**1. Introduction**

98 Induction Motors – Modelling and Control

Ameziquita-Brook, L., Liceaga-Castro, J., Liceaga-Castro, E., (2009). Induction Motor Identification for High Performance Control Design, *International Review of Electrical* 

Cohen, V. (1995). Induction motors-protection and starting. *Elektron Journal-South African* 

Erceg, G., Tesnjak, S. & Erceg, R., (1996). Modelling and Simulation of Diesel Electrical Aggregate Voltage Controler with Current Sink, *Proceedings of the IEEE International* 

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Maljković, Z., Cettolo, M., Pavlica, M. (2001). The Impact of the Induction Motor on Short-Circuit Current, *IEEE Industry Applications Magazine*. Vol. 7, No. 4; pp. (11-17). McElveen, R., Toney, M., Autom, R., & Mountain, K. (2001). Starting high-inertia loads, *IEEE* 

Mirošević, M., Maljković, Z., Milković, M., (2002), Torsional Dynamics of Generator-Units for Feeding Induction Motor Drives, *Proceedings of EPE-PEMC 9th International Conference on Power Electronics and Motion Control*, Cavtat, Dubrovnik, Croatia, Sept.

Mirošević, M., Sumina, D., & Bulić, N. (2011) Impact of induction motor starting on ship power network, *International Review of Electrical Engineering*, Vol. 6, No. 1; (February

Tolšin,V. & Kovalevskij, E., Prethodnie procesi u dizel generatorov. (Mašinostroenie, 1977). Vas, P. (1996). *Electrical Machines and Drives A Space Vector Theory Approach*, Oxford Science

*Engineering*, Vol. 4, No. 5; (October 2009), pp. (825-836) ISSN 1827-6660.

Jones, C. V. (1967). *The Unified Theory of Electrical Machines*, Butterworths, London. Krause, P. C. (1986). *Analysis of Electric Machinery*, McGrawHill, Inc. New York, N.Y.

*Institute of Electrical Engineers.12*, pp. (5-10) Citeseer.

Kundur, P. (1994). *Power System Stability and Control*, McGraw-Hill

*Transactions on Industry Applications, 37* (1), pp. (137-144).

*conference on industrial technology*, 1996

2011), pp. (186-197) ISSN 1827-6660.

**8. References** 

2002., T8-069.

Publications

Induction motors are by far the most used electro-mechanical device in industry today. Induction motors hold many advantages over other types of motors. They are cheap, rugged, easily maintainable and can be used in hazardous locations. Despite its advantages it has one major disadvantage. It draws reactive power from the source to be able to operate and therefore the power factor of the motor is inherently poor especially under starting conditions and under light load (Jimoh and Nicolae, 2007). Poor power factor adversely affects the economics of distribution and transmission systems and therefore may lead to higher electricity charges (Muljadi et al., 1989). At starting, power drawn by the motor is mainly reactive and it can draw up to 8 times its rated current at a power factor of about 0.2 until it reaches rated speed after which the power factor will increase to more than 0.6 if the motor is properly loaded and depending on the size of the motor.

To improve the power factor, reactive power compensation is needed where reactive power is injected. Several techniques have been suggested including synchronous compensation which is complex and expensive. Switched capacitor banks which requires expensive switchgear and may cause voltage regeneration, over voltage and high inrush currents (El-Sharkawi et al., 1985).

In this chapter another approach for power factor correction is explored where the stator of an induction motor has two sets of three phase windings which is electrically isolated but magnetically coupled. The main winding is connected to the three phase supply and the auxiliary winding connected to fixed capacitors for reactive power injection.

The first part of this chapter focuses on the development of a mathematical model for a normal three phase induction motor, the second part of the chapter focuses on the

$$\begin{aligned} \{C\} = \frac{2}{3} \begin{bmatrix} \cos\theta & \cos\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta - \frac{4\pi}{3}\right) \\ \sin\theta & \sin\left(\theta - \frac{2\pi}{3}\right) & \sin\left(\theta - \frac{4\pi}{3}\right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \end{aligned} \tag{1}$$

$$[C]^{-1} = \begin{bmatrix} \cos \theta & \sin \theta & 1\\ \cos \left(\theta - \frac{2\pi}{3}\right) & \sin \left(\theta - \frac{2\pi}{3}\right) & 1\\ \cos \left(\theta - \frac{4\pi}{3}\right) & \sin \left(\theta - \frac{4\pi}{3}\right) & 1 \end{bmatrix} \tag{2}$$


