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The basic voltage and torque equations for squirrel-cage induction motors allowing for all MMF and permeance harmonics

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Starting from a mathematical model for the harmonic fluxes generated by a single coil in a doubly-slotted air-gap, voltage and torque equations are derived for a squirrel-cage induction motor taking cognizance of all MMF and permeance harmonics. The analysis is in fact an extension of the space vector component machine theory allowing only for MMF harmonics. It is shown that permeance effects do not introduce new stator or rotor frequencies, but only additional mutual inductances. The iron is considered to be infinitely permeable.

Übersicht

Ausgehend von einem mathematischen Modell der Feldharmonischen, erregt durch eine einzige Spule in einem doppelseitig genuteten Luftspalt, wird ein Satz von Spannungs- und Drehmomentgleichungen für einen Asynchronmotor abgeleitet, der alle Wicklungs- und Nutungsoberwellen berücksichtigt. Die Ableitung ist jedoch eine Erweiterung der Raumzeigertheorie für Maschinen mit glattem Luftspalt. Es wird gezeigt, daß die Nutungseffekte keine neuen Frequenzen im Ständer oder Läufer erregen, sondern nur zusätzliche Koeffizienten der Gegeninduktivität hervorbringen. Die Permeabilität des Eisens wird als unendlich hoch angenommen.

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Abbreviations

B n/k :

k-th rotational harmonic ofn-th flux density harmonic

[C]:

matrixconnecting coils to pole-phase groups

[i], [u]:

matrices containing pole-phase group currents, voltages

[i s ], [u s ]:

matrices containing the space vector component currents, voltages derived from the polephase group variables. When derived from coil variable, subscripts′ is used.

k :

indicatesk-th rotational harmonic

k ′dx :

complexx-th order distribution factor (stator only)=(ɛ 0s′ +ɛ xs′ +ɛ 2xs′ +...+ɛ (q−1)xs′ )/q

k pn :

n-th order pitch factor=sin (nθ s/2)

K, y :

any integers, including zero

n :

indicate harmonic order

N :

number of pole-phase groups

N s′ :

number of stator slots

p :

=d/dt, the time differential operator

P :

pole-pair number

q :

slots per pole per phase (stator)

Q :

turns per coil

r, s :

used as superscripts indicate rotor, stator

s′ :

indicates that stator coil currents andN s′ are involved

sr, rs :

superscripts indicating stator/rotor, rotor/stator mutual flux densities, inductances and reactances.

[S] and [S′]:

space vector transformation matrix and its pseudo-inverse

t :

time

T :

denotes transpose

x, y, z :

indices for matrix multiplication

ξ:

axial iron length

ϱ:

air-gap radius

δ:

Kronecker delta

0:

mechanical angular position in air-gap

θ r :

rotor position relative to stator=ω γ t

θ s :

coil pitch angle

ω:

radian frequency of supply voltage

ω r :

radian speed of rotor

ε:

=ej2θ/N and\(\varepsilon _{s'} = e^{j2\pi /N^{s'} } \)

\(\mathop \Sigma \limits_{n, k} \) :

short notation for\(\mathop {\mathop \Sigma \limits^\infty }\limits_{n = - \infty } \cdot \mathop {\mathop \Sigma \limits^2 }\limits_{k = - 2} \)

*:

denotes the conjugate

References

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, University of Stellenbosch, 7600, Stellenbosch, Republic of South Africa

    F. S. van der Merwe

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  1. F. S. van der Merwe
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van der Merwe, F.S. The basic voltage and torque equations for squirrel-cage induction motors allowing for all MMF and permeance harmonics. Archiv f. Elektrotechnik 64, 251–261 (1982). https://doi.org/10.1007/BF01574755

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  • DOI: https://doi.org/10.1007/BF01574755

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