Armature Reaction Reactances of Alternating-Pulsating Machines
Abstract Generated abstract
This paper derives expressions for the armature reaction inductive reactances of alternating pulsating machines, whose magnetic systems combine permanent magnet poles, field winding poles, and axial magnetic paths. Using an equivalent magnetic circuit and superposition for a linearized system, the authors formulate the direct axis reactance with allowance for unequal pole arcs, saturation coefficients, permanent magnet reluctance, and axial magnetic circuit reluctance, and adapt an empirical relation for magnetoelectric machines to estimate the transverse axis reactance. The calculations indicate that the direct axis reactance can decrease substantially as magnet and axial circuit reluctances increase, while pole arc inequality has comparatively little effect, and that the transverse axis reactance depends more strongly on pole shoe presence and pole arc ratio.
Full Text
Inductive Reactances of Armature Reaction in Alternating-Pulsating Machines
V. B. Gomzyakov, Yu. N. Kroneberg, A. S. Zhivinov
Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering
The distinctive feature of machines of the alternating-pulsating type [1] consists in the fact that poles of one polarity are excited by permanent magnets, while poles of the opposite polarity are excited predominantly by the field winding, and the difference of the fluxes of poles of opposite polarity is closed through axial magnetic cores. Usually, in order to increase the depth of regulation, the poles excited by permanent magnets are made with a smaller pole arc. For these reasons the armature-reaction fluxes under poles of unlike polarity prove to be different, while the armature-reaction reactances \(x_{ad}\), \(x_{aq}\) are determined by their combined action.
Although, because of the complexity of the direct-axis circuit, the use of \(x_{ad}\) in machines of the alternating-pulsating type for calculating operating characteristics is inexpedient [2, 3], knowledge of this parameter is sometimes nevertheless necessary (operation together with regulators, analyses of the behavior of machines in systems with several sources and loads of different character, etc.). In determining \(x_{ad}\), the specifics of the structure of the machine’s direct magnetic circuit must be taken into account.
Direct Axis
The inductive reactance of the armature reaction along the direct axis, as is known, is determined by the expression [4]
\[ x_{ad}=2mf\frac{w_{\mathrm{э}}^{2}}{p}\frac{\Phi_{\Sigma ad1}}{F_{ad1}}, \tag{1} \]
where \(f\) is the frequency; \(p, m, w_{\mathrm{э}}\) are the numbers of pole pairs, phases, and effective turns; \(\Phi_{\Sigma ad1}\) is the resultant flux of the first harmonic from the magnetomotive force (m.m.f.) of the armature reaction with amplitude \(F_{ad1}\). To determine \(\Phi_{\Sigma ad1}\), let us use the equivalent circuit of the magnetic circuit (Fig. 1), whose reluctances are expressed in relative units with the base reluctance of the air gap
\[ R_{\delta}=\frac{1}{\mu_{0}}\frac{\delta k_{\delta}}{\tau l}, \tag{2} \]
where \(\tau, l\) are the pole pitch and armature length;
\(\delta, k_{\delta}\) are the air gap and Carter coefficient.
Fig. 1. Equivalent circuit of the magnetic circuit along the longitudinal axis and the components of the inductions in the air gap: 1 — m.m.f. of armature reaction along the longitudinal axis; 2 — armature-reaction field; 3 — excitation field; \(R_y\) — reluctance of the permanent magnet, taking account of its demagnetization; \(R_j\) — reluctance of the axial magnetic conductor
For a linear magnetic circuit, its final state may be regarded as the result of superposition of the action of two sources: the m.m.f. of armature reaction, sinusoidally distributed over the stator bore, and the m.m.f. from the drops of magnetic potential across the reluctances \(R_y\) and \(R_j\), the latter acting analogously to the m.m.f. of the excitation winding, creating the fluxes of the wide and narrow poles \((\Phi_{\mathrm{v\,sh},y})\) and air-gap fields with inductions along the pole axes
\[ B_{\mathrm{v\,sh},y}=\frac{\Phi_{\mathrm{v\,sh},y}}{\alpha_{i\,\mathrm{sh},y}\tau l}, \tag{3} \]
where \(\alpha_{i\,\mathrm{sh},y}\) are the coefficients of the calculated pole arc [5].