$$
\sigma\_{as} = r\_s i\_{as} + \frac{d\lambda\_{as}}{dt} \tag{3}
$$

$$
\upsilon\_{bs} = r\_s i\_{bs} + \frac{d\lambda\_{bs}}{dt} \tag{4}
$$

$$
\upsilon\_{cs} = r\_s i\_{cs} + \frac{d\lambda\_{cs}}{dt} \tag{5}
$$

$$
\omega\_{ar} = r\_r i\_{ar} + \frac{d\lambda\_{ar}}{dt} \tag{6}
$$

$$
\omega\_{br} = r\_r i\_{br} + \frac{d\lambda\_{br}}{dt} \tag{7}
$$

$$
\sigma\_{cr} = r\_r i\_{cr} + \frac{d\lambda\_{cr}}{dt} \tag{8}
$$

$$i'\_{abcr} = \frac{N\_r}{N\_s} i\_{abcr} \tag{9}$$

$$\boldsymbol{\upsilon}'\_{abcr} = \frac{\boldsymbol{\upsilon}\_s}{\boldsymbol{\upsilon}\_r} \boldsymbol{\upsilon}\_{abcr} \tag{10}$$

$$
\lambda'\_{abcr} = \frac{N\_s}{N\_r} \lambda\_{abcr} \tag{11}
$$

$$
\begin{bmatrix}
\lambda\_{as} \\
\lambda\_{bs} \\
\lambda\_{cs} \\
\lambda\_{ar} \\
\lambda\_{br} \\
\lambda\_{cr}
\end{bmatrix} = \begin{bmatrix}
L\_{asas} & L\_{asbs} & L\_{ascs} & L\_{asar} & L\_{asbr} & L\_{ascr} \\
L\_{bsas} & L\_{bssb} & L\_{bscss} & L\_{bsar} & L\_{bssbr} & L\_{bscr} \\
L\_{csas} & L\_{csbs} & L\_{cscs} & L\_{csar} & L\_{csbr} & L\_{cscr} \\
L\_{aras} & L\_{arbs} & L\_{arcs} & L\_{arar} & L\_{arbr} & L\_{arcr} \\
L\_{bras} & L\_{brbss} & L\_{brcs} & L\_{brar} & L\_{brbr} & L\_{brcr} \\
L\_{cras} & L\_{crbss} & L\_{crcs} & L\_{crar} & L\_{crbr} & L\_{crcr}
\end{bmatrix} \times \begin{bmatrix}
l\_{as} \\
l\_{bs} \\
l\_{cs} \\
l\_{ar} \\
l\_{br} \\
l\_{cr}
\end{bmatrix} \tag{12}
$$

$$L\_{asas} = L\_{bs\text{obs}} = L\_{c\text{sccs}} = L\_{ms} + L\_{ls} \tag{13}$$

$$L\_{ms} = \frac{\mu\_0 \ell r N\_s^2 \pi}{4g} \tag{14}$$

$$L\_{arar} = L\_{brbr} = L\_{crcr} = L\_{mr} + L\_{lr} \tag{15}$$

$$L\_{mr} = \frac{\mu\_{\theta} \ell r N\_{r}^{2} \pi}{4g} \tag{16}$$

$$L\_{\rm xsys} = \frac{\mu\_0 \ell r N\_s^2 \pi}{4g} \cos \theta\_{\rm xsys} \tag{17}$$

$$L\_{\rm xsys} = L\_{\rm ms} \cos \theta\_{\rm xsys} \tag{18}$$

$$
\cos \theta\_{\text{xsys}} = \cos \{ \pm 120^{\circ} \} = \cos \{ \pm 240^{\circ} \} = -\frac{1}{2} \tag{19}
$$

$$L\_{\rm asbs} = L\_{\rm ascs} = L\_{\rm bscs} = L\_{\rm bsas} = L\_{\rm cscas} = L\_{\rm cabs} = -\frac{1}{2} L\_{\rm ms} \tag{20}$$

$$L\_{arbr} = L\_{arcr} = L\_{brcr} = L\_{brar} = L\_{crar} = L\_{crbr} = -\frac{1}{2}L\_{mr} \tag{21}$$

$$L\_{\text{xsyr}} = L\_{\text{sr}} \cos \theta\_{\text{xsyr}} \tag{2}$$

$$L\_{sr} = \left(\frac{N\_s}{2}\right) \left(\frac{N\_r}{2}\right) \frac{\mu\_0 \pi r \ell}{g} \tag{23}$$

$$L\_{asar} = L\_{bsbr} = L\_{cscr} = L\_{sr} \cos \theta\_r \tag{24}$$

$$L\_{asbr} = L\_{bscr} = L\_{csar} = L\_{sr} \cos\left(\theta\_r + \frac{2\pi}{3}\right) \tag{25}$$

$$L\_{ascr} = L\_{bsar} = L\_{csbr} = L\_{sr} \cos\left(\theta\_r - \frac{2\pi}{3}\right) \tag{26}$$

