**6.2. Asymmetric supply system**

34 Induction Motors – Modelling and Control

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

Rotor phase current iar [A]


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

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

50

0

100

0

Electromagnetic torque Te [Nm]

200

Rotational pulsatance ω

R [rad/s]

100

150

0.2

0 0.4 0.6 Time t [s]

0.2 0.4 0.6

Time t [s]

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

0.2 0.4 0.6

Time t [s]

can be considered the locked-rotor (starting) torque of the machine, Fig. 17.

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

eventually) is possible by using the same mathematical model described by the equation system (26-1…8). The values of the resistant torques and the expressions of the instantaneous phase voltages have to be stated. Since the rotor winding is short-circuited, the supply rotor voltages are uar=ubr=ucr=0. On this basis, the structural diagram has been put into effect in the Matlab-Simulink environment. As regards the unbalanced three-phase supply system, it has to be mentioned that the phase voltages are no more equal in amplitude and the angles of phase difference may have other values than 2π/3 rad. In any event, the sum of the instantaneous values of the applied voltages must be zero, that is uas+ubs+ucs=0. As an argument for this seemingly constraint stands the fact that the vast majority of the three-phase induction machines are connected to the industrial system via three supply leads (no neutral).

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 37

**Figure 20.** Time variation of rotational pulsatance – *RNS-1 (start-up + sudden load)*

50

0

100

Rotational pulsatance ω

R [rad/s]

150

**Figure 21.** Time variation of rotational pulsatance – *RNS-2 (start-up + sudden load)*

50

0

100

Rotational pulsatance ω

R [rad/s]

150

**Figure 22.** Time variation of electromagnetic torque – *RNS-1*

0

100 150

50 Electromagnetic torque Te [Nm]

The inspection of the electromagnetic torque variation (Fig. 22 and 23) shows the presence of a variable oscillating torque, whose frequency is twice the supply voltage frequency (in our case 100 Hz) and overlaps the average torque. *This oscillating component is demonstrated by the analytic expression of the instantaneous torque*, *which is written using nothing but total flux linkages* (25). The symmetric components theory, for example, is not capable to provide information about these oscillating torques. At the most, this theory evaluates the average torque, probably with inherent errors. Coming back to the torque variations, one can see that the amplitude

0.1 0.3 0.5

Time t [s]

0.1 0.3 0.5

Time t [s]

oscillations increase with the asymmetry degree, but their frequency keeps unchanged.

Time t [s]

0.1 0.3 0.5

The simulation presented here takes into discussion an induction machine with the same parameters as above that is: Rs=Rr=2; Lhs=0,09; Lσs= Lσr=0,01; J=0,05; p=2; kz=0,02; ω1=314,1 (SI units). Consequently, the equations (73-1) - (73-8) keep unchanged. The expressions (73-9) have to be modified in accordance with the asymmetry degree.

Two varying duties under unbalanced condition have been simulated. The first (denoted *RNS-1*) is characterized by an asymmetry degree, un = 16,5% and the following supply voltages:

$$\overline{u}\_{\rm abs} \leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t)}; \overline{u}\_{\rm bs} \leftrightarrow \frac{375}{\sqrt{2}} e^{j(314, 1t - 1, 96)}; \overline{u}\_{\rm cs} \leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t - 3, 927)}; \underline{u}\_{\rm n} = 16, 5\% \tag{74}$$

The simulation results are presented in Fig. 20, 22, 24, 25 and 28. The second study simulation (denoted *RNS-2*) has an asymmetry degree of un = 27% given by the following stator voltages:

$$\overline{u}\_{\text{abs}} \leftrightarrow \frac{490}{\sqrt{2}} e^{j(314, 1t)}; \overline{u}\_{\text{abs}} \leftrightarrow \frac{346, 43}{\sqrt{2}} e^{j(314, 1t - 2, 357)}; \overline{u}\_{\text{sc}} \leftrightarrow \frac{346, 43}{\sqrt{2}} e^{j(314, 1t - 3, 295)}; \overline{u}\_{\text{n}} = 27\% \tag{75}$$

The simulation results are presented in Fig. 21, 23, 26, 27 and 29. The varying duties are similar to those discussed above and consist 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.

