Some Results on the Study of Factors Determining the Gain Coefficient in Brushless AC Electric Machine Amplifiers
Abstract Generated abstract
This paper analyzes the factors governing the gain coefficient of brushless AC electric machine amplifiers under the assumption of an unsaturated magnetic system. It derives expressions relating gain to machine geometry, air gap, rotational speed, winding parameters, reactances, rectifier circuit coefficients, load current, and feedback conditions, including formulas for maximum gain and optimum rotor winding turns. Experimental results for a 4 kW, 220 V, 100 Hz amplifier show low gain without feedback, substantial improvement with voltage feedback, and further increases obtained by changing the rotor rectifier from a Larionov circuit to a three-phase half-wave circuit and reducing rotor winding reactance. The study reports a maximum observed gain of 2688 and argues that further increases are feasible through parameter optimization.
Full Text
SOME RESULTS ON THE STUDY OF FACTORS DETERMINING THE GAIN IN BRUSHLESS AC EMU
A. I. SKOROSPLESHKIN, E. N. PODBORSKII
(Recommended by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The gain coefficient is the principal indicator of any amplifier. It is evaluated by the ratio of the input and output powers of the amplifier:
\[ K_y=\frac{m_s U_s\cdot I_s}{U_y\cdot I_y} = m_s\frac{U_s}{U_y}\cdot\frac{I_s}{I_y}, \tag{1} \]
where
\(m_s\) — number of phases of the output winding of the brushless EMU;
\(U_s\) — phase voltage at the output;
\(U_y\) — control voltage;
\(I_y\) — control current;
\(I_s\) — current in a phase of the output winding;
\[ K_u=\frac{U_s}{U_y} \]
— voltage gain coefficient;
\[ K_I=\frac{I_s}{I_y} \]
— current gain coefficient.
Let us express the gain coefficient of the amplifier through its parameters, assuming here that the magnetic system of the machine is not saturated. This assumption is quite acceptable, since the operating point of all amplifiers is usually chosen on the linear part of the no-load characteristic.
The e.m.f. of a phase of the rotor winding is
\[ E_{p1}=4.44 f_{p1}\cdot W_{p1}\cdot \Phi_y K_{\mathrm{ob}.p1}, \tag{2} \]
where \(f_{p1}=\dfrac{p_1 n}{60}\) — frequency of the e.m.f. of the rotor winding;
\(W_{p1}\) — number of turns of a phase of the rotor winding of the first cascade;
\(\Phi_y\) — control flux;
\(K_{\mathrm{ob}.p1}\) — winding factor of winding \(W_{p1}\);
\(n\) — rotor speed in rpm.
The flux produced by the current of the control winding is equal to
\[ \Phi_y=\alpha_1\tau_1 l i B_{\delta y}, \tag{3} \]
where \(\alpha_1\) — pole-arc coefficient of the first cascade;
\(\tau_1\) — pole pitch of the first cascade;
\(li\) — calculated length of the machine;
\(B_{\delta y}\) — induction in the air gap due to the control flux.
To conduct the control flux through the air gap, an m.m.f. across the gap is necessary:
\[ F_{\delta y}=1.6B_{\delta y}\cdot\delta'. \tag{4} \]
From expression (4),
\[ B_{\delta y}=\frac{F_{\delta y}}{1.6\delta'}=\frac{F_y}{1.6\delta'K_{\mu1}}= \frac{I_y W_y K_{fy}}{1.6p_1\delta'K_{\mu1}}, \tag{5} \]
where \(\delta'\) is the calculated air gap;
\(p_1\) is the number of pole pairs of the first stage;
\(W_y\) is the number of series-connected turns of the control winding;
\(K_{\mu1}=\dfrac{F_{\delta y}}{F_y}\) is the saturation coefficient of the first stage;
\(K_{fy}\) is the field-form coefficient of the control winding;
\(F_y\) is the m.m.f. of the control winding.