$$L\_{arsa} = L\_{brbs} = L\_{crcs} = L\_{sr} \cos(-\theta\_r) \tag{27}$$

$$L\_{arbs} = L\_{brcs} = L\_{cras} = L\_{sr} \cos\left(\frac{2\pi}{3} - \theta\_r\right) \tag{28}$$

$$L\_{\rm arccs} = L\_{\rm bras} = L\_{\rm crbs} = L\_{\rm sr} \cos\left(\frac{4\pi}{3} - \theta\_{\rm r}\right) \tag{29}$$

$$L = \begin{bmatrix} L\_{asas} & L\_{asbss} & L\_{ascs} & L\_{asar} & L\_{asbr} & L\_{ascr} \\ L\_{bsas} & L\_{bssbs} & L\_{bscss} & L\_{bsar} & L\_{bssbr} & L\_{bscr} \\ L\_{csas} & L\_{csbs} & L\_{cscs} & L\_{csar} & L\_{csbr} & L\_{cscr} \\ L\_{aras} & L\_{arbs} & L\_{arcs} & L\_{arar} & L\_{arbr} & L\_{arcr} \\ L\_{bras} & L\_{brbs} & L\_{brcs} & L\_{brar} & L\_{brbr} & L\_{brrcr} \\ L\_{cras} & L\_{crbs} & L\_{crcs} & L\_{crar} & L\_{crbr} & L\_{crrr} \end{bmatrix} \tag{30}$$

$$L = \begin{bmatrix} L\_S & L\_{SR} \\ L\_{RS} & L\_r \end{bmatrix} \tag{31}$$

$$L\_s = \begin{bmatrix} L\_{ms} + L\_{ls} & -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & L\_{ms} + L\_{ls} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} & L\_{ms} + L\_{ls} \end{bmatrix} \tag{32}$$

$$L\_{r} = \begin{bmatrix} L\_{mr} + L\_{lr} & -\frac{1}{2}L\_{mr} & -\frac{1}{2}L\_{mr} \\ -\frac{1}{2}L\_{mr} & L\_{mr} + L\_{lr} & -\frac{1}{2}L\_{mr} \\ -\frac{1}{2}L\_{mr} & -\frac{1}{2}L\_{mr} & L\_{mr} + L\_{lr} \end{bmatrix} \tag{33}$$

$$L\_{SR} = L\_{sr} \begin{bmatrix} \cos \theta\_r & \cos(\theta\_r + \frac{2\pi}{3}) & \cos(\theta\_r - \frac{2\pi}{3}) \\ \cos(\theta\_r - \frac{2\pi}{3}) & \cos \theta\_r & \cos(\theta\_r + \frac{2\pi}{3}) \\ \cos(\theta\_r + \frac{2\pi}{3}) & \cos(\theta\_r - \frac{2\pi}{3}) & \cos \theta\_r \end{bmatrix} \tag{34}$$

$$L\_{RS} = (L\_{SR})^T = L\_{sr} \begin{bmatrix} \cos \theta\_r & \cos(\theta\_r - \frac{2\pi}{3}) & \cos(\theta\_r + \frac{2\pi}{3}) \\ \cos(\theta\_r + \frac{2\pi}{3}) & \cos \theta\_r & \cos(\theta\_r - \frac{2\pi}{3}) \\ \cos(\theta\_r - \frac{2\pi}{3}) & \cos(\theta\_r + \frac{2\pi}{3}) & \cos \theta\_r \end{bmatrix} \tag{35}$$

$$L\_{ms} = \left(\frac{N\_s}{N\_r}\right) L\_{sr} \tag{36}$$

$$L'\_{sr} = \frac{N\_s}{N\_r} \langle L\_{sr} \rangle \tag{37}$$

$$L\_{sr}' = L\_{ms} \tag{38}$$

$$L'\_{SR} = L\_{ms} \begin{bmatrix} \cos \theta\_r & \cos (\theta\_r + \frac{2\pi}{3}) & \cos (\theta\_r - \frac{2\pi}{3}) \\ \cos (\theta\_r - \frac{2\pi}{3}) & \cos \theta\_r & \cos (\theta\_r + \frac{2\pi}{3}) \\ \cos (\theta\_r + \frac{2\pi}{3}) & \cos (\theta\_r - \frac{2\pi}{3}) & \cos \theta\_r \end{bmatrix} \tag{39}$$

$$L'\_{r} = \begin{bmatrix} L'\_{lr} + L\_{ms} & -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & L'\_{lr} + L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} & L'\_{lr} + L\_{ms} \end{bmatrix} \tag{40}$$

