Stator MMF Harmonics at Non-sinusoidal Machine Power Supply (for M ≥ 1, Q ≥ 1)

Appendices

Appendix 3.1

See Table 3.7.

Table 3.7 Field rotation speeds ${\text{V}}_{\text{BOR}}$ in the air gap relative to stator winding

List of Symbols

a:

Number of parallel branches in phase

b_P :

Width of pole shoe

AS:

Stator current load

D:

Stator boring diameter

${\text{F}}_{1} , \ldots ,{\text{F}}_{\text{m}}$ :

Amplitudes of spatial harmonics of stator winding phase MMF

${\text{F}}_{\text{ST}} ( {\text{m}}_{\text{EL}} ,{\text{Q)}}$, ${\text{F}}_{{{\text{ST}},{\text{RES}}}} ( {\text{x}},{\text{m}}_{\text{EL}} ,{\text{Q)}}$ :

Respectively, complex amplitude (phasor) and amplitude (modulus) of spatial harmonic of order of m_EL and time harmonic of order Q of all winding phases m_PH

I_Q :

Harmonic amplitude of order Q for stator phase current

K_W(m):

Winding factor for m order field harmonic

k_Q :

Relation of stator current harmonic amplitude of order Q > 1 to harmonic amplitude of order Q = 1

m, m_EL :

Orders of MMF spatial harmonics of MMF and stator fields

m_PH :

Number of stator winding phases

Q:

Order of time harmonics

Q_DIR, m_DIR :

Order of time and spatial harmonics of direct field

Q_AD, m_AD :

Order of time and spatial harmonics of additional field

n, n_EL :

Orders of spatial harmonics of MMF and rotor fields

S_N :

Number of effective conductors in slot

T:

Time

P:

Number of machine pole pairs

Q:

Number of slots per pole and phase of stator winding

T, T_EL :

Expansion periods of MMF and mutual induction field to harmonic series

${\text{U}}_{ 1} , \ldots ,{\text{U}}_{\text{Q}}$ :

Amplitudes of time harmonics

W_PH :

Number of turns in stator winding phase

${\text{Z}}_{ 1}$ :

Number of stator slots

$\upbeta$ :

Pitch shortening of stator winding

${\upvarphi}_{{ 1 , {\text{U}}}}, \ldots,{\upvarphi}_{{{\text{Q}},{\text{U}}}}$ :

Initial phase angles of voltage harmonics

τ:

Pole pitch

$\upomega_{ 1} , \ldots ,\upomega_{\text{Q}}$ :

Stator current circular frequencies

$\upomega_{\text{REV}}$ :

Angular rotation speed of rotor

$\upomega_{\text{BOR}}$ :

Angular rotation speed of MMF and stator field in air gap

$\upomega_{\text{ROT}}$ :

Circular frequency of EMF and rotor current

Brief Conclusions

• 1. A number of mutual induction fields different in their periods and rotation speeds occur in A.C. machine air gap at non-sinusoidal power supply. The fields inducing EMF of the same frequency in rotor loops can be determined in them. Loops generating these fields are magnetically coupled, for instance, loops of rotor and stator (in Table 3.4). Except them, there are mutual induction fields inducing EMF of identical frequency in stator loops, but differing in order of spatial harmonics; loops corresponding to them are also magnetically coupled, for instance loops of rotor and stator (in Table 3.5). The machine processes are determined taking into account the interaction of these loops in the general system of Eqs. 

• 2. Two fields determined by time harmonics of various orders Q containing in expansion (3.1) and spatial harmonics of order m in expansion (3.4) at certain combination of these orders induce in rotor loops EMF of identical frequency, for example, at m_AD = 5: Q_DIR = 1 and Q_AD = 11, and also Q_DIR = 7 and Q_AD = 17.

• 3. Unlike fields in air gap of three-phase machine, fields in air gap of six-phase machine, determined by time harmonics of orders Q = 3 and spatial of order ${\text{m}}_{\text{EL}} = 3$ form rotating (but not pulsating) fields; in that specific case, the angular rotation speed of this fields is equal to rotor synchronous rotation speed. Consequently, this field does not induce any EMF in synchronous machine rotor loops in continuous operation mode $({\upomega}_{\text{REV}} =\upomega_{1}/{\text{p)}}$.

• 4. For non-sinusoidal power supply it is possible to avoid the influence of series higher time harmonics on rotor loops using the stator winding scheme with phase number ${\text{m}}_{\text{PH}} > 3$. However, there are some drawbacks in winding with increased phase numbers.