Substituting expression (5) into (2), taking (3) into account, we obtain
\[ E_{p1}= \frac{4.44nW_{p1}K_{\mathrm{ob}.p1}\alpha_1 r_1 l_i W_y K_{fy}} {60\cdot1.6\cdot K_{\mu1}\cdot\delta'}\,I_y =R_{a1}\cdot I_y, \tag{6} \]
where
\[ R_{a1}= \frac{44.4nW_{p1}K_{\mathrm{ob}.p1}\alpha_1 r_1 l_i W_y K_{fy}} {60\cdot1.6\cdot K_{\mu1}\cdot\delta'} \tag{7} \]
is the magnetization resistance of the first stage. Since the magnetic circuit of the amplifier is not saturated, \(K_\mu=\mathrm{const}\) and \(R_{a1}=\mathrm{const}\), because all quantities entering into (7) are constant for a given amplifier.
The e.m.f. of the stator winding is
\[ E_s=4.44f_s W_s\Phi_{\delta2}K_{\mathrm{ob}.s}, \tag{8} \]
where
\[ f_s=\frac{p\cdot n}{60} \]
is the frequency of the e.m.f. of the stator output winding;
\(p_2\) is the number of pole pairs of the second stage;
\(W_s\) is the number of turns per phase of the output winding;
\(\Phi_{\delta2}\) is the flux per pole pair of the second stage;
\(K_{\mathrm{ob}.s}\) is the winding factor of the output winding.
By analogy with the expression for \(E_{p1}\), we write
\[ E_s=R_{a2}\cdot I_{p2}, \tag{9} \]
where
\[ R_{a2}= \frac{4.4\cdot44nW_s\cdot K_{\mathrm{ob}.s}\alpha_2 r_2 l_i W_{p2}K_{f2}} {60\cdot1.6\cdot K_{\mu2}\cdot\delta'} \tag{10} \]
is the magnetization resistance;
\(I_{p2}\) is the excitation current of the second stage.
The first stage of a brushless electric-machine amplifier is a three-phase synchronous generator operating into a rectifier load. The power factor in this case is usually very close to unity [1]; therefore, for simplicity, we shall assume that it is equal to unity. This case corresponds to the vector diagram shown in Fig. 1.
Fig. 1. Vector diagram of the first stage of the amplifier.
From the diagram one may write
\[ E_{p1}^{2}=(U_{p1}+I_{p1}r_{p1})^{2}+I_{p1}^{2}X_{dp}^{2} \]
or
\[ E_{p1}^{2}=K_{up}^{2}I_{p2}^{2}r_{p2}^{2}+K_{Ip}^{2}I_{p2}^{2}r_{p1}^{2} +2K_{up}I_{p2}^{2}r_{p2}K_{Ip}r_{p1} +K_{Ip}^{2}I_{p2}^{2}X_{dp}^{2}. \]
Hence we find the excitation current of the second stage
\[ I_{p2}= \frac{E_{p1}} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}+K_{Ip}^{2}r_{p1}^{2} +2K_{up}K_{Ip}r_{p2}r_{p1}}}\,, \tag{11} \]
\[
K_{Ip}=\frac{I_{p1}}{I_{p2}}
\]
is the ratio of the phase current in the rotor winding to the rectified current;
\[
K_{up}=\frac{U_{p1}}{U_{p2}}
\]
is the ratio of the phase voltage to the rectified voltage;
\(X_{dp}\) is the synchronous reactance of the rotor winding of the first stage;
\(r_{p1}\) is the active resistance of winding \(W_{p1}\);
\(r_{p2}\) is the active resistance of winding \(W_{p2}\).
By analogy with expression (6), one may write
\[ I_{p2}= \frac{R_{\alpha1}\cdot I_y} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}+K_{Ip}^{2}r_{p1}^{2} +2K_{up}K_{Ip}r_{p1}r_{p2}}}\,. \tag{12} \]
Since the parameters are considered constant, then
\[ I_{p2}=I_y\cdot K_{11}, \tag{13} \]
where
\[ K_{11}= \frac{R_{\alpha1}} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}+K_{Ip}^{2}r_{p1}^{2} +2K_{up}K_{Ip}r_{p1}r_{p2}}} \tag{14} \]
is the current gain coefficient of the first stage.
