## 4 Stable _V/f_ Control with Constant Power Factor Loop
### Stabilizing Loop  
The stabilizing loop can be implemented in the \(V/f\) control to prevent rotor oscillations, thus stabilizing the system. The principle of the stabilizing loop lies in the superimposition of the small frequency signal onto the required electrical angular velocity. The superimposed frequency signal must be of opposite polarity to the experienced perturbation. The perturbation can be extracted, for example, from:  
* DC link current;
* Actual rotor speed;
* Input active power.  
Each of the components requires some extra features to be employed or implemented. For the perturbation extraction from DC link current, a current sensor in DC bus is necessary, leading to higher additional costs. The extraction from the actual rotor speed can be considered as the most reliable. However, a position sensor is required for this purpose, removing advantages associated with sensorless control. The most practical with respect to the final cost is the solution that uses an input active power component. This strategy requires phase currents measurements, which is the common feature in the motor control drives. The transformation of the three phase currents from \(abc\) to \(\alpha\beta\) reference frame must be performed to calculate the active power \(P_{e}\) as:  
\[P_{e}=\frac{3}{2}\left(v_{\alpha}^{*}i_{\alpha}+v_{\beta}^{*}i_{\beta}\right) \tag{33}\]  
where \(v_{\alpha}^{*}\), \(v_{\beta}^{*}\) are the required \(\alpha\beta\) voltages and \(i_{\alpha}\), \(i_{\beta}\) are the measured \(\alpha\beta\) currents. The input active power perturbation component \(\Delta P_{e}\) can be extracted from origin signal with a first-order high-pass filter. It eliminates a DC component from the active power \(P_{e}\), and the useful signal contains only higher-frequency components:  
\[\Delta P_{e}=\frac{s}{s+\frac{1}{v_{\mu}}}P_{e} \tag{34}\]  
Figure 14: Block diagram of the stable \(V/f\) control with constant power factor loop.  
where \(\tau_{H}\) is the time constant of the high-pass filter. This constant can be experimentally determined regarding the speed oscillations [49]. Afterward, it is necessary to amplify the AC component of active power with proportional gain of the stabilizing loop \(K_{S}\):  
\[K_{s}=\frac{C_{1}}{\omega_{e}^{*}},\omega_{e}^{*}\
eq 0 \tag{35}\]  
where \(C_{1}\) is the stabilizing loop constant, which must be divided by the reference electrical angular velocity \(\omega_{e}^{*}\) in order to maintain almost constant damping, particularly at higher frequencies. Then, the output of the stabilizing loop is the speed perturbation component \(\Delta\omega_{e}\), which is superimposed onto the required velocity \(\omega_{e}^{*}\) to stabilize scalar control:  
\[\Delta\omega_{e}=-K_{s}\Delta P_{e} \tag{36}\] \[\omega_{v}^{*}=\omega_{e}^{*}+\Delta\omega_{e}\]  
where \(\omega_{v}^{*}\) is the voltage vector reference angular velocity. Integrating this variable, the voltage vector position \(\theta_{v}^{*}\) is generated to perform Park transformation in feedforward path and constant power factor loop. The block diagram of the stabilizing loop is presented in Figure 15[42; 43]: