# Double Dead-Time Signal Injection Strategy for Stator Resistance Estimation of Induction Machines
## 2 Signal Injection Principle and Considerations  
It is based on the relationship between temperature and resistance. As the temperature of the stator windings is related to the stator resistance by the Equation (1) from [9]. \[T_{s}=T_{s0}+\frac{R_{s}-R_{s0}}{\propto\cdot R_{s0}} \tag{1}\]  
where \(\propto\) is the winding material thermal coefficient. Copper is the most common winding material in electric machines, being \(\propto\approx 0.0039\). \(T_{s}\) and \(T_{s0}\) are the actual stator temperature and the cold state temperature, respectively. \(R_{s}\) and \(R_{s0}\) are the stator resistance and the cold state stator resistance, respectively. Thus, based on this relation, an online estimation of the stator resistance value allows to calculate the temperature. From the stator voltage equation of the induction machine, the DC equation in steady-state results in the simple Equation (2). \[R_{s}=\frac{V_{sDC}}{I_{sDC}} \tag{2}\]  
Thus, the stator resistance only depends on the DC voltage and the DC current values. The DC offset of the current can be obtained from the sensors installed generally in the control systems. However, the voltage signal is generally not available unless an extra hardware is implemented, so it needs to be estimated. An injection strategy is presented in this paper, where a controlled DC current is injected and the needed voltage to generate this DC current is analyzed. There are two main reasons for using a controlled current injection. From one side, the voltage drop in the inverter depends on the current DC level. The DC level will change with the temperature, so the adjustment of any estimation algorithm will be also dependent on the temperature. From the other side, if a constant voltage is injected, the current will decrease in function of the temperature. At medium/high size machines, the stator resistance is usually very small. This supposes that a considerable change in temperature can cause a small DC current level change. At high temperatures, the current will decrease, so the precision will be smaller. In order to have a constant DC current, a controllable injection strategy has been implemented, where the injected offset is controlled, as it is shown in Figure 1. The offset current of the phases \"a\" and \"b\" is fixed using a PI controller. To control the current offset, the voltage signal introduced in the modulator is adjusted. In this implementation, the current of the phases \"a\" and \"b\" is obtained with a low-pass filter (LPF). From one side, a current reference is injected in \"a\" phase and the same but negative reference is injected in phase \"b\". Making this, an offset is generated only in the phases \"a\" and \"b\", making the offset of the phase \"c\" equal to 0. In this way, the three-phase system can be considered as a two-phase system. Furthermore, as the DC voltage level at two phases is equal, it is enough to take into account just one phase, considerably simplifying the developed equations. In order to equilibrate the injection in the first two phases, a voltage reference equal to 0 is injected in the phase \"c\" so it can be ensured that the middle point voltage will be equal to 0. The controller will detect the introduced offset as a perturbation, and it will try to eliminate it. In order to hide the injected offset to the controller, the feedback current signals are filtered with an LPF. In order to avoid the use of extra hardware, the offset injection is performed through the inverter. However, the introduced DC voltage will be affected by different non-idealities present in the inverter. These non-idealities are identified as the dead-time and the voltage drop in semiconductors. Aside from the voltage drop in the inverter, there is also a voltage drop in the cable that connects the inverter and the machine. All these voltage drops should be compensated for in order to estimate the stator resistance value using the Equation (2).