Allowance for Iron Saturation in Calculating the Slot Leakage Permeance of the Stator Winding of a Surge Generator
Abstract Generated abstract
The paper addresses the calculation of stator slot leakage permeance in shock generators under severe transient currents, where iron saturation substantially reduces winding reactances and makes conventional unsaturated or current-based methods cumbersome or unreliable. It proposes a simpler calculation based on the leakage flux linkage of the stator winding during sudden short circuit, using the constancy of flux linkage in the initial transient to estimate tooth-zone flux, determine local induction and permeability from the magnetization curve, and compute a sectional saturated slot permeance. An example for the TI-25-2 shock generator shows reduced slot leakage reactance compared with the unsaturated value and results comparable to existing methods, while avoiding iterative recalculation of short-circuit current.
Full Text
ACCOUNTING FOR IRON SATURATION IN CALCULATING THE SLOT LEAKAGE PERMEANCE OF THE STATOR WINDING OF A SHOCK GENERATOR
K. A. KHOR'KOV, V. S. BAKLIN
(Presented by the scientific seminar of the Department of Electrical Machines and by the Department of Electrical Machines of the Research Institute of Automation and Electromechanics)
In calculating the permeance coefficient of slot leakage, in a first approximation an infinitely large magnetic permeability of the steel is assumed. The slot-leakage permeance then becomes a function only of the slot geometry. For example, for open slots with a single-layer winding,
\[ \lambda_n=\frac{h_1}{3b_n}+\frac{h_k}{b_n}, \tag{1} \]
where \(h_1\) is the height of the winding copper,
\(h_k\) is the height of the wedge and of the insulation between the wedge and the copper,
\(b_n\) is the slot width.
However, when calculating transient processes in electrical machines, especially sudden short circuits, when the current in the circuit exceeds the rated current of the machine by tens of times, it becomes impossible to neglect the influence of steel saturation on the parameters of the stator winding. Experimental values of the subtransient and transient reactances of synchronous machines are considerably smaller than their calculated values obtained without accounting for the influence of saturation. Thus, in [1 and 2] dependences are given of
\[ \frac{X''_{dn}}{X''_d} \quad \text{and} \quad \frac{X'_{dn}}{X'_d} \]
on the multiple of the short-circuit current. As follows from the curves, the average saturated value of the subtransient reactance along the longitudinal axis is, for two-pole turbogenerators, \(0.65\) of its unsaturated value; for four-pole turbogenerators, \(0.77\); and for salient-pole synchronous machines, \(0.88\).
In [3], saturated and unsaturated values of the subtransient reactance of shock generators are given: for TI-25-2,
\[ \frac{X''_{dn}}{X''_d}=0.6; \]
for TI-75-2,
\[ \frac{X''_{dn}}{X''_d}=0.67. \]
In [4], experimental dependences of \(X''_{dn}\) as a function of the multiple of the short-circuit current are given for TI-75-2. At a sudden-short-circuit current equal to \(15I_n\), the value of \(X''_{dn}\) is \(0.7\) of the unsaturated value \(X''_d\).
Known methods for calculating parameters with allowance for saturation are either too approximate or are based on the method of successive approximations and require a large amount of computation.
In [5] it is proposed to take into account the effect of saturation on the slot-leakage permeance by increasing the slot width by a certain amount equal to the tooth width divided by the effective magnetic permeability
\[ b_n' = b_n + \frac{b_z}{\mu'} \tag{2} \]
\[ \mu'=\frac{B_{zk}}{\mu_0\left(H_{zk}+A_N I\right)}, \tag{3} \]
where
\(B_{zk}\) and \(H_{zk}\) are the “critical” induction and field intensity on the magnetization curve,
\[ B_{zk}=2.04\cdot 10^{-4}\ \frac{\mathrm{Wb}}{\mathrm{cm}^2},\qquad H_{zk}=300\ \mathrm{A/cm}; \]
\(A_N\) is the rated linear loading;
\(I\) is the expected current multiple.
For shock generators the short-circuit current multiple considerably exceeds 10 times the rated current, and calculations with the above \(B_{zk}\) and \(H_{zk}\) give underestimated values of the slot-leakage permeance. For example, for the shock generator GI-25-2, under a sudden short circuit at zero emf, the linear loading \(A_N I\), taking into account the aperiodic component of the current, is obtained as equal to \(18.0\ \mathrm{kA/cm}\); consequently, \(\mu = 0.87\);
\[ b_n' = b_n + \frac{b_z}{0.87} > t_z, \]
which contradicts the physical meaning and gives clearly overestimated values of \(b_n'\).
