Calculation of the Optimal Forcing Coefficient under the Condition of Minimum No-Load Losses of a Shock Generator
Abstract Generated abstract
This paper examines how to choose the excitation and de-excitation forcing coefficient of a shock generator so as to minimize no-load energy losses per stator-current pulse. Using previously derived expressions for steel losses and rotor copper losses over an operating cycle, the authors formulate a transcendental condition for the optimal forcing coefficient in terms of the ratio of steel to excitation-winding copper losses and the rotor time constants. They derive comparison formulas for intermittent excitation with forcing, intermittent excitation without forcing, and constant excitation, and identify the pause-duration condition under which intermittent excitation is advantageous. The analysis indicates that operation at the optimal forcing coefficient can nearly halve losses relative to constant excitation over the same time and reduce losses by about three to five times relative to intermittent excitation without forcing.
Full Text
CALCULATION OF THE OPTIMAL FORCING COEFFICIENT FROM THE CONDITION OF MINIMUM NO-LOAD LOSSES OF A SHOCK GENERATOR
G. A. Sipailov, A. I. Verkhoturov
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
Shock generators have higher no-load losses in comparison with turbogenerators of the same rated power. This is due to the greater saturation of sections of the magnetic circuit of shock generators, and also to the fact that, because of the reduced air gap, additional losses acquire a large specific weight. The main part of the no-load losses of a shock generator consists of losses in steel, which exceed by several times the losses in the copper of the excitation winding. Therefore, in shock generators the question of reducing energy losses calculated per one pulse of stator current acquires especially great importance. For this purpose, in shock generators, for a certain duration of pauses between individual pulses, it is expedient to use intermittent excitation with forced buildup of the magnetic flux to the required value before each switching-on of the generator and subsequent suppression of the magnetic field after the stator current pulse [1].
In the present article the problem is considered of the calculated determination of the optimal forcing coefficient of excitation and de-excitation from the condition of minimum no-load losses of the generator, calculated per one pulse of stator current; a comparison is given of intermittent excitation with other types of excitation, and the limits of its application are established.
In [2, 3], dependences were obtained for the energy of losses in the steel of the machine and in the copper of the rotor windings at no load over the operating cycle of a shock generator for any forcing coefficient \(K\). For the steel:
\[ Q_{\mathrm{st}} = P_0^1 T \left(k^2 \ln \frac{k+1}{k-0.95} - 1.95k + 0.05\right). \tag{1} \]
For the copper of the rotor windings:
\[ Q_{\mathrm{cu}} = \frac{u_2^2 T}{r_2} \left[ k^2 \ln \frac{k+1}{k-0.95} + B(0.05 - 1.5k) \right], \tag{2} \]
where
\[ \begin{aligned} P_0^1 &\text{— no-load losses in the steel of the machine,}\\ T = T_2 + T_3 &\text{— total time constant of the excitation and damping windings,}\\ u_2 &\text{— voltage at the terminals of the excitation winding,}\\ r_2 &\text{— active resistance of the excitation winding,} \end{aligned} \]
\[ B=\frac{T_2}{T} \]
is the ratio of the time constant of the field winding to the total time constant of the rotor circuits.
Let us denote \(k_c=\dfrac{Q_{\text{st}}}{Q'_{\text{st}}}\), where \(Q'_{\text{st}}\) is the energy loss in the steel over the operating cycle of the shock generator without forcing; then
\[ k_c=0.556\left(k^2\ln\frac{k+1}{k-0.95}-1.95k+0.05\right). \tag{3} \]
The dependence of \(k_c\) on the forcing coefficient is shown in Fig. 1, \(a\).
Fig. 1. Dependence of \(k_c\) and \(k_m\) on the forcing coefficient.
By analogy with the preceding, let us denote by \(k_m\) the ratio of the losses in the copper of the field and damper windings under forcing over the operating cycle to the loss energy at \(k=1\); then
\[ k_m=\frac{k^2\ln\dfrac{k+1}{k-0.95}+B(0.05-1.95k)}{3.69-1.9B}. \tag{4} \]
The results of calculating the influence of forcing on the magnitude of the energy losses in the copper of the rotor windings as a function of \(k\), for various values of \(\dfrac{T_3}{T_2}\), are presented in Fig. 1b.
From an analysis of Figs. 1a and 1b it is seen that there exists an optimal forcing coefficient at which the energy of electromagnetic losses will be minimal. The optimal forcing coefficient is determined from the condition of equality of the energy of losses in the steel of the machine (1) and in the copper of the rotor windings (2) over the operating cycle. We first simplify expressions (1) and (2). In deriving these formulas it was assumed that the voltages and currents reach their steady-state values in the absence of forcing in a time \(t = 3T\); however, theoretically this time is equal to infinity, and then (1), (2) may be written as
\[ Q_{\mathrm{st}} = P_0^{\,1} T k \left( k \ln \frac{k+1}{k-1} - 2 \right), \tag{5} \]
\[ Q_{\mathrm{cu}} = \frac{u_2^{\,2} T k}{r_2} \left( k \ln \frac{k+1}{k-1} - 2B \right). \tag{6} \]
The equations have meaning for \(k > 1\).
