**4.4 Corrections of asynchronous electric drive**

It is interesting to analyze the stability of AED, the block diagram of which contains several nonlinear dynamic links described above. As shown in **Figure 1**, vector control "linearizes" the drive converting it to a linear structure. As was shown above, this linearization is very conditional and not too accurate mathematically. However, even if we make assumptions about Laplace transformations with variable coefficients, it is impossible to avoid discrepancies in the parameters (and structure) of the model and the real motor. This will lead to the fact that their serial connection will produce a disproportionate linear link and a floating dynamic link that will be close to the resonant link. Even if we assume that this link is close to linear, for stability it is necessary to apply the stability criterion for nonlinear systems (a variant of the Popov criterion) and, therefore, to consider the entire frequency range, and not just the cutoff frequency. **Figures 28** and **29** show how applying the Nyquist criterion gives an incorrect result (and the Popov criterion is not satisfied) for a vector-controlled drive closed in speed with a PID controller. As the experiments showed, the applied algorithms are not very effective, and the hard positive feedback on the stator current breaks the stability. In works [6, 15, 18] the positive dynamic connection on current is described. Without compromising

#### **Figure 28.**

**Figure 25.**

*Control Theory in Engineering*

**Figure 26.**

**Figure 27.**

**192**

*Stability assessment for servo drive according to the Popov criterion (not satisfied).*

*Stability assessment for servo drive with stabilizing rate feedback according to the Popov criterion (satisfied).*

*Block diagram of servo drive with stabilizing rate feedback.*

*Assessment of stability in the AC drive according to the Popov criterion (not satisfied).*

#### **Figure 29.**

*Stabilization of the drive by positive feedback of the stator current with a first-order filter. Verification by Popov criterion.*

stability (and in some cases improving it), this relationship provides full compensation for the load torque and not only in statics. The effect of such a relationship is described in [5, 6] and the patent [9, 11] and also confirmed by numerous experiments. The effectiveness of this connection is also confirmed by the calculation of the transfer function and frequency characteristics of the link for the formation of the torque with a positive feedback of the stator current (PDF—positive dynamic feedback).

slip. The sign (�) in front of the formula means that the relationship must be positive. We call this relationship positive dynamic feedback (PDF). It should be noted that the correction of the coefficient from the frequency is very easy to implement in frequency converters; several options for partial (indirect) adaptation of the connection to the load will be discussed below. Thus, the proposed positive feedback, selected by condition (15), makes it possible to compensate for the effect of the external load and the nonlinearity of the AED, spreading the transfer function of the motor as a link of the first order for any values of β. In addition, the block diagram (**Figure 13**) and the transfer function of the torque formation link (6) connecting the torque and slip allow us to offer an estimate of the efficiency of the torque formation algorithm: the algorithm that generates the necessary torque with

*Nonlinear Dynamics of Asynchronous Electric Drive: Engineering Interpretation and Correction…*

Next, we consider the correction of the AED with the parameters corresponding

As expected, the frequency characteristics of the AED with the structural correction proposed in the work are close to the frequency characteristics of the first-

In widely used AEDs, it is very difficult to realize the mechanical torque feedback. Given that the electromagnetic torque is equal to *I1\*Ψ<sup>2</sup>* and in almost all calculations it is assumed that the rotor flux linkage is constant, you can replace the original signal in this local connection with the effective value of the stator current

To communicate with the stator current, the linearization conditions will vary

This expression shows that when controlling the flux linkage, the linearization

On the other hand, it is easy to show that with some inaccuracy in the fulfillment

<sup>2</sup>*<sup>p</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>W</sup>*<sup>1</sup> Ψ<sup>2</sup>

<sup>2</sup>*<sup>p</sup>* <sup>þ</sup> <sup>1</sup> *WDF*

*Ψ*2

6¼ 0 ¼ Δ*,* (18)

**Dynamic feedback transfer function,** *WDF*

> 3*:*128 0*:*017pþ1

> 16*:*99 0*:*017pþ1

> 0*:*09 0*:*017pþ1

> 0*:*497 0*:*017pþ1

(17)

or its active component, which is calculated in all the inverters.

*<sup>ω</sup>*1*β*<sup>2</sup> ¼ �2*MkSk <sup>T</sup>*<sup>0</sup>

conditions can be refined, thereby providing high-quality regulation.

� <sup>2</sup>*Mk <sup>T</sup>*<sup>0</sup>

**Slip\* Electromagnetic torque transfer function, W p**ð Þ

0*:*0002p2þ0*:*0229pþ1*:*38

0*:*0002p2þ0*:*0229pþ4*:*52

0*:*006p2þ0*:*628pþ20*:*56

0*:*006p2þ0*:*628pþ21*:*19

*The transfer functions of the channel of formation of the electromagnetic torque and dynamic feedback for*

of the linearization condition, that is,

*β Sk*

β<sup>2</sup> <sup>0</sup>*:*038pþ0*:*<sup>226</sup>

β<sup>2</sup> <sup>0</sup>*:*027pþ1*:*<sup>548</sup>

*β*<sup>1</sup> *corresponds to slip at low load, and β*<sup>2</sup> *corresponds to slip at nominal load.*

10 β<sup>1</sup> <sup>0</sup>*:*038pþ0*:*<sup>226</sup>

50 β<sup>1</sup> <sup>0</sup>*:*027pþ1*:*<sup>548</sup>

*different values of the frequency of the stator voltage and slip.*

to the frequencies of the supply voltage of 10 and 50 Hz. The initial frequency characteristics are shown in **Figure 19**. The transfer functions of the original AED with such parameters and the transfer functions of the corrective units are given in **Table 3**, and the initial and corrected frequency characteristics of the AED are shown in **Figures 31** and **32** for the frequencies of the supply voltage of 10 and

the smallest absolute slip will be more effective.

*DOI: http://dx.doi.org/10.5772/intechopen.88223*

50 Hz, respectively.

order linear link.

somewhat:

**Frequency, Hz**

*\**

**195**

**Table 3.**
