Modelling generator saturation

Small-signal models of synchronous generators, FACTS devices and the power system

4.2 Small-signal models of synchronous generators

4.2.8 Modelling generator saturation

118 Generators, FACTS devices & system models

Figure 4.5 Interface between the generator and exciter model taking account of the con- version between the reciprocal and non-reciprocal per-unit systems in the

respective models.

In the reciprocal per-unit system it follows from (4.2) on page 103 that under steady-state condition . By applying the conversion in (4.65) to this relationship it follows that in the non-reciprocal per-unit system the steady-state value of the field voltage and cur- rent are equal (i.e. ).

Some of the reasons why the non-reciprocal per-unit system is preferred [10] are:

• The measured generator open-circuit characteristic (O.C.C.) rarely extends beyond a stator voltage of 1.1 per-unit and never to . Thus, direct graphical determination of the base field current from the measured O.C.C. is straight-forward in the non-reciprocal system whereas supplementary calculation is required to determine the base field current in the reciprocal system.

• The numerical value of per-unit field-voltage in the reciprocal per-unit system is very small whereas, under steady-state conditions, in the non-reciprocal sys- tem.

Although IEEE Std. 421.5 recommends the use of the non-reciprocal per-unit system for modelling of the excitation system, it is sometimes the case that vendors or testing contrac- tors provide excitation system model parameters on a different per-unit system. For exam- ple, sometimes the base value of field current is defined as that field current, in Amperes, that is required to produce rated stator voltage when the generator is operating at ^{rated out-} put and frequency. Therefore, it is essential that those who are entering data into simulation programs understand the basis on which model parameters are supplied and, if necessary, adjust parameter values to comply with the per-unit system assumed by the simulation pro- gram being used.

Sec. 4.2 Modelling generator saturation 119 methods are strictly equivalent and yield identical results. Both methods are employed in dif- ferent simulation packages. In the first method the airgap mutual inductances in the respec- tive axes are assumed to be subject to magnetic saturation. The objective in this case is to show that the perturbations in these inductances, and , about their steady-state saturated values and can be expressed in terms of the perturbations in the airgap flux linkage components in the respective axes. In the second method, the component of excitation current necessary to account for the demagnetizing effect of magnetic saturation is deducted from the respective axes. The d-and q-axis components of the “saturation de- magnetizing current” are referred to as and respectively. The objective in the follow- ing is show that the perturbations in the saturation demagnetizing currents and about their steady-state values of and can also be expressed in terms of the pertur- bations in the airgap mutual flux-linkages. Provision is made for these two representations when formulating the generator equations in Section 4.2.3.4 by including the perturbations in either the mutual inductances or saturation demagnetizing currents depending on the method employed.

In both methods, the saturation level is determined from the user-supplied, open-circuit sat- uration characteristic(s). The user may choose to supply only the d-axis characteristic and select one of several functions for determining the ^q-axis characteristic from the ^d-axis char- acteristic. Alternatively, the manufacturer or testing contractor may supply a separate char- acteristic for each axis.

Figure 4.6 Open-circuit characteristic of the generator in the non-reciprocal per-unit sys- tem. (Note that the per-unit values of terminal voltage and airgap flux linkages are equal

when the machine is open-circuit and rotating steadily at base rotor speed.)

L_ad L_aq L_ad

0 L_aq

0

i_sd i_sq

i_sd i_sq i_sd

0 i_sq

0

Airgap line

Open-circuit characteristic

Excitation (pu) Terminal

voltage (pu) V_a V_t

(unity slope)

V_a V_t

120 Generators, FACTS devices & system models An open-circuit saturation characteristic is shown in Figure 4.6 in the non-reciprocal or uni- ty-slope per-unit system. If is the terminal voltage and is the corresponding voltage on the airgap line, then the saturation function is defined as

. (4.66)

For the given open-circuit characteristic of the generator, saturation is characterized by the values of and such that together with the selection of a function to interpolate between the latter two points on the saturation characteristic. Typically, pu and pu. Several commonly employed interpolation functions are detailed in Table 4.7 although other functions may be used.

Table 4.7 Interpolation functions for saturation characteristics.

