ for urethane on asphalt
• $C_{rr}=0.040$ for penumatics on asphalt

### Aerodynamic drag

This is the air resistance that opposed to the rider motion and is proportional to the square of the velocity and the frontal area. The frontal area is the area of the rider when looked from the front.

The formula for the aerodynamic drag in $N$ is:

$F_a = \frac{1}{2}.C_d.A.\rho.V^2$

where:

• $C_d$ is the drag coefficient
• $A$ is your frontal area in $m^2$
• $\rho$ is the air density, equals to $1.2kg/m^3$
• $V$ is your desired speed in $m/s^2$

For a rider, $C_d.A$ is estimated equals to $0.6m^2$

### Acceleration phase

During this phase, that we consider on flat, three major forces are presents: rolling resistance, aerodynamic drag, inertia. The resistive force formula in $N$ is:

$F = F_{ie}+F_r+F_a$

### Constant velocity phase

During this phase, the inertia is replaced by gravity when you are going uphill. The resistive force is:

$F = F_g+F_r+F_a$

### Power

Power is proportional to the resistive forces, but also to the velocity. Its formula is:

$P_w=F.V$

where:

• $P_w$ is the power on your wheels in $W$
• $F$ is the resistive forces, during acceleration or constant velocity phase
• $V$ is your desired speed in $m/s$

### Power losses

As you all know, our world is not perfect, a part of the motor power is dissipated by your drivetrain, but also by your electrical components, like motor resistors, windings. We can consider $20\%$ losses for pulley and belt system.

The electrical losses depends on your parts, the suppliers, etc. If you have a lot of confidence in your parts specifications, you could chose a low value of electrical losses, between $0\%$ to $20\%$. On contrary, if you don't trust the specifications, you could consider losses around $50\%$.

The formula for motor power is:

$P_m = \large{ \frac{P_w}{( 1 - loss_{mecha} ) ( 1 - loss_{elec})} }$

where:

• $P_m$ is the power provided by the motor in $W$
• $P_w$ is the power provided to the wheel in $W$
• $loss_{mecha}$ is the drivetrain losses in percentage
• $loss_{elec}$ is the electrical losses in percentage

$( 1 - loss_{elec})$ is also expressed as electrical efficiency.

### Dual/Single

If you use dual motor, $P_m$ is divided by two.

### Capacity

Battery capacity is the amount of charge available. To determine the required capacity, we need first to calculate the average current consumed by your motor, and your travelling time.

The formula for current in $A$ is:

$I = \large{\frac{P_m}{U}}$

where:

• $P_m$ is motor power in $W$
• $U$ is the battery voltage in $V$

The formula for the travelling time is:

$t= \large{ \frac{d}{V} }$

where:

• $d$ is your travelling distance in $km$
• $V$ is your speed in $kph$

The formula for capacity in $Ah$ is:

$C = I.t$

• $I$ is the average current in $A$
• $t$ is the travelling time in $h$

## Drivetrain

### Motor speed

The motor speed is directly proportional to the motor voltage.

The formula for motor speed in $RPM$ is:

$N_{m} = kV.U$

where:

• $kV$ is the motor velocity constant in $RPM/V$
• $U$ is the battery voltage in V

### Drivetrain ratio

The ratio is represented in reference to the number 1.

The formula for ratio is:

$Ratio= \large { \frac{ T_{w} }{ T_{m} } }$

where:

• $T_{m}$ is the number of teeth of the motor pulley
• $T_{w}$ is the number of teeth of the wheel pulley

### Wheel speed

The wheel speed is obtained by divided the motor speed by the ratio.

The formula for wheel speed in $RPM$ is:

$N_w = \large{ \frac{N_m}{Ratio} }$

where:

• $N_m$ is the motor speed in $RPM$
• $N_w$ is the wheel speed in $RPM$

To convert $RPM$ into $rad/s$:

$\omega_w = \large{ \frac{2\pi}{60} }.\normalsize{N_w}$

### Board speed

The wheel converts rotational motion to linear motion.

The formula for linear speed in $m/s$ is:

$V_{max} = \omega_w.r$

where:

• $\omega_w$ is the rotational speed in $rad/s$
• $r$ is the wheel diameter in $m$

### Belt length

The belt length depends on the pulley diameters and center to center distance.

The formula for pulley diameter in $mm$ is:

$D = \large{ \frac{Z}{\pi} }T$

where:

• $Z$ is the pitch size, equals to $5mm$ for HTD5M
• $T$ is the number of teeth

We need to calculate the belt contact angle:

$\beta = \text{sin}^{-1} \left( \frac{D_w - D_m}{2C} \right)$

where:

• $D_w$ is the pulley wheel diameter in $mm$
• $D_m$ is the pulley motor diameter in $mm$
• $C$ is the center distance or distance between the pulleys in $mm$

The contact angle of the pulley motor is:

$\theta_m=\pi-2\beta$

where:

• $\beta$ is the belt contact angle

The contact angle of the pulley wheel is:

$\theta_w=\pi+2\beta$

where:

• $\beta$ is the belt contact angle

Finally, the formula for belt length is:

$L=\sqrt{ 4C^2-(D_m - D_w)^2 } + \large{ \frac{1}{2} } \normalsize{ (D_w\theta_w + D_m\theta_m) }$

where:

• $D_w$ is the pulley wheel diameter in $mm$
• $D_m$ is the pulley motor diameter in $mm$
• $C$ is the center-to-center distance between pulleys in $mm$
• $\theta_w$ is the contact angle of the pulley wheel
• $\theta_m$ is the contact angle of the pulley motor

### Teeth in Mesh

A minimal number of 5 teeth in mesh for the pulley motor assures you best performance and prolonging your belt longevity.

The formula for number of teeth in mesh is:

$N_{mesh} = \large { \frac{ D_m \theta_m}{ 2Z } }$

where

• $D_m$ is the pulley motor diameter in $mm$
• $\theta_m$ is the contact angle of the pulley motor
• $Z$ is the pitch size, equals to $5mm$ for HTD5M