Equating the right-hand sides of equations (5), (6):
\[ P_0^{\,1} T k \left( k \ln \frac{k+1}{k-1} - 2 \right) = \frac{u_2^{\,2} T k}{r_2} \left( k \ln \frac{k+1}{k-1} - 2B \right). \tag{7} \]
After simplification:
\[ (k_0 - 1)\, k \ln \frac{k+1}{k-1} + 2(B - k_0) = 0, \tag{8} \]
where \(k_0 = \dfrac{P_0^{\,1} r_2}{u_2^{\,2}}\) is the ratio of the no-load losses in the steel of the machine to the losses in the copper of the excitation winding.
For known \(k_0\) and \(B\), this transcendental equation is solved approximately with any desired degree of accuracy with respect to \(k\). Fig. 2 presents the dependence of the optimal forcing coefficient \(k_{\mathrm{op}}\) as a function of \(\dfrac{T_3}{T_2}\) for \(k_0 = 6 \div 20\).
The dependence of the total electromagnetic energy of losses on the forcing coefficient can be obtained by adding the right-hand sides of equations (5) and (6):
\[ Q_{\Sigma k} = \frac{u_2^{\,2} T k}{r_2} \left[ (k_0 + 1) k \ln \frac{k+1}{k-1} - 2(k_0 + B) \right]. \tag{9} \]
Let us denote \(k_2 = \dfrac{Q_{\Sigma k}}{Q_{\Sigma 1}}\), where \(Q_{\Sigma 1}\) is the total loss at \(k = 1\); then
\[ k_2 = \frac{ k \left[ k(k_0 + 1)\ln \dfrac{k+1}{k-1} - 2(k_0 + B) \right] }{ 3.69 + 1.69 k_0 - 2B }. \tag{10} \]
If in (10) \(k_{\mathrm{op}}\) is substituted for \(k\), then this equation gives the ratio of the loss energy at the optimal forcing coefficient to the loss energy at \(k = 1\). The dependence of \(k_2\) as a function of the ratio \(\dfrac{T_3}{T_2}\) is presented in Fig. 3a.
Fig. 2. For calculating the optimal forcing coefficient.
Let us, by analogy, denote by \(k_z\) the ratio of the energy losses per cycle at the optimal forcing coefficient to the energy losses under constant excitation over the same time:
\[ k_z = \frac{ u_2^2 T k \left[(k_0+1)k\ln\frac{k+1}{k-1}-2(k_0+B)\right] }{ r_2\left(P_0^1+\frac{u_2^2}{r_2}\right)t_c }, \tag{11} \]
where
\[ t_c = T\ln\frac{k+1}{k-1} \]
is the cycle time.
Or, finally:
\[ k_z = \frac{ k\left[(k_0+1)k\ln\frac{k+1}{k-1}-2(k_0+B)\right] }{ (k_0+1)\ln\frac{k+1}{k-1} }, \tag{12} \]
The results of the calculation according to (12) are presented in Fig. 3b. Intermittent excitation becomes advisable when the electromagnetic energy losses per cycle under intermittent excitation are less than the energy losses over the same time under constant excitation:
\[ \frac{ \left(P_0^1+\frac{u_2^2}{r_2}\right)t_c }{ \frac{u_2^2Tk}{r_2} \left[(k_0+1)k\ln\frac{k+1}{k-1}-2(k_0+B)\right] } =1. \tag{13} \]
After simplification:
\[ (k_0+1)(k^2-1)\ln\frac{k+1}{k-1}-2k(k_0+B)=0, \tag{14} \]
From this equation one can find \(k\), or \(t_c=T\ln\frac{k+1}{k-1}\), at which the energy losses under constant excitation will be equal to the energy losses under intermittent excitation. Intermittent excitation may be used when the durations of the pauses between stator-current pulses are greater than the cycle durations \(t_c\), calculated from (14) for the specified values of \(k_0\) and \(B\).
Thus, intermittent excitation at the optimal forcing coefficient reduces the no-load energy losses, calculated per one stator-current pulse, by almost a factor of 2 in comparison with constant excitation over the same time, and reduces by a factor of 3–5
Fig. 3. Comparison of intermittent excitation at \(k=k_{\mathrm{op}}\) with intermittent excitation at \(k=1\) and with constant excitation.
the energy of losses in comparison with intermittent excitation without forcing.
The use of intermittent excitation becomes expedient when the pause durations are greater than the cycle durations calculated from (14).
REFERENCES
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I. M. Postnikov. On the design of a shock generator, Proceedings of the LPI, No. 209, Gosenergoizdat, 1960.
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G. A. Sipailov, A. I. Verkhoturov. On the forcing of the excitation of a shock generator, Proceedings of the TPI, vol. 160, 1966.
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G. A. Sipailov, A. I. Verkhoturov. Experimental investigation of intermittent excitation on a model of a shock generator, Proceedings of the TPI, vol. 162, 1966.