Linkage magnetic flux equations of the synchronous machine in the general theory of electrical machines

In Eq. (65) of linkage magnetic fluxes, expressions for inductances as they were derived in Section 16 are introduced. For example, for field winding with a subscript “f,” the equation for linkage magnetic flux is written as follows:

ψ_f¼L_fai_aþL_fbi_bþL_fci_cþL_ffi_fþL_fDi_DþL_fQi_Q, (351) ψ_f ¼L_afmax iacosϑ_rþibcos ϑ_r�2π

3

þiccos ϑ_rþ2π 3

þL_ffi_fþL_fDiD

þL_fQiQ:

(352) If this equation is compared with Eq. (124), written for a general variable x, it is seen that the expression in the square bracket is equal to id/kd,, ifϑ_k¼ϑ_r:

iacosϑ_rþibcos ϑ_r�2π 3

þiccos ϑ_rþ2π 3

¼ 1

k_did: (353) After this modification in Eq. (352), the following is obtained:

ψ_f¼ 1

k_dL_afmaxi_dþL_ffi_fþL_fDiDþL_fQiQ ¼L_fdi_dþL_ffi_fþL_fDiD, (354)

where it was taken into account that mutual inductance of two perpendicular windings f and Q is zero.

On the same basis, the linkage magnetic flux for damping rotor windings D and Q is received:

ψ_D¼ 1

k_dLaDmaxidþLDfifþLDDiDþLDQiQ ¼LDdidþLDfifþLDDiD, (355) ψ_Q ¼ �LaQmax iasinϑ_rþibsin ϑ_r�2π

3

þicsin ϑ_rþ2π 3

þLQfifþLQDiD

þLQQiQ

¼ 1

kqLaQmaxiqþL_Qfi_fþLQDiDþLQQiQ ¼LQqiqþLQQiQ

(356) A derivation for the stator windings is made in the like manner. It is started with a formal transformation equation from system a, b, c into the d-axis and then into the q-axis. The equation in the d-axis is as follows:

ψ_d¼k_d ψ_acosϑ_rþψ_bcos ϑ_r�2 3π

þψ_ccos ϑ_rþ2 3π

: (357)

If into this equation expressions from Eq. (321), for linkage magnetic fluxes of a, b, c phases, are introduced, and for inductances appropriate expressions from Section 16 are introduced, then after widespread modifications of the goniometrical functions and for a rotor position in d-axis, i.e., if

ϑ_k¼ϑ_r¼0, the following is received:

ψ_d¼L_di_dþ3

2k_dL_afmaxi_fþ3

2k_dLaDmaxiD¼L_di_dþL_dfi_fþL_dDiD: (358) Here:

L_d¼L_a0þL_ab0þ3

2L₂ (359)

is a direct synchronous inductance. The other symbols are for mutual induc- tances between the stator windings transformed into the d-axis and rotor windings, which are also in the d-axis:

L_df¼3

2k_dL_afmax, (360)

L_dD¼3

2k_dLaDmax: (361)

The linkage magnetic flux in the q-axis is derived in a similar way, which results in:

ψ_q¼L_qi_qþ3

2k_qL_aQmaxi_Q ¼L_qi_qþL_qQi_Q, (362)

u_d¼R_si_dþdψ_d

dt �ω_kψ_q, (345)

u_q¼R_si_qþdψ_q

dt þω_kψ_d, (346)

u₀¼R_si₀þdψ₀

dt : (347)

Equations (160)–(162) are voltage equations of the three-phase stator windings, in this case synchronous machine but also asynchronous machine, as it was men- tioned in Section 13. They are equations transformed into reference k-system rotating by angular speedω_k, with the axes d, q, 0. As it is seen, they are the same equations as in Section 2.1, which were derived for universal configuration of an electrical machine.

