Asynchronous motor and its equations in the system αβ 0

ψ_q¼L_di_qþL_dDi_Q, (266) and also

L_dD¼L_qQ¼3

2M¼L_μ: (267)

Zero component is as follow:

ψ₀¼L₀i₀, (268)

where:

L0¼Ls�2Ms¼L_σsþM�2M

2 ¼L_σs: (269)

The fact that the zero-component inductance L0is equal to the stator leakage inductance L_σscan be used with a benefit if L_σsshould be measured. All three phases of the stator windings are connected together in a series, or parallelly, and fed by a single-phase voltage. In this way a pulse, non-rotating, magnetic flux is created. Thus a zero, non-rotating, component of the voltage, current, and impedance is measured.

Linear transformation is employed also at rotor linkage magnetic flux deriva- tions in the system DQ0:

ψ_D¼LDiDþL_Ddi_d, (270) ψ_Q ¼L_Qi_QþL_Qqi_q, (271) eventually considering that the air gap is constant and the parameters in the d-axis and q-axis are equal:

ψ_Q ¼LDiQþL_Ddiq: (272) The meaning of the rotor parameters is as follows:

LD¼LQ¼LrþMr¼L_σrþMþM

2 ¼L_σrþ3

2M¼L_σrþL_μ: (273) Similarly, for the zero rotor component can be written as:

ψ_O¼L_Oi_O, (274)

where:

L_O¼L_r�2Mr¼L_σrþM�2M

2 ¼L_σr: (275)

Take into account that all rotor variables are referred to the stator side; eventu- ally they are measured from the stator side.

2.13 Voltage equations of the asynchronous machine after

ψ_q¼L_di_qþL_dDi_Q, (266) and also

L_dD¼L_qQ ¼3

2M¼L_μ: (267)

Zero component is as follow:

ψ₀¼L₀i₀, (268)

where:

L0¼Ls�2Ms¼L_σsþM�2M

2 ¼L_σs: (269)

The fact that the zero-component inductance L0is equal to the stator leakage inductance L_σscan be used with a benefit if L_σsshould be measured. All three phases of the stator windings are connected together in a series, or parallelly, and fed by a single-phase voltage. In this way a pulse, non-rotating, magnetic flux is created. Thus a zero, non-rotating, component of the voltage, current, and impedance is measured.

Linear transformation is employed also at rotor linkage magnetic flux deriva- tions in the system DQ0:

ψ_D¼LDiDþL_Ddi_d, (270) ψ_Q ¼L_Qi_QþL_Qqi_q, (271) eventually considering that the air gap is constant and the parameters in the d-axis and q-axis are equal:

ψ_Q ¼LDiQþL_Ddiq: (272) The meaning of the rotor parameters is as follows:

LD¼LQ ¼LrþMr¼L_σrþMþM

2 ¼L_σrþ3

2M¼L_σrþL_μ: (273) Similarly, for the zero rotor component can be written as:

ψ_O¼L_Oi_O, (274)

where:

L_O¼L_r�2Mr¼L_σrþM�2M

2 ¼L_σr: (275)

Take into account that all rotor variables are referred to the stator side; eventu- ally they are measured from the stator side.

2.13 Voltage equations of the asynchronous machine after transformation into k-system with d-axis and q-axis

Voltage equations of the asynchronous machines in the dq0 system are obtained by a procedure described in Section 7. There are equations for the stator terminal voltage in the form:

u_d¼R_si_dþdψ_d

dt �ω_kψ_q, (276)

u_q¼R_si_qþdψ_q

dt þω_kψ_d, (277)

u₀ ¼R_si₀þdψ₀

dt : (278)

The rotor voltage equations are derived in a similar way as the stator ones but with a note that the rotor axis is shifted from the k-system axis about the angle

ϑ_k�ϑ_r

ð Þ; thus in the equations there are members with the angular speedðω_k�ω_rÞ:

u_D¼R_ri_Dþdψ_D

dt �ðω_k�ω_rÞψ_Q, (279) u_Q ¼R_ri_Qþdψ_Q

dt þðω_k�ω_rÞψ_D, (280) uO¼RriOþdψ_O

dt : (281)

These six equations create a full system of the asynchronous machine voltage equations. Rotor variables are referred to the stator side; expressions for the linkage magnetic flux are shown in Section 12.

