Investigation Methods of Currents Distribution in Irregular Damper Windings

Abstract

In this Chapter, investigation methods of asymmetrical active recurrent circuits (open and closed) developed in Chap. 7, are used for calculation of currents distribution in irregular damper windings of salient pole machines.

Appendices

Appendix 11.1

Values of coefficients A, V, H, S, M, P, Q, F, B, C, L, K, G, R, T, W for determination of constants from systems (11.17) and (11.18).

$$ \begin{array}{*{20}l} {\text{A = a}_{1}^{{\text{N}_{\text{B}} + 1}} ;\text{V} = - \text{a}_{1}^{ - 1} ;\text{H = a}_{1}^{ - 1} \left( {\text{a}_{1} - 1} \right)\text{Z}_{\text{B}} ;} \hfill \\ {\text{S}_{0} = \text{a}_{1}^{{\text{N}_{\text{B}} }} \left( {1 - \text{a}_{1} } \right)\text{Z}_{\text{B}} - 2\text{a}_{1} \text{Z}_{\text{F}} ;\text{M = a}_{1}^{{\text{N}_{\text{P}} }} \left( {1 + \frac{{{\Delta Z}}}{{\text{Z}_{\text{B}} }}} \right);} \hfill \\ {\text{P = a}_{1}^{{\text{N}_{\text{P}} - 2}} - \text{a}_{1}^{{\text{N}_{\text{P}} - 1}} \left( {2 +\upsigma + \frac{{{\Delta Z}}}{{\text{Z}_{\text{B}} }}} \right);} \hfill \\ {\text{Q = a}_{1}^{{\text{N}_{\text{P}} }} \left[ {\text{a}_{1} - \left( {2 +\upsigma + \frac{{{\Delta Z}}}{{\text{Z}_{\text{B}} }}} \right)} \right];\text{F = a}_{1}^{{\text{N}_{\text{P}} - 1}} \left( {1 + \frac{{{\Delta Z}}}{{\text{Z}_{\text{B}} }}} \right).} \hfill \\ \end{array} $$

Coefficients B, C, L, K, G, R, T, W can be obtained respectively from coefficients A, V, H, S₀, M, P, Q, F by replacement of $\text{a}_{1} \,\text{by}\,\text{a}_{2}$.

Appendix 11.2

Investigation of currents in damper winding elements of SDSZ type motor; a comparison with experimental data (Table 11.1).

Table 11.1 Calculated val ∆R = −0.2875 × 10⁻³ ues of resulting currents distribution with frequency f = 6.25 Hz; a comparison with experimental data

Research objective—determine the currents distribution in damper winding elements of SDSZ type motor (2000 kW, 6 kV, 100 rpm, 50 Hz.) with power supply from a frequency converter. Brief description of damper winding construction for this motor is given in para 11.1.1. Investigations were performed at voltage U supplied to a motor from frequency converter in the range of 5 Hz < f < 20 Hz under the condition: $\frac{\text{U}}{\text{f}} = \text{const,where}$ f—frequency. Investigation results are given in Table 11.1 for frequency f = 6.25 Hz; this frequency is selected proceeding from the following assumptions:

• for determining bar impedances $\text{Z}_{\text{B}}$ and bar impedance $\text{Z}_{{\text{N}_{\text{P}} }}$, skin effect with this rather low frequency can be neglected that eliminates possible additional calculation errors in results;

• for low frequency, signal transmission to “analog-digital” converter is connected with smaller distortions.

Investigations were carried out within development of the USSR’s first high-power frequency controlled motor of SDT type [10] for gearless drive (Fig. 11.1) of cement mill with productive capacity of 100 tons of cement per hour (5500 kW, 900 V, 16 rpm, 8 Hz; shaft toque apprx. 350 T m). SDSZ type motor was used as the model. In bench conditions (Fig. 11.2), at investigation of SDSZ type motor, the following problems were solved:

Fig. 11.1

The USSR’s first high-power low-frequency SDT type motor for gearless drive of cement mill with productive capacity of 100 tons of cement per hour (5500 kW, 16 rpm, 8 Hz, shaft toque apprx. 350 tm) at “Elektrosila” Work, Stock Company “Power Machines” St. Petersburg

Fig. 11.2

SDSZ type motor in bench; investigations of electromagnetic processes at operation from frequency converter at LEZ, Stock Company “Ruselprom”

• choice of optimum rotor sensor design (sensor of torque angle Θ in salient-pole machine);

• working set of electric equipment with frequency converter, exciter and rotor sensor; purpose: minimization of current and voltage distortion factors;

• investigations of additional losses from higher time harmonics in resulting field and currents distribution in damper winding construction elements, when operating from frequency converter.

