2. Model and characteristics (mathematical model)

2.1 Linearized model of a single phase referred to the stator

We refer the elements of the rotor to the stator by dividing them by m², m: being the transformation ratio per phase with an open rotor. We present the model of a phase referred to the stator in figure 5.

Figure 5: Model of a phase referred to the stator.

We define the different elements of the model as follows:

R₁ : resistance of stator winding

L₁ and R_f : iron core coil

l₂ : leakage inductance

R₂/g : fictitious resistance representing the transmitted power

R₂ : resistance of the rotor referred to the stator

 

We take: P_Tr = P_M + p_Jr, we show the resistance of the rotor referred to the stator by decomposing the fictitious resistance as follows: 

 

This makes it possible to separate the copper losses (dissipated in R₂) from the mechanical power dissipated in (1-g)R₂/g.

2.2 Determination of the model elements

- A no-load test at synchronism makes it possible to determine R_F and L₁ by measuring:

- A no-load test with locked rotor under reduced voltage such that I_1eff = I_1Neff makes it possible to determine R₂ and l₂ by measuring:

- R₂ is measured by a hot-line continuous test.

If we cannot measure the resistance R₁ of a winding, we measure the resistance R between two phases. We recall that R = 2.R₁ (in star coupling) and R = 2.R₁/3 (in triangle coupling).

 

2.3 Electromagnetic torque characteristic

The curve of the electromagnetic torque (T_em or C_em) as a function of the motor speed n at constant voltage and frequency, is given in Figure 6.

Figure 6: Electromagnetic torque characteristic

 

We notice that:

- The moment of the electromagnetic torque is non zero at start-up which means that the motor can start on its own.

- In the operating zone (between the breakdown torque point and the full load point), Tem = -an+b (which means that the variation is substantially linear).

2.4 Characteristic T_U  and I against speed n

In Figure 7, the curve of the torque and current intensity characteristic is plotted knowing that:

- Start-up current is high.

- The no-load current is not negligible.

Figure 7: Curves of torque and current characteristics

 

3. Startup

At start-up operation, the fact that the current is high and the torque low implies that we should not start under load at rated (nominal) voltage, except for low power motors. To do this, we perform the startup:

- By actions on the stator, the current is reduced by reducing the voltage (star-delta coupling, dimmer, rheostat or starting coil, etc.). These techniques, also reducing starting torque, are generally used for starting without load.
- By acting on the rotor, the torque is increased: rheostat for wound rotor motors, and for others, creation of a motor with a double cage (two concentric cages) or with deep slots. 

The current solution uses soft starters which act on the supply voltage and frequency.

 

4. Reversibility

Driven beyond synchronism (g < 0), the induction machine operates as a generator. It restores active power, but still consumes reactive power.

 

5. Braking

There are several possibilities depending on the type of load driven by the induction machine. Let us cite:

- Hypersynchronous braking. Based on Lenz's law, if the machine is driven beyond synchronism, the resulting rotating magnetic field (RMF) opposes this overspeed and the machine is braked. If the supply frequency is fixed, the speed of the rotating field is fixed, so this type of braking does not allow the machine to stop. On the other hand, fed by a power static converter providing the motor with a gradually decreasing frequency, the rotating field slows down and so does the rotor.

- Counter-current braking. It consists of inverting two power phases. In this case, the rotation direction of the rotating field reverses, and the motor brakes.

 

6. Single-phase induction motor

The starting torque being zero, a single-phase induction motor does not start alone, without special devices to create a rotating field such as the use of a capacitor or an auxiliary winding, Frager turns, etc. Compared to a three-phase motor, their efficiency is lower and their torque lower. They are mainly used in places not supplied with three-phase network (domestic installations) and for low-power driven machines.