2.14 Asynchronous motor and its equations in the system

αβ

0

According to Sections 7 and 10, the reference k-system can be positioned arbi- trarily, but some specific positions can simplify solutions; therefore, they are used with a benefit. One of such cases happens if the d-axis of the k-system is identified with the axis of the stator a-phase; it meansϑ_k¼0,ω_k¼0. This system is in this book marked asαβ0 system.

This system is obtained by phase variable projection into stationary reference system, linked firmly with a-phase. It is a two-axis system, and zero components are identical with the non-rotating components known from the theory of symmetrical components.

The original voltage equations of asynchronous machine derived in Sections 7 and 13 are as follows:

u_d¼R_si_dþdψ_d

dt �ω_kψ_q, (282)

uq¼Rsiqþdψ_q

dt þω_kψ_d, (283)

u0¼Rsi0þdψ₀

dt , (284)

uD¼RriDþdψ_D

dt �ðω_k�ω_rÞψ_Q, (285) uQ ¼RriQþdψ_Q

dt þðω_k�ω_rÞψ_D, (286) uO¼RriOþdψ_O

dt , (287)

where:

ψ_d¼L_di_dþL_dDiD, (288) ψ_q¼LqiqþLqQiQ, (289)

ψ₀¼L₀i₀, (290)

L_d¼L_q¼L_σsþL_μ, (291)

L_dD¼L_Dd¼3

2M¼L_μ, (292)

LqQ¼LQq¼LdD¼3

2M¼L_μ, (293)

ψ_D¼L_Di_DþL_Ddi_d, (294) ψ_Q ¼L_Qi_QþL_Qqi_q, (295)

ψ_O¼LOiO, (296)

L_D¼L_Q ¼L_rþM_r¼L_σrþMþM

2 ¼L_σrþ3

2M¼L_σrþL_μ, (297) L_O¼L_r�2M_r¼L_σrþM�2M

2 ¼L_σr: (298)

Now new subscripts the following are introduced:

For currents and voltages:

d¼αs, q¼βs, D¼αr, Q ¼βr, (299) For inductances:

L_d¼Lq¼L_σsþL_μ ¼LS, (300) L_D¼L_Q ¼L_σrþL_μ ¼L_R, (301)

L_dD¼L_qQ¼3

2M¼L_μ: (302)

The original equations, rewritten with the new subscripts, with the fact that ϑ_k¼0 andω_k¼0 and with an assumption that the three-phase system is symmet- rical, meaning the zero components are zero, are as follows:

u_αs¼Rsi_αsþLSdi_αs

dt þL_μdi_αr

dt (303)

u_βs¼R_si_βsþL_Sdi_βs

dt þL_μdi_βr

dt (304)

u_αr¼R_ri_αrþω_rL_Ri_βrþω_rL_μi_βsþL_Rdi_αr

dt þL_μdi_αs

dt (305)

u_βr¼Rri_βr�ω_rLRi_αr�ω_rL_μi_αsþLRdi_βr

dt þL_μdi_βs

dt (306)

If transients are solved for motoring operation, then stator terminal voltages on the left side of the equations are known variables and are necessary to introduce derived expressions for sinusoidal variables transformed into dq0 system, nowα, β-axes ((196) for udand (199) for uq). Rotor voltages are zero, if there is squirrel

cage rotor. If there is wound rotor, here is a possibility to introduce a voltage applied to the rotor terminals, as in the case of asynchronous generator for wind power stations, where the armature winding is connected to the frequency converter. If the rotor winding is short circuited, then the rotor voltages are also zero.

