Design Issues of DC BEMUs
Abstract Generated abstract
This paper analyzes design choices for a DC brushless electric machine amplifier formed by two contactless DC machines sharing a magnetic circuit. It derives relationships for distributing electric and magnetic loadings between the two stages, including formulas for output power and gain as functions of the loading distribution coefficient and machine parameters. The analysis shows that electromagnetic load distribution strongly affects both maximum output power and amplification factor, and that the coefficient values giving these optima do not coincide. The paper proposes selecting the magnetic loading distribution within a range bounded by the conditions for maximum gain and maximum power, with efficiency maximum coinciding with the power maximum in the examined case.
Full Text
Design Issues of a DC Brushless Electric Machine Amplifier
A. I. Skorospelkin, E. N. Podborskii, E. K. Dergobuzova
(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
This article examines questions of selecting electromagnetic loadings, determining the maximum output power, and the maximum amplification factor in DC brushless electric machine amplifiers. According to [1], DC brushless electric machine amplifiers are two electric contactless DC machines combined in a single magnetic circuit.
In designing electric machines combined in one magnetic circuit, the rational distribution of electromagnetic loadings between stages is of great importance. Moreover, if the magnetic inductions of the stages \(B_{\delta 1}\) and \(B_{\delta 2}\) are specified, then the linear loadings must be completely determined.
According to the basic design equation of an electric machine [2], the quantity \(D^2 l\), proportional to the weight of the active materials, may be expressed, on the one hand, through the power and electromagnetic loadings of the first stage:
\[ D^2 l = \frac{P_1}{\pi^2 \alpha_1 n A_1 B_{\delta 1}} = \frac{P_1}{\pi^2 \alpha_1 n(1-c_A)\sum A(1-c_B)\sum B_{\delta}}, \tag{1} \]
and, on the other hand, through the power and electromagnetic loadings of the second stage:
\[ D^2 l = \frac{P_3}{\pi^2 \alpha_2 n A_2 B_{\delta 2}} = \frac{P_3}{\pi^2 \alpha_2 n c_A \sum A c_B \sum B_{\delta}} . \tag{2} \]
Here
\(P_1\) and \(P_3\) are the powers, respectively, of the output winding of the first stage \(W_{p1}\) and of the output winding of the second stage \(W_3\);
\(n\) is the rotational speed of the rotor of the brushless electric machine amplifier;
\(\alpha_1\) and \(\alpha_2\) are the pole-arc coefficients of both stages;
\(A_1\) is the linear loading of winding \(W_{p1}\) in \(a/cm\);
\(A_2\) is the linear loading of winding \(W_3\), equal to the linear loading of the compensation winding located on the rotor, in \(a\,cm\);
\(\sum A = A_1 + A_2\) is the total linear loading of windings \(W_{p1}\) and \(W_3\);
\(B_{\delta 1}; B_{\delta 2}\) are the magnetic inductions in the air gap of the magnetic fields \(\Phi_1\) and \(\Phi_2\) in \(g\);
\(\sum B_{\delta} = B_{\delta 1} + B_{\delta 2}\) is the total magnetic induction;
\[ c_B = \frac{B_{\delta 2}}{\sum B_{\delta}}, \qquad c_A = \frac{A_2}{\sum A} \]
are coefficients characterizing the distribution of the electric and magnetic loadings between the windings.
Equations (1) and (2) make it possible to obtain the relationship between the coefficients \(C_A\) and \(C_B\). Equating the right-hand sides of equations (1) and (2), after several transformations we obtain
\[ c_A=\frac{1-c_B}{1-c_B(1-k)}, \tag{3} \]
where
\[ k=\frac{P_1\alpha_2}{P_3\alpha_1}. \tag{4} \]
Expression (3) is analogous to the equation in [3], but the coefficient \(k\) has a different meaning.
Let us consider how the distribution of electromagnetic loads affects the power, gain coefficient, and efficiency of the DC brushless electric machine amplifier (BEMU).