For the inductions \((B_{ad\,\mathrm{sh},y})\) and the fluxes of the first harmonics \((\Phi_{1ad\,\mathrm{sh},y})\) of armature reaction, using the coefficients of the air-gap field [6], we obtain
\[ B_{ad\,\mathrm{sh},y}=F_{ad1}\frac{\mu_0}{\delta k_\delta k_{\mu\,\mathrm{sh},y}}, \tag{4} \]
\[ \Phi_{1ad\,\mathrm{sh},y}=\frac{2}{\pi}\tau l B_{ad\,\mathrm{sh},y} k_{d\,\mathrm{sh},y}. \tag{5} \]
Taking into account that the ratio of the first harmonics to the total armature-reaction fluxes for alternating-pulsating machines has the form
\[ \frac{\Phi_{1ad\,\mathrm{sh},y}}{\Phi_{ad\,\mathrm{sh},y}} = \frac{2}{\pi}\frac{k_{d\,\mathrm{sh},y}}{k_{f\,\mathrm{sh},y}}, \tag{6} \]
we find
\[ \Phi_{ad\,\mathrm{sh},y} = \frac{F_{ad1}}{R_\delta} \frac{k_{f\,\mathrm{sh},y}}{k_{\mu\,\mathrm{sh},y}}, \tag{7} \]
where \(k_{f\mathrm{sh},y}\), \(k_{d\mathrm{sh},y}\) are the form factors of the excitation and armature-reaction fields; \(k_{\mu\mathrm{sh},y}\) are the saturation coefficients of the stator steel.
For the independent loops of the equivalent circuit one can write equations for determining the unknown excitation fluxes
\[ R_\delta k_{\mu\mathrm{sh}} \left( \frac{F_{ad1}}{R_\delta k_{\mu\mathrm{sh}}} - \frac{\Phi_{\mathrm{vsh}}}{a_{i\mathrm{sh}}} \right) + \left(\Phi_{ad\mathrm{sh}}-\Psi_{ady}-\Phi_{\mathrm{vsh}}+\Phi_{\mathrm{v}y}\right)R_jR_\delta = F_{ad1}, \tag{8} \]
\[ R_\delta k_{\mu y} \left( \frac{F_{ad1}}{R_\delta k_{\mu y}} - \frac{\Phi_{\mathrm{v}y}}{a_{iy}} \right) + \left(\Phi_{ady}-\Phi_{\mathrm{v}y}\right)R_yR_\delta - \left(\Phi_{ad\mathrm{sh}}-\Phi_{\mathrm{vsh}}-\Phi_{ady}+ \Phi_{\mathrm{v}y}\right)R_jR_\delta = F_{ad1}, \tag{9} \]
solving which, we find
\[ \Phi_{\mathrm{vsh}} = \frac{a_{i\mathrm{sh}}}{M} \frac{F_{ad1}}{R_\delta} R_j \left[ \frac{k_{f\mathrm{sh}}}{k_{\mu\mathrm{sh}}} \left(k_{\mu y}+a_{iy}R_y\right) - k_{fy} \right], \tag{10} \]
\[ \Phi_{\mathrm{v}y} = \frac{a_{iy}}{M} \frac{F_{ad1}}{R_\delta} \left\{ \frac{k_{fy}}{k_{\mu y}}k_{\mu\mathrm{sh}}R_y + R_j \left[ \frac{k_{fy}}{k_{\mu y}} \left(k_{\mu\mathrm{sh}}+a_{i\mathrm{sh}}R_y\right) - k_{f\mathrm{sh}} \right] \right\}, \tag{11} \]
where
\[ M = k_{\mu\mathrm{sh}}\left(k_{\mu y}+a_{iy}R_y\right) + R_j\left[ a_{iy}k_{\mu\mathrm{sh}} + a_{i\mathrm{sh}}\left(k_{\mu y}+a_{iy}R_y\right) \right], \tag{12} \]
and the coefficients \(a_{i\mathrm{sh},y}\) entering into (10)--(12) must be determined with allowance for the ratio of the inductions \((\beta)\) under the wide and narrow poles [5]
\[ \beta = \frac{a_{iy}}{a_{i\mathrm{sh}}} \frac{\Phi_{\mathrm{vsh}}}{\Phi_{\mathrm{v}y}} . \tag{13} \]
With known \(\Phi_{\mathrm{v}\mathrm{sh},y}\), the first harmonics of the resultant fluxes of the wide and narrow poles can be found:
\[ \Phi_{1\mathrm{sh},y} = \frac{2}{\pi} \left( \frac{F_{ad1}}{R_\delta} \frac{k_{d\mathrm{sh},y}}{k_{\mu\mathrm{sh},y}} - \Phi_{\mathrm{vsh},y} \frac{k_{f\mathrm{sh},y}}{a_{i\mathrm{sh},y}} \right), \tag{14} \]
and, substituting their sum into (1), we obtain
\[ x_{ad} = 2mf \frac{\mu_0 Dl}{\delta k_\delta} \frac{w_3^2}{p^2} \frac{C}{M}, \tag{15} \]
where
\[ C = M \left( \frac{k_{d\mathrm{sh}}}{k_{\mu\mathrm{sh}}} + \frac{k_{dy}}{k_{\mu y}} \right) - R_y \left[ k_{fy}^{\,2} \frac{k_{\mu\mathrm{sh}}}{k_{\mu y}} + R_j \left( a_{iy}\frac{k_{f\mathrm{sh}}^{\,2}}{k_{\mu\mathrm{sh}}} + a_{i\mathrm{sh}}\frac{k_{fy}^{\,2}}{k_{\mu y}} \right) \right] - R_j \frac{k_{\mu\mathrm{sh}}}{k_{\mu y}} \left( k_{fy} - k_{f\mathrm{sh}}\frac{k_{\mu y}}{k_{\mu\mathrm{sh}}} \right)^2 . \tag{16} \]
Transverse Axis
The first harmonic of the transverse armature-reaction flux can likewise be obtained by summing the first harmonics of the fluxes of the narrow and wide poles; moreover, in the absence of pole shoes, the armature-reaction fluxes under the narrow poles (from permanent magnets) are substantially reduced, which leads to a decrease in \(x_{aq}\). To account for this fact, we shall use the empirical formula for \(x_{aq}\) of magnetoelectric machines [7].
\[ x_{aq\mathrm{M}}=x_{aq\mathrm{э}}\frac{1}{1+\dfrac{b_{\mathrm{M}}}{2\delta\rho}}, \tag{17} \]
where \(b_{\mathrm{M}}, \rho\) are the magnet width and the recoil permeability. Then, for the inductive reactance along the transverse axis of an alternating-pulsating machine, we have
\[ x_{\mathrm{э}q}=2mf\,\frac{\mu_0 Dl}{k_c\delta}\,\frac{w_{\mathrm{э}}^2}{l^2} \left( \frac{k_{q\mathrm{ш}}}{k_{\mu\mathrm{ш}}} + \frac{k_{qy}}{k_{\mu y}}\, \frac{1}{1+\dfrac{b_{\mathrm{M}}}{2\delta\rho}} \right), \tag{18} \]
whence, for a large recoil permeability \((\rho\sim\infty)\), we obtain the reactance \(x_{aq}\) for a machine with pole shoes.
Calculations carried out by (15) show that the inductive reactance of armature reaction along the direct axis decreases markedly as the resistances of the magnet and of the axial magnetic circuit increase, falling, for large values of the latter, to \(20\text{–}30\%\) of the reactance of the base machine with electromagnetic excitation and a pole arc \(\alpha=0.5(\alpha_y+\alpha_{\mathrm{ш}})\), whereas inequality of the pole arcs \((\alpha_y/\alpha_{\mathrm{ш}}=0.5\div1)\) has little effect on the value of \(x_{ad}\).
The inductive reactance of armature reaction along the transverse axis of a machine with pole pieces increases as \(\alpha_y/\alpha_{\mathrm{ш}}\) decreases, to approximately \(1.1\) times the reactance of the base machine; in the absence of shoes, \(x_{aq}\) grows more intensively (\(\sim 0.6\,x_{aq\,\mathrm{б}}\) for \(\alpha_y/\alpha_{\mathrm{ш}}=1\) and \(\sim 0.9\,x_{aq\,\mathrm{б}}\) for \(\alpha_y/\alpha_{\mathrm{ш}}=0.5\)).
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