Three-phase power and torque in the system dq0 .1 Three-phase power in the system dq0

The instantaneous value of the input power in a three-phase system is a sum of instantaneous values of power in each phase (see also Section 2.2):

p_in¼u_ai_aþu_bi_bþu_ci_c: (163) Instantaneous values of uaand iawill be introduced into this equation. These were derived in the inverse transformation chapter. They are Eqs. (133)–(135). It means at first u is introduced for x, and it must be multiplied by expression, where i was introduced, and then the further phases in the same way are adapted. At the end all expressions are summed:

p_in¼ 2 3

1

kdudcosϑ_k�2 3

1

kquqsinϑ_kþ1 3

1 k0u0

� �

2 3

1

kdidcosϑ_k�2 3

1

kqiqsinϑ_kþ1 3

1 k0i0

� �

þ 2 3

1

kdu_dcos ϑ_k�2π 3

� �

�2 3

1

kqu_qsin ϑ_k�2π 3

� �

þ1 3

1 k0u₀

� �

: 2 3

1

kdi_dcos ϑ_k�2π 3

� �

�2 3

1

kqi_qsin ϑ_k�2π 3

� �

þ1 3

1 k0i₀

� �

þ 2 3

1

k_du_dcos ϑ_kþ2π 3

� �

�2 3

1

k_qu_qsin ϑ_kþ2π 3

� �

þ1 3

1 k₀u₀

� �

: 2 3

1

k_di_dcos ϑ_kþ2π 3

� �

�2 3

1

k_qiqsin ϑ_kþ2π 3

� �

þ1 3

1 k₀i0

� �

(164) Now it is necessary to multiply all members with each other, including the goniometrical functions, and after a modification the result is:

p_in¼ 2 3

1

k²_du_di_dþ2 3

1

k²_qu_qi_qþ1 3

1 k²₀u₀i₀

" #

: (165)

Variables ud, uq, and u0are given by Eqs. (160) to (162), which were introduced above, and after a modification, the result is:

p_in¼ 2 3 1

k²_d Rsi²_dþiddψ_d

dt �ω_kψ_qid

� �

þ2 3 1

k²_q Rsi²_qþiq

dψ_q

dt þω_kψ_diq

� �

þ1 3 1

k²₀ Rsi²₀þi0dψ₀ dt

� �

" #

(166) If an analysis in greater details is made, it is seen that an input power on the left side must be in equilibrium with the right side. It is supposed to be motoring operation. Therefore the input power applied to the terminals of the three-phase motor is distributed between the Joule’s resistance loss�ΣRi²�

, time varying of the field energy stored in the investigated circuitΣi^d_dt^ψ, and the rest of the members’

mean conversion of electrical to mechanical energy and eventually to mechanical output power. If the resistance loss and power of magnetic field are subtracted from the input power on the terminals, the result is an air gap electromagnetic power, which is given by the difference of two rotating voltages in both axes:

p_e¼2 3

1

k²_qω_kψ_di_q� 1 k²_qω_kψ_qi_d

!

: (167)

Here it is seen that it is advantageous to choose the same proportional constants:

kd¼kq, to be able to set out it in front of the brackets, together with the angular speed:

p_e¼2 3

1

k²_qω_k ψ_diq�ψ_qid

� �

(168)

and, eventually,

p_e¼2 3

1

k_dk_qω_k�ψ_diq�ψ_qi_d�

: (169)

The expressions from the inverse transformation are introduced also for uaand ia, in Eq. (156):

dψ_a dt ¼2

3 1

k_du_dcosϑ_k�2 3

1

k_qu_qsinϑ_kþ1 3

1 k₀u₀

�Rs 2 3

1

k_didcosϑ_k�2 3

1

kqiqsinϑ_kþ1 3

1 k0i0

(158) The left sides of Eqs. (155) and (158) are equal; therefore, also right sides will be equal. Now the members with the same goniometrical functions and members without goniometrical functions will be selected and put equal, e.g., members at

cosϑ_kyield:

2 3

1 kd

dψ_d dt �2

3 1

kqω_kψ_q¼2 3

1

kdu_d�R_s2 3

1

kdi_d: (159) The equation for udis gained after modification, and in a very similar way, also the two other equations are obtained:

u_d¼R_si_dþdψ_d

dt �ω_kψ_q, (160)

u_q¼R_si_qþdψ_q

dt þω_kψ_d, (161)

u₀¼R_si₀þdψ₀

dt : (162)

Equations (160)–(162) are the voltage equations for the stator windings of the three-phase machines, such as asynchronous motors in k-reference frame, rotating by the angular speedω_kwith the dq0-axis. As it can be seen, they are the same equations as the voltage equations in Section 2.1, which were derived for universal arrangement of the electrical machine. Here general validity of the equations is seen: if windings of any machine are arranged or are transformed to the arrange- ment with two perpendicular axes to each other, the same equations are valid. Of course, the parameters, mainly inductances, of the machine are different, and it is necessary to know how to get them.

2.8 Three-phase power and torque in the system dq0

2.8.1 Three-phase power in the system dq0

The instantaneous value of the input power in a three-phase system is a sum of instantaneous values of power in each phase (see also Section 2.2):

p_in¼u_ai_aþu_bi_bþu_ci_c: (163) Instantaneous values of uaand iawill be introduced into this equation. These were derived in the inverse transformation chapter. They are Eqs. (133)–(135). It means at first u is introduced for x, and it must be multiplied by expression, where i was introduced, and then the further phases in the same way are adapted. At the end all expressions are summed:

p_in¼ 2 3

1

kdudcosϑ_k�2 3

1

kquqsinϑ_kþ1 3

1 k0u0

� �

2 3

1

kdidcosϑ_k�2 3

1

kqiqsinϑ_kþ1 3

1 k0i0

� �

þ 2 3

1

kdu_dcos ϑ_k�2π 3

� �

�2 3

1

kqu_qsin ϑ_k�2π 3

� �

þ1 3

1 k0u₀

� �

: 2 3

1

kdi_dcos ϑ_k�2π 3

� �

�2 3

1

kqi_qsin ϑ_k�2π 3

� �

þ1 3

1 k0i₀

� �

þ 2 3

1

k_du_dcos ϑ_kþ2π 3

� �

�2 3

1

k_qu_qsin ϑ_kþ2π 3

� �

þ1 3

1 k₀u₀

� �

: 2 3

1

k_di_dcos ϑ_kþ2π 3

� �

�2 3

1

k_qiqsin ϑ_kþ2π 3

� �

þ1 3

1 k₀i0

� �

(164) Now it is necessary to multiply all members with each other, including the goniometrical functions, and after a modification the result is:

p_in¼ 2 3

1

k²_du_di_dþ2 3

1

k²_qu_qi_qþ1 3

1 k²₀u₀i₀

" #

: (165)

Variables ud, uq, and u0are given by Eqs. (160) to (162), which were introduced above, and after a modification, the result is:

p_in¼ 2 3 1

k²_d Rsi²_dþiddψ_d

dt �ω_kψ_qid

� �

þ2 3 1

k²_q Rsi²_qþiq

dψ_q

dt þω_kψ_diq

� �

þ1 3 1

k²₀ Rsi²₀þi0dψ₀ dt

� �

" #

(166) If an analysis in greater details is made, it is seen that an input power on the left side must be in equilibrium with the right side. It is supposed to be motoring operation. Therefore the input power applied to the terminals of the three-phase motor is distributed between the Joule’s resistance loss�ΣRi²�

, time varying of the field energy stored in the investigated circuitΣi^d_dt^ψ, and the rest of the members’

mean conversion of electrical to mechanical energy and eventually to mechanical output power. If the resistance loss and power of magnetic field are subtracted from the input power on the terminals, the result is an air gap electromagnetic power, which is given by the difference of two rotating voltages in both axes:

p_e¼2 3

1

k²_qω_kψ_di_q� 1 k²_qω_kψ_qi_d

!

