MMF of Damper Winding, Squirrel Cage (at Asymmetry in Them or at Its Absence) and Excitation Winding. Representation of MMF in the Form of Harmonic Series in Complex Plane

Abstract

This chapter deals with investigation methods of MMF harmonics of spatial orders n ($\left| {\text{m}} \right|{ = }\left| {\text{n}} \right| \ge 1$) excited by the currents with frequency $\upomega_{\text{ROT}}$ in rotor loops, included in short-circuited windings: damper winding of synchronous machines and squirrel cage of induction machines (also in asymmetrical).

Appendices

Appendix 12.1: Accounting the Finite Width of Rotor Slots in the Calculation of Damper Winding MMF (Regular and Irregular) or Squirrel Cage (Symmetrical and Asymmetrical)

Figure 12.1 represents an example of MMF distribution in rotor slots. Numbers of slots with currents ${\text{J}}_{1} ,{\text{J}}_{2} ,{\text{J}}_{3} , \ldots ,{\text{J}}_{\text{N}} , \ldots$ are designated respectively as 1;2;3; …; N. Generally, currents are not equal in amplitude and are shifted in time by some phase angles not equal to each other ($\left. {{\upvarphi }_{1} \ne {\upvarphi }_{2} \ne {\upvarphi }_{3} \ne \cdots \ne {\upvarphi }_{6} } \right).$ Therefore, MMF distribution diagram in Fig. 12.1 has only qualitative character; actually, currents in slots are complex, but not real values, therefore, to represent MMF distribution diagram reflecting its quantitative variation along the boring the following is required:

Fig. 12.1

MMF distribution diagram with account of finite width of slots

• to consider the current phase angle in the bar of each ${\text{N}}_{0}$ slot;

• to represent the distribution diagram separately for its real and imaginary part of this current.

However, it is not required for illustrating a method of accounting the finite slot width because we apply a symbolic method for representing currents.

Let us use the general method stated in para 12.3; Let us find the complex amplitude C(−n), using the ratio (12.4). It is convenient to superimpose the origin of coordinates with the side of the first slot.

One proceeds to the mathematical statement of this general problem: We represent MMF of rotor short-circuited winding with account of the finite slot width in the form of step function, similar to Table 12.1. Then, using the ratios (12.4), it is easy to represent it in the form of several rotating waves. For calculation of separate steps we will determine, as previously, the concept of slot current estimated density: ${\text{S}}_{\text{N}} = \frac{{{\text{J}}_{\text{N}} }}{{{\text{b}}_{\text{SLT}} {\text{h}}}};\,{\text{here}}\,{\text{J}}_{\text{N}} ,{\text{b}}_{\text{SLT}} ,{\text{h}}$ respectively, current amplitude of N^th slot, its width and height. We will designate the slot pitch as ${\text{t}}_{\text{SLT}}$. By means of this calculation current density, it is easy to calculate the current function of separate steps.

Table 12.4 represents all necessary parameters for MMF investigation, including its harmonic analysis.

Table 12.4 MMF step function of short-circuited rotor winding with account of finite slot width (Fig. 12.1)

Brief Conclusions

• 1. Investigation methods of damper winding, squirrel cages and excitation winding MMFs are based on representation of periodic (with ${\text{T}}\,{\text{period}}\, =\uppi{\text{D}}$) distribution curve of current load in the form of harmonic series in complex plane. Thus, it is convenient to present currents in these windings in the form of time dependent complexes.

• 2. As a result of such representation, the expressions for rotor MMF complex amplitudes acquire a simple physical sense: they correspond to two components of rotor field in the air gap, which differ in amplitude and rotate relative to the rotor in opposite directions. It allows one to develop a general method of MMF investigation for the whole class of short-circuited rotor windings of A.C. machines without restrictions in their construction (for example, for irregular and regular damper winding, asymmetrical squirrel cages).

• 3. By means of this general method, calculation expressions for MMF complex amplitudes are obtained:

  • irregular and regular damper windings;

  • asymmetrical squirrel cage (general problem);

  • asymmetrical squirrel cage with one damaged bar;

  • asymmetrical squirrel cage with three adjacent damaged bars;

  • asymmetrical squirrel cage with three damaged bars; two bars nearby, the third—next but one.

  • asymmetrical squirrel cage with three damaged bars; three bars, not adjacent.

  • squirrel cage without damages;

  • excitation winding of salient pole machine;

  • electromagnetic screen on pole of high-power low-speed frequency controlled motor.

Analysis of obtained calculation expressions validated the results.