$$
\begin{bmatrix}
\dot{\lambda}\_{abcs} \\
\lambda'\_{abcr}
\end{bmatrix} = \begin{bmatrix}
L\_s & L'\_{SR} \\
(L'\_{SR})^T & L'\_r
\end{bmatrix} \times \begin{bmatrix}
i\_{abcs} \\
i'\_{abcr}
\end{bmatrix} \tag{41}
$$

$$
\begin{bmatrix}
\lambda\_{q\,d0s} \\
\lambda'\_{q\,dor}
\end{bmatrix} = \begin{bmatrix}
K\_s L\_s K\_s^{-1} & K\_s L'\_{SR} K\_r^{-1} \\
K\_r (L'\_{SR})^T K\_s^{-1} & K\_r L'\_r K\_r^{-1}
\end{bmatrix} \times \begin{bmatrix}
l\_{q\,d0s} \\
l'\_{q\,d0r}
\end{bmatrix} \tag{42}
$$

$$K\_S = \frac{2}{3} \begin{bmatrix} \cos \theta & \cos \left(\theta - \frac{2\pi}{3}\right) & \cos \left(\theta - \frac{4\pi}{3}\right) \\ \sin \theta & \sin \left(\theta - \frac{2\pi}{3}\right) & \sin \left(\theta - \frac{4\pi}{3}\right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \tag{43}$$

$$K\_s^{-1} = \begin{bmatrix} \cos \theta & \sin \theta & 1\\ \cos \left(\theta - \frac{2\pi}{3}\right) & \sin \left(\theta - \frac{2\pi}{3}\right) & 1\\ \cos \left(\theta - \frac{4\pi}{3}\right) & \sin \left(\theta - \frac{4\pi}{3}\right) & 1 \end{bmatrix} \tag{44}$$

$$K\_{r} = \frac{2}{3} \begin{bmatrix} \cos \beta & \cos \left(\beta - \frac{2\pi}{3}\right) & \cos \left(\beta - \frac{4\pi}{3}\right) \\ \sin \beta & \sin \left(\beta - \frac{2\pi}{3}\right) & \sin \left(\beta - \frac{4\pi}{3}\right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \tag{45}$$

$$\begin{aligned} \, \_K \, \_r \, ^{-1} = \begin{bmatrix} \cos \beta & \sin \beta & 1\\ \cos \left( \beta - \frac{2\pi}{3} \right) & \sin \left( \beta - \frac{2\pi}{3} \right) & 1\\ \cos \left( \beta - \frac{4\pi}{3} \right) & \sin \left( \beta - \frac{4\pi}{3} \right) & 1 \end{bmatrix} \end{aligned} \tag{46}$$

$$
\beta = \theta - \theta\_r \tag{47}
$$

$$
\begin{bmatrix}
\lambda\_{qs} \\
\lambda\_{ds} \\
\lambda\_{qs} \\
\lambda'\_{qr} \\
\lambda'\_{dr} \\
\lambda'\_{qr}
\end{bmatrix} = \begin{bmatrix}
L\_{ls} + \frac{3}{2}L\_{ms} & 0 & 0 & \frac{3}{2}L\_{ms} & 0 & 0 \\
0 & L\_{ls} + \frac{3}{2}L\_{ms} & 0 & 0 & \frac{3}{2}L\_{ms} & 0 \\
0 & 0 & L\_{ls} & 0 & 0 & 0 \\
\frac{3}{2}L\_{ms} & 0 & 0 & L'\_{lr} + \frac{3}{2}L\_{ms} & 0 & 0 \\
0 & \frac{3}{2}L\_{ms} & 0 & 0 & L'\_{lr} + \frac{3}{2}L\_{ms} & 0 \\
0 & 0 & 0 & 0 & 0 & L'\_{lr}
\end{bmatrix} \times \begin{bmatrix}
i\_{qs} \\
i\_{ds} \\
i\_{qr} \\
i'\_{qr} \\
i'\_{dr} \\
i'\_{qr}
\end{bmatrix} \tag{48}
$$

$$L\_m = \frac{3}{2} L\_{ms} \tag{49}$$

$$
\sigma\_{abcs} = \tau\_s i\_{abcs} + \frac{d\lambda\_{abcs}}{dt} \tag{50}
$$

$$
\sigma'\_{abcr} = r'\_r l'\_{abcr} + \frac{d\lambda\nu\_{abcr}}{dt} \tag{51}
$$

$$
\sigma\_{qd0s}^{res} = K\_s r\_s K\_s^{-1} i\_{qd0s} \tag{52}
$$

$$\begin{aligned} \, \, \_K \mathbf{r}\_{\mathbf{S}} \mathbf{r}\_{\mathbf{S}}^{-1} &= \begin{bmatrix} r\_{\mathbf{S}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & r\_{\mathbf{S}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & r\_{\mathbf{S}} \end{bmatrix} \end{aligned} \tag{53}$$