: (167)

Here it is seen that it is advantageous to choose the same proportional constants:

kd¼kq, to be able to set out it in front of the brackets, together with the angular speed:

p_e¼2 3

1

k²_qω_k ψ_diq�ψ_qid

� �

(168)

and, eventually,

p_e¼2 3

1

k_dk_qω_k�ψ_diq�ψ_qi_d�

: (169)

This is the base expression for the power, which is converted from an electrical to a mechanical form in the motor or from a mechanical to an electrical form in the case of the generator. Next an expression for the electromagnetic torque is derived.

2.8.2 Electromagnetic torque of the three-phase machines in the dq0 system As it is known, an air gap power can be expressed by the product of the devel- oped electromagnetic torque and a mechanical angular speed, now in the k-system:

p_e¼t_eΩ_k (170)

or by means of electrical angular speed:

p_e¼teω_k

p (171)

where p is the number of pole pairs. An instantaneous value of the developed electromagnetic torque is valid:

t_e¼ p

ω_kp_e¼ p ω_k

2 3

1

kdkqω_k�ψ_di_q�ψ_qi_d�

(172)

and after a reduction the torque is:

te¼p2 3

1

k_dkq ψ_diq�ψ_qid

� �

: (173)

This is the base expression for an instantaneous value of developed electromag- netic torque of a three-phase machine. It is seen that its concrete form will be modified according to the chosen proportional constants. The most advantageous choice seems to be the next two possibilities:

(1) k_d¼k_q¼²₃, k₀¼₃¹. Then:

te¼p2 3 1

2233

ψ_diq�ψ_qi_d

� �

¼p3

2�ψ_diq�ψ_qi_d�

(174)

(2) kd¼kq¼ ffiffi

23

q, k0¼ ffiffi

13

q. Then:

t_e¼p2 3

ffiffi1

23

q ffiffi

23

q �ψ_di_q�ψ_qi_d�

¼p�ψ_di_q�ψ_qi_d�

: (175)

It will be shown later that the first choice is more advantageous for asynchro- nous machines and the second one for synchronous machine.

A developed electromagnetic torque in the rotating electrical machines directly relates with equilibrium of the torques acting on the shaft. During the transients in motoring operation, i.e., when the speed is changing, developed electromagnetic torque tecovers not only load torque tL, including loss torque, but also load created by the moment of inertia of rotating mass J^d_dt^Ω. Therefore, it is possible to write:

t_e¼JdΩ

dt þt_L: (176)

Unknown variables in motoring operation are obviously currents and speed, which can be eliminated from Eqs. (176) and (175). The mechanical angular speed is valid:

dΩ dt ¼1

Jðte�tLÞ (177)

and electrical angular speed is:

dω dt ¼p

Jðte�tLÞ: (178)

The final expression for the time changing of the speed will be gotten, if for te

Eqs. (174) and (175) according to the choice of the constants kdand kqare introduced:

(1) k_d¼kq¼²₃, k0¼¹₃ Then

t_e¼p3

2�ψ_di_q�ψ_qi_d�

(179) dω

dt ¼p J p3

2�ψ_di_q�ψ_qi_d�

�t_L

� �

(180)

(2) k_d¼k_q¼ ffiffi

23

q, k₀¼ ffiffi

13

q.

Then

te¼p ψ_diq�ψ_qid

� �

(181) dω

dt ¼p

J p ψ_diq�ψ_qid

� �

�tL

� �

: (182)

If there is a steady-state condition,^d_dt^ω¼0, and electromagnetic and load torque are in balance:

t_e¼t_L: (183)

2.8.3 Power invariance principle

The expression for the three-phase power in dq0 system is:

p_in¼ 2 3

1

k²_du_di_dþ2 3

1

k²_qu_qi_qþ1 3

1 k²₀u₀i₀

" #

(184) which was derived from the original expression for the three-phase power in abc system:

p_in¼½uaiaþubibþucic�: (185) The expression can be modified by means of the constants kdand kq:

This is the base expression for the power, which is converted from an electrical to a mechanical form in the motor or from a mechanical to an electrical form in the case of the generator. Next an expression for the electromagnetic torque is derived.