List of symbols

b:

Pitch between axes of adjacent bars of short-circuited rotor winding;

b(x, t):

Flux density instantaneous values of resulting field in air gap;

${\text{b}}_{\text{SLT}} ,{\text{h}}$ :

Width and height of rectangular slot;

${\text{b}}_{\text{F}}$ :

Distance between axes of edge bars on adjacent poles $\left( {\frac{{{\text{b}}_{\text{F}} }}{\text{b}} > 1} \right)$;

${\text{b}}_{\text{OP}} ,{\text{t}}_{\text{SLT}}$ :

Opening width in slot and slot pitch;

${\text{C}}_{1} ,{\text{C}}_{2} ,{\text{C}}_{3} ,{\text{C}}_{4} , \ldots$ :

Constants for calculation of additional currents caused by asymmetry (damage) of squirrel cage;

C(−n), C(n):

MMF harmonic amplitudes of n order in expanding step current function ${\text{F}}_{\text{ROT}} ({\text{x}})$;

D:

Stator boring diameter;

${\text{F}}_{\text{ST}} ({\text{m}},{\text{Q}}),{\text{F}}_{\text{ST}} ( - {\text{m}},{\text{Q}})$ :

MMF harmonic amplitudes of order m corresponding to currents in stator winding of time order ${\text{Q}}$;

${\text{F}}_{\text{ROT}} ({\text{n}},\upomega_{\text{ROT}} ),\,{\text{F}}_{\text{ROT}} ( - {\text{n}},\upomega_{\text{ROT}} )$ :

Harmonic amplitudes of spatial order n of direct and additional fields of MMF corresponding to rotor currents with frequency $\upomega_{\text{ROT}}$;

$\underline{\underline{\text{F}}}_{{{\text{RES}},{\text{CAG}}}} ({\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{CAG}} ({\text{n}},\upomega_{\text{ROT}} ),\Delta {\text{F}}_{\text{CAG}} ({\text{n}},\upomega_{\text{ROT}} )$ :

Harmonic amplitude of direct field of order n of MMF of asymmetrical (with damage) squirrel cage, corresponding to resulting current in bars, to the main current of symmetrical cage and additional current, caused by asymmetry (damage) of cage;

$\underline{\underline{\text{F}}}_{{{\text{RES}},{\text{CAG}}}} ( - {\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{CAG}} ( - {\text{n}},\upomega_{\text{ROT}} ),\Delta {\text{F}}_{\text{CAG}} ( - {\text{n}},\upomega_{\text{ROT}} )$ :

The same for harmonics of additional field;

${\text{F}}_{\text{ROT}} ({\text{x}})$ :

Current function of short-circuited rotor winding, dependence of MMF from on spatial coordinate x along the boring periphery (“MMF step heights under teeth”);

${\text{f}}_{\text{ROT}} ({\text{x}},{\text{t}})$ :

Instantaneous values of step function rotor currents

${\text{F}}_{\text{D}} ({\text{x}})$ :

Similar to ${\text{F}}_{\text{ROT}} ({\text{x}})$, but for damper winding MMF;

${\text{F}}_{\text{D}} ({\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{D}} ( - {\text{n}},\upomega_{\text{ROT}} )$ :

Similar ${\text{F}}_{\text{ROT}} ({\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{ROT}} ( - {\text{n}},\upomega_{\text{ROT}} )$, but for damper winding MMF;

${\text{F}}_{\text{EX}} ({\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{EX}} ( - {\text{n}},\upomega_{\text{ROT}} )$ :

Similar ${\text{F}}_{\text{ROT}} ({\text{n}},\upomega_{\text{ROT}} ),{\text{F}}_{\text{ROT}} ( - {\text{n}},\upomega_{\text{ROT}} )$, but for excitation winding MMF;

${\text{F}}_{\text{SCR}} ({\text{x}})$ :

Similar to ${\text{F}}_{\text{ROT}} ({\text{x}})$, but for screen on pole;

${\text{I}}_{\text{N}} ,{\text{J}}_{\text{N}}$ :

Currents in ring portions (segments) and damper winding bars;

${\text{L}}_{\text{COR}}$ :

Active length of stator core;

N:

Bar number of short-circuited rotor winding;

n:

Number of spatial harmonic of rotor MMF;

${\text{N}}_{0}$ :

Number of bars of damper winding per pole (squirrel cage on rotor);

р:

Number of machine pole pairs;

${\text{q}}_{\text{B}}$ :

Section of damper winding bar;

${\text{q}}_{\text{R}}$ :

Section of ring (segment) between poles;

${\text{S}}_{\text{N}}$ :

Calculation current density in bar with number N;

t:

Time

Т:

Expansion period of MMF and mutual induction flux;

${\text{t}}_{\text{S}}$ :

Slot pitch;

${\text{V}}_{{{\text{ROT}},1}} ,{\text{V}}_{{{\text{ROT}},2}}$ :

Linear speeds of direct and additional harmonic field of order n;

${\text{W}}_{\text{PH}} ,{\text{K}}_{\text{W}} \left( {{\text{m}}_{\text{EL}} } \right)$ :

Number of turns in stator winding phase and its winding factor for harmonic of order ${\text{m}}_{\text{EL}}$;

x:

Coordinate along stator boring (in tangential direction);

${\text{x}}_{1,0} ;{\text{x}}_{2,1} ; \ldots ;{\text{x}}_{{{\text{N}},{\text{N}} - 1}}$ :

Distance between axes of adjacent bars;

${\text{x}}_{0}$ :

Distance from the origin of coordinates to axis of the rotor first slot (bar);

$\updelta$ :

Air gap;

${\Delta \varphi }$ :

Phase angle of EMF between two adjacent loops of short-circuited rotor winding;

$\uptau$ :

Pole pitch;

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

Frequency of EMF and current in rotor loops;

${\Delta \varphi }$ :

Phase angle of EMF between two adjacent loops of short-circuited rotor winding.