$$
\sigma\_{qd0s}^{res} = r\_s i\_{qd0s} \tag{54}
$$

$$r\_s = \begin{bmatrix} r\_s & 0 & 0 \\ 0 & r\_s & 0 \\ 0 & 0 & r\_s \end{bmatrix} \tag{55}$$

$$\upsilon\_{qd0s}^{ind} = K\_s \frac{d}{dt} \left[ K\_s^{-1} \lambda\_{qd0s} \right] \tag{56}$$

$$
\sigma\_{qd0s}^{ind} = K\_s \frac{d}{dt} [K\_s^{-1}] \lambda\_{qd0s} + K\_s K\_s^{-1} \frac{d}{dt} \lambda\_{qd0s} \tag{57}
$$

$$
\theta = \int w(\xi) \, d\xi + \theta(0) \tag{58}
$$

$$\frac{d}{dt}[K\_s^{-1}] = \omega \begin{bmatrix} -\sin\theta & \cos\theta & 0\\ -\sin\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta - \frac{2\pi}{3}\right) & 0\\ -\sin\left(\theta + \frac{2\pi}{3}\right) & \cos\left(\theta + \frac{2\pi}{3}\right) & 0 \end{bmatrix} \tag{59}$$

$$K\_s \frac{d}{dt} [K\_s^{-1}] = \begin{bmatrix} 0 & \omega & 0 \\ -\omega & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \tag{60}$$

$$K\_S K\_S^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \tag{61}$$

$$\nu\_{qd0s}^{ind} = \omega \begin{bmatrix} \lambda\_{ds} \\ -\lambda\_{qs} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{vmatrix} \lambda\_{qs} \\ \lambda\_{ds} \\ \lambda\_{0s} \end{vmatrix} \tag{62}$$

$$w\_{qd\otimes s} = \begin{bmatrix} r\_s & 0 & 0 \\ 0 & r\_s & 0 \\ 0 & 0 & r\_s \end{bmatrix} \begin{bmatrix} i\_{qs} \\ i\_{ds} \\ i\_0 \end{bmatrix} + \omega \begin{bmatrix} \lambda\_{ds} \\ -\lambda\_{qs} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \lambda\_{qs} \\ \lambda\_{ds} \\ \lambda\_{0s} \end{bmatrix} \tag{63}$$

$$\boldsymbol{v}'\_{qd0r} = \begin{bmatrix} r'\_r & 0 & 0 \\ 0 & r'\_r & 0 \\ 0 & 0 & r'\_r \end{bmatrix} \begin{bmatrix} l'\_{qr} \\ l'\_{dr} \\ l'\_{0r} \end{bmatrix} + \{\boldsymbol{\omega} - \boldsymbol{\omega}\_r\} \begin{bmatrix} \boldsymbol{\lambda}'\_{dr} \\ -\boldsymbol{\lambda}'\_{qr} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \boldsymbol{\lambda}'\_{qr} \\ \boldsymbol{\lambda}'\_{dr} \\ \boldsymbol{\lambda}\_{0r} \end{bmatrix} \tag{64}$$

### **3.4. Electromagnetic torque**

The torque equation for a three phase induction machine is well known and is not derived in this section. The torque equation of the three phase machine with auxiliary winding is derived in Section 4.

$$T\_e = \left(\frac{3}{2}\right) \left(\frac{p}{2}\right) \left(\lambda\_{ds}\iota\_{qs} - \lambda\_{qs}\iota\_{ds}\right) \tag{65}$$

Modelling and Analysis of Squirrel Cage Induction Motor with Leading Reactive Power Injection 111

winding is labelled with the subscript '*abc*'. The remaining winding is treated as the auxiliary winding. The auxiliary winding is connected to static capacitors for reactive power injection. The injection of reactive power will improve the power factor of the machine. The auxiliary winding is labelled with the subscript '*xyz*'. The winding arrangement is as shown

The assumptions in developing the equations which describe the behaviour of this machine are the same as the assumptions mentioned in Section 3 with one addition. It is assumed that the main and auxiliary winding is identical. It has the same conductor cross section and

The voltage equations for this machine are developed in the same way as described in Section 3.1. There are three additional voltage equations because of the additional set of three phase windings. For simplicity the voltage equations are represented in matrix format. It is assumed that the main and auxiliary windings are identical and will therefore have the

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Because of the addition of the auxiliary winding the dimensions of the inductance matrix will increase. The dimension of the inductance matrix is equal to the number of windings, in

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**Figure 4.** Winding arrangement of Main and Auxiliary Windings

in Figure 4.

the same number of turns.

**4.1. Voltage equations** 

same resistance.