Substituting (12) into (9), we obtain
\[ E_s=R_{\alpha2}I_{p2}= \frac{R_{\alpha1}R_{\alpha2}I_y} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}+K_{Ip}^{2}r_{p1}^{2} +2K_{up}K_{Ip}r_{p1}r_{p2}}} \tag{15} \]
or
\[ E_s=R_a\cdot I_y, \tag{16} \]
where
\[ R_a= \frac{R_{\alpha1}R_{\alpha2}} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}+K_{Ip}^{2}r_{p1}^{2} +2K_{up}K_{Ip}r_{p2}r_{p1}}}\,. \tag{17} \]
If the active resistance of the rotor winding is neglected, \(r_{p1}=0\), then
\[ K_{11}= \frac{R_{\alpha1}} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}}} \tag{18} \]
and
\[ I_{p2}= \frac{R_{\alpha1}I_y} {\sqrt{K_{Ip}^{2}X_{dp}^{2}+K_{up}^{2}r_{p2}^{2}}}\,. \tag{19} \]
When the amplifier operates with a symmetrical load and with a constant control signal, the output voltage varies according to the equation
\[ U_s= \sqrt{E_s^{2}-\left(I_sX_{ds}\cos\varphi-I_sr_s\sin\varphi\right)^{2}} -\left(I_sr_s\cos\varphi+I_sX_{ds}\sin\varphi\right), \tag{20} \]
where \(X_{ds}\) is the synchronous reactance of the output winding;
\(r_s\) is the active resistance of a phase of the output winding;
\(\cos\varphi\) is the power factor.
Substituting (19) into (1), we obtain an expression for the gain of a brushless electromechanical amplifier:
\[ K_y= \frac{ m_s I_s \left[ \sqrt{E_s^2-\left(I_sX_{ds}\cos\varphi-I_s r_s\sin\varphi\right)^2} -I_s r_s\cos\varphi+I_sX_{ds}\sin\varphi \right] }{ U_y I_y }. \tag{21} \]
Taking (16) into account, expression (21) takes the form
\[ K_y= \frac{ m_s I_s \left[ \sqrt{R_a^2 i_y^2-\left(I_sX_{ds}\cos\varphi-I_s r_s\sin\varphi\right)^2} -I_s r_s\cos\varphi+I_sX_{ds}\sin\varphi \right] }{ U_y I_y }. \tag{22} \]
Expression (22), obtained for the gain under a symmetrical load, does not take into account losses in the steel. Therefore, the actual gain will in practice differ from that calculated by the formula given.
When a brushless electromechanical amplifier operates in the external-characteristic mode, the control power remains constant. Therefore, the gain will depend primarily on the magnitude of the output power, increasing as the latter grows. However, the growth of the gain is not unlimited, just as the output power is not unlimited; it is determined by the geometry of the machine and by its electromagnetic loadings.
The gain depends strongly on the stiffness of the amplifier’s external characteristics: the stiffer the external characteristic, the greater the gain. A brushless electromechanical amplifier without feedback has steeply drooping external characteristics and, consequently, a low gain.
With the control signal held constant, as the load in a brushless electromechanical amplifier without feedback increases, the gain increases with the growth of the power delivered up to a maximum value, and then begins to decrease. The greatest value of the gain in this case can be determined by taking the derivative of expression (21) with respect to \(I_s\) and setting it equal to zero. If the active resistance of the stator winding is neglected \((r_s=0)\), then, after simple transformations, one can obtain that the critical load current at which the gain has its greatest value is equal to
\[ I_{s\mathrm{cr}}=\frac{E_s}{\sqrt{2}\,X_{ds}}, \tag{23} \]
and the output voltage is then equal to
\[ U_{s\mathrm{cr}}=\frac{E_s}{\sqrt{2}}. \tag{24} \]
The greatest value of the gain is then determined as
\[ K_{y\max}=\frac{m_sE_s^2}{2U_y I_y X_{ds}}. \tag{25} \]
If expression (16) is substituted into (25), then the maximum gain of a brushless electromechanical amplifier without feedback will be equal to
\[ K_{y\max}=\frac{m_sR_a^2}{2r_yX_{ds}}. \tag{26} \]
This expression is convenient in that it contains only the parameters of the amplifier.