So far only the open-circuit characteristic has been considered. However, when the genera- tor is loaded the open-circuit characteristics no longer apply. A common approximation is that the resultant airgap flux is indicative of the level of saturation when the generator is on-load. This is based on the fact that when the generator is on open-circuit and rotating at one per-unit speed the terminal voltage and airgap flux are equal in the per-unit system used. Other approximations for the level of saturation which are employed in widely-used software packages are described in Section 4.2.13.2.

Note that in Table 4.7 the value of to be used depends on the context. When determining the parameters A and B of the interpolation function the values of are the o.c. flux-link-

Type A, B

Exponential

Quadratic if

otherwise , where

Ontario Hydro [10]

if otherwise;

Linear

, , where

V_t V_a

S V _t S V _t = V_a–V_tV_t

S ₁ S ₂ ₂₁0

₁ = 1.0 ₂ = 1.2

S  = f A B   S

A^B

B ln ₂S ₂

₁S ₁ ---

 

  ln ₂

₁ ---

  

 

 



=

A = ₁S ₁  ₁^B

A B –1^B–2

B–A² A 0

A =₂–a₁1–a B = ₁S  ₁  ₁–A²

a = ₂S ₂  ₁S ₁

B 1.0 A

---

  ²

 – 

 

Ae^{B – I}   _I 0

_I₁

B ln ₂S ₂

₁S ₁ ---

 

   ₂–₁

=

A = ₁S e₁ ^{–B1–I}

S  B 1

--- –

B–A ---

A = ₂–a₁1–a B =₁S ₁  ₁–A

a = ₂S ₂  ₁S ₁

AB

² ---

_ag





Sec. 4.2 Modelling generator saturation 121 ages (equivalently o.c. voltages) obtained from the open-circuit-characteristic. When evalu- ating the saturation function when the machine is loaded then is the value of the selected saturation level indicator such as the resultant airgap flux ( ), the resultant k^th-transient flux-linkages ( ), etc.

The resultant airgap flux linkages are defined as:

. (4.67)

To determine the steady-state operating value of the airgap flux, it is noted that is also equal to the voltage behind the stator resistance and leakage inductance in the per-unit sys- tem used, taking into account any difference between the synchronous speed and base fre- quency of the generator (i.e. to account for the situation when ).

, (4.68)

in which , and are respectively the initial steady-state values of the generator sta- tor terminal voltage, and real and reactive power output.

The perturbations in the resultant airgap flux linkages about the operating point de- fined in (4.68) and in which the corresponding steady-state values of the d- and q-axis flux linkages are and respectively are obtained by linearizing equation (4.67) to give:

. (4.69)

It should be noted that the saturation characteristic interpolation functions listed in Table 4.7 are intended to be used when the machine is operating within its normal range of steady-state operating conditions, i.e. is expected to range at most between about 0.8 and 1.3 pu. The interpolation functions may require modification at higher flux levels that may occur under some large disturbance conditions.

The d- and q-axis saturation characteristics are denoted by and respectively. In the situation where a q-axis saturation characteristic is not provided one of the rules in Table 4.8 can be used to derive the q-axis characteristic from the d-axis characteristic provided.



_ag

^k

_ag = _ad² +_aq²

_ag

₀1

_ag

0

v_ag

0

₀ --- 1

₀ ---

   V_t

0

r_sP₀+₀L_lQ₀ V_t

0

---

 

 

 

 + 

 

 ² ₀L_lP₀–r_sQ₀ V_t

0

---

 

 

 ²

+

= =

V_t

0 P₀ Q₀

_ag

0

_ad

0 _aq

0

_ag _ad

0

_ag

0

---

 

 

 

_ad _aq

0

_ag

0

---

 

 

 

_aq

+ 1

_ag

0

---

 

  _ad

0 _aq

0 

˜^adq

= =

_ag

0

S_d S_q

122 Generators, FACTS devices & system models Table 4.8 q-Axis saturation characteristics as a function of those for the d-axis

4.2.8.1 Method 1: Non-linear airgap mutual inductances

As mentioned earlier the first method for representing generator magnetic saturation is to treat the d- and q-axis airgap mutual inductances as non-linear parameters that depend on the resultant airgap flux linkages. It is assumed that the leakage inductances are not subject to magnetic saturation and are thus assumed to be constant parameters.

The values of the d- and q-axis airgap mutual inductances are expressed in terms of their re- spective saturation characteristics by:

and . (4.70)

The steady-state saturated values of the airgap mutual inductances and are ob- tained by substituting in (4.70).

