**6. Simulation study upon some transient duties of the three-phase induction machine**

### **6.1. Symmetric supply system**

28 Induction Motors – Modelling and Control

torque is equal to 124 Nm and obviously *Uas(-)* = 0). If the voltage on phase *b* keeps the same amplitude as the voltage in phase *a*, for example, but the angle of phase difference changes with *π/24=7,5* degrees (from *2π/3=16π/24* to *17π/24* rad.) then the new characteristic is the B curve. The pull-out torque value decreases with approx. 12% but the pull-out slip keeps its

Usually, the *unbalance degree* of the supply voltage is defined as the ratio of inverse and

Slip s [-]


1 2 sin( / 6) 100[%]

(70)

 A B

 C D

 

 

1 2 sin( / 6)

The curves A, B, C, and D from Fig. 7 correspond to the following values of the unbalance degree: *un*= 0; 8%; 16% and 27%. The highest unbalance degree (27% - curve D) causes a

b. The second approach takes into consideration the following reasoning. When the amplitudes of the three-phase supply system and/or the angles of the phase difference are not equal to 2π/3 then the *unbalanced system* can be replaced by two *balanced threephase* systems that act in opposition. One of them is the *direct sequence* system and has higher voltages and the other is the *inverse sequence* system and has lower voltages. A transformation of the unbalanced voltages and total fluxes into two symmetric systems is again necessary. In other words, there is an unbalanced voltage system (*Uas, Ubs, Ucs*),

2

value. Other two unbalance degrees are presented in Fig. 7 as well.

k=1, β=16π/24; un=0% k=1, β=17π/24; un=8% k=1, β=18π/24; un=16% k=0.71, β=2π/3; un=27%

**Figure 7.** Te=f(s) characteristic for different unbalance degrees

*n*

*u*


Electromagnetic torque Te [Nm]

decrease of the pull-out torque by 40%.

( )

 

*s*

*s*

<sup>2</sup> ( )

*U k k*

*U k k*

direct components:

The mathematical model described by the equation system (26-1…8) allows a complete simulation study of the operation of the three-phase induction machine, which include startup, any sudden change of the load and braking to stop eventually. To this end, the machine parameters (resistances, main and leakage phase inductances, moments of inertia corresponding to the rotor and the load, coefficients that characterize the variable speed and torque, etc.) have to be calculated or experimentally deduced. At the same time, the values of the load torque and the expressions of the instantaneous voltages applied to each stator phase winding are known, as well. The rotor winding is considered short-circuited. Using the above mentioned equation system, the structural diagram in the Matlab-Simulink environment can be carried out. Additionally, for a complete evaluation, virtual oscillographs for the visualization of the main physical parameters such as voltage, current, magnetic flux, torque, speed, rotation angle and current or specific characteristics (mechanical characteristic, angular characteristic or flux hodographs) fill out the structural diagram.

The study of the *symmetric three-phase* condition in the Matlab-Simulink environment takes into consideration the following parameter values: three identical supply voltages with the amplitude of 490 V (Uas=346.5V) and 2π/3 rad. shifted in phase; uar=ubr=ucr=0 since the rotor winding is short-circuited; Rs=Rr=2; Lhs=0,09; Lσs= Lσr=0,01; J=0,05; p=2; kz=0,02; ω1=314,1 (SI units). The equation system becomes:

$$\begin{aligned} \left(\overline{\text{s}} + 135, 71\right) \overline{\nu}\_{\text{as}} &= \overline{\text{u}}\_{\text{as}} - 32, 14\left(\overline{\nu}\_{\text{bs}} + \overline{\nu}\_{\text{cs}}\right) + 32, 14\left(2\overline{\nu}\_{\text{ar}} - \overline{\nu}\_{\text{br}} - \overline{\nu}\_{\text{cr}}\right) \cos\theta\_{\text{R}} + 55, 67(\overline{\nu}\_{\text{cr}} - \overline{\nu}\_{\text{br}}) \sin\theta\_{\text{R}} \\ \left(\overline{\text{s}} + 135, 71\right) \overline{\nu}\_{\text{bs}} &= \overline{\text{u}}\_{\text{bs}} - 32, 14\left(\overline{\nu}\_{\text{cs}} + \overline{\nu}\_{\text{as}}\right) + 32, 14\left(2\overline{\nu}\_{\text{br}} - \overline{\nu}\_{\text{cr}} - \overline{\nu}\_{\text{ar}}\right) \cos\theta\_{\text{R}} + 55, 67(\overline{\nu}\_{\text{ar}} - \overline{\nu}\_{\text{cr}}) \sin\theta\_{\text{R}} \end{aligned}$$

 135,71 32,14 *cs cs as bs* 32,14 2 *cr ar br* cos 55,67( )sin *R R br ar s u* 135,71 0 32,14 *ar br cr* 32,14 2 *as bs cs* cos 55,67( )sin *R R bs cs s* 135,71 0 32,14 *br cr ar* 32,14 2 *bs cs as* cos 55,67( )sin *R R cs as s* 135,71 0 32,14 *cr ar br* 32,14 2 *cs as bs* cos 55,67( )sin *R R as bs s* 0,4 40 32,14 sin 2 2 2 3 cos *R R as ar br cr bs br cr ar cs cr ar br R as br cr bs cr ar cs ar br st s T* (73-1-7) 1

$$
\boldsymbol{\theta}\_R = \boldsymbol{\phi}\_R \underset{\mathbf{S}}{=} \tag{73-8}
$$

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 31

**Figure 8.** Time variation of rotational pulsatance – *RS-50*

50

0

100

Rotational pulsatance

ω

R [rad/s]

150

**Figure 9.** Time variation of rotational pulsatance – *RS-120*

50

0

100

Rotational pulsatance

ω

R [rad/s]

150

operation), S points on the hodographs.

duties.

In the first moments of the start-up, the electromagnetic torque oscillates around 100 Nm and after the load torque enforcement, it gets to approx. 53 Nm for *RS-50*, Fig. 10 and to approx. 122 Nm for *RS-120*, Fig. 11. The operation of the motor remains stable for the both

Time t [s] 0.1 0.3 0.5

Time t [s] 0.1 0.3 0.5

The behavior of the machine is very interesting described by the hodograph of the resultant rotor flux (the locus of the head of the resultant rotor flux phasor), Fig. 12 and 13. With the connecting moment, the rotor fluxes start from 0 (O points on the hodograph) and track a corkscrew to the maximum value that corresponds to synchronism (ideal no-load

$$\begin{aligned} \stackrel{\text{u}}{u}\_{\text{as}} &\leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t)}; \stackrel{\text{u}}{u}\_{\text{bs}} \leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t - 2, 094)}; \stackrel{\text{u}}{u}\_{\text{cs}} &\leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t - 4, 188)}; \\\ &\cup \quad \cup \quad \cup\_{\text{as}\,\text{max}} = \text{U}\_{\text{bs}\,\text{max}} = \text{L}\mathbf{0}\_{\text{cs}\,\text{max}} = \text{490} \end{aligned} \tag{73-9}$$

It has to be mentioned again that the above equation system allows the analysis of the threephase induction machine under any condition, that is transients, steady state, symmetric or unbalanced, with one or both windings (from stator and rotor) connected to a supply system. Generally, a supplementary requirement upon the stator supply voltages is not mandatory. The case of short-circuited rotor winding, when the rotor supply voltages are zero, include the wound rotor machine under rated operation since the starting rheostat is short-circuited as well.