To study the currents distribution in damper winding construction elements, its bars from both end faces at slot exit were equipped with Rogowski sensor and thermocouples. Bars were previously drilled for thermocouples, after installation of thermocouples, holes were thoroughly calked. Signals from sensors arrived at the “analog-digital” converter, and then were processed in real time in Argus system (ICL Company, England). The decipherment of thermocouples data is carried out according to Ross method [12].

Results

Distribution of resulting currents in bars $\underline{\underline{\text{J}}}_{{(\text{N})}}$ for the mode f = 6.25 Hz at $\Delta \text{R =}-0 \text{.2875}\text{.10}^{ - 3} \text{Ohms}$ $\left( {\frac{{\Delta \text{R}}}{{\text{R}_{\text{B}} }} = - 0.75} \right)$ are specified in Table 11.1. Current in middle bar (N = 2) is taken for unity. These values of currents $\underline{\underline{\text{J}}}_{{(\text{N})}}$ compared with experimental $\text{J}_{{(\text{N})\text{EXP}}}$, are also given in Table 11.1. Here is given the difference $\upvarepsilon_{1}$ between $\text{J}_{{(\text{N})}} \,\text{and}\,\text{J}_{{(\text{N}),\text{EXP}}}$.

Appendix 11.3

Values of coefficients A, B, E, F, G, H, K, L for determination of constants $\text{C}^{{(\text{S})}} \,,\text{D}^{{(\text{S})}} \,,\text{C}_{\text{R}}^{{(\text{q})}} \,\text{C}_{\text{L}}^{{(\text{q})}} \,,\text{D}_{\text{R}}^{{\text{(q)}}} \,,\text{D}_{\text{L}}^{{\text{(q)}}}$ from systems (11.27) to (11.32).

$$ \begin{array}{*{20}l} {\text{A = a}_{1}^{{\text{N}_{\text{P}} }} \left( {1 + \frac{{\Delta \text{Z}}}{{\text{Z}_{\text{B}} }}} \right);\text{E} = - \text{a}_{1}^{{\text{N}_{\text{P}} - 1}} \left( {2 +\upsigma + \frac{{\Delta \text{Z}}}{{\text{Z}_{\text{B}} }}} \right);} \hfill \\ {\text{G = a}_{1}^{{\text{N}_{\text{P}} + 1}} - \text{a}_{1}^{{\text{N}_{\text{P}} }} \left( {2 +\upsigma + \frac{{\Delta \text{Z}}}{{\text{Z}_{\text{B}} }}} \right);\text{K = a}_{1}^{{\text{N}_{\text{P}} - 1}} \left( {1 + \frac{{\Delta \text{Z}}}{{\text{Z}_{\text{B}} }}} \right);} \hfill \\ {\text{M = (1 - a}_{2} \text{)Z}_{\text{B}} ;\;\text{Q = }\left( {\text{a}_{1}^{{\text{N}_{\text{B}} }} - \text{a}_{1}^{{\text{N}_{\text{B}} - 1}} } \right)\text{Z}_{\text{B}} - 2\text{a}_{1}^{{\text{N}_{\text{B}} + 1}} \text{Z}_{\text{F}} .} \hfill \\ \end{array} $$

Coefficients B, F, H, L, P, R can be obtained respectively from coefficients A, E, G, K, M, Q replacing $\text{a}_{1} \,\text{by}\,\text{a}_{2}$ and, respectively, $\text{a}_{2} \,\text{by}\,\text{a}_{1}$.

Appendix 11.4

Calculation example: dependence of additional current in the damaged bar of SBGD type generator (6 MW, 1000 rpm, 10.5 kV “Elektrosila” Work, Stock Company “Power Machines” St. Petersburg) on impedance $\Delta \text{Z}_{\text{B}}$.

$$ \begin{aligned} & {\text{Given:}}\,{\text{Z}}_{{\text{B}}} = \left( {1.62 + {\text{j}}1.995} \right) \times 10^{{ - 4}} ,{\text{Z}}_{{\text{R}}} = \left( {1.814 + {\text{j}}2.165} \right) \times 10^{{ - 4}} \\ & {\text{Z}}_{{\text{F}}} = \left( {7.514 + {\text{j}}8.967} \right) \times 10^{{ - 6}} ;\,{\text{N}}_{0} = 8;\,{\text{N}}_{{\text{P}}} = 3;\,{\text{J}}_{3} = 4{\text{e}}^{{ - {\text{j}}25.5}} \\ \end{aligned} $$