A 1

Note A:

B

N/A N/A

Note B: The points and on the q-axis open-circuit saturation charac- teristic are specified. The same interpolation function used for the d-axis character- istic is employed.

C Note C: . This characteristic is based on empirical results reported by Shackshaft [19]; the variation of and with rotor position is neglected.

D 0 0

Note D: The q-axis is unsaturated. Useful in modelling salient pole machines.

E Note E: This option is employed in the saturation model of some software.

F

, where

Note F: This option is employed in the saturation model of some software.

S_qS_d   S_qS_d

S_d  L_aqL_ad= L_aquL_adu

S_q1.0 S_q1.2

S_d

S_dL_aquL_adu–1+L_aquL_adu

--- S_d S_q

--- 1 S_dS_q L_aqu L_adu --- 1–

 

 

–

L_ad–L_aq = L_adu–L_aqu =L_sal

L_ad L_aq

L_duL_qu

 S_d L_duL_qu

L_duL_qu

 ^ZS_d 

Z= lnS_d1.2S_d1.0ln 1.2  L_duL_qu^Z

L_ad L_ad

u

1+S_d_ag ---

= L_aq L_aq

u

1+S_q_ag ---

=

L_ad

0 L_aq

0

_ag _ag

= 0

Sec. 4.2 Saturation modelling method 2: Saturation demagnetization current 123 The perturbations in the airgap mutual inductances about the operating point defined in (4.68) are obtained by linearization of the equations for the non-linear airgap mutual in- ductances in (4.70) to yield:

(4.71)

Substituting for the perturbations in the resultant airgap flux linkages from equation (4.69) into the preceding equation results in the following expression for the perturbations in the airgap mutual inductances in terms of the perturbations in the airgap flux-linkages:

4.2.8.2 Method 2: Saturation demagnetization current

The second method for representing the effects of magnetic saturation involves deducting non-linear components of d- and q-axis saturation demagnetization current and from the excitation of the d- and q-axis windings respectively. In this formulation the model utilizes the fixed unsaturated airgap mutual inductances. This is especially advantageous when representing saturation in the Classical Parameter Formulation of the generator model because it is unnecessary to adjust the classically-defined standard parameters to account for the effects of saturation.

To determine the expression for it can be deduced from (4.12) on page 105 that:

. (4.74)

_ag

0

L_ad

L_aq

L_ad

0 2

L_ad

u

---

 

 

 S_d_ag

_ag ---

0

L_aq

0 2

L_aq

u

---

 

 

 S_q_ag

_ag ---

0

_ag –

=

Saturation Method 1:

Perturbations in airgap mutual inductances

, (4.72)

in which

(4.73)

L˜^adq C_ladq

0

˜^adq

=

C_ladq

0

1_ag

0

---

 

 

L_ad

0 2

L_ad

u

---

 

 

 S_d_ag

_ag ---

0

L_aq

0 2

L_aq

u

---

 

 

 S_q_ag

_ag ---

0

_ad

0 _aq – 0

=

i_sd i_sq

i_sd i_sd u^Ti

˜^rd–i_d

  _ad

L_ad

u

--- –

=

124 Generators, FACTS devices & system models Alternatively, if the effects of magnetic saturation are represented by non-linear mutual air- gap inductances according to (4.70) (i.e. by Method 1) then from (4.12) with and with defined according to (4.70) it follows that:

. (4.75)

Substituting from the preceding equation for into (4.74) yields:

. (4.76)

The q-axis saturation demagnetization current component is similarly derived:

. (4.77)

Equations (4.76) and (4.77) are combined to yield:

in which . (4.78)

Linearizing the preceding equation about the operating point yields the following expression for the perturbations in the saturation demagnetizing current compo- nents in terms of the perturbations in the airgap flux-linkages.

s₂ = 0 L_ad

u^Ti

˜^rd–i_d

  1+S_d_ag_ad L_ad

u

---

=

u^Ti

˜^rd–i_d

 

i_sd S_d_ag_ad L_ad

u

---

=

i_sq S_q_ag_aq L_aq

u

---

=

˜i^sdq S_dq_ag

˜^adq

= S_dq_ag S_d_ag

L_ad

u

--- S_q_ag L_aq

u

---

  

 

 

= D

_ag

0 _ad

0 _aq

  0

 

Saturation Method 2:

