Calculation of the Characteristics of an Asynchronous Generator with Capacitor Excitation
Abstract Generated abstract
This paper presents a calculation method for the external and regulating characteristics of a self-excited induction generator with capacitor excitation and frequency stabilization by varying the active resistance of the rotor circuit. Using a T-shaped equivalent circuit with constant parameters and neglected iron losses, the method derives relationships among circuit conductances and susceptances, magnetizing reactance, load parameters, slip, and added rotor resistance. The procedure permits calculation of load voltage, currents, powers, and required rotor resistance for varying load, excitation capacitance, compounding capacitance, power factor, and drive speed. A worked comparison for a 1200 VA, 400 Hz generator shows good agreement between calculated and experimental characteristics, with a reported maximum error not exceeding 10 percent.
Full Text
Calculation of the Characteristics of an Induction Generator with Capacitor Excitation
V. F. Sergeev
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
An induction generator with self-excitation from a bank of capacitors connected in parallel with the load may be used as a source of alternating current of stable frequency when the drive speed varies. Stabilization of the frequency at the generator output is accomplished by regulating the slip through changing some parameter of the primary or secondary circuits.
Despite the fact that the use of such systems appears promising with respect to reducing the weight and dimensions of power sources used for similar purposes at the present time, they have not yet become at all widespread owing to insufficient development of their theory and of engineering methods of calculation and design.
The best-known method of slip regulation is the introduction of additional active resistance into the secondary circuit. In this case the generator slip is directly proportional to the sum of the additional resistance and the active resistance of the rotor winding.
In the present work a method is proposed for calculating the external and regulating characteristics of an induction generator with frequency stabilization by varying the active resistance of the rotor.
The basis is the method proposed in [1], which, in contrast to that considered here, assumes frequency stabilization by a corresponding change in the drive speed at constant rotor resistance.
The calculation is carried out on the basis of a T-shaped equivalent circuit (Fig. 1). The circuit parameters are assumed to be constant and independent of the saturation of the magnetic circuits; iron losses are neglected.
Fig. 1. T-shaped equivalent circuit of an induction generator with capacitor excitation
The equations for the equivalent circuit have the following form:
\[ \left. \begin{aligned} \dot{E}y_1 &= \dot{I}_1,\\ \dot{E}y'_2 &= \dot{I}'_2,\\ \dot{E}y_m &= \dot{I}_m,\\ \dot{I}_1+\dot{I}'_2 &= \dot{I}_m \end{aligned} \right\} \tag{1} \]
For the case of an active-inductive load in the presence of compounding capacitances \((x_{\mathrm{ck}})\), the conductivities of the machine circuits may be represented by the following expressions.
The complex conductivity of the primary circuit
\[ y_1=g_1-jb_1, \]
where
\[ g_1= \frac{ r_1(z_{\mathrm{H}}^2+x_c^2)+z_{\mathrm{H}}x_c(x_c\cos\varphi-2r_1\sin\varphi) }{ \{z_{\mathrm{H}}[(x_c+x_{\mathrm{ck}}-x_1)\sin\varphi+r_1\cos\varphi]+x_c(x_1-x_{\mathrm{ck}})\}^2+\ \to } \tag{2} \]
\[ \to\ +\{z_{\mathrm{H}}[r_1\sin\varphi-(x_c+x_{\mathrm{ck}}-x_1)\cos\varphi]-r_1x_c\}^2 \]
\[ b_1= \frac{ z_{\mathrm{H}}^2(x_c+x_{\mathrm{ck}}-x_1)+x_c(x_1-x_{\mathrm{ck}})(x_c-2z_{\mathrm{H}}\sin\varphi) }{ \{z_{\mathrm{H}}[(x_c+x_{\mathrm{ck}}-x_1)\sin\varphi+r_1\cos\varphi]+x_c(x_1-x_{\mathrm{ck}})\}^2+\ \to } \]
\[ \to\ +\{z_{\mathrm{H}}[r_1\sin\varphi-(x_c+x_{\mathrm{ck}}-x_1)\cos\varphi]-r_1x_c\}^2 . \tag{3} \]
The complex conductivity of the secondary circuit, including the additional active resistance,
\[ y'_2=g'_2-jb'_2, \]
where
\[ g'_2=\frac{R'_2/s}{\left(\dfrac{R'_2}{S}\right)^2+(x'_2)^2} \tag{4} \]
\[ b'_2=\frac{x'_2}{\left(\dfrac{R'_2}{s}\right)^2+(x_2)^2} \tag{5} \]
\[ R'_2=r'_2+r'_g . \tag{6} \]
The complex conductivity of the magnetizing circuit
\[ y_m=-jb_m, \tag{7} \]
where
\[ b_m=1/x_m . \]
From the equivalent circuit one can obtain the following equalities [1, 2]:
\[ g_1=-g'_2, \tag{8} \]
\[ b_m=-(b_1+b'_2). \tag{9} \]
Having the magnetization characteristic of the machine, one can construct the dependence
\[ E=f(b_m). \]
This dependence and the relations given above make it possible to determine the output parameters of the generator for specified values of the load parameters, and also to construct the external and regulation characteristics of the generator at constant frequency in the primary circuit and variable drive speed.
It is characteristic of generator operation that the rotational speed of its rotor is not a function of the load, but is varied forcibly in accordance with the rotational speed of the drive. At the same time, we have the known relation [2]
\[ \omega=\frac{p\omega_{\mathrm{vr}}}{1-s}, \tag{10} \]
where \(p\omega_{\mathrm{vr}}\) is the angular speed of rotation of the rotor,
\(\omega\) is the angular frequency of the stator, which establishes the nature of the dependence of the stator angular frequency on slip.
