Investigation Methods of Currents Distribution in Regular Damper Windings and Squirrel Cages

Abstract

Chapter 9 presents methods of investigation of currents in regular damper winding elements of various design (complete, incomplete) caused by influence of the first and higher harmonics (Q ≥ 1, |m| = |n| ≥ 1) of resulting field in air gap. They are based on investigation results for currents in U-shaped recurrent circuits of various structure. Obtained here are regularities of currents distribution in construction elements of regular damper winding in operational modes of salient pole machine.

Appendices

Appendix 9.1: Method of Calculation of Overheats of Short-Circuited Rotor Winding Elements at Start-Up with Account for Change of the Main and Additional Losses in it from Temperature (with Account of Skin Effect)

In calculation practice, when determining admissible short-term overloads of stator winding made of copper elementary conductors, the following relation is usually applied [3, 4, 10]:

$$ \frac{{{\text{d}}\Theta \,({\text{t}})}}{\text{dt}} = \frac{{{\text{j}}_{\text{EQ}}^{2} {\text{K}}_{\text{F}} }}{175}. $$

(A.9.1.1)

Here Θ(t)—winding temperature (overheat); t—time; ${\text{j}}_{\text{EQ}}$—mean square value of current density in winding element during ${\text{T}}_{\text{TR}}$ of transient thermal performance (${\text{A/mm}}^{ 2}$), ${\text{K}}_{\text{F}}$—Field’s factor [3, 4, 7]:

$$ {\text{j}}_{\text{EQ}}^{2} = \frac{1}{{{\text{T}}_{\text{TR}} }}\int\limits_{0}^{{{\text{T}}_{\text{TR}} }} {{\text{j}}^{2} } ({\text{t}})\,{\text{dt}} $$

and ${\text{j}}_{\text{EQ}}$—corresponds to the effective value of current. From this ratio the stator winding temperature is obtained:

$$ \Theta ( {\text{t)}} =\Theta (\text{t} = 0) + {\text{j}}_{\text{EQ}}^{2} {\text{K}}_{\text{F}} \frac{{{\text{T}}_{\text{TR}} }}{175} \cdot $$

(A.9.1.2)

Here Θ(t = 0)—stator winding temperature in the set mode before overload, for example, at no-load or before machine commissioning.

This expression is an approximate solution of thermal balance equation for adiabatic heating [10]:

$$ {\text{Pdt = CGd}}\,\Theta \, ( {\text{t)}} \cdot $$

(A.9.1.3)

Here designated: P—electric losses dissipated in winding; Θ(t)—temperature (overheat); C—specific heat; G—winding weight.

Calculation Equation (A.9.1.2) is obtained from (A.9.1.3) without account of temperature for conductor specific impedance; it is true for copper elementary conductors with resistivity calculated at their average temperature in transient thermal mode, equal to $\Theta \approx 50\,{^\circ }{\text{C}}$.

However, unlike admissible temperatures of stator winding [3, 7] temperature of short-circuited rotor winding bars of A.C. machines (damper winding, squirrel cages) in the operational modes reaches $\Theta < 300\,{^\circ }{\text{C}}$ [3, 4, 7]; in such modes for calculation of bar temperature it is also necessary to consider an increase in conductor resistivity with temperature increase.

At first, let us consider an increase from temperature of D.C. bar losses (ohmic). For small power machines (with bar height in rotor slot at most 15–20 mm) we can approximately consider that ${\text{K}}_{\text{F}}$ factor with temperature increase remains invariable; then from the solution of differential equation (A.9.1.3) we obtain:

$$ \Theta ({\text{t}}) \approx \frac{{[\Theta ({\text{t}} = 0)\uprho^{\text{ * }} + 1]{\text{e}}^{{{\text{tA}}\uprho^{\text{ * }} }} - 1}}{{\uprho^{\text{ * }} }}. $$

(A.9.1.4)

Here ρ*—temperature factor of conductor resistivity, ${\text{A}} = {\text{j}}_{\text{EQ}}^{2} \frac{{{\text{K}}_{\text{F}} }}{\text{B}}$; values B and ρ* are given in Table 9.2.

Table 9.2 Values B and ρ*

Let us consider an increase in not only D.C. losses (ohmic) in bar, but also additional losses from temperature: for high power A.C. machines (with bar height in slot over 25–30 mm) it is necessary to consider additionally a change of ${\text{K}}_{\text{F}}$ factor with temperature increase. Let us note that for each bar shape (rectangular, round, trapezoidal, bottle, etc.) this relation for specified dimensions is considered only by conductor “reduced height” [1, 2].

As a result, for high-power A.C. machines fr ${\text{K}}_{\text{F}}$ om the solution of Equation (A.9.1.3) we obtain [23]:

$$ {\text{T}}_{\text{TR}} \approx \frac{2}{{{\text{A}}_{1}\uprho^{\text{ * }} }}\left[ {\sqrt {1 +\uprho^{\text{ * }}\Theta (\text{t})} - \sqrt {1 +\uprho^{\text{ * }}\Theta ({\text{t}} = 0)} } \right]. $$

(A.9.1.5)

Here,${\text{A}}_{1} = {\text{j}}_{\text{EQ}}^{2} \frac{{{\text{K}}_{\text{F,0}} }}{\text{B}}$. ${\text{K}}_{\text{F,0}}$ —value of bar Field’s factor at resistivity for temperature Θ(t = 0).

