# Adaptive Cruise Control in Electric Vehicles with Field-Oriented Control
## 3 Field-Oriented Control
For a squirrel cage induction machine, the rotor windings are short-circuited and \(\mathrm{v_{dr}}=\mathrm{v_{qr}}=0\); thus, the expressions of the rotor voltages can be written:  
\[\mathrm{v_{qr}}=\frac{\mathrm{d}}{\mathrm{dt}}\Psi_{qr}+\mathrm{R_{r}i_{qr}}+( \omega_{\mathrm{e}}-\omega_{\mathrm{r}})\Psi_{dr}=0 \tag{4}\]  
\[\mathrm{v_{dr}}=\frac{\mathrm{d}}{\mathrm{dt}}\Psi_{dr}+\mathrm{R_{r}i_{dr}}-( \omega_{\mathrm{e}}-\omega_{\mathrm{r}})\Psi_{qr}=0 \tag{5}\]  
Using the stator currents, the rotor flux linkages, magnetizing, and rotor inductions, even though the rotor currents cannot be measured, their expressions are:  
\[\mathrm{i_{qr}}=\frac{1}{\mathrm{L_{r}}}\Psi_{qr}-\frac{\mathrm{L_{m}}}{ \mathrm{L_{r}}}\mathrm{i_{qs}} \tag{6}\]  
\[\mathrm{i_{dr}}=\frac{1}{\mathrm{L_{r}}}\Psi_{dr}-\frac{\mathrm{L_{m}}}{ \mathrm{L_{r}}}\mathrm{i_{ds}} \tag{7}\]  
The rotor currents in (4) and (5) can be eliminated by the expressions from (6) and (7); thus, the following equations are obtained:  
\[\frac{\mathrm{d}}{\mathrm{dt}}\Psi_{qr}+\frac{\mathrm{R_{r}}}{\mathrm{L_{r}}} \Psi_{qr}-\frac{\mathrm{L_{m}}}{\mathrm{L_{r}}}\mathrm{R_{r}i_{qs}}+\omega_{ sl}\Psi_{dr}=0 \tag{8}\]  
\[\frac{\mathrm{d}}{\mathrm{dt}}\Psi_{dr}+\frac{\mathrm{R_{r}}}{\mathrm{L_{r}}} \Psi_{dr}-\frac{\mathrm{L_{m}}}{\mathrm{L_{r}}}\mathrm{R_{r}i_{ds}}+\omega_{ sl}\Psi_{qr}=0 \tag{9}\]  
To assure the decoupling between flux and torque, the rotor flux \(\Psi_{r}\) will be only on the d axis, so the q component of the rotor flux will be zero:  
\[\Psi_{qr}=0 \tag{10}\]  
If \(\Psi_{qr}\) is zero, then its derivative is also zero:  
\[\frac{\mathrm{d}}{\mathrm{dt}}\Psi_{qr}=0 \tag{11}\]  
Using the information from (10) and (11) in Equation (8), the expression of the slip frequency is the following:  
Figure 5:0  
Additionally, the rotor flux is aligned with \(\mathrm{i_{ds}}\) current and has the following expression:  
Figure 5:1  
All the assumptions made also affected the torque expression, and it can be characterized by the following equation:  
Figure 5:2  
When working with electric machines, as in this case with an induction machine, it is important to also know the mechanical behavior of the machine. The rotor speed depends on the mechanical speed (\(\omega_{m}\)) and the number of poles in the machine. The mechanical speed is the actual speed of the machine and is influenced by the load torque (\(\mathrm{T_{L}}\)) and rotor inertia (J) based on the following equation:  
Figure 5:3To develop a vector control loop for an induction machine, Equations (12)-(14) and the stator current and mechanical speed measurements are required. Figure 6 presents a block diagram of an induction machine drive system with field-oriented control, where the iqs current is controlled in a closed loop. The set value of iqs is obtained based on Equation (14), where the torque is determined from the speed error, and for the flux, Equation (13) is used and the ids current, all this using the speed and stator current feedback. Ids current is controlled in an open loop, using the rated current of the motor, to obtain a maximum flux. The paper's goal is to illustrate the relation between motor control and cruise control; therefore, Figure 7 presents the control block of an electric vehicle considering the cruise control system, vector control system, torque load disturbance, sensors, and vehicle transmission system. Figure 5:4 Vector-controlled induction machine system. Reproduced from [18] with permission from IEEE.