2.8.2 Electromagnetic torque of the three-phase machines in the dq0 system As it is known, an air gap power can be expressed by the product of the devel- oped electromagnetic torque and a mechanical angular speed, now in the k-system:

p_e¼t_eΩ_k (170)

or by means of electrical angular speed:

p_e¼teω_k

p (171)

where p is the number of pole pairs. An instantaneous value of the developed electromagnetic torque is valid:

t_e¼ p

ω_kp_e¼ p ω_k

2 3

1

kdkqω_k�ψ_di_q�ψ_qi_d�

(172)

and after a reduction the torque is:

te¼p2 3

1

k_dkq ψ_diq�ψ_qid

� �

: (173)

This is the base expression for an instantaneous value of developed electromag- netic torque of a three-phase machine. It is seen that its concrete form will be modified according to the chosen proportional constants. The most advantageous choice seems to be the next two possibilities:

(1) k_d¼k_q¼²₃, k₀¼¹₃. Then:

te¼p2 3 1

2233

ψ_diq�ψ_qi_d

� �

¼p3

2�ψ_diq�ψ_qi_d�

(174)

(2) kd¼kq¼ ffiffi

23

q, k0¼ ffiffi

13

q. Then:

t_e¼p2 3

ffiffi1

23

q ffiffi

23

q �ψ_di_q�ψ_qi_d�

¼p�ψ_di_q�ψ_qi_d�

: (175)

It will be shown later that the first choice is more advantageous for asynchro- nous machines and the second one for synchronous machine.

A developed electromagnetic torque in the rotating electrical machines directly relates with equilibrium of the torques acting on the shaft. During the transients in motoring operation, i.e., when the speed is changing, developed electromagnetic torque tecovers not only load torque tL, including loss torque, but also load created by the moment of inertia of rotating mass J^d_dt^Ω. Therefore, it is possible to write:

t_e¼JdΩ

dt þt_L: (176)

Unknown variables in motoring operation are obviously currents and speed, which can be eliminated from Eqs. (176) and (175). The mechanical angular speed is valid:

dΩ dt ¼1

Jðte�tLÞ (177)

and electrical angular speed is:

dω dt ¼p

Jðte�tLÞ: (178)

The final expression for the time changing of the speed will be gotten, if for te

Eqs. (174) and (175) according to the choice of the constants kdand kqare introduced:

(1) k_d¼kq¼²₃, k0¼₃¹ Then

t_e¼p3

2�ψ_di_q�ψ_qi_d�

(179) dω

dt ¼p J p3

2�ψ_di_q�ψ_qi_d�

�t_L

� �

(180)

(2) k_d¼k_q¼ ffiffi

23

q, k₀ ¼ ffiffi

13

q.

Then

te¼p ψ_diq�ψ_qid

� �

(181) dω

dt ¼p

J p ψ_diq�ψ_qid

� �

�tL

� �

: (182)

If there is a steady-state condition,^d_dt^ω¼0, and electromagnetic and load torque are in balance:

t_e¼t_L: (183)

2.8.3 Power invariance principle

The expression for the three-phase power in dq0 system is:

p_in¼ 2 3

1

k²_du_di_dþ2 3

1

k²_qu_qi_qþ1 3

1 k²₀u₀i₀

" #

(184) which was derived from the original expression for the three-phase power in abc system:

p_in¼½uaiaþubibþucic�: (185) The expression can be modified by means of the constants kdand kq:

1. If kd¼kq¼²₃, k0¼¹₃, then

p_in¼3

2udidþ3

2uqiqþ3u0i0, (186) in which the principle of power invariance is not fulfilled, because the members in dq0 axes are figures, although it was derived from Eq. (163), where no figures were employed.

2. If kd¼k_q¼ ffiffi

23

q, k0¼ ffiffi

13

q, then

p_in¼u_di_dþuqiqþu0i0, (187) in which the principle of power invariance is fulfilled.