In the motoring operation, the terminal voltages are known variables, and unknown variables are currents and speed. Therefore it is suitable to accommodate the previous equations in the form where the unknown variables are solved. From Eq. (303), the following is obtained:

L_Sdi_αs

dt ¼u_αs�R_si_αs�L_μdi_αr

dt , (307)

and from Eq. (305): di_αr

dt ¼ 1

LR u_αr�Rri_αr�ω_rLRi_βr�ω_rL_μi_βs�L_μdi_αs dt

� �

: (308)

This equation is introduced into Eq. (307). Then it is possible to eliminate a time variation of the stator current in theα-axis:

di_αs

dt ¼ LR

LSLR�L²_μ u_αs�R_si_αsþL_μ

L_RR_ri_αrþω_rL²_μ

L_Ri_βsþω_rL_μi_βr�L_μ L_Ru_αr

!

: (309) The same way is applied for the other current components:

di_αr

dt ¼ L_S

LSLR�L²_μ u_αr�R_ri_αrþL_μ

LSR_si_αs�ω_rL_μi_βs�ω_rL_Ri_βr�L_μ LSu_αs

� �

, (310) di_βs

dt ¼ L_R

LSLR�L²_μ u_βs�Rsi_βsþL_μ

LRRri_βr�ω_rL²_μ

LRi_αs�ω_rL_μi_αr�L_μ LRu_βr

!

, (311) di_βr

dt ¼ L_S

LSLR�L²_μ u_βr�R_ri_βrþL_μ

LSR_si_βsþω_rL_μi_αsþω_rL_Ri_αr�L_μ LSu_βs

� �

: (312) The last equation is for time variation of the speed. On the basis of Section 8, if in the equation for the electromagnetic torque the constants kd= kq= 2/3 are introduced and after changing the subscripts, the torque is in the form:

te¼p2 3

1

k_dk_q�ψ_diq�ψ_qi_d�

¼p3

2�ψ_αsi_βs�ψ_βsi_αs�

¼p3

2L_μ�i_αri_βs�i_βri_αs� , te¼p3

2L_μ�i_αri_βs�i_βri_αs�

: (313)

After considering Eq. (176), the electrical angular speed is obtained in the form:

dωr

dt ¼p J p3

2L_μ�i_αri_βs�i_βri_αs�

�tL

� �

: (314)

Mechanical angular speed is linked through the number of the pole pairsΩ_r¼^ω_p^r, which directly corresponds to the revolutions per minute.

For Eqs. (309)–(312), the next expressions are introduced for the voltages (see Sections 9 and 10):

where:

ψ_d¼L_di_dþL_dDiD, (288) ψ_q¼LqiqþLqQiQ, (289)

ψ₀¼L₀i₀, (290)

L_d¼L_q¼L_σsþL_μ, (291)

L_dD¼L_Dd¼3

2M¼L_μ, (292)

LqQ ¼LQq¼LdD¼3

2M¼L_μ, (293)

ψ_D¼L_Di_DþL_Ddi_d, (294) ψ_Q ¼L_Qi_QþL_Qqi_q, (295)

ψ_O¼LOiO, (296)

L_D¼L_Q ¼L_rþM_r¼L_σrþMþM

2 ¼L_σrþ3

2M¼L_σrþL_μ, (297) L_O¼L_r�2M_r¼L_σrþM�2M

2 ¼L_σr: (298)

Now new subscripts the following are introduced:

For currents and voltages:

d¼αs, q¼βs, D¼αr, Q¼βr, (299) For inductances:

L_d¼Lq¼L_σsþL_μ¼LS, (300) L_D¼L_Q ¼L_σrþL_μ¼L_R, (301)

L_dD¼L_qQ ¼3

2M¼L_μ: (302)