By using feedbacks it is possible to maintain a constant voltage at the output when the load changes up to the rated value. In this case, for the gain factor one may write the following expression:
\[ K_y=\frac{m_s E_s}{U_y I_y} I_{sн} =\frac{m_s R_\alpha I_y}{U_y I_y} =\frac{m_s R_\alpha}{U_y} I_{sн}, \tag{27} \]
where \(I_{sн}\) is the rated phase current.
We have constructed and investigated a brushless electric-machine amplifier with fixed frequency, with the following data: \(P_n=4\ \text{kW}\), \(U_{sн}=220\ \text{V}\), \(f_s=100\ \text{Hz}\), \(n=3000\ \text{rpm}\).
Figure 2 shows the dependences of the gain factor on the load current at a constant control signal, obtained experimentally for the given amplifier, whose rotor rectifier is assembled according to the Larionov circuit.
Fig. 2. Gain factor in the absence of feedback. The rotor rectifier is assembled according to the Larionov circuit.
As can be seen from this figure, the gain factor of the BEMA without feedback is very small; the maximum gain factor is \(K_{y\max}=30\). The maximum gain factor calculated by formula (26) has approximately the same value.
To increase the gain factor, feedbacks were used, which made it possible to maintain a constant voltage at the output. Figure 3 gives the dependence of the gain factor on the load current for this case (straight line 1).
Thus, by means of feedbacks, it was possible to increase the gain factor by approximately ten times and bring it up to \(K_y=280\).
A special feature of this amplifier is that the rated load current exceeds the current of the established symmetrical short circuit without feedbacks by \(K=4 \div 5\) times. Therefore the rated load current may be written in the following form:
\[ I_{sн}=K\cdot\frac{E_s}{X'_{ds}} =\frac{K R_\alpha I_y}{X'_{ds}}. \tag{28} \]
Substituting this expression into (27), we obtain
\[ K_{yн}=\frac{m_s R_\alpha^2}{r_y X'_{ds}}K. \tag{29} \]
Comparing (29) and (26), we see that the feedbacks make it possible to increase the gain by approximately 8–10 times. However, the obtained gain coefficient \(K_y = 280\) is insufficient; to increase it, it is necessary to reduce the control power. This can be achieved by increasing the demagnetization resistance.
\[ R_a = \frac{R_{a1} R_{a2}} {\sqrt{ K_{lp}^{2} X_{dp}^{2} + K_{up}^{2} r_{p2}^{2} + r_{p1}^{2} K_{lp}^{2} + 2K_{lp}K_{up}r_{p1}r_{p2} }} . \]
Fig. 3. Gain coefficient in the presence of feedbacks.
In this expression, the first term under the radical is the largest, and the remaining terms may, in a first approximation, be neglected in comparison with it; therefore
\[ R_a = \frac{R_{a1}R_{a2}}{K_{lp}X_{dp}} . \tag{30} \]
From (30) it is evident that \(R_a\), and hence also the gain coefficient, can be increased by reducing: 1) the coefficient \(K_{lp}\), 2) the inductive resistance of the rotor winding.
Let us consider what the coefficient \(K_{lp}\) depends on. For a three-phase rectifier,
\[ K_{lp} = \frac{I_{p1}}{I_{p2}}, \]
i.e., it is equal to the ratio of the rms value of the current in the rotor phase to the rectified current and depends on the rectifier circuit.