The presented simulation takes into discussion a varying duty, which consists in a *no-load* start-up (the load torque derives of frictions and ventilation and is proportional to the angular speed and have a steady state rated value of approx. 3 Nm) followed after 0,25 seconds by a sudden loading with a constant torque of 50 Nm. The simulation results are presented in Fig. 8, 10, 12, 14 and 15 and denoted by the symbol *RS-50.* A second simulation iterates the presented varying duty but with a load torque of 120 Nm, symbol *RS-120*, Fig. 9, 11 and 13. Finally, a third simulation takes into consideration a load torque of 125 Nm, which is a value over the pull-out torque. Consequently, the falling out and the stop of the motor in t≈0,8 seconds mark the varying duty (symbol *RS-125*, Fig. 16, 17, 18 and 19).

The *RS-50* simulation shows an upward variation of the angular speed to the no-load value (in t ≈ 0,1 seconds), which has a weak overshoot at the end, Fig. 8. The 50 Nm torque enforcement determines a decrease of the speed corresponding to a slip value of s ≈ 6,5%. In the case of the *RS-120* simulation, the start-up is obviously similar but the loading torque determines a much more significant decrease of the angular speed and the slip value gets to s ≈ 25%, Fig. 9.

**Figure 8.** Time variation of rotational pulsatance – *RS-50*

*s*

short-circuited as well.

> 

135,71 32,14 *cs cs as bs* 32,14 2 *cr ar br* cos 55,67( )sin *R R br ar s u*

135,71 0 32,14 *ar br cr* 32,14 2 *as bs cs* cos 55,67( )sin *R R bs cs s*

135,71 0 32,14 *br cr ar* 32,14 2 *bs cs as* cos 55,67( )sin *R R cs as s*

135,71 0 32,14 *cr ar br* 32,14 2 *cs as bs* cos 55,67( )sin *R R as bs s*

*cs cr ar br R as br cr bs cr ar cs ar br st*

 

*R R s*

max max max

It has to be mentioned again that the above equation system allows the analysis of the threephase induction machine under any condition, that is transients, steady state, symmetric or unbalanced, with one or both windings (from stator and rotor) connected to a supply system. Generally, a supplementary requirement upon the stator supply voltages is not mandatory. The case of short-circuited rotor winding, when the rotor supply voltages are zero, include the wound rotor machine under rated operation since the starting rheostat is

The presented simulation takes into discussion a varying duty, which consists in a *no-load* start-up (the load torque derives of frictions and ventilation and is proportional to the angular speed and have a steady state rated value of approx. 3 Nm) followed after 0,25 seconds by a sudden loading with a constant torque of 50 Nm. The simulation results are presented in Fig. 8, 10, 12, 14 and 15 and denoted by the symbol *RS-50.* A second simulation iterates the presented varying duty but with a load torque of 120 Nm, symbol *RS-120*, Fig. 9, 11 and 13. Finally, a third simulation takes into consideration a load torque of 125 Nm, which is a value over the pull-out torque. Consequently, the falling out and the stop of the

motor in t≈0,8 seconds mark the varying duty (symbol *RS-125*, Fig. 16, 17, 18 and 19).

more significant decrease of the angular speed and the slip value gets to s ≈ 25%, Fig. 9.

The *RS-50* simulation shows an upward variation of the angular speed to the no-load value (in t ≈ 0,1 seconds), which has a weak overshoot at the end, Fig. 8. The 50 Nm torque enforcement determines a decrease of the speed corresponding to a slip value of s ≈ 6,5%. In the case of the *RS-120* simulation, the start-up is obviously similar but the loading torque determines a much

2 3 cos

(73-1-7)

1

(314,1 ) (314,1 2,094) (314,1 4,188)

;;;

 

> 

490 490 490

*u eu e u e*

*UUU*

22 2

*as bs cs*

*j t j t j t as bs cs*

 

*R R as ar br cr bs br cr ar*

0,4 40 32,14 sin 2 2

   

 

 

 

 

490

 

 

 

 

 

(73-8)

*T*

 

 

 

 

(73-9)

**Figure 9.** Time variation of rotational pulsatance – *RS-120*

In the first moments of the start-up, the electromagnetic torque oscillates around 100 Nm and after the load torque enforcement, it gets to approx. 53 Nm for *RS-50*, Fig. 10 and to approx. 122 Nm for *RS-120*, Fig. 11. The operation of the motor remains stable for the both duties.