Variants for

$$ \Delta \text{Z}_{\text{B}} :\text{A}\,\text{)}\text{.}\,\Delta \text{Z}_{\text{B}} \, = 2\text{Z}_{\text{B}} ;\text{B}\,).\,\Delta \text{Z}_{\text{B}} = 20\text{Z}_{\text{B}} ;\text{C}\,\text{)}\text{.}\,\Delta \text{Z}_{\text{B}} = 80\text{Z}_{\text{B}} ;\text{D}\,\text{)}\text{.}\,\Delta \text{Z}_{\text{B}} = 100\text{Z}_{\text{B}} . $$

To find for $\text{N}_{\text{P}} = 3:\Delta \text{J}_{3} ,\text{J}_{3} ,\frac{{\left| {\underline{\underline{\text{J}}}_{3} } \right|}}{{\text{J}_{3} }}$

The solution is obtained from the system (11.27) to (11.32) and Eq. (11.37) and tabulated in Table 11.2

Table 11.2 Dependence of additional current in damaged bar of damper winding on impedance $\Delta \text{Z}_{\text{B}}$ of SBGD type generator

Note. When designing powerful synchronous motors with severe operating conditions (load change, etc.), the values of overheating of damper winding bars and Field’s factor may be clarified by the method stated in Appendix 1 of Chap. 23. This method provides the calculation of skin effect in the bars of the rotor, taking into account the temperature distribution on its height.

Brief Conclusions

• 1. If in damper winding on each pole there is one bar with impedance differing from the others, calculation of currents distribution is reduced to the solution of two problems: for main and additional currents.

• 2. For this design of damper winding, calculating distribution of additional currents (the second problem) is reduced to the computation of only four constants as part of difference equations. The order of this system does not depend on the number of bars $\text{N}_{0}$ on the pole and is equal to r = 4. In solving the problem by a numerical method the system order is determined by the product $\text{r}^{{\prime }} = 2\text{pN}_{0}$.

• 3. At emergence of damage in damper winding bar on one of poles, calculation of currents distribution in its elements is reduced, as well as in previous design, to the solution of two problems. Constants of calculation expressions for additional currents are calculated from the system of linear equations whose order is determined only by the number of machine poles and does not depend on the number of bars per pole: r ≤ 4p + 2. In solving the problem the system order is determined as well as in previous design by product $\text{r}^{{\prime }} = 2\text{pN}_{0}$. Note that the methods presented in [13–20] do not provide a solution to this problem.

• 4. The method allows one to calculate the distribution of currents in the presence of several bars or ring portions whose impedances differ from the others.

List of Symbols

$\text{D}_{1} ,\,\text{D}_{2} ,\,\text{D}_{3} ,\,\text{D}_{4} ,\,\text{D}_{1}^{{\prime }} ,\,\text{D}_{2}^{{\prime }} ,\,\text{D}_{3}^{{\prime }} ,\,\text{D}_{4}^{{\prime }}$ :

Constants for determination of currents of both poles of opposite polarity

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

EMF amplitude induced by air gap resulting field in rotor loops on pole

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

EMF in loop formed by edge bars of adjacent poles

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

Currents in ring portions (segments) and in damper winding bars at emergence of asymmetry (damage)

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

Currents in ring portions (segments) and in damper winding bars before asymmetry emergence

$\Delta \text{I}_{{(\text{N})}} ,\Delta \text{J}_{{(\text{N})}}$ :

Additional currents in bars and ring portions of asymmetrical damper winding

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

Additional currents of asymmetrical (damaged) bar and ring portions in the loop containing this bar

N:

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

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

Number of damper winding bars on pole

N_p :

Asymmetrical bar number

р:

Number of pole pairs

q:

Pole number with asymmetrical (damaged) bar

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

Expansion period of MMF and mutual fields in air gap to harmonic series

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

Impedances of bar, ring portion between bars on pole, ring portion between bars of adjacent poles

$\Delta \text{Z}_{{\text{N}_{\text{P}} }}$ :

Additional impedance of asymmetrical (damaged) bar

$\Delta {\kern 1pt} \upvarphi$ :

Phase angle between EMF of adjacent bars on pole