Terminal voltage equations of the synchronous machine rotor windings are not needed to transform in the d-axis and q-axis, because the rotor windings are embedded in these axes, as it is seen inFigure 34, and are written directly in the two-axis system d, q, 0:

u_f¼R_fi_fþdψ_f

dt , (348)

u_D¼R_Di_Dþdψ_D

dt , (349)

u_Q ¼R_Qi_Qþdψ_Q

dt : (350)

The next the expressions for linkage magnetic flux are investigated.

2.18 Linkage magnetic flux equations of the synchronous machine in the general theory of electrical machines

In Eq. (65) of linkage magnetic fluxes, expressions for inductances as they were derived in Section 16 are introduced. For example, for field winding with a subscript “f,” the equation for linkage magnetic flux is written as follows:

ψ_f ¼L_fai_aþL_fbi_bþL_fci_cþL_ffi_fþL_fDi_DþL_fQi_Q, (351) ψ_f ¼L_afmax iacosϑ_rþibcos ϑ_r�2π

3

þiccos ϑ_rþ2π 3

þL_ffi_fþL_fDiD

þL_fQiQ:

(352) If this equation is compared with Eq. (124), written for a general variable x, it is seen that the expression in the square bracket is equal to id/kd,, ifϑ_k¼ϑ_r:

iacosϑ_rþibcos ϑ_r�2π 3

þiccos ϑ_rþ2π 3

¼ 1

k_did: (353) After this modification in Eq. (352), the following is obtained:

ψ_f ¼ 1

k_dL_afmaxi_dþL_ffi_fþL_fDiDþL_fQiQ ¼L_fdi_dþL_ffi_fþL_fDiD, (354)

where it was taken into account that mutual inductance of two perpendicular windings f and Q is zero.

On the same basis, the linkage magnetic flux for damping rotor windings D and Q is received:

ψ_D¼ 1

k_dLaDmaxidþLDfifþLDDiDþLDQiQ ¼LDdidþLDfifþLDDiD, (355) ψ_Q ¼ �LaQmax iasinϑ_rþibsin ϑ_r�2π

3

þicsin ϑ_rþ2π 3

þLQfifþLQDiD

þLQQiQ

¼ 1

kqLaQmaxiqþL_Qfi_fþLQDiDþLQQiQ ¼LQqiqþLQQiQ

(356) A derivation for the stator windings is made in the like manner. It is started with a formal transformation equation from system a, b, c into the d-axis and then into the q-axis. The equation in the d-axis is as follows:

ψ_d¼k_d ψ_acosϑ_rþψ_bcos ϑ_r�2 3π

þψ_ccos ϑ_rþ2 3π

: (357)

If into this equation expressions from Eq. (321), for linkage magnetic fluxes of a, b, c phases, are introduced, and for inductances appropriate expressions from Section 16 are introduced, then after widespread modifications of the goniometrical functions and for a rotor position in d-axis, i.e., if

ϑ_k¼ϑ_r¼0, the following is received:

ψ_d¼L_di_dþ3

2k_dL_afmaxi_fþ3

2k_dLaDmaxiD¼L_di_dþL_dfi_fþL_dDiD: (358) Here:

L_d¼L_a0þL_ab0þ3

2L₂ (359)

is a direct synchronous inductance. The other symbols are for mutual induc- tances between the stator windings transformed into the d-axis and rotor windings, which are also in the d-axis:

L_df ¼3

2k_dL_afmax, (360)

L_dD¼3

2k_dLaDmax: (361)

The linkage magnetic flux in the q-axis is derived in a similar way, which results in:

ψ_q¼L_qi_qþ3

2k_qL_aQmaxi_Q ¼L_qi_qþL_qQi_Q, (362)

where:

Lq¼La0þL_ab0�3

2L2 (363)

is a quadrature synchronous inductance and L_qQ ¼3

2k_qL_aQmax, (364)

is the mutual inductance between the stator winding transformed into the q-axis and rotor winding which is in the q-axis . From Eqs. (359) and (363), it is seen that if there is a cylindrical rotor, then L2= 0, and inductances in both axes are equal:

Ld¼Lq, (365)

which is a known fact.