Let us note that for motor mode the overheat is calculated at each step of numerical integration of start-up differential equation; thus, dynamic moment of inertia and torque on the shaft of mechanism are considered. The refined method for calculating the overheating at the start, or due to sudden change of the mode of cage rotor cores with the account of the skin effect is set out in the appendix to Chap. 23. It can be used as an element of CAD when calculating of the powerful induction motors (see Chap. 4).

Brief Conclusions

• 1. Processes of currents distribution in active recurrent circuits (open and closed) are identical to those in construction elements of short-circuited rotor windings used in modern practice (damper winding, squirrel cages). Methods of investigation of these circuits developed in previous chapters allow obtaining regularities of currents distribution, important for practice, in these windings. In particular, we obtain the following regularity of currents distribution in construction elements of regular damper winding in asynchronous operating mode of salient pole machine and for operation in nonlinear network. Current contains three components which vary with the number N as follows:

• one of them varies depending on bar or ring portion number under the harmonic law;

• two others—under the aperiodic law, one of which with increase in this number fades in amplitude, the other—decreases.

• The solution obtained for currents distribution in regular damper winding elements is confirmed:

• by coincidence of calculation expressions for currents in elements of complete and incomplete damper winding (in limit case at $Z_{\text{F}} \to \infty$);

• by coincidence of calculation expressions for currents in elements of complete and squirrel cage (in limit case at ${\text{b}}_{\text{F}} = {\text{b}}$ and respectively, $Z_{\text{F}} = Z_{\text{R}}$);

• 2. With sharp load variation in frequency-controlled salient pole motors or industrial motors, the current components vary with damper winding loop number under the aperiodic law, and result in uneven distribution of resulting current in bars: pole middle bars pass current considerably of smaller amplitude, than edge ones; also currents in ring portions (segments) are respectively distributed. Uneven distribution of currents in elements of damper winding causes uneven distribution of their temperature, appearance of additional temperature deformations and corresponding stresses, and sometimes damages (breaks) in bars.

Stator winding field, when operating in non-linear network at T = idem, is created by two “adjacent” harmonics and has two components differing in amplitude; they rotate relative to rotor in opposite direction, but induce in its loops EMF and currents of equal frequency $\upomega_{\text{ROT}}$. A result of influence of these harmonics, the processes of currents distribution and losses in bars and damper winding ring portions can differ from those in asynchronous mode.

Representation of squirrel cage in the form of symmetrical multiphase winding with “reduction” of short-circuited ring element impedance to bar impedance is a special case of its representation in the form of symmetrical active closed recurrent circuit. In case of damages in cage, investigation of currents in it by the method of symmetrical components (as in asymmetrical multiphase system) meets considerable computing difficulties [21] and is unpromising; at the same time, the methods of investigation of asymmetrical active recurrent circuits developed in the monograph allow us to investigate also more difficult rotor windings constructions: irregular damper winding, squirrel cages with damage.

List of symbols

${\text{b}}$ :

Pitch between pole slot axes;

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

Width of pole;

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

Distance between axes of edge slots on adjacent poles;

B(m, Q):

Flux density amplitude of resulting field in air gap;

C:

Specific heat;

D:

Stator boring diameter;

${\text{E}}_{\text{M}}$ :

Rotor loop EMF amplitude;

G:

Winding weigth;

${\text{j}}_{\text{EQ}}$ :

Mean square value of current density in rotor winding element;

J, I:

Currents in bars and ring portions of symmetrical short-circuited rotor winding;

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

Field’s factor;

${\text{K}}_{\text{F,0}}$ :

Value of bar Field’s factor at resistivity for temperature Θ(t = 0);

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

Active length of stator core;

${\text{m, m}}_{\text{EL}}$ :

Spatial harmonic order;

N:

Bar (ring portion) number of damper winding or squirrel cage;

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

Number of bars of damper winding on pole or squirrel cage on the rotor;

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

Asymmetrical bar number;

p:

Number of pole pairs;

P:

Electric losses dissipated in winding;

Q:

Time harmonic order;

t:

Time;

${\text{T, T}}_{\text{EL}}$ :

Serial expansion periods of MMF and of mutual field to harmonic series;

${\text{T}}_{\text{TR}}$ :

Time of transient thermal performance;

x:

Current coordinate along stator boring (in tangential direction);

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

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

${\text{Z}}_{\text{B}} ,{\text{Z}}_{\text{R}} ,{\text{Z}}_{\text{F}}$ :

Impedances of bar, ring portion between pole bars, of ring portion between edge bars of adjacent poles;

${\text{Z}}_{{{\text{N}}_{\text{p}} }}$ :

Impedance of asymmetrical bar;

∆Z:

Additional impedance of asymmetrical bar;

α:

Pole arc $(\upalpha = {\text{b}}_{\text{p}} /\uptau) ;$

$\upsigma ,\upsigma^{\prime }$ :

Relations determined by impedances of symmetrical recurrent circuit elements;

τ:

Pole pitch;

$\Phi _{1,0} , \ldots ,\Phi _{{{\text{N}} + 2,{\text{N}} + 1}}$ :

Mutual resulting fluxes linked accordingly with loop (1, 0) and loop (N + 2, N + 1);

φ_N :

Phase angle of loop EMF with number N;

Θ(t):

Winding temperature (temperature rise);

ρ*:

Temperature factor of conductor resistivity;