When you are building a DIY electric vehicle, upgrading a ride-on toy, or designing an autonomous rover, guessing your motor size is a fast track to melted wires and tripped battery management systems. You need to translate the electrical power from your battery into the mechanical power at the wheel. The bridge between those two domains is the power wheel formula.
The direct answer for steady-state cruising power at the wheel is P = (τ × v) / r. When you factor in motor and drivetrain efficiency to find your required battery current, the master equation becomes V × I × η = (τ × v) / r. Below, we break down exactly how to use this equation, track the units so you don't blow up your ESC, and walk through real-world bench scenarios.
The Core Power Wheel Equation
The fundamental relationship linking electrical input to mechanical wheel output accounts for voltage, current, system efficiency, wheel torque, linear velocity, and wheel radius. This is the master equation you will use for 90% of your DC motor sizing tasks.
V × I × η = (τ × v) / r
| Symbol | Parameter | Standard SI Unit | Typical DIY Range |
|---|---|---|---|
| V | Battery / Supply Voltage | Volts (V) | 12V - 48V |
| I | Continuous Current Draw | Amperes (A) | 2A - 40A |
| η | System Efficiency (Motor + Gears) | Decimal (0.0 - 1.0) | 0.60 - 0.85 |
| τ | Torque at the Wheel | Newton-meters (N·m) | 0.5 - 15 N·m |
| v | Linear Velocity of the Vehicle | Meters per second (m/s) | 1 - 15 m/s |
| r | Wheel Radius | Meters (m) | 0.05 - 0.25 m |
Rearranged Forms for Motor Sizing
Depending on what you are trying to figure out on the bench, you will need to isolate different variables. Here are the rearranged forms of the master equation:
- Solving for Current (I):
I = (τ × v) / (r × V × η)— Use this to size your battery wires, ESC, and BMS. - Solving for Velocity (v):
v = (V × I × η × r) / τ— Use this to find your top speed given a known load torque. - Solving for Wheel Torque (τ):
τ = (V × I × η × r) / v— Use this to determine if your motor can push the vehicle up a specific hill. - Solving for Wheel Radius (r):
r = (τ × v) / (V × I × η)— Use this to find the maximum wheel size your motor can spin without stalling under load.
When the Formula Applies (And When It Breaks)
This formula is derived from the conservation of energy, equating electrical power input (P = V × I) to mechanical power output (P = F × v, where Force F = τ / r). However, it relies on strict assumptions.
Assumptions and Applicability
The equation assumes steady-state cruising on a rigid surface. It calculates the continuous power required to overcome rolling resistance, aerodynamic drag (if bundled into your torque figure), and gravitational forces on an incline. It applies perfectly to brushed and brushless DC motors operating in their continuous duty cycle.
Unit Mistakes That Break the Math
The most common reason DIYers get wildly incorrect numbers is unit mismatch.
- The RPM Trap: Do not plug motor RPM into the velocity (v) slot. Velocity must be linear (m/s). If you only have motor RPM, you must convert it using
v = (RPM × 2 × π × r) / (60 × Gear_Ratio). - The Inch/Newton Clash: If your torque is in Newton-meters, your radius must be in meters. Mixing inches for radius and Newtons for force will skew your power calculation by a factor of 39.37.
- Efficiency Ignorance: Assuming η = 1.0 (100% efficiency). DC motors drop to 40-50% efficiency at low speeds and high torque. Always use 0.75 as a baseline unless you have the motor's dyno chart.
Realistic Answer Magnitudes
What should your final numbers look like? For a standard 12V kids' ride-on toy (Power Wheels style), expect 30W to 80W per wheel. For a 24V DIY electric skateboard or heavy rover, expect 250W to 800W per wheel. If your math says you need 50W to move a 100kg go-kart, you forgot to account for the starting torque multiplier or incline.
Worked Problem 1: Sizing a Motor for a Custom Robot Chassis
Let's size the drive motors for a 30kg indoor delivery robot rolling on hard concrete. We need to find the continuous current draw per motor to maintain a speed of 2 m/s.