Finally, a linkage magnetic flux for zero axis is derived in a similar way:

ψ₀¼L0i0, (366)

where:

L0 ¼La0�2L_ab0 (367)

is called zero inductance. It is seen that this linkage magnetic flux and induc- tance are linked only with variables with the subscript 0 and do not have any relation to the variables in the other axes. Additionally, also here a knowledge from the theory of the asynchronous machine can be applied that zero impedance is equal to the leakage stator inductance that can be used during the measurement of the leakage stator inductance. Here can be reminded equation Section 12:

L0¼Ls�2Ms¼L_σsþM�2M

2 ¼L_σs: (368)

Namely, if all three phases of the stator windings (in series or parallel connec- tion) are fed by a single-phase voltage, it results in the pulse, non-rotating magnetic flux (see [8]).

If there is a request to make equations more simple, then it is necessary to ask for equality of mutual inductances of two windings, for example, inductance Lfdfor the current idshould be equal to the inductance Ldffor the current if:

L_fd¼L_df: (369)

Therefore from Eq. (354) forψ_f, take the expression at the current id, which was marked as L_fdand put into the equality with the expression at the current ifin Eq. (358) forψ_d, which was marked as L_df:

1

k_dLafmax¼3

2kdLafmax: (370)

Then:

k²_d¼2

3, (371)

and

k_d¼ ffiffiffi2 3 r

: (372)

The same value is obtained if expressions forψ_dat the current iD(358) andψ_Dat the current id(355) are put into the equality, to get LDd¼LdD. Then:

1

kdL_aDmax¼3

2k_dL_aDmax, (373)

which results in kd¼ ffiffi

23

q.

In the q-axis it is done at the same approach: The expression at the current i_qin the equation forψ_Q(356) and the expression at the current iQin equation forψ_q (362), put into equality to get LQq¼LqQ, are as follows:

1

k_qLaQmax¼3

2kqLaQ max: (374)

It results in the value:

kq¼ ffiffiffi2 3 r

: (375)

Hence a choice for the coefficients suitable for synchronous machines flows:

k_d¼k_q¼ � ffiffiffi2 3 r

, but it is better to use the positive expression:

kd¼kq¼ ffiffiffi2 3 r

: (376)

If the next expressions are introduced:

Ldf¼3 2

ffiffiffi2 3 r

Lafmax¼ ffiffiffi3 2 r

Lafmax¼Lfd, (377)

LdD¼3 2

ffiffiffi2 3 r

LaDmax¼ ffiffiffi3 2 r

LaDmax¼LDd, (378)

LqQ ¼3 2

ffiffiffi2 3 r

LaQmax¼ ffiffiffi3 2 r

LaQmax¼LQq, (379)

then equations for linkage magnetic fluxes of the synchronous machines in the d, q, 0 system have the form as follows:

ψ_d¼L_di_dþL_dfi_fþL_dDi_D, see 358ð Þ ψ_q¼LqiqþLqQiQ, see 362ð Þ

ψ₀ ¼L0i0, see 366ð Þ (380)

where:

Lq¼La0þL_ab0�3

2L2 (363)

is a quadrature synchronous inductance and L_qQ¼3

2k_qL_aQmax, (364)

is the mutual inductance between the stator winding transformed into the q-axis and rotor winding which is in the q-axis . From Eqs. (359) and (363), it is seen that if there is a cylindrical rotor, then L2= 0, and inductances in both axes are equal:

Ld¼Lq, (365)

which is a known fact.