In comparison to symmetric supply, the unbalanced voltage system causes a longer start-up time with approx. 20% for *RNS-1* (Fig. 20) and with 50% for *RNS-2* (Fig. 21). Moreover, the higher asymmetry degree of *RNS-2* leads to the cancelation of the overshoot at the end of the start-up process. At the same time, significant speed oscillations are noticeable during the operation (no matter the load degree), which are higher with the increase of the asymmetry degree. These oscillations have a constant frequency, which is twice of the supply voltage frequency. They represent the main cause that determines the specific noise of the machines with unbalanced supply system.

Mathematical Model of the Three-Phase Induction Machine for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 37

**Figure 20.** Time variation of rotational pulsatance – *RNS-1 (start-up + sudden load)*

36 Induction Motors – Modelling and Control

three supply leads (no neutral).

voltages:

stator voltages:

constant torque of 50 Nm.

of the machines with unbalanced supply system.

eventually) is possible by using the same mathematical model described by the equation system (26-1…8). The values of the resistant torques and the expressions of the instantaneous phase voltages have to be stated. Since the rotor winding is short-circuited, the supply rotor voltages are uar=ubr=ucr=0. On this basis, the structural diagram has been put into effect in the Matlab-Simulink environment. As regards the unbalanced three-phase supply system, it has to be mentioned that the phase voltages are no more equal in amplitude and the angles of phase difference may have other values than 2π/3 rad. In any event, the sum of the instantaneous values of the applied voltages must be zero, that is uas+ubs+ucs=0. As an argument for this seemingly constraint stands the fact that the vast majority of the three-phase induction machines are connected to the industrial system via

The simulation presented here takes into discussion an induction machine with the same parameters as above that is: Rs=Rr=2; Lhs=0,09; Lσs= Lσr=0,01; J=0,05; p=2; kz=0,02; ω1=314,1 (SI units). Consequently, the equations (73-1) - (73-8) keep unchanged. The expressions (73-9)

Two varying duties under unbalanced condition have been simulated. The first (denoted *RNS-1*) is characterized by an asymmetry degree, un = 16,5% and the following supply

The simulation results are presented in Fig. 20, 22, 24, 25 and 28. The second study simulation (denoted *RNS-2*) has an asymmetry degree of un = 27% given by the following

The simulation results are presented in Fig. 21, 23, 26, 27 and 29. The varying duties are similar to those discussed above and consist 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

In comparison to symmetric supply, the unbalanced voltage system causes a longer start-up time with approx. 20% for *RNS-1* (Fig. 20) and with 50% for *RNS-2* (Fig. 21). Moreover, the higher asymmetry degree of *RNS-2* leads to the cancelation of the overshoot at the end of the start-up process. At the same time, significant speed oscillations are noticeable during the operation (no matter the load degree), which are higher with the increase of the asymmetry degree. These oscillations have a constant frequency, which is twice of the supply voltage frequency. They represent the main cause that determines the specific noise

<sup>490</sup> (314,1 ) <sup>375</sup> (314,1 1,96) <sup>490</sup> (314,1 3,927) ; ; ; 16,5%

<sup>490</sup> (314,1 ) 346,43 (314,1 2,357) 346,43 (314,1 3,295) ; ; ; 27%

*j t j t j t as bs cs <sup>n</sup> u eu e u e u* (75)

*j t j t j t as bs cs <sup>n</sup> u eu e u e u* (74)

have to be modified in accordance with the asymmetry degree.

