The Double Fed Induction Machine (DFIM) is advantageous in applications that have limited speed range, allowing the reduction in size of the supplying power electronic converter as, for instance, in variable-speed generation, water pumping and so on.
The stator is supplied by three-phase voltages directly from the grid at constant amplitude and frequency.
The rotor is also supplied by three-phase voltages that take a different amplitude and frequency at steady-state to reach different operating conditions of the machine (speed, torque, etc.).
This is achieved by using a back-to-back three-phase converter, which generally consists of the rotor-side converter (RSC) and Grid-Side Converter (GSC).
In this laboratory, DFIM with only the RSC is investigated, and the typical configuration of the studied system is shown in figure 1.


Figure 1: DFIM System with Rotor Side Converter

Modes of Operation

The relationships between the frequencies of the machine are basics that must be known prior to study of electrical equations of the DFIM.
Thus, the equation that relates, ω_s (frequency of stator voltages and currents), ω_r (frequency of rotor voltages and currents) and ω_m (rotor electrical speed) are given in (1):

ω_r obviously depends on the shaft’s electrical speed ω_m, which leads to three operating modes of the machine dependent on the speed, as shown in figure 2:

Figure 2: DFIM Operating Speed Modes

The basic active power balance of the DFIM release adopting a motor convention, the addition of the stator active power 

and rotor active power  is equal to mechanical power in the shaft  minus the copper losses in the stator and rotor.
The positive values of  and  are interpreted as the power absorbed by the machine, whereas a positive value of  means mechanical power developed by the motor through the shaft.

The mechanical power and the stator electric power output are computed as follows:

In steady state at fixed speed for a lossless generator,

It is followed that:

>0 for negative slip and <0 for positive slip.

Generally, the absolute value of slip is much lower than 1 and, consequently,

is only a fraction of .
is positive for negative slip (speed greater than synchronous speed) and it is negative for positive slip (speed lower than synchronous speed).
For super-synchronous speed operation, is transmitted to DC bus capacitor and tends to raise the DC voltage. For subsynchronous speed operation, is taken out of DC bus capacitor and tends to decrease the DC voltage.
Therefore, it is possible to distinguish four combinations of torque (positive and negative) and speed (sub-synchronism and hyper-synchronism), which lead to four quadrants operation modes of the DFIM, as presented in figure 3.
 

Figure 3: Four Quadrant Operation Modes of DFIM

Dynamic model of DFIM

Figure 4 presents the different rotating reference frames typically used to develop space-vector-based models of the DFIM.
The stationary reference frame (α - β) is the stationary reference frame, the rotor reference frame (

) rotates at the ω_m and the synchronous reference frame () rotates at ω_s.

Figure 4: Different References for Space Vectors Representation of DFIM

The space vector model of the DFIM represented in a synchronously rotating frame is obtained through the following equations:

Similarly, the flux yields:

Hence, from the above equations, the equivalent electric circuit in 

is obtained, as shown in figure 5.

Figure 5: DFIM Modeling in 

Frame

Vector control of DFIM

The vector control of DFIM is performed in a synchronously rotating 

frame, in which the d-axis is aligned, in this case, with the stator space vector as illustrated in figure 6 below.
As seen in this figure, only rotor current control and rotor speed control loops are implemented in this case.
The current loops work with the rotor currents referred to the stator side, while the conversion to rotor-referred quantities is performed at the measurement stage for the currents and before the creation of the pulses for the converter for the voltages.
It is possible to obtain the angle of the stator voltage space vector, then subtract 90° from this estimated angle, and then obtain θ_s.

Figure 6: Overall Vector Control Structure

The analytical model of the closed loop current controls with PI regulator is presented in figure 7.

Figure 7: Analytical Model of the Closed Loop PI Controllers