In [6] the equivalent slot permeance is found on the assumption that the leakage reactance is proportional to the series connection of the slot permeance and the tooth permeance:
\[ \lambda_n=\frac{\lambda_n\cdot\lambda_z}{\lambda_n+\lambda_z} =\lambda_n\cdot \frac{1}{1+\dfrac{b_z}{b_n}\cdot\dfrac{1}{\mu_e}}, \tag{4} \]
where \(\mu_e\) is the relative magnetic permeability of the tooth steel at the saturation under consideration. The value \(\mu_e\) is determined from the magnetization curve from the known linear loading \(A_N i\).
The difference between this method and the preceding one is only that \(B_{zk}\) is not limited; in principle, the methods are similar.
The method of [7] takes into account the nonuniform distribution of flux over the slot height, but it is even more cumbersome than those considered above and finds no application in practice.
A common shortcoming of the existing methods for calculating saturated values of winding parameters is that all of them are based on knowing the current flowing in the winding during a sudden short circuit. A more or less reliable value of this current can be established only after several recalculations of the electromagnetic design of a shock generator, since the magnitude of the current depends to a substantial degree on the value of the leakage reactance of the stator winding, and the leakage flux and current are related to each other nonlinearly. From what has been said it is clear that it is impossible to construct a simple method for calculating winding parameters, taking into account the influence of iron saturation, on the basis considered.
We propose to base the calculation not on the value of the sudden short-circuit current, but on the value of the leakage flux linkage of the winding.
of the stator. It is known that an increase in the short-circuit current is accompanied by an increase in the leakage fluxes and, consequently, by an increase in the saturation of the steel. The latter leads to a decrease in the slot-leakage permeance, which in turn leads to an even greater increase in the current. At the same time, the product of these nonlinearly varying quantities during a sudden short circuit remains constant in the first moments of time:
\[ L_s \cdot i_k = \Psi_{sk} = \mathrm{const}. \tag{5} \]
The condition of constancy of the flux linkage makes it possible to determine at once the magnitude of the leakage flux of the stator winding and, consequently, to determine immediately the value of the relative magnetic permeability in any section of the tooth.
In the case of a sudden short circuit of the stator winding at the instant when the generator emf passes through its maximum (the phase angle of the emf is equal to \(\frac{\pi}{2}\)), the flux linkage of the stator winding is equal to zero. When the rotor is turned through an angle \(\frac{\pi}{2}\), the excitation flux tends to penetrate into the circuit of the stator winding. However, by virtue of the law of electromagnetic inertia, the short-circuited circuit of the stator winding tends to keep its flux linkage constant. A current flows in the stator winding, creating a flux equal to the excitation flux but oppositely directed. This flux passes along the leakage paths of the stator winding, mainly through the tooth layer. Taking into account the branching of the flux along two parallel paths, we obtain the flux across the stator slots equal to \(\frac{\Phi_0}{2}\).
In the case of a sudden short circuit of the stator winding at the instant when the emf of the winding passes through zero (the phase angle of the emf is \(\gamma_0 = 0\)), the flux linkage of the stator winding is equal to \(\Psi_0\). The current of the sudden short circuit reaches a maximum when the rotor is turned through an angle \(\pi\). In this case the excitation flux tends to penetrate into the circuit of the stator winding
Fig. 1. Distribution of the fluxes \(\Phi_0\) and \(\Phi_{zn}\) during a sudden short circuit at “0” emf for \(\gamma_0 = 0\) (a), \(\gamma_0 = \pi\) (b)
in the direction opposite to the initial one. A current flows in the stator winding, creating a leakage flux equal to twice the value of the excitation flux. The flux across the stator slots is equal to \(\Phi_0\), Fig. 1.