Perturbations in saturation demagnetizing currents

, (4.79)

where

(4.80)

and . (4.81)

i

˜^sdq C_madq

0

˜^adq

=

C_madq

0 S_dq

0

_ad

0

L_ad

u

--- S_d_ag

_ag ---

 

 

0

_aq

0

L_aq

u

--- S_q_ag

_ag ---

 

 

0

_ad

0

_ag

0

---

 

 

  aq₀

_ag

0

---

 

 

 

+

=

S_dq

0

S_d _ag

 0 L_ad

u

--- S_q _ag

 0 L_aq

u

---

  

 

 

= D

Sec. 4.2 Steady-state conditions, coupled-circuit model 125 4.2.9 Balanced steady-state operating conditions of the coupled-circuit model The initial steady-state values of the coupled-circuit generator model variables when oper- ating under balanced conditions are now calculated. In the following analysis the subscript

‘0’ denotes the steady-state value of the variable.

It is assumed that the steady-state generator stator voltage magnitude, , and the real and reactive power output and of the generator are given, in per-unit on the gen- erator base quantities. These initial values are usually obtained from the power flow solution on which the dynamic analysis is to be based.

Under steady-state conditions so from (4.56) on page 114 it follows that

per-unit; and so from (4.57) on page 114 it follows that . Note, that nor- mally the generator rated frequency is the same as the system nominal frequency and so nor- mally . However, if, for example, a generator rated for 60 Hz is connected to a 50 Hz system and is chosen to be , then per-unit.

Under balanced steady-state operating conditions the stator voltage is represented as a pha- sor in the complex plane in which the d-axis corresponds to the real axis of the complex plane and the q-axis to that of the imaginary axis so that:

. (4.82)

where is the angle by which the voltage phasor leads the d-axis. (Note that is defined differently than the “load angle” which is the angle by which the voltage phasor lags the q- axis. The use of is convenient analytically and the results are consistent.).

The phasor representing the generator current output is:

, (4.83)

where

(4.84) is the magnitude of the current; and

1 (4.85)

is the angle by which the current phasor leads that of the voltage.

V_t0 P₀ P_e

= 0 Q₀

p = 0  = ₀

p = 0 T_m = T_g

₀ = 1

_b 2 60 ₀ = 5 6

Vˆ

t

Vˆ

t = v_d0+jv_q0 = V_t0e^j0 = V_t0cos₀+jV_t0sin₀

₀ ₀

₀

Iˆ i_d0+ji_q0 P₀–jQ₀ V_t0e^–j0

--- I₀e^{j0+0}

= = =

I₀ P₀²+Q₀² V_t0 ---

=

₀ = atan2–Q₀P₀

126 Generators, FACTS devices & system models If the effects of magnetic saturation are being represented then, for the purpose of calculat- ing the initial steady-state operating conditions, the following steady-state saturated values of the d- and q-axis airgap mutual inductances obtained from (4.70) are used. This applies to both of the methods of representing the effects of magnetic saturation in the dynamic model of the machine.

and , (4.86)

in which the resultant airgap flux-linkages, , are obtained from (4.68) on page 121.

The saturated values of the d- and q-axis synchronous inductances at the steady-state oper- ating point are:

and . (4.87)

Since, under steady-state conditions, and since it is deduced from (4.5) on page 104 that . Consequently, from equation (4.12) on page 105,

(4.88) and then from (4.22) on page 106 it follows that:

, (4.89)

a result that is independent of the number of rotor windings.

From (4.24) and (4.89), under steady-state conditions,

; (4.90)

which again is independent of the number of rotor windings.

The d-axis damper winding currents are zero in the steady-state so from (4.9) on page 104, (4.22) on page 106 and (4.65) on page 117 it follows that:

and (4.91)

1. Definition of :

If then arbitrarily define ;

else if then if , ; else ,

otherwise let and define then

First quadrant: and then ; Second quadrant: and then ; Third quadrant: and then ; and Fourth quadrant: and then .

