OpenSpecEngineering

Finds the back-EMF of a DC motor: the voltage its spinning armature generates, which opposes the supply. Work it out from the motor’s voltage constant and speed, or from a measurement of the terminal voltage and armature current with the armature resistance. Engineers and hobbyists use it to estimate motor speed, check a motor’s constant against its datasheet and size drives and controllers for brushed and brushless DC motors.

Equations

E = K_b·ω     ω = 2πN/60

E = V − I·R_a

E is the back-EMF in volts. K_b is the voltage (back-EMF) constant in volts per rad/s, and ω the shaft speed in rad/s, found from the speed N in rpm [1]. In the second equation, V is the voltage at the motor’s terminals, I the armature current and R_a the armature resistance, including the brushes. The supply voltage drives the current against E, and the difference is lost in R_a [1].

In SI units, K_b in V·s/rad has the same value as the torque constant K_t in N·m/A. Because E is proportional to speed, a motor runs up until E almost equals V; its no-load speed is about V/K_b.

Limits and common mistakes

• Datasheets often give the constant in V per 1000 rpm. Divide by 104.72 to get V per rad/s; 10 V/krpm is 0.0955 V·s/rad.
• Measure R_a at the motor’s running temperature. Copper resistance rises about 0.4 % per °C, so a cold reading overstates E.
• The equations are for a DC motor with a fixed field (a permanent-magnet or separately excited motor). In a series or shunt motor the field changes with load and K_b changes with it.
• Enter a negative current when the machine is generating (being driven by its load). E then comes out larger than V.

Worked example

A 24 V brushed gearmotor runs at 2500 rpm on its test load and draws 3.2 A. Its armature resistance, measured warm, is 0.8 Ω. Find its back-EMF.

1. Choose Method From terminal voltage, current & Ra.
2. Enter Terminal voltage (V) 24 V and Armature current (I) 3.2 A.
3. Enter Armature resistance (Ra) 0.8 Ω and press Solve. The Solution shows: Back-EMF (E): 21.44 V

Check: the resistance takes 3.2 × 0.8 = 2.56 V, and 24 − 2.56 = 21.44 V. At 2500 rpm, ω = 261.8 rad/s, so the motor’s K_b is 21.44 / 261.8 = 0.0819 V·s/rad. With the other method, a K_b of 0.0819 at 2500 rpm gives 21.4414 V, the same to the precision of the rounded constant.