The behavior of the machine is very interesting described by the hodograph of the resultant rotor flux (the locus of the head of the resultant rotor flux phasor), Fig. 12 and 13. With the connecting moment, the rotor fluxes start from 0 (O points on the hodograph) and track a corkscrew to the maximum value that corresponds to synchronism (ideal no-load operation), S points on the hodographs.

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 33

2 [Wb]

The enforcement of the load torque determines a decrease of the resultant rotor flux, which is proportional to the load degree, and is due to the rotor reaction. The locus of the head of the phasor becomes a circle whose radius is proportional to the amplitude of the resultant rotor flux. The speed on this circle is given by the rotor frequency that is by the slip value. It is interesting to notice that the load torque of 50 Nm causes a unique rotation of the rotor flux whose amplitude becomes equal to the segment ON (Fig. 12) whereas the 120 Nm torque causes approx. 4 rotations of the rotor flux and the amplitude OF is significantly smaller (Fig. 13).

O


F

S

2 [Wb]


If the expressions (1) and (2) are also used in the structural diagram then both stator and rotor phase currents can be plotted. The stator current corresponding to *as* phase has the frequency f1=50 Hz and gets a start-up amplitude of approx. 70 A. This value decreases to approx. 6 A (no-load current) and after the torque enforcement (50 Nm) it rises to a stable value of approx. 14 A, Fig. 14. The rotor current on phase *ar*, which has a frequency value of f2 = s· f1, gets a similar (approx. 70 A) start-up variation but in opposition to the stator current, *ias*. Then, its value decrease and the frequency go close to zero. The loading of the machine has as result an increase of the rotor current up to 13 A and a frequency value of f2≈3Hz, Fig. 15. The fact that the current variations are sinusoidal and keep a constant

frequency is an argument for a stable operation under symmetric supply conditions.

0.2

0 0.4 0.6 Time t [s]

**Figure 13.** Hodograph of resultant rotor flux – *RS-120*

**Figure 14.** Time variation of stator phase current – *RS-50*

Stator phase current ias [A]

80


**Figure 10.** Time variation of electromagnetic torque – *RS-50*

**Figure 11.** Time variation of electromagnetic torque – *RS-120*

**Figure 12.** Hodograph of resultant rotor flux – *RS-50*

**Figure 13.** Hodograph of resultant rotor flux – *RS-120*

**Figure 10.** Time variation of electromagnetic torque – *RS-50*

100

0

Electromagnetic torque Te [Nm]

200

Time t [s] 0.1 0.3 0.5

Time t [s] 0.1 0.3 0.5

S

O

0



2 [Wb]

N

[Wb]

**Figure 11.** Time variation of electromagnetic torque – *RS-120*

100

0

Electromagnetic torque Te [Nm]

200

**Figure 12.** Hodograph of resultant rotor flux – *RS-50*

The enforcement of the load torque determines a decrease of the resultant rotor flux, which is proportional to the load degree, and is due to the rotor reaction. The locus of the head of the phasor becomes a circle whose radius is proportional to the amplitude of the resultant rotor flux. The speed on this circle is given by the rotor frequency that is by the slip value. It is interesting to notice that the load torque of 50 Nm causes a unique rotation of the rotor flux whose amplitude becomes equal to the segment ON (Fig. 12) whereas the 120 Nm torque causes approx. 4 rotations of the rotor flux and the amplitude OF is significantly smaller (Fig. 13).

If the expressions (1) and (2) are also used in the structural diagram then both stator and rotor phase currents can be plotted. The stator current corresponding to *as* phase has the frequency f1=50 Hz and gets a start-up amplitude of approx. 70 A. This value decreases to approx. 6 A (no-load current) and after the torque enforcement (50 Nm) it rises to a stable value of approx. 14 A, Fig. 14. The rotor current on phase *ar*, which has a frequency value of f2 = s· f1, gets a similar (approx. 70 A) start-up variation but in opposition to the stator current, *ias*. Then, its value decrease and the frequency go close to zero. The loading of the machine has as result an increase of the rotor current up to 13 A and a frequency value of f2≈3Hz, Fig. 15. The fact that the current variations are sinusoidal and keep a constant frequency is an argument for a stable operation under symmetric supply conditions.