 = atan2y x 

x = y = 0  = 0

x = 0 y0  = 2  = –2 z = y x  = atan z

x0 y0  =  x0 y0  =  – x0 y0  = –+

x0 y0  = – L_ad

0 L_ad

u 1 S_d _ag

 0

 + 

=  L_aq

0 L_aq

u 1 S_q _ag

 0

 + 

= 

_ag

0

L_d

0 L_ad

0+L_l

= L_q

0 L_aq

0+L_l

=

p˜^rq = 0 v

˜^rq 0

= ˜ i˜^rq0 0

= ˜

_aq

0 L_aq

0i_q – 0

=

_q

0 _aq

0 L_li_q

– 0 L_aq

0+L_l

 i_q

– 0 L_q

0i_q – 0

= = =

v_d

0 r_si_d

– 0 ₀_q

– 0 r_si_d

– 0 ₀L_q

0i_q + 0

= =

_ad

0 L_ad

0 i_fd

0 i_d – 0

  L_ad

0 L_ad

 u

 I_fd

0 L_ad

0i_d – 0

= =

Sec. 4.2 Steady-state conditions, coupled-circuit model 127

. (4.92)

From (4.24) on page 106 and (4.92) the following expression for the steady-state ^q-axis volt- age is obtained:

. (4.93)

The stator voltage phasor is obtained by combining (4.90) and (4.93) to yield:

(4.94)

in which (4.95)

Rearranging (4.94) yields:

. (4.96)

The artificial voltage phasor is aligned with q-axis and corresponds to the voltage

behind the impedance .

Now, substituting for and from equations (4.82) and (4.83) into the preceding equation yields:

. (4.97)

By equating the arguments of both sides of (4.97) yields , the angle by which voltage pha- sor leads the d-axis:

_d

0 _ad

0 L_li_d

– 0 L_d

0i_d

– 0 L_ad

0i_fd

+ 0 L_d

0i_d

– 0 L_ad

0

L_ad

u

---

 

 

 

I_fd + 0

= = =

v_q

0 r_si_q

– 0 ₀_d

+ 0 r_si_q

– 0 ₀L_d

0i_d

– 0 ₀ L_ad

0

L_ad

u

---

 

 

 

I_fd + 0

= =

Vˆ

t v_d

0 jv_q

+ 0 r_s j₀L_q

+ 0

 Iˆ

– j₀ L_ad

0

L_ad

u

---

 

 

 

I_fd

0 L_d

0 L_q – 0

 i_d

– 0

 

 

 

+

= =

Iˆ i_d

0 ji_q + 0

 

=

Eˆ

q Vˆ

t r_s j₀L_q + 0

 Iˆ

+ j₀ L_ad

0

L_ad

u

---

 

 

 

I_fd

0 L_d

0 L_q – 0

 i_d

– 0

 

 

 

jE_q

= = =

Eˆ

q = jE_q Z r_s j₀L_q

+ 0

 

=

Vˆ

t Iˆ

E_qe^{j2} V_t

0 r_s j₀L_q + 0

 I₀e^j0

 + 

 e^j0

= E_qe^{j2–0}

 V_t

0 r_sI₀cos₀ ₀L_q

0I₀sin₀ –

 +  j r_sI₀sin₀ ₀L_q

0I₀cos₀

 + 

+

=

₀

128 Generators, FACTS devices & system models

(4.98)

This result is independent of the number of ^d- or ^q-axis rotor-windings or of the representa- tion of coupling between the d-axis rotor windings. Figure 4.7 is a phasor diagram showing the computation of , and the associated location of the d- and q-axes with respect to the voltage phasor. The ^d- and ^q-axis components of the voltage and current phasors are also shown in this diagram.

Figure 4.7 Phasor diagram showing the computation of , and the location of the d- and q-axes in relation to the voltage and current phasors. The d- and q-axis components of

the voltage and current are also shown.

in which, from equations (4.84) & (4.85) respectively:

and .

₀ 

2--- 2 I₀ r_ssin₀ ₀L_q

0cos₀

 +  V_t

0 I₀ r_scos₀ ₀L_q

0sin₀

 – 

 + 



 

atan –

=

I₀ P₀²+Q₀² V_t0 ---

= ₀ = atan2–Q₀P₀

Eˆ

q ₀

Vˆt

Iˆ β⁰

γ⁰

Eˆ_q= ˆV_t+ (r_s+jω₀L_q0) ˆI

rsIˆ

jω₀L_q0Iˆ

d−axis

q−axis

vd₀

vq₀

id₀

i_q0

Eˆ

q ₀

Sec. 4.2 Steady-state conditions, coupled-circuit model 129 Having calculated the steady-state saturated values of the airgap mutual inductances and syn- chronous inductances according to (4.86) and (4.87) and the values of , and in (4.98) the initial steady-state values of those generator variables that are independent of the rotor winding structure are readily found to be:

(4.99)

The calculation of , the initial steady-state value of the angle by which the d-axis leads the R-axis of the synchronously rotating network reference frame is deferred until Section 4.2.10.