Finally, a linkage magnetic flux for zero axis is derived in a similar way:

ψ₀¼L0i0, (366)

where:

L0¼La0�2L_ab0 (367)

is called zero inductance. It is seen that this linkage magnetic flux and induc- tance are linked only with variables with the subscript 0 and do not have any relation to the variables in the other axes. Additionally, also here a knowledge from the theory of the asynchronous machine can be applied that zero impedance is equal to the leakage stator inductance that can be used during the measurement of the leakage stator inductance. Here can be reminded equation Section 12:

L0¼Ls�2Ms¼L_σsþM�2M

2 ¼L_σs: (368)

Namely, if all three phases of the stator windings (in series or parallel connec- tion) are fed by a single-phase voltage, it results in the pulse, non-rotating magnetic flux (see [8]).

If there is a request to make equations more simple, then it is necessary to ask for equality of mutual inductances of two windings, for example, inductance Lfdfor the current idshould be equal to the inductance Ldffor the current if:

L_fd¼L_df: (369)

Therefore from Eq. (354) forψ_f, take the expression at the current id, which was marked as L_fdand put into the equality with the expression at the current ifin Eq. (358) forψ_d, which was marked as L_df:

1

k_dLafmax¼3

2kdLafmax: (370)

Then:

k²_d¼2

3, (371)

and

k_d¼ ffiffiffi2 3 r

: (372)

The same value is obtained if expressions forψ_dat the current iD(358) andψ_Dat the current id(355) are put into the equality, to get LDd¼LdD. Then:

1

kdL_aDmax¼3

2k_dL_aDmax, (373)

which results in kd¼ ffiffi

23

q.

In the q-axis it is done at the same approach: The expression at the current i_qin the equation forψ_Q (356) and the expression at the current iQ in equation forψ_q (362), put into equality to get LQq¼LqQ, are as follows:

1

k_qLaQmax¼3

2kqLaQ max: (374)

It results in the value:

kq¼ ffiffiffi2 3 r

: (375)

Hence a choice for the coefficients suitable for synchronous machines flows:

k_d¼k_q¼ � ffiffiffi2 3 r

, but it is better to use the positive expression:

kd¼kq¼ ffiffiffi2 3 r

: (376)

If the next expressions are introduced:

Ldf¼3 2

ffiffiffi2 3 r

Lafmax¼ ffiffiffi3 2 r

Lafmax¼Lfd, (377)

LdD¼3 2

ffiffiffi2 3 r

LaDmax¼ ffiffiffi3 2 r

LaDmax¼LDd, (378)

LqQ¼3 2

ffiffiffi2 3 r

LaQmax¼ ffiffiffi3 2 r

LaQmax¼LQq, (379)

then equations for linkage magnetic fluxes of the synchronous machines in the d, q, 0 system have the form as follows:

ψ_d¼L_di_dþL_dfi_fþL_dDi_D, see 358ð Þ ψ_q¼LqiqþLqQiQ, see 362ð Þ

ψ₀¼L0i0, see 366ð Þ (380)

ψ_f ¼L_fdi_dþL_ffi_fþL_fDi_D, see 354ð Þ ψ_D¼LDdidþLDfifþLDDiD, see 355ð Þ ψ_Q ¼LQqiqþLQQiQ, see 356ð Þ where:

Ld¼La0þLab0þ3

2L2, see 359ð Þ L_q¼L_a0þL_ab0�3

2L₂, see 363ð Þ L₀¼L_a0�2Lab0, see 367ð Þ L_df¼L_fd¼

ffiffiffi3 2 r

L_afmax, see 377ð Þ (381)

L_dD¼L_Dd¼ ffiffiffi3 2 r

L_aDmax, see 378ð Þ L_qQ ¼L_Qq¼

ffiffiffi3 2 r

L_aQmax, see 379ð Þ:

By this record it was proven that not only terminal voltage equations of the stator and rotor windings but also equations of the linkage magnetic fluxes are identical with the equations of the universal electrical machine. Of course, expres- sions for inductances and a mode of their measurements are changed according the concrete electrical machine.

To complete a system of equations, it is necessary to add the equation for angular speed and to derive expression for the electromagnetic torque.

2.19 Power and electromagnetic torque of the synchronous machine