Rotor Windings:

**4.2. Inductances** 

Main Stator Windings:

Auxiliary Stator Windings:

An electric motor is an electro-mechanical device and needs an equation that couples the electrical and mechanical systems.

$$T\_{em} = f\left(\frac{2}{p}\right)\frac{d\omega\_r}{dt} + T\_L \tag{66}$$

Where P is number of poles, J is moment of inertia, TL is torque connected to the shaft and ߱ is the angular rotational speed of the rotor.

### **3.5. Equivalent circuit**

The full mathematical model of the three phase induction machine is given by Equations (48),(63)&(64). These equations are used to develop the equivalent circuits for the three phase induction machine as in Figure 3

**Figure 3.** Equivalent Circuits

### **4. Modelling of three-phase with auxiliary winding**

This machine consists of two three phase windings arranged on top of each other in the same slots. This means that there is no displacement between the two windings. These two windings are electrically isolated but magnetically connected. One of these windings is treated as the main winding and will be supplied with a three phase voltage. The main winding is labelled with the subscript '*abc*'. The remaining winding is treated as the auxiliary winding. The auxiliary winding is connected to static capacitors for reactive power injection. The injection of reactive power will improve the power factor of the machine. The auxiliary winding is labelled with the subscript '*xyz*'. The winding arrangement is as shown in Figure 4.

**Figure 4.** Winding arrangement of Main and Auxiliary Windings

The assumptions in developing the equations which describe the behaviour of this machine are the same as the assumptions mentioned in Section 3 with one addition. It is assumed that the main and auxiliary winding is identical. It has the same conductor cross section and the same number of turns.

### **4.1. Voltage equations**

110 Induction Motors – Modelling and Control

**3.4. Electromagnetic torque** 

electrical and mechanical systems.

**3.5. Equivalent circuit** 

**Figure 3.** Equivalent Circuits

**4. Modelling of three-phase with auxiliary winding** 

߱ is the angular rotational speed of the rotor.

phase induction machine as in Figure 3

derived in Section 4.

The torque equation for a three phase induction machine is well known and is not derived in this section. The torque equation of the three phase machine with auxiliary winding is

An electric motor is an electro-mechanical device and needs an equation that couples the

 <sup>ቁ</sup> ௗఠ

Where P is number of poles, J is moment of inertia, TL is torque connected to the shaft and

The full mathematical model of the three phase induction machine is given by Equations (48),(63)&(64). These equations are used to develop the equivalent circuits for the three

This machine consists of two three phase windings arranged on top of each other in the same slots. This means that there is no displacement between the two windings. These two windings are electrically isolated but magnetically connected. One of these windings is treated as the main winding and will be supplied with a three phase voltage. The main

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ௗ௧ ܶ (66)

ܶ ൌ ቀ<sup>ଷ</sup> ଶ ቁ ቀ ଶ

> The voltage equations for this machine are developed in the same way as described in Section 3.1. There are three additional voltage equations because of the additional set of three phase windings. For simplicity the voltage equations are represented in matrix format. It is assumed that the main and auxiliary windings are identical and will therefore have the same resistance.

Main Stator Windings:

$$
\omega\_{abcs} = \tau\_s l\_{abcs} + \frac{d\lambda\_{abcs}}{dt} \tag{67}
$$

Auxiliary Stator Windings:

$$
\sigma\_{\text{xyzs}} = \tau\_s i\_{\text{xyzs}} + \frac{d\lambda\_{\text{xyzs}}}{dt} \tag{68}
$$

Rotor Windings:

$$\mathbf{r}'\_{abcr} = r'\_{r}i'\_{abcr} + \frac{d\lambda\nu\_{abcr}}{dt} \tag{69}$$

### **4.2. Inductances**

Because of the addition of the auxiliary winding the dimensions of the inductance matrix will increase. The dimension of the inductance matrix is equal to the number of windings, in

$$
\begin{bmatrix} L\_{abc\,\,s} \\ L\_{\rm xyz\,\,s} \\ L\_{abc\,\,r} \end{bmatrix} = \begin{bmatrix} L\_{abc\,\,cs\,} & L\_{abc\,\,cxy\,\,z\,\,s} & L\_{abc\,\,sab\,\,cbr} \\ L\_{\rm xyz\,\,sab\,\,c\,s} & L\_{\rm xyz\,\,s} & L\_{\rm xyz\,\,sab\,\,cr} \\ L\_{ab\,\,crab\,\,c\,\} & L\_{ab\,\,cray\,\,z\,\,s} & L\_{ab\,\,cr} \end{bmatrix} \tag{70}
$$

$$L\_{abcs} = L\_{\rm xyzs} = \begin{bmatrix} L\_{ms} + L\_{ls} & -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & L\_{ms} + L\_{ls} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} & L\_{ms} + L\_{ls} \end{bmatrix} \tag{71}$$