For the ^3d3q-c1 model the steady-state values of the following ^d- and ^q-axis rotor winding current and flux-linkage and variables are:

(4.100)

For generator models which neglect unequal coupling between the ^d-axis rotor windings . For models with only one d-axis damper winding the variables and do not exist and the non-existence of is represented by setting its value to zero in the above equations. Similar trivial modifications are made to (4.100) so they can be applied to coupled-circuit models with other rotor structures.

₀ I₀ ₀

v_d

0 V_t

0cos₀

= v_q

0 V_t

0sin₀

= i_d

0 = I₀cos₀+₀ i_q

0 = I₀sin₀+₀

_d

0 v_q

0 r_si_q + 0

   ₀

=

_ad

0 _d

0 L_li_d + 0

=

_q

0 v_d

0 r_si_d + 0

   ₀

–

=

_aq

0 _q

0 L_li_q + 0

=

i_fd

0 _d

0 L_d

0i_d + 0

  L_ad

 0

= I_fd

0 L_ad

ui_fd

= 0

e_fd

0 r_fdi_fd

= 0

E_fd

0 I_fd

= 0

T_g

0 _d

0i_q

0 _q

0i_d

– 0 P₀+r_sI₀²  ₀

= =

T_m

0 T_g

0 P_m

0₀

= =

₀

i_1d

0 i_2d

0 0

= =

_fd

0 _ad

0 L_fd+L_c1+L_c2i_fd + 0

=

_1d

0 _ad

0 L_c1+L_c2i_fd + 0

=

_2d

0 _ad

0 L_c2i_fd + 0

=

i_1q

0 i_2q

0 i_3q

0 0

= = =

_1q

0 _2q

0 _3q

0 _aq

= = = 0

L_c1 = L_c2 = 0 i_2d _2d

L_c2

130 Generators, FACTS devices & system models 4.2.10 Interface between the generator Park/Blondel reference frame and the syn- chronous network reference frame

The generator equations are developed in the Park/Blondel co-ordinate system in which, as mentioned earlier, the ^d-axis is aligned with the magnetic axis of the rotor field winding and the q-axis leads the d-axis by 90 electrical degrees. The dq reference frame rotates in an anti- clockwise direction at the speed of the generator rotor per-unit. As explained in the de- velopment of the generator equations of motion in [12] the generator rotor angle (elec.

rad) is measured relative to a synchronously rotating reference. In the analysis of multi-ma- chine systems the R-axis of the synchronously rotating network RI reference frame is chosen as the reference for the rotor angle of each generator.

To facilitate the analysis of multi-machine systems it is necessary to transform the stator cur- rent and voltage at the machine terminals between the generator dq reference frame and net- work ^RI reference frame.

In Figure 4.8 the stator current phasor, , is shown at the instant t. The superscript (g) denotes that the current phasor is in per-unit of the generator base quantities. At this instant the d-axis leads the R-axis by (rad) and the current phasor leads the d-axis by (rad).

In the generator ^dq reference frame the current phasor is expressed as:

, and in the RI reference frame it is:

.

Expanding the preceding equation yields the following relationship between the current phasor components in the dq and RI reference frames.

Equating respectively the real and imaginary components in the above equation yields the following matrix relationship between the current components in the respective reference frames:

, (4.101)

In compact matrix form this equation is denoted as:

, (4.102)



 t

Iˆ^{ g}

 t  t

Iˆ^{ g} Iˆ^{ g} = i_d+ji_q = Iˆ^{ g} e^{j t}

Iˆ^{ g} = i_R^{ g} +ji_I^{ g} = Iˆ^{ g} e^{j  t +  t } = e^{j t} Iˆ^{ g} e^{j t} = e^{j t} i_d+ji_q

i_R^{ g} +ji_I^{ g} = i_d+ji_qe^j = i_d+ji_qcos  +jsin  



cos i_d–sin  i_q

 +jsin  i_d+cos  i_q

=

i_R^{ g} i_I^{ g}



cos  –sin 



sin  cos  i_d i_q

=

˜i^RI

 g

R  i

˜^dq

=