The original equations, rewritten with the new subscripts, with the fact that ϑ_k¼0 andω_k¼0 and with an assumption that the three-phase system is symmet- rical, meaning the zero components are zero, are as follows:

u_αs¼Rsi_αsþLSdi_αs

dt þL_μdi_αr

dt (303)

u_βs¼R_si_βsþL_Sdi_βs

dt þL_μdi_βr

dt (304)

u_αr¼R_ri_αrþω_rL_Ri_βrþω_rL_μi_βsþL_Rdi_αr

dt þL_μdi_αs

dt (305)

u_βr¼Rri_βr�ω_rLRi_αr�ω_rL_μi_αsþLRdi_βr

dt þL_μdi_βs

dt (306)

If transients are solved for motoring operation, then stator terminal voltages on the left side of the equations are known variables and are necessary to introduce derived expressions for sinusoidal variables transformed into dq0 system, nowα, β-axes ((196) for udand (199) for uq). Rotor voltages are zero, if there is squirrel

cage rotor. If there is wound rotor, here is a possibility to introduce a voltage applied to the rotor terminals, as in the case of asynchronous generator for wind power stations, where the armature winding is connected to the frequency converter. If the rotor winding is short circuited, then the rotor voltages are also zero.

In the motoring operation, the terminal voltages are known variables, and unknown variables are currents and speed. Therefore it is suitable to accommodate the previous equations in the form where the unknown variables are solved. From Eq. (303), the following is obtained:

L_Sdi_αs

dt ¼u_αs�R_si_αs�L_μdi_αr

dt , (307)

and from Eq. (305):

di_αr dt ¼ 1

LR u_αr�Rri_αr�ω_rLRi_βr�ω_rL_μi_βs�L_μdi_αs dt

� �

: (308)

This equation is introduced into Eq. (307). Then it is possible to eliminate a time variation of the stator current in theα-axis:

di_αs

dt ¼ LR

LSLR�L²_μ u_αs�R_si_αsþL_μ

L_RR_ri_αrþω_rL²_μ

L_Ri_βsþω_rL_μi_βr�L_μ L_Ru_αr

!

: (309) The same way is applied for the other current components:

di_αr

dt ¼ L_S

LSLR�L²_μ u_αr�R_ri_αrþL_μ

LSR_si_αs�ω_rL_μi_βs�ω_rL_Ri_βr�L_μ LSu_αs

� �

, (310) di_βs

dt ¼ L_R

LSLR�L²_μ u_βs�Rsi_βsþL_μ

LRRri_βr�ω_rL²_μ

LRi_αs�ω_rL_μi_αr�L_μ LRu_βr

!

, (311) di_βr

dt ¼ L_S

LSLR�L²_μ u_βr�R_ri_βrþL_μ

LSR_si_βsþω_rL_μi_αsþω_rL_Ri_αr�L_μ LSu_βs

� �

: (312) The last equation is for time variation of the speed. On the basis of Section 8, if in the equation for the electromagnetic torque the constants kd= kq= 2/3 are introduced and after changing the subscripts, the torque is in the form:

te¼p2 3

1

k_dk_q�ψ_diq�ψ_qi_d�

¼p3

2�ψ_αsi_βs�ψ_βsi_αs�

¼p3

2L_μ�i_αri_βs�i_βri_αs� , te¼p3

2L_μ�i_αri_βs�i_βri_αs�

: (313)

After considering Eq. (176), the electrical angular speed is obtained in the form:

dωr

dt ¼p J p3

2L_μ�i_αri_βs�i_βri_αs�

�tL

� �

: (314)

Mechanical angular speed is linked through the number of the pole pairsΩ_r¼^ω_p^r, which directly corresponds to the revolutions per minute.

For Eqs. (309)–(312), the next expressions are introduced for the voltages (see Sections 9 and 10):

u_αs¼U_msinωst¼u_a, (315)

u_βs¼ �Umcosωst, (316)

which is displaced about 90° with regard to the u_αs. Rotor voltages in the most simple case for the squirrel cage rotor are zero:

u_αr¼u_βr¼0: (317)

In the next chapter, solving of the transients in a concrete asynchronous motor with squirrel cage rotor and wound rotor is shown.

2.15 Simulation of the transients in asynchronous motors