For the Larionov circuit in an idealized rectifier, according to [1] it is equal to \(K_{lp} = 0.82\). For a single-half-period three-phase circuit, \(K_{lp} = 0.58\). Comparing the values of this coefficient for the Lari-
… and the single-half-period circuit, we see that for the latter it is approximately 1.4 times smaller than for the Larionov circuit. Therefore it may be concluded that the gain can be increased if the rotor rectifier is assembled according to a single-half-period three-phase circuit. In this case the number of diodes is reduced by half.
Figure 4 presents the dependences of the gain on the load current for various values of the control current of the BEMA, whose rotor rectifier is assembled according to a three-phase single-half-period circuit.
Fig. 4. Gain in the absence of feedback. The rotor rectifier is assembled according to a three-phase single-half-period circuit.
Comparing Fig. 4 and Fig. 3, one can verify that the gain has increased. The maximum gain is equal to \(K_{y\max}=96\).
By maintaining a constant voltage at the output by means of feedbacks, it was possible to increase the output power significantly; in this case the gain increased to \(K_y=1050\). The dependence of the gain on the load current in this case is given in Fig. 3 (straight line 2).
A further increase in the gain is possible by reducing \(X_{dp}\). The inductive reactance of the rotor winding was reduced by decreasing the number of turns \(W_{p1}\) when switching it into two parallel branches.
For this case, Fig. 5 presents the dependences of the gain on the output current for various values of the control current. Comparing Fig. 5 and Fig. 4, we see that the gain increased to \(K_y=240\). The feedbacks made it possible to maintain a constant voltage when the load changed. In Fig. 3, straight line 3 presents the dependence of the gain on the load current. The maximum value of the gain in this case is equal to \(K=2688\).
It should be noted, however, that reducing the number of turns leads to an increase in \(R_a\) only up to certain limits, at which the current gain of the first stage has a maximum value. The maximum of \(K_{n}\) can be found by taking its derivative with respect to \(W_{p1}\) and setting it equal to zero,
\[ \frac{dK_{n}}{dW_{p1}} = \frac{d}{dW_{p1}} \left[ \frac{R_a^{*}\cdot W_{p1}} {\sqrt{aW_{p1}^{4}+b+cW_{p1}^{2}+eW_{p1}}} \right] =0, \tag{31} \]
where
\[ R_{\alpha}^{1}= \frac{4.44\, n K_{\mathrm{ob},p1} a_{1}\tau_{1} l i W_{y} K_{fy}} {60\cdot 1.6\cdot K_{\mu 1}\delta^{1}}, \]
\[ a=\left(2m f_{p1}\frac{\mu_{0}D l i K_{\mathrm{ob},p1}} {K_{\mu 1}\delta^{1}p_{1}^{2}}\right)^{2}K_{\Gamma p}^{2}, \]
\[ b=K_{\mathrm{up}}^{2}r_{p2}^{2}, \]
\[ c=K_{\Gamma p}\cdot K^{1}, \]
\[ r_{p1}=K^{1}W_{p1}, \]
\[ e=2K_{\mathrm{up}}K_{\Gamma p}r_{p2}\cdot K^{1}. \]
Fig. 5. Gain coefficient in the absence of feedback. The rotor rectifier is assembled according to a three-phase half-wave circuit, and the number of turns in the rotor winding \(W_{p1}\) is reduced by half.
After transformations, equation (31) will take the form
\[ 2aW_{p1}^{4}-eW_{p1}-2b=0. \tag{32} \]
Solving this equation with respect to \(W_{p1}\), we obtain the optimum number of turns at which the gain coefficient will be maximal.
Conclusions
-
As a result of the study it has been established that the gain coefficient is decisively affected by the air gap, the rotational speed, and the parameters of the amplifier, and a regular relationship among them has been established.
-
Formulas for the gain coefficient have been derived; they can be used in the design of brushless electric-machine amplifiers.
-
The obtained maximum value of the gain coefficient \(K_y=2688\) is not limiting. There are real possibilities for increasing it by 5–8 times.
LITERATURE
- G. G. Fedoseev. Rectifiers and Stabilizers. Gosizdat, Iskusstvo, 1960.