- Calculate Normal Force (Weight):
F_normal = mass × gravity
F_normal = 30 [kg] × 9.81 [m/s²] = 294.3 [N] - Calculate Required Linear Force:
Using a rolling resistance coefficient (C_rr) of 0.015 for rubber on concrete (Engineering ToolBox).
F_drive = F_normal × C_rr
F_drive = 294.3 [N] × 0.015 = 4.41 [N] - Calculate Wheel Torque (τ):
Assume a wheel radius (r) of 0.1 m (100mm diameter wheels).
τ = F_drive × r
τ = 4.41 [N] × 0.1 [m] = 0.441 [N·m] - Calculate Mechanical Power:
P_mech = (τ × v) / r(which simplifies back to F × v)
P_mech = 4.41 [N] × 2 [m/s] = 8.82 [W] - Calculate Electrical Current (I):
Assume a 12V system with 75% efficiency (η = 0.75).
I = P_mech / (V × η)
I = 8.82 [W] / (12 [V] × 0.75) = 0.98 [A]
Result: Each motor will draw roughly 1A continuously. A 5A ESC per motor is more than sufficient for cruising.
Worked Problem 2: Calculating Battery Current Draw on an Incline
Now, let's take that same 30kg robot and push it up a 5% grade (a 5-meter rise over 100 meters of travel). How much current will the 12V motor pull now?
- Calculate Grade Force:
A 5% grade means sin(θ) ≈ 0.05.
F_grade = mass × gravity × sin(θ)
F_grade = 30 [kg] × 9.81 [m/s²] × 0.05 = 14.71 [N] - Calculate Total Linear Force:
F_total = F_drive (flat) + F_grade
F_total = 4.41 [N] + 14.71 [N] = 19.12 [N] - Calculate New Wheel Torque:
τ = F_total × r
τ = 19.12 [N] × 0.1 [m] = 1.912 [N·m] - Calculate New Current Draw:
Maintaining 2 m/s velocity.
P_mech = F_total × v = 19.12 [N] × 2 [m/s] = 38.24 [W]
I = 38.24 [W] / (12 [V] × 0.75) = 4.25 [A]
Result: The current draw jumps from 1A to 4.25A just to climb a mild 5% hill. This is why sizing your BMS based only on flat-ground cruising is a critical mistake.
Real-World Scenario Walkthrough: The 24V Go-Kart Build That Tripped the BMS
Formulas on a whiteboard are clean; the jobsite is not. Here is a scenario that highlights the gap between steady-state math and transient reality.
The Setup
A hobbyist was building a 24V electric go-kart for their child using two surplus 250W scooter motors. The total weight (kart + child) was 60kg. The wheels were 0.15m in radius. They used the power wheel formulas to calculate the continuous current draw on flat grass and sized a 20A BMS for the 24V LiFePO4 battery pack.
The Numbers
Using a rolling resistance coefficient of 0.05 for rubber on short grass, the math looked solid. The required continuous force was roughly 30N per wheel. At a target speed of 3 m/s, the mechanical power was 90W per wheel. Factoring in 70% efficiency on a 24V system, the continuous current draw calculated out to roughly 5.3A per motor (10.6A total). The 20A BMS seemed to have plenty of headroom.
The Outcome
The child got in, pressed the pedal, and the kart jerked forward exactly two inches before the BMS clicked off and killed all power. Resetting the BMS resulted in the exact same immediate trip.
What Went Wrong
The Lesson: The power wheel formula calculates steady-state cruising power. It completely ignores starting torque and the static friction breakout threshold.
When a DC motor starts from zero RPM, it draws stall current. Furthermore, the static rolling resistance of a heavy kart sinking into soft grass is vastly higher than the dynamic rolling resistance once it is already moving. The initial breakout force required was nearly 4x the calculated cruising force. The motors pulled over 35A each for the first 500 milliseconds to break static friction, instantly tripping the 20A BMS's short-circuit/overcurrent protection.
The Fix: The builder swapped to a 40A continuous / 80A peak BMS (following Battery University guidelines for high-inrush DC loads) and added a soft-start controller to ramp the PWM duty cycle over 1.5 seconds, keeping the inrush current below the BMS trip threshold while still providing enough torque to break static friction. Always size your BMS and wiring for the stall or breakout current, not just the cruising current derived from the power wheel formula.