According to [2], the output power of the BEMU is
\[ P_{\text{out}}=\pi^2 \alpha_2 nD^2 lB_{\delta 2}A_2 =\pi^2D^2 l\alpha_2 n\sum B_{\delta}\cdot c_B\cdot A_2 . \tag{5} \]
Here it must be taken into account that on the rotor, in addition to the output winding of the first stage \(W_{p1}\) and the compensating winding \(W_k\), which create the linear loads \(A_1\) and \(A_2\), there is also the excitation winding of the second stage \(W_{p2}\), which creates its own linear load \(A_{p2}\).
Consequently, the total linear load determining the thermal state of the machine, which is specified in the design, is equal to
\[ \sum A^1=A_1+A_2+A_{p2}=\sum A+A_{p2}. \]
To determine the influence of the coefficients \(c_A\) and \(c_B\) on the power of the BEMU, it is necessary to express \(A_2\) through \(\sum A^1\).
\[ A_2=c_A(\sum A^1-A_{p2}). \]
Let us express the magnetic inductions of the stages through the linear loads of the excitation windings:
\[ B_{\delta 1}=\frac{\lambda_1 A_y}{2p_1\alpha_1 l k_{\mu 1}k_{E1}}, \tag{7a} \]
\[ B_{\delta 2}=\frac{\lambda_2 A_{p2}}{2p_2\alpha_2 l k_{\mu 2}k_{E2}}, \tag{7b} \]
where
\(p_1\) and \(p_2\) are the numbers of pole pairs of the first and second stages, respectively;
\(A_y\) is the linear load of the control winding;
\(k_{\mu 1}\) and \(k_{\mu 2}\) are the saturation coefficients of the first and second stages;
\[ \lambda_1=\frac{\alpha_1\tau_1 l}{\delta'}\mu_0,\qquad \lambda_2=\frac{\alpha_2\tau_2 l}{\delta'}\mu_0 \]
is the permeance of the air gap for the fluxes \(\Phi_1\) and \(\Phi_2\).
The ratio of the inductions of the first and second stages, taking into account the values of \(\lambda_1\) and \(\lambda_2\), is
\[ \frac{B_{\delta 1}}{B_{\delta 2}} =\frac{1-c_B}{c_B} =\frac{A_y p_2 k_{\mu 2}k_{E2}}{A_{p2}p_1 k_{\mu 1}k_{E1}}. \]
Hence
\[ A_{p2}=A_y\frac{c_Bp_2k_{\mu 2}k_{E2}}{(1-c_B)p_1k_{\mu 1}k_{E1}}. \tag{8} \]
From the equation of the external characteristic of the first stage of the BEMU [1]
\[ U_2+\frac{c_1\lambda_1W_{p1}l_2^2}{16F_y}=E_2 \]
we obtain the value of \(A_y\).
\[ A_y=A_1\sqrt{\frac{k_{E1}}{2k_{E1}(k_{E1}-1)}}. \tag{9} \]
Here
\[ E_2 \]
is the emf of the first stage;
\[ U_2=\frac{E_2}{k_{E1}} \]
is the voltage of the first stage;
\[ c_1=4f_1w_{p1}, \]
\(F_y\) is the m.m.f. of the excitation winding of the first stage;
\(I_2\) is the current of the excitation winding of the second stage.
Substituting (9) into (8), we obtain an expression for \(A_{p2}\)
\[ A_{p2}=A_1\frac{c_B k_{\mu2}k_{E2}p_2}{(1-c_B)k_{\mu1}\sqrt{2k_{E1}(k_{E1}-1)}\cdot p_1}. \tag{10} \]
From (3) and (4) we obtain
\[ A_1=\frac{c_B k}{1-c_B}A_2. \tag{11} \]
Finally, taking into account (3), (6), and (11), the formula for \(A_2\) takes the form
\[ A_2=\sum A'\frac{(1-c_B)^2}{(1-c_B)[1-c_B(1-k)]+c_B^2mk}, \tag{12} \]
where
\[ m=\frac{p_2k_{\mu2}k_{E2}}{p_1k_{\mu1}\sqrt{2k_{E1}(k_{E1}-1)}}. \]
Substituting (12) into (15), we obtain the power formula
\[ P_{\text{out}}=\pi^2D^2ln\,a_2\sum B_\delta\sum A'\cdot \frac{c_B(1-c_B)^2}{(1-c_B)[1-c_B(1-k)]+c_B^2mk}. \tag{13} \]
Equation (13) is one of the principal design equations of a dc brushless electric machine amplifier.