If \(\omega\) is stabilized by varying the active resistance of the rotor, then, with constant parameters of the external circuit and stator frequency \(\omega=\mathrm{const}\), the output voltage of the generator remains practically constant over the entire range of variation of the drive speed. This is explained by the unchanged value of the conductivity of the secondary circuit due to the linear dependence \(R'_2=f(s)\). It follows from this that the form of the external characteristics under the given conditions does not depend on the rotor speed, which makes it possible to calculate them for \(p\omega_{\mathrm{vr}}=\mathrm{const}\), subsequently taking into account the change in speed when calculating the regulating characteristics.
The calculation of the external characteristics under these conditions consists of the following.
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For the minimum load resistance \(z_{\mathrm{n}}=z_{\min}\), the value \(g_1\) is determined from (2) for the specified value of the power factor.
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From (4), taking (8) into account, the nominal slip value \(s=s_{\mathrm{nom}}\) is determined from the value of \(g_1\) at \(r'_g=0\).
At the minimum drive rotation speed, the current frequency in the stator must not be lower than the specified one. Otherwise, a corresponding recalculation of the generator parameters is necessary.
- A series of values of \(z_{\mathrm{n}}\) is specified within the range of load variation. From expressions (2) and (8), \(g'_2\) is determined for these values. Since \(\omega=\mathrm{const}\), for any load the equality \(s=s_{\mathrm{nom}}\) must be satisfied. The active resistance of the rotor required for this can be determined from (4):
\[ R'_2=\frac{s_{\mathrm{nom}}}{2g'_2} \left(1+\sqrt{1-4(g'_2)^2(x'_2)^2}\right). \tag{11} \]
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From the calculated values \(R'_2\), using (5), the values \(b'_2\) are determined for \(s=s_{\mathrm{nom}}\).
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\(b_1\) is determined according to (3) for the selected values of \(z_{\mathrm{n}}\); after this, \(b_m\) is determined from (9).
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From the values of \(b_m\), using the characteristic \(E=f(b_m)\), the value of the electromotive force \(E\) is determined.
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From the vector diagram of the induction generator, using the known values of \(E\) and the parameters of the external circuit, the value of the phase voltage across the load can be determined:
\[ U=-I_1\left[r_1\cos\psi_1+(x_1-x_{\mathrm{ck}})\sin\psi_1\right] +\sqrt{E^2-I_1^2\left[(x_1-x_{\mathrm{ck}})\cos\psi_1-r_1\sin\psi_1\right]^2}, \]
where
\[ I_1=E\sqrt{g_1^2+b_1^2} \]
for the corresponding values of \(z_{\mathrm{n}}\). The angle \(\psi_1\) between the current \(I_1\) and the emf \(E\) can be determined from the expression
\[ \psi_1=2\psi-\gamma-\operatorname{arc\,tg} \left[ \frac{I_1 z_{\mathrm{n}}+E}{I_1 z_{\mathrm{n}}-E} \,\operatorname{tg}\left(\frac{\psi-\gamma}{2}\right) \right], \]
\[ \gamma=\operatorname{arctg}\frac{x_1}{r_1}, \]
\[ \psi=\operatorname{arctg} \frac{ z_{\mathrm{n}}^2x_c-z_{\mathrm{n}}x_c^2\sin\varphi -(x_1-x_{\mathrm{ck}})(z_{\mathrm{n}}^2-2z_{\mathrm{n}}x_c\sin\varphi+x_c^2) }{ r_1(z_{\mathrm{n}}^2-2z_{\mathrm{n}}x_c\sin\varphi+x_c^2)+z_{\mathrm{n}}x_c^2\cos\varphi }. \]
- From the values of \(U\) and the parameters of the external circuit, the currents and powers of the load and excitation capacitors can be determined.
Thus, as a result of the calculation, the external characteristics of the generator can be constructed for various excitation capacitances, compounding capacitances, and load power factors.
The additional active resistance in the rotor compensates for the change in slip corresponding both to a change in load and to a forced change in the drive speed. The control characteristics of the machine \(R'_2 = f(s)\) can be calculated from (11), if instead of \(s_{\mathrm{nom}}\) one substitutes the value \(s = s_{\mathrm{nom}} + s_{\Delta}\), where \(s_{\Delta}\) is the component of the slip corresponding to the relative change in the drive speed.
Using this method, the characteristics of an induction generator were calculated for the following parameters: rated output voltage (line-to-line) \(133\ \mathrm{V}\), power \(1200\ \mathrm{VA}\), frequency \(400\ \mathrm{Hz}\), rated rotational speed \(8000\ \mathrm{rpm}\), maximum speed \(12000\ \mathrm{rpm}\). Figures 2 and 3 show the calculated and experimental characteristics of the generator.
As follows from the figures, the proposed method ensures good agreement between calculated and experimental data. The maximum error does not exceed \(10\%\).
Fig. 2. External characteristics of an induction generator:
\(1\) — calculated,
\(2\) — experimental
Fig. 3. Control characteristics of an induction generator:
\(1\) — calculated,
\(2\) — experimental
REFERENCES
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O. V. Kekot, R. V. Filts. Calculation of the characteristics of a three-phase induction machine with capacitor excitation. Collection of Scientific Works of Postgraduate Students of the Lviv Polytechnic Institute, 1, 1961.
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Yu. D. Zubkov. Induction Generators with Capacitor Excitation. Publishing House of the Academy of Sciences of the Kazakh SSR, Alma-Ata, 1949.