22 2

22 2

**Figure 21.** Time variation of rotational pulsatance – *RNS-2 (start-up + sudden load)*

The inspection of the electromagnetic torque variation (Fig. 22 and 23) shows the presence of a variable oscillating torque, whose frequency is twice the supply voltage frequency (in our case 100 Hz) and overlaps the average torque. *This oscillating component is demonstrated by the analytic expression of the instantaneous torque*, *which is written using nothing but total flux linkages* (25). The symmetric components theory, for example, is not capable to provide information about these oscillating torques. At the most, this theory evaluates the average torque, probably with inherent errors. Coming back to the torque variations, one can see that the amplitude oscillations increase with the asymmetry degree, but their frequency keeps unchanged.

**Figure 22.** Time variation of electromagnetic torque – *RNS-1*

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 39

The stator currents variation, Fig. 24 and 26, have a sinusoidal shape and an unmodified frequency of 50 Hz. Their amplitude increases however with the asymmetry degree (approx. 18 A for *RNS-1* and approx. 32 A for *RNS-2*). As a consequence of this fact, both power factor and efficiency decrease. The rotor currents (Fig. 25 and 27) include besides the main component of f2=s· f1 frequency a second oscillating component of high frequency, f'2=(2-s)f1, which is responsible for parasitic torques and vibrations of the rotor. The amplitude of these

2 [Wb]



2 [Wb]

0



0 0.4 0.6 Time t [s]

0.2

**Figure 27.** Time variation of rotor phase current – *RNS-2*


Rotor phase current iar [A]

oscillating currents increases with the asymmetry degree.

**Figure 28.** Hodograph of resultant rotor flux – *RNS-1*

**Figure 29.** Hodograph of resultant rotor flux – *RNS-2*

**Figure 23.** Time variation of electromagnetic torque – *RNS-2*

**Figure 24.** Time variation of stator phase current – *RNS-1*

**Figure 25.** Time variation of rotor phase current – *RNS-1*

**Figure 26.** Time variation of stator phase current – *RNS-2*

**Figure 27.** Time variation of rotor phase current – *RNS-2*

**Figure 23.** Time variation of electromagnetic torque – *RNS-2*

0



Stator phase current ias [A]

Rotor phase current iar [A]

Stator phase current ias [A]

100

 50 Electromagnetic torque Te [Nm]

Time t [s]

0.2

0.2

0.2

0 0.4 0.6 Time t [s]

0 0.4 0.6 Time t [s]

0 0.4 0.6 Time t [s]

0.1 0.3 0.5

**Figure 24.** Time variation of stator phase current – *RNS-1*

**Figure 25.** Time variation of rotor phase current – *RNS-1*

**Figure 26.** Time variation of stator phase current – *RNS-2*

The stator currents variation, Fig. 24 and 26, have a sinusoidal shape and an unmodified frequency of 50 Hz. Their amplitude increases however with the asymmetry degree (approx. 18 A for *RNS-1* and approx. 32 A for *RNS-2*). As a consequence of this fact, both power factor and efficiency decrease. The rotor currents (Fig. 25 and 27) include besides the main component of f2=s· f1 frequency a second oscillating component of high frequency, f'2=(2-s)f1, which is responsible for parasitic torques and vibrations of the rotor. The amplitude of these oscillating currents increases with the asymmetry degree.

**Figure 28.** Hodograph of resultant rotor flux – *RNS-1*

**Figure 29.** Hodograph of resultant rotor flux – *RNS-2*

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 41

The hodographs of the resultant rotor flux show a very interesting behavior of the unbalanced machines, Fig. 28 and 29. In comparison to the symmetric supply cases where the hodograph is a circle under steady state, the asymmetric system distort the curve into a "gear wheel" with a lot of teeth placed on a mean diameter whose magnitude depends inverse proportionally with the asymmetry degree. Generally, these curves do not overlap and prove that during the operation the interaction between stator and rotor fluxes is not constant in time since the rotor speed is not constant. Consequently, the rotor vibrations are