In the general case, for a sudden short circuit at the instant when the phase angle of the voltage is equal to \(\gamma_0\), the flux through the tooth layer across the stator slots is equal to
\[ \Phi_{sz}=\frac{\Phi_0(1+\cos\gamma_0)}{2}. \tag{6} \]
Taking into account the displacement of the leakage flux into the air gap of the machine by the coefficients \(k_v\), \(k_l\), and the attenuation of the leakage flux by the coefficient \(k_z\), we obtain
\[ \Phi_{sz}=\frac{\Phi_0(1+\cos\gamma_0)}{2}\cdot k_v\cdot k_z\cdot k_l . \tag{7} \]
The coefficient of displacement of the flux into the air gap of the machine depends on the magnitude of the machine flux, the geometry of the tooth zone, and the size of the air gap. For shock generators the value of \(k_v\) lies within the limits \(0.9 \div 0.95\).
The leakage-flux attenuation coefficient has the same physical meaning as the attenuation coefficient of the sudden-short-circuit current. For the first half-period after the instant of short circuit, the attenuation coefficient may be taken equal to \(0.95 \div 0.98\), depending on the parameters and design of the shock generator.
If the stator winding of the shock generator is closed on an inductive load, the effect of the change in operating condition on the magnitude of the leakage flux can be taken into account by the load coefficient \(k_n\). Then
\[ \Phi_{sz}=\frac{\Phi_0}{2}(1+\cos\gamma_0)\cdot k_v\cdot k_z\cdot k_n\cdot k_l . \tag{8} \]
The dependence of the load coefficient on the ratio between the parameters of the load and of the generator is presented in Fig. 2. For the sudden-short-circuit condition \(k_n=1\); when the generator operates on a matched load, \(k_n=0.5\). Knowing the magnitude of the flux passing through the tooth layer of the stator, it is not difficult to determine the induction in any section along the height of the tooth. The distribution of induction over the height of the stator tooth is shown in Fig. 3.
The maximum induction over the height of the tooth is the induction in the wedge region. Having calculated the flux \(\Phi_{sz}\) per unit length of the active part of the machine, we obtain:
\[ B_{\mathrm{kl}}=2\Phi_{sz}\frac{1}{h_n+h_k}. \tag{9} \]
The induction in any other section of the tooth at height \(h_x\) is equal to:
\[ B_x=B_{\mathrm{kl}}\frac{h_x}{h_n-h_k}. \tag{10} \]
From the magnetization curve, for the induction values found, it is not difficult to determine the relative magnetic permeability of the iron in the section under consideration:
\[ \mu_x=\frac{B_x}{\mu_0 H_x}. \tag{11} \]
The slot leakage permeance of the stator winding, taking account of iron saturation, is determined as
\[ \lambda_{pn}=\frac{h_k}{b_n+\dfrac{b_z}{\mu_k}} +\frac{1}{3}\sum_i \frac{h_i}{b_n+\dfrac{b_{zi}}{\mu_i}}, \tag{12} \]
where \(h_i\) is the height of the \(i\)-th section of the tooth,
\(b_{zi}\) is the average width of the tooth in the \(i\)-th section,
\(\mu_i\) is the average value of the relative magnetic permeability in the \(i\)-th section.
It is advisable to divide the stator slot by height into three or four sections.
As an example, we give the results of calculating the saturated value of the inductive reactance of slot leakage of the stator winding
Fig. 2. Values of \(k_n\) as a function of the ratio
\[
\frac{L_n}{L_g}
\]
Fig. 3. Distribution of induction over the slot height
of the stator of the shock generator TI-25-2 under a sudden short circuit at the instants \(\gamma_0=\dfrac{\pi}{2}\) and \(\gamma_0=0\), Table 1.
Table 1
| \(\gamma_0\) | \(\Phi_{sz}\) (Wb/m) | Section No. | \(h_i\) (cm) | \(b_{zi}\) (cm) | \(b_p\) (cm) | \(\mu_i\) (rel. units) | \(\lambda_i\) | \(\lambda_{pn}\) | \(X_{pn}\) (ohm) |
|---|---|---|---|---|---|---|---|---|---|
| \(\dfrac{\pi}{2}\) | 0.410 | 1 | 2.60 | 4.50 | 3.2 | 1.42 | 0.408 | 0.94 | 0.0450 |
| \(\dfrac{\pi}{2}\) | 0.410 | 2 | 2.46 | 4.94 | 3.2 | 1.52 | 0.127 | 0.94 | 0.0450 |
| \(\dfrac{\pi}{2}\) | 0.410 | 3 | 2.46 | 5.35 | 3.2 | 2.3 | 0.148 | 0.94 | 0.0450 |
| \(\dfrac{\pi}{2}\) | 0.410 | 4 | 2.46 | 5.80 | 3.2 | 1600 | 0.256 | 0.94 | 0.0450 |
| 0 | 0.762 | 1 | 2.60 | 4.50 | 3.2 | 1.19 | 0.373 | 0.84 | 0.040 |
| 0 | 0.762 | 2 | 2.46 | 4.94 | 3.2 | 1.24 | 0.114 | 0.84 | 0.040 |
| 0 | 0.762 | 3 | 2.46 | 5.35 | 3.2 | 1.47 | 0.120 | 0.84 | 0.040 |
| 0 | 0.762 | 4 | 2.46 | 5.80 | 3.2 | 19.1 | 0.234 | 0.84 | 0.040 |
The slot-leakage permeance without accounting for saturation is \(\lambda_p=1.53\); correspondingly, \(\chi_p=0.0735\ \Omega\).