$$L\_{abcr} = \begin{bmatrix} L\_{mr} + L\_{lr} & -\frac{1}{2}L\_{mr} & -\frac{1}{2}L\_{mr} \\ -\frac{1}{2}L\_{mr} & L\_{mr} + L\_{lr} & -\frac{1}{2}L\_{mr} \\ -\frac{1}{2}L\_{mr} & -\frac{1}{2}L\_{mr} & L\_{mr} + L\_{lr} \end{bmatrix} \tag{72}$$

$$L\_{abc\text{-}abcr} = L\_{\text{xyz}\text{-}abcr} = L\_{\text{ms}} \begin{bmatrix} \cos\theta\_r & \cos(\theta\_r + \frac{2\pi}{3}) & \cos(\theta\_r - \frac{2\pi}{3})\\ \cos(\theta\_r - \frac{2\pi}{3}) & \cos\theta\_r & \cos(\theta\_r + \frac{2\pi}{3})\\ \cos(\theta\_r + \frac{2\pi}{3}) & \cos(\theta\_r - \frac{2\pi}{3}) & \cos\theta\_r \end{bmatrix} \tag{73}$$

$$L\_{abcrabcs} = L\_{abcrxyzs} = L\_{abcsabcr}^T \tag{74}$$

$$L\_{\text{abcscxyz}} = \begin{bmatrix} L\_{\text{asxs}} & L\_{\text{asys}} & L\_{\text{asxs}} \\ L\_{\text{bsxs}} & L\_{\text{bsys}} & L\_{\text{bszs}} \\ L\_{\text{csxs}} & L\_{\text{cyss}} & L\_{\text{cszs}} \end{bmatrix} \tag{75}$$

$$L\_{\rm asxs} = L\_{\rm bsys} = L\_{\rm cszs} = L\_{\rm ms} \cos \left( 0^{\circ} \right) = L\_{\rm ms} \tag{76}$$

$$L\_{\rm asys} = L\_{\rm aszs} = L\_{\rm bsxs} = L\_{\rm bsxs} = L\_{\rm csxs} = L\_{\rm csxs} = L\_{\rm ms} \cos\left(\pm 120^{\circ}\right) = -\frac{1}{2} L\_{\rm ms} \tag{77}$$

$$L\_{\text{abcscxyz}} = \begin{bmatrix} L\_{ms} & -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} & L\_{ms} \end{bmatrix} \tag{78}$$

$$L\_{\text{xyzsabcs}} = L\_{\text{abcsxyzs}} \tag{79}$$

$$
\begin{bmatrix} L\_{11s} \\ L\_{1d1s} \\ L\_{01s} \\ L\_{01s} \\ L\_{02s} \\ L\_{02s} \\ L\_{qr} \\ L\_{dr} \\ L\_{0r} \\ L\_{0r} \end{bmatrix} = \begin{bmatrix} L\_{SS} & 0 & 0 & L\_m & 0 & 0 & L\_m & 0 & 0 \\ 0 & L\_{SS} & 0 & 0 & L\_m & 0 & 0 & L\_m & 0 \\ 0 & 0 & L\_{ls} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & L\_m & 0 & 0 & L\_{SS} & 0 & 0 & L\_m & 0 \\ 0 & 0 & 0 & 0 & 0 & L\_{ls} & 0 & 0 & 0 \\ L\_m & 0 & 0 & L\_m & 0 & 0 & L\_{rr} & 0 & 0 \\ 0 & L\_m & 0 & 0 & L\_m & 0 & 0 & L\_{rr} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & L\_{lr} \end{bmatrix} \tag{80}
$$

$$L\_{\rm ss} = L\_{\rm ls} + L\_{\rm m} \tag{81}$$

$$L\_{rr} = L\_{lr}' + L\_m \tag{82}$$

$$\mathbf{y} = \mathbf{L}\mathbf{I} \tag{83}$$

$$
\boldsymbol{v}\_{q\,d0\,1\,s} = \begin{bmatrix} r\_s & 0 & 0 \\ 0 & r\_s & 0 \\ 0 & 0 & r\_s \end{bmatrix} \begin{bmatrix} i\_{q1s} \\ i\_{d1s} \\ i\_{01s} \end{bmatrix} + \boldsymbol{\omega} \begin{bmatrix} \lambda\_{d1s} \\ -\lambda\_{q1s} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \lambda\_{q1s} \\ \lambda\_{d1s} \\ \lambda\_{01s} \end{bmatrix} \tag{84}$$