**Figure 14.** Time variation of stator phase current – *RS-50*

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 35

The described critical duty that involves no-load start-up and operation, overloading, falling out and stop is plotted in terms of resultant rotor flux and angular speed versus electromagnetic torque. The hodograph (Fig. 18) put in view a cuasi corkscrew section, corresponding to the start-up, characterized by its maximum value represented by the segment OS. The falling out tracks the corkscrew SP with a decrease of the amplitude, which is proportional to the deceleration of the rotor. The point P corresponds to the locked-rotor position (s=1). Fig. 19 presents the dynamic mechanical characteristic, which shows the variation of the electromagnetic torque under variable operation condition. During the noload start-up, the operation point tracks successively the points O, M, L and S, that is from locked-rotor to synchronism with an oscillation of the electromagnetic torque inside certain limits (≈+200Nm to ≈-25Nm). The enforcement of the overload torque leads the operation point along the *downward* curve SK characterized by an *oscillation* section followed by the unstable falling out section, KP. The PKS curve, together with the marked points (Fig. 19)

can be considered the *natural mechanical characteristic* under motoring duty.

**Figure 18.** Hodograph of resultant rotor flux – *RS-125 (start-up to locked-rotor)*

S

100

50

O

Rotational pulsatance

ω

R [rad/s]

**Figure 19.** Rotational pulsatance vs. torque – *RS-125 (start-up to locked-rotor)*

A simulation study of the three-phase induction machine under unbalanced supply condition and varying duty (start-up, sudden torque enforcement and braking to stop

0 200

100

P

Electromagnetic torque Te [Nm]

S

K

L

[Wb]

M


O P

+2 [Wb]


**6.2. Asymmetric supply system** 

**Figure 15.** Time variation of rotor phase current – *RS-50*

**Figure 16.** Time variation of rotational pulsatance – *RS-125 (start-up to locked-rotor)*

**Figure 17.** Time variation of electromagnetic torque – *RS-125*

The third simulation, *RS-125*, has a similar start-up but the enforcement of the load torque determines a fast deceleration of the rotor. The pull-out slip (s≈33%) happens in t≈0,5 seconds after which the machine falls out. The angular speed reaches the zero value in t≈0,8 seconds, Fig. 16, and the electromagnetic torque get a value of approx. 78 Nm. This value can be considered the locked-rotor (starting) torque of the machine, Fig. 17.

The described critical duty that involves no-load start-up and operation, overloading, falling out and stop is plotted in terms of resultant rotor flux and angular speed versus electromagnetic torque. The hodograph (Fig. 18) put in view a cuasi corkscrew section, corresponding to the start-up, characterized by its maximum value represented by the segment OS. The falling out tracks the corkscrew SP with a decrease of the amplitude, which is proportional to the deceleration of the rotor. The point P corresponds to the locked-rotor position (s=1). Fig. 19 presents the dynamic mechanical characteristic, which shows the variation of the electromagnetic torque under variable operation condition. During the noload start-up, the operation point tracks successively the points O, M, L and S, that is from locked-rotor to synchronism with an oscillation of the electromagnetic torque inside certain limits (≈+200Nm to ≈-25Nm). The enforcement of the overload torque leads the operation point along the *downward* curve SK characterized by an *oscillation* section followed by the unstable falling out section, KP. The PKS curve, together with the marked points (Fig. 19) can be considered the *natural mechanical characteristic* under motoring duty.

**Figure 18.** Hodograph of resultant rotor flux – *RS-125 (start-up to locked-rotor)*

**Figure 19.** Rotational pulsatance vs. torque – *RS-125 (start-up to locked-rotor)*
