#### 1. How is back EMF generated?

For a permanent magnet synchronous motor, its coil is fixed on the stator (conductor) and the permanent magnet is fixed on the rotor (magnetic field). When the rotor rotates, the magnetic field generated by the permanent magnet on the rotor will rotate and be cut by the coil on the stator, generating back EMF in the coil. Why is it called back EMF? As the name suggests, the direction of the back EMF E is opposite to the direction of the terminal voltage U (as shown in Figure 1).

Figure 1

#### 2.What is the relationship between back EMF and terminal voltage?

From Figure 1, it can be seen that the relationship between the back EMF and the terminal voltage under load is:

For the test of back EMF, it is generally tested under no-load state, no current, and a speed of 1000rpm. Generally, the value of 1000rpm is defined, and the back EMF coefficient = average back EMF/speed. The back EMF coefficient is a relatively important parameter of the motor. It should be noted here that the back EMF under load is constantly changing before the speed is stable.

From formula (1), it can be seen that the back EMF under load is less than the terminal voltage. If the back EMF is greater than the terminal voltage, it becomes a generator and outputs voltage to the outside. Since the resistance and current in actual work are small, the value of the back EMF is approximately equal to the terminal voltage and is limited by the rated value of the terminal voltage.

#### 3. The physical meaning of back EMF

Imagine what would happen if back EMF did not exist? From equation (1), we can see that without back EMF, the entire motor is equivalent to a pure resistor, which becomes a device that generates particularly severe heat. This is contrary to the motor’s conversion of electrical energy into mechanical energy.

In the electrical energy conversion equation, UIt=E_{反}It+I^{2}Rt

UIt is the input electrical energy, such as the input electrical energy to the battery, motor or transformer; I^{2}Rt is the heat loss energy in each circuit, which is a kind of heat loss energy, the smaller the better; the difference between the input electrical energy and the heat loss electrical energy is the part of useful energy E_{反}It corresponding to the back EMF. In other words, the back EMF is used to generate useful energy and is inversely correlated with the heat loss. The greater the heat loss energy, the smaller the achievable useful energy.

Objectively speaking, the back EMF consumes the electrical energy in the circuit, but it is not a “loss”. The part of the electrical energy corresponding to the back EMF will be converted into useful energy for electrical equipment, such as the mechanical energy of the motor and the chemical energy of the battery.

It can be seen from this that the size of the back electromotive force means the ability of the electrical equipment to convert the total input energy into useful energy, which reflects the level of the electrical equipment’s conversion ability.

#### 4. What does the magnitude of the back EMF depend on?

First, the calculation formula for the back EMF is given:

E is the coil electromotive force, ψ is the flux, f is the frequency, N is the number of turns, and Φ is the magnetic flux.

According to the above formula, we can know the factors that affect the magnitude of the back EMF:

(1) The back EMF is equal to the rate of change of the flux. The higher the speed, the greater the rate of change, and the greater the back EMF;

(2) The flux itself is equal to the number of turns multiplied by the single-turn flux. Therefore, the higher the number of turns, the greater the flux and the greater the back EMF;

(3) The number of turns is related to the winding scheme, such as star-delta connection, the number of turns per slot, the number of phases, the number of teeth, the number of parallel branches, and the full-pitch or short-pitch scheme;

(4) The single-turn flux is equal to the magnetomotive force divided by the magnetic resistance. Therefore, the greater the magnetomotive force, the smaller the magnetic resistance in the direction of the flux, and the greater the back EMF;

(5) The magnetic resistance is related to the air gap and the pole-slot coordination. The larger the air gap, the greater the magnetic resistance and the smaller the back EMF. The pole-slot coordination is more complicated and needs to be analyzed in detail;

(6) The magnetomotive force is related to the residual magnetism of the magnet and the effective area of the magnet. The larger the residual magnetism, the higher the back EMF. The effective area is related to the magnetizing direction, size and placement of the magnet, and requires specific analysis;

(7) Remanence is also related to temperature. The higher the temperature, the smaller the back EMF

In summary, the factors affecting back EMF include speed, number of turns per slot, number of phases, number of parallel branches, full-spacing and short-spacing, motor magnetic circuit, air gap length, pole-slot matching, magnet remanence, magnet placement and magnet size, magnetizing direction, and temperature.

#### 5. How to select the back EMF in motor design?

In motor design, back EMF E is very important. If the back EMF is well designed (appropriate size and low waveform distortion rate), the motor is good.

The back EMF has several main effects on the motor:

1. The back EMF determines the weak magnetic point of the motor, and the weak magnetic point determines the distribution of the motor efficiency map.

2. The back EMF waveform distortion rate affects the motor ripple torque and the smoothness of the torque output when the motor is running.

3. The size of the back EMF directly determines the torque coefficient of the motor, and the back EMF coefficient is proportional to the torque coefficient. From this, the following contradictions can be drawn in motor design:

a. If the back EMF is large, the motor can maintain high torque at the controller limit current in the low-speed operation area, but it cannot output torque at high speed, and even cannot reach the expected speed;

b. If the back EMF is small, the motor still has output capacity in the high-speed area, but the torque cannot be achieved at the same controller current at low speed.

Therefore, the design of the back EMF size depends on the actual needs of the motor. For example, in the design of a small motor, if it is required to output sufficient torque at low speed, the back EMF must be designed to be larger.