For comparison, we give the results of calculating \(\chi_{pn}\), \(\chi_{sn}\), and \(\chi_{dn}\) by three methods in Table 2.
In Table 2 the following notation is adopted:
\(\chi_{le}\) — inductive leakage reactance of the end part of the stator winding, taking into account the damping action of nonmagnetic cones;
\(\chi_{kn}\) — inductive leakage reactance along the tooth crowns, taking into account the influence of steel saturation;
\(\chi_{\mathrm{dn}}\) — differential leakage inductive reactance, taking into account the effect of steel saturation;
\(\chi_{\mathrm{sh}}\) — inductive reactance of the stator-winding leakage, taking into account the effect of steel saturation and damping of the leakage fluxes of the end parts;
Table 2
| \(\gamma_0\) | Calculation method | \(X_{\mathrm{gn}}\) (ohm) | \(\dfrac{X_{\mathrm{pn}}}{X_{\mathrm{p}}}\) | \(X_{\mathrm{le}}\) (ohm) | \(X_{\mathrm{kn}}\) (ohm) | \(X_{\mathrm{dn}}\) (ohm) | \(X_{\mathrm{sh}}\) | \(X''_{\mathrm{dn}}\) (ohm) | \(\dfrac{X''_{\mathrm{dn}}}{X''_{d}}\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\dfrac{\pi}{2}\) | [5] | 0,043 | 0,585 | 0,012 | 0,129 | 0,157 | 0,58 | ||
| \(\dfrac{\pi}{2}\) | [6] | 0,050 | 0,680 | 0,066 | 0,014 | 0,008 | 0,138 | 0,166 | 0,615 |
| \(\dfrac{\pi}{2}\) | proposed | 0,045 | 0,613 | 0,013 | 0,132 | 0,160 | 0,59 | ||
| 0 | [5] | 0,028 | 0,386 | 0,008 | 0,110 | 0,138 | 0,51 | ||
| 0 | [6] | 0,043 | 0,590 | 0,063 | 0,012 | 0,008 | 0,130 | 0,158 | 0,59 |
| 0 | proposed | 0,040 | 0,55 | 0,011 | 0,126 | 0,154 | 0,57 |
\(\chi''_{\mathrm{dn}}\) — subtransient inductive reactance of the generator, taking into account the effect of steel saturation and damping of the leakage fluxes of the end parts;
\(\chi''_{d}\) — subtransient inductive reactance of the generator without taking into account the effect of steel saturation and damping of the leakage fluxes of the end parts.
Thus, the proposed method for taking into account the effect of saturation on the permeance of slot leakage is not inferior to existing methods in calculation accuracy, but is considerably simpler, since it does not require a large amount of computation by the method of successive approximations.
References
-
M. P. Kostenko. Electric Machines, special course, GEI, 1949.
-
L. A. Kilgore. “Effects of Saturation ou Machine Reactaces,” El. Engineering, vol. 54, no. 5, 1935.
-
I. K. Fedchenko. High-Voltage Engineering. Publishing House “Vyshcha Shkola,” Kiev, 1969.
-
A. I. Vorontsov, G. M. Khutoretskii. Test generators for apparatus laboratories, Proceedings of VNIIEM, vol. 14, 1963.
-
I. M. Postnikov. Design of Electrical Machines, GITL Ukrainian SSR, Kiev, 1960.
-
V. V. Titov, G. M. Khutoretskii, et al. Turbogenerators: Calculation and Design, Energiya, 1967.
-
R. Richter. Electric Machines, vol. 1, ONTI, 1935; vol. 2, ONTI, 1936.