# Double Dead-Time Signal Injection Strategy for Stator Resistance Estimation of Induction Machines
## 3 Optimization of the Injection Using Double Dead-Time Strategy  
In order to eliminate the effect of the dead time, a new injection strategy has been designed. This strategy is based on the injection of a controlled current at two different dead times. Using the current control injection strategy, previously presented in Figure 1, the current DC value is maintained constant during the injection. In Figure 3, the double dead-time strategy is described. First, a controlled current signal is injected with the first dead-time value (\(T_{tm1}\)). As soon as the current arrives to the steady state, the injected value \(V_{inj1}\) is stored. The dead-time value is changed to \(T_{tm2}\) and the injected voltage \(V_{inj2}\) is stored after the transition time. The injected voltage eliminating the dead-time effect is calculated. Once this value is obtained, the stator current and the temperature in the semiconductors is measured and the voltage drop in the semiconductors is estimated. In the other side, the voltage drop in the cable is estimated and the voltage at machine terminals is calculated. The mathematical approach is hereinafter described. First, the dead-time voltage drop will only be considered. As it has been mentioned, the injection is performed considering the controlled way; the estimation of the three-phase system can be made considering only one phase. Thus, the equation of the injected voltage in machine terminals can be expressed as (3). \[V_{DC\_out}=V_{inj}-N_{DT}\cdot D_{tm}\cdot V_{bus}=V_{inj}-N_{DT}\cdot\frac{T _{tm}}{T}\cdot V_{bus} \tag{3}\]  
where \(V_{DC\_out}\) is the voltage at the machine terminals; \(V_{inj}\) represents the injected voltage by inverter; \(N_{DT}\) is the difference between the dead-time cycle number in the positive side and the negative side of the current; and \(D_{tm}\) is the dead-time duty cycle. The value \(N_{DT}\) could change in function of the working point of the machine. This value is dependent on the voltage, current and phase of the feeding signal of the machine. The on-line estimation of this value during the control is very complicated due to the difficulty of detecting the zero crossing and the delay in the switching of the IGBTs of the inverter. Figure 3: Double dead-time injection strategy to compensate dead-time effect. For this reason, a double injection strategy has been designed to compensate the dead-time voltage drop. This strategy consists in the injection of two different signals at two different dead-time values. These two injections are made at the same working point, so the \(N_{DT}\) will not change in most of the cases. If the dead-time suffers a small change, the current, voltage and phase of the feeding signal will barely change. In (4), the equations of the double injection are shown. \[ \begin{array}{l}V_{DC_{out1}}=V_{inj1}-N_{DT}.\frac{T_{out1}}{T}.V_{bus}\\ V_{DC_{out2}}=V_{inj2}-N_{DT}.\frac{T_{out2}}{T}.V_{bus}\end{array}  \tag{4}\]  
where \(V_{DCout1}\) and \(V_{DCout2}\) are the voltage at the terminals of the machine for the two different injections respectively; \(V_{inj1}\) and \(V_{inj2}\) are the injected voltages for the two different dead-times; \(T_{tnt1}\) and \(T_{tnt2}\) are the two dead-time values. Considering that the two injections are made with the same stator resistance value and the same DC current offset, the \(V_{DCout1}\) and \(V_{DCout2}\) values will be identical. If the injections are made with a short time interval, the resistance will be practically the same due to the low dynamic of the temperature. As \(V_{DCout1}\) and \(V_{DCout2}\) are equal, the Equation (4) transforms in (5). \[ \begin{array}{l}V_{DC_{out}}=V_{inj1}-N_{DT}.\frac{T_{tnt1}}{T}.V_{bus}\\ V_{DC_{out}}=V_{inj2}-N_{DT}.\frac{T_{tnt2}}{T}.V_{bus}\end{array}  \tag{5}\]  
The system is formed by two equations and two unknown values. Resolving the system, the value \(V_{DC_{out}}\) could be obtained with the simple Equation (6). \[ \begin{array}{l}V_{DC_{out}}=\frac{T_{tnt2}\cdot V_{inj1}-T_{tml}\cdot V_{inj2 }}{T_{tml}-T_{tml}}\end{array}