$$
\boldsymbol{\upsilon}'\_{qd0r} = \begin{bmatrix} r'\_r & 0 & 0 \\ 0 & r'\_r & 0 \\ 0 & 0 & r'\_r \end{bmatrix} \begin{bmatrix} l'\_{qr} \\ l'\_{dr} \\ l'\_{0r} \end{bmatrix} + \{\boldsymbol{\omega} - \boldsymbol{\omega}\_r\} \begin{bmatrix} \boldsymbol{\lambda}'\_{dr} \\ -\boldsymbol{\lambda}'\_{qr} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \boldsymbol{\lambda}'\_{qr} \\ \boldsymbol{\lambda}'\_{dr} \\ \boldsymbol{\lambda}\_{0r} \end{bmatrix} \tag{85}
$$

$$V c\_{qd02s} = \frac{1}{c} \int [i\_{qd0s2}] dt + \omega \begin{bmatrix} V c\_{d02} \\ -V c\_{q02} \\ 0 \end{bmatrix} \tag{86}$$

$$
\boldsymbol{\upsilon}\_{qd02s} = \mathbf{0} = \begin{bmatrix} r\_s & 0 & 0 \\ 0 & r\_s & 0 \\ 0 & 0 & r\_s \end{bmatrix} \begin{bmatrix} i\_{q2s} \\ i\_{d2s} \\ i\_{02s} \end{bmatrix} + \boldsymbol{\omega} \begin{bmatrix} \lambda\_{d2s} \\ -\lambda\_{q2s} \\ 0 \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \lambda\_{q2s} \\ \lambda\_{d2s} \\ \lambda\_{02s} \end{bmatrix} + \begin{bmatrix} V c\_{q2} \\ V c\_{d2} \\ 0 \end{bmatrix} \tag{87}
$$

$$\mathcal{W}\_f = \frac{1}{2} (\mathfrak{l}\_{\rm abscs})^T (L\_s - L\_{\rm ls}I)\mathfrak{l}\_{\rm abscs} + (\mathfrak{l}\_{\rm abscs})^T L'\_{sr} \mathfrak{l}'\_{\rm abscr} + \frac{1}{2} (\mathfrak{l}'\_{\rm abscr})^T (L'\_r - L'\_{lr}I) \mathfrak{l}'\_{\rm abscr} \tag{88}$$

$$dW\_m = T\_{em} d\theta\_{rm} \tag{89}$$

$$
\theta\_r = \left(\frac{p}{2}\right)\theta\_{rm} \tag{90}
$$

$$d\mathcal{W}\_m = T\_{em} \, \frac{z}{p} d\theta\_r \tag{91}$$

$$T\_{em} \left( \mathfrak{i}\_f, \theta\_r \right) = \frac{P}{2} \frac{\partial W\_{c \left( \mathfrak{i}\_f, \theta\_r \right)}}{\partial \theta\_r} \tag{92}$$

$$\frac{P}{2}\frac{t\_T}{2}\overline{t\_S}^T\frac{\partial\left[L\_{ss}\right]}{\partial\theta\_r}\text{ is}\tag{93}$$

$$\mathfrak{i}\_{\mathfrak{s}} = \left[ \begin{array}{c} \iota\_{\text{labcs}} \\ \end{array} \right] \qquad \qquad \iota\_{\text{kxyzs}} \left[ \begin{array}{c} \\ \end{array} \right] \tag{94}$$

$$L\_s = \begin{bmatrix} L\_{abccs} & L\_{abcxyzs} \\ L\_{\text{xyzabcs}} & L\_{\text{xyzas}} \end{bmatrix} \tag{95}$$

$$L\_{\rm ss} = \{L\_{\rm s} - L\_{\rm ls}\} \tag{96}$$

$$T\_{em} = \frac{p}{2} \{ \{ \begin{array}{c} \quad \text{labcs} \end{array} \} ^T \frac{\partial [L\_{\text{labcs}}]}{\partial \theta\_r} \quad \text{labcs} $$

$$+ \{ \begin{array}{c} \quad \text{labcs} \end{array} \} ^T \frac{\partial [L\_{\text{labcs}}]\_{\text{x}}}{\partial \theta\_r} \quad \text{lazyz} + \left( \begin{array}{c} \quad \text{lcyzs} \end{array} \right) ^T \frac{\partial [L\_{\text{qzyz}}]}{\partial \theta\_r} \quad \text{lcyzs} +$$

$$+ \left( \begin{array}{c} \quad \text{lcyzs} \end{array} \right) ^T \frac{\partial [L\_{\text{qzyz}}]\_{\text{x}}}{\partial \theta\_r} \quad \text{labor} \right) \tag{97}$$

$$T\_{em} = \frac{3}{2} \frac{p}{2} \left(\lambda\_{d1s}\dot{\iota}\_{q1s} - \lambda\_{q1s}\dot{\iota}\_{d1s} + \lambda\_{d2s}\dot{\iota}\_{q1s} - \lambda\_{q2s}\dot{\iota}\_{d2s}\right) \tag{98}$$