In order to point out the superiority of the proposed mathematical model, Fig. 30 shows the structural diagram used in Simulink environment. The diagram is capable to simulate any steady-state and transient duty under balanced or unbalanced state of the induction machine including doubly-fed operation as generator or motor by simple modification of the input data. To prove this statement, a simulation of an unbalanced doubly-fed operation has been performed. The operation cycle involves: I. A no-load start-up (the wound rotor winding is short-circuited); II. Application of a supplementary output torque of (-70) Nm (at the moment t=0.4 sec.) which leads the induction machine to the generating duty (over synchronous speed); III. Supply of two series connected rotor phases with d.c. current (Uar=+40V, Ubr= −40V, Ucr=0V), at the moment time t=0.6 sec., which change the operation of

Fig. 31 and 32 show the dynamic mechanical characteristic, Te=f(ΩR) and the hodograph of the resultant rotor flux respectively. The start-up corresponds to A-S1 curve, the over synchronous acceleration is modeled by S1-S curve and the operation under SIG duty corresponds to S-S2 curve. A few observations regarding Fig. 32 are necessary as well. The rotor flux hodograph is rotating in a *counterclockwise direction* corresponding to motoring duty, in a *clockwise direction* for generating duty and stands still at synchronism. The "in time" modification and the position of the hodograph corresponding to SIG duty depend on

S1 S2


Electromagnetic torque Te [Nm]

100

M

usually propagated to the mechanical components and working machine.

the induction generator into a synchronized induction generator (SIG).

the moment of d.c. supply and the load angle of the machine.

100

Rotational pulsatance

ω

R [rad/s]

200 S

**Figure 31.** Dynamic mechanical characteristic

**Figure 30.** Structural diagram of the three-phase induction machine

The hodographs of the resultant rotor flux show a very interesting behavior of the unbalanced machines, Fig. 28 and 29. In comparison to the symmetric supply cases where the hodograph is a circle under steady state, the asymmetric system distort the curve into a "gear wheel" with a lot of teeth placed on a mean diameter whose magnitude depends inverse proportionally with the asymmetry degree. Generally, these curves do not overlap and prove that during the operation the interaction between stator and rotor fluxes is not constant in time since the rotor speed is not constant. Consequently, the rotor vibrations are usually propagated to the mechanical components and working machine.

In order to point out the superiority of the proposed mathematical model, Fig. 30 shows the structural diagram used in Simulink environment. The diagram is capable to simulate any steady-state and transient duty under balanced or unbalanced state of the induction machine including doubly-fed operation as generator or motor by simple modification of the input data. To prove this statement, a simulation of an unbalanced doubly-fed operation has been performed. The operation cycle involves: I. A no-load start-up (the wound rotor winding is short-circuited); II. Application of a supplementary output torque of (-70) Nm (at the moment t=0.4 sec.) which leads the induction machine to the generating duty (over synchronous speed); III. Supply of two series connected rotor phases with d.c. current (Uar=+40V, Ubr= −40V, Ucr=0V), at the moment time t=0.6 sec., which change the operation of the induction generator into a synchronized induction generator (SIG).

Fig. 31 and 32 show the dynamic mechanical characteristic, Te=f(ΩR) and the hodograph of the resultant rotor flux respectively. The start-up corresponds to A-S1 curve, the over synchronous acceleration is modeled by S1-S curve and the operation under SIG duty corresponds to S-S2 curve. A few observations regarding Fig. 32 are necessary as well. The rotor flux hodograph is rotating in a *counterclockwise direction* corresponding to motoring duty, in a *clockwise direction* for generating duty and stands still at synchronism. The "in time" modification and the position of the hodograph corresponding to SIG duty depend on the moment of d.c. supply and the load angle of the machine.

**Figure 31.** Dynamic mechanical characteristic

40 Induction Motors – Modelling and Control

**Figure 30.** Structural diagram of the three-phase induction machine

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 43

parameters (electromagnetic torque, resultant rotor and stator fluxes). They put in view the behavior of the induction machine for different transient duties. In particular, they prove that any unbalance of the supply system generates important variations of the

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**Author details** 

**8. References** 

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**Figure 32.** Hodograph of resultant rotor flux
