# Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems
## 3 Converter System Description and Analysis
### System Description
#### 3.1.2 Load Model  
The converter load is described by an ohmic-inductive filter and a voltage source. The ohmic-inductive filter part shown in Figure 9 can be used to model the simple output filter plus grid impedance of an electrical power supply system or the stator windings of an electrical machine, where the voltage source corresponds to the grid voltage or the electromagnetic force (EMF), respectively; see, e.g., [7; 13; 23]. Figure 8: Hexagon of a two-level converter and stationary reference frames. Figure 9: Circuit diagram of the converter load. In this paper, the load should be given by a symmetrical permanent magnetic synchronous machine (PMSM), which can be modeled by the normalized \((\alpha,\beta)\) frame voltage equations from [24]  
\[\frac{d^{\prime}_{\alpha}(t)}{dt} =\frac{-1}{T_{\rm s}}\bigg{(}i^{\prime}_{\alpha}(t)-\frac{u^{ \prime}_{\alpha}(t)}{R^{\prime}}+\frac{u_{\rm EMF,\alpha}}{R^{\prime}}\bigg{)} =\frac{-1}{T_{\rm s}}\bigg{(}i^{\prime}_{\alpha}(t)-\frac{u^{ \prime}_{\alpha}(t)}{R^{\prime}}-c_{\rm EMF}\,\omega^{\prime}(t)\,\sin( \varepsilon(t))\bigg{)} \tag{27}\] \[\frac{d^{\prime}_{\beta}(t)}{dt} =\frac{-1}{T_{\rm s}}\bigg{(}i^{\prime}_{\beta}(t)-\frac{u^{ \prime}_{\beta}(t)}{R^{\prime}}+\frac{u_{\rm EMF,\beta}}{R^{\prime}}\bigg{)} =\frac{-1}{T_{\rm s}}\bigg{(}i^{\prime}_{\beta}(t)-\frac{u^{ \prime}_{\beta}(t)}{R^{\prime}}+c_{\rm EMF}\,\omega^{\prime}(t)\,\cos( \varepsilon(t))\bigg{)} \tag{28}\]  
using the normalized values  
\[ \begin{array}{cccccccc}u^{\prime}&=&\frac{u}{u_{\rm nom}}&,&i^{\prime}&=& \frac{j}{u_{\rm nom}}&,&\omega^{\prime}&=&\frac{\omega}{\omega_{\rm nom}}\\ R^{\prime}&=&\frac{R_{\rm s}\cdot u_{\rm nom}}{u_{\rm nom}}&,&L^{\prime}&=& \frac{L_{\rm s}\cdot u_{\rm nom}}{u_{\rm nom}}&,&T_{\rm s}&=&\frac{L}{R^{\prime }}\end{array}  \tag{29}\]  
where \(u_{\rm nom}\) is the machine's nominal voltage, \(i_{\rm nom}\) is the machine's nominal current, and \(c_{\rm EMF}\) defines the EMF constant  
\[c_{\rm EMF}=\frac{z_{p}\,\Phi_{\rm F}\,\omega_{\rm nom}}{L^{\prime}\,u_{\rm nom }}\,T_{\rm s}. \tag{30}\]  
Here, \(R_{\rm s}\) and \(L_{\rm s}\) are the resistance and inductance of a stator winding, respectively, \(\omega_{\rm nom}\) specifies the machine's nominal angular frequency, \(\Phi_{\rm F}\) is the permanent magnetic excitation, and \(z_{p}\) is the number of pole pairs. The EMF constant \(c_{\rm EMF}\) in combination with normalized angular frequency \(\omega^{\prime}(t)\), i.e., the last term in (27) and (28), respectively, defines the normalized current caused by the EMF. The electrical angle, and thus, the position of the rotor, are represented by \(\varepsilon(t)\), specified by  
\[\frac{d\omega^{\prime}(t)}{dt} =\frac{2\,\Phi_{\rm F}\,z_{p}\,i_{\rm nom}}{3\,\Theta\,\omega_{ \rm nom}}\,\Big{(}\cos(\varepsilon(t))\,i^{\prime}_{\beta}(t)-\sin( \varepsilon(t))\,i^{\prime}_{\alpha}(t)\Big{)}-\frac{m_{\rm L}(t)}{\Theta\omega _{\rm nom}} \tag{31}\] \[\frac{d\varepsilon(t)}{dt} =z_{p}\,\omega_{\rm nom}\,\omega^{\prime}(t), \tag{32}\]  
where \(m_{\rm L}(t)\) and \(\Theta\) are the load torque and rotor inertia, respectively. Since the dynamics of the mechanical machine part are slow compared with the dynamics of the electrical part, the EMF is usually considered to be a slowly varying but independent disturbance in the current control loop; see, e.g., [4]. Moreover, (31) and (32) are not considered in the following. In this paper, the load and the VSC are analyzed as one system.