Gain factor of a brushless electric machine amplifier
\[ k_y=\frac{P_{\text{out}}}{i_y^2r_y}. \]
Expressing the current and resistance of the control winding through the specific loadings and dimensions of the machine, we obtain
\[ k_y=\frac{P_{\text{out}}}{\pi(l+\tau_1)D\,j_yA_y\rho}, \tag{14} \]
where
\[ \tau_1=\frac{\pi D}{2p_1} \]
is the pole pitch of the first stage of the brushless electric machine amplifier,
\(j_y\) is the current density in the control winding.
From (7a) we have
\[ A_y=\frac{2k_{\mu1}k_{E1}\delta'}{\mu_0\tau_1}(1-c_B)\sum B_\delta. \]
Substituting the value of \(A_y\) into (14), we obtain the final expression for \(k_y\)
\[ K_y = \frac{\pi^2 D^2 n \alpha_y \sum A' \mu_0} {4\rho(1+\beta_1)p_1 i_y k_{\mu 1} k_{E1}\delta'} \cdot \frac{c_B(1-c_B)} {(1-c_B)[1-c_B(1-k)] + c_B^2 k m}, \tag{15} \]
where
\[ \beta_1=\frac{\tau_1}{l}. \]
From equations (13) and (15) it is evident that the power and gain coefficient of the BEMA depend substantially on the distribution of electromagnetic loads between the stages.
Figures 1 and 2 present the dependences of the BEMA power and gain coefficient on \(c_B\) for different \(k\). The calculation was carried out for a
Fig. 1. Dependence of the output power on \(c_B\) for various \(k\)
BEMA having the following data:
\[ \begin{aligned} D &= 12.2\ \text{cm}, & \sum A' &= 140\ \text{A/cm},\\ l &= 11.5\ \text{cm}, & n &= 3000\ \text{rpm},\\ \sum B_\delta &= 3200\ \text{G}, & 2p_1 &= 2,\quad 2p_2 = 8. \end{aligned} \]
As can be seen from the figures, when \(k\) decreases, the power and gain coefficient of the BEMA increase; and when \(c_B\) changes from zero to unity, the values \(P_{\text{out}}\) and \(K_y\) change from zero to a certain maximum value, and then again, as \(c_B \to 1\), approach zero. Moreover, the values
Fig. 2. Dependence of the gain coefficient on \(c_{\mathrm{B}}\) for different \(k\)
The \(c_{\mathrm{B}}\) values corresponding to the maximum value of the power and of the gain coefficient do not coincide.
The value of \(c_{\mathrm{B}}\) corresponding to the maximum gain coefficient, for specified values of the other quantities entering into equation (15), can be obtained from the equation
\[ \frac{\partial k_y}{\partial c_{\mathrm{B}}}=0. \tag{16} \]
Substituting into (16) the value of \(k_y\) expressed by equation (15), after some transformations we obtain
\[ c_{\mathrm{Bm}}=\frac{1-\sqrt{mk}}{1-mk}. \tag{17} \]
In Fig. 3, within the limits of values of \(k\) possible in a brushless electric machine amplifier, the dependences \(c_{\mathrm{Bm}}=f(k)\) are presented: from the condition of the maximum gain coefficient (curve 1) and from the condition of the maximum output power (curve 2).
When \(k\) changes from 0.005 to 0.02, the value of \(c_{\mathrm{B}}\) corresponding to the maximum gain coefficient changes within the limits \(0.925\)–\(0.83\).
The case of maximum output power, when \(K\) varies within the same limits, corresponds to the values \(c_{\mathrm{B}}=0.75—0.67\).
Thus, \(c_{\mathrm{B}}\) should be chosen in the region bounded by curves 1 and 2 in Fig. 3, either on the basis of the requirement for the maximum amplification factor or for maximum power.
Fig. 3
Fig. 4. Dependence \(\eta=f(c_{\mathrm{B}})\) and \(P_{\mathrm{out}}=f(c_{\mathrm{B}})\) for \(K=0.009\)
Figure 4 shows the dependence of the efficiency as a function of \(c_{\mathrm{B}}\) for \(k=0.009\), whose maximum coincides with the maximum of the output power, which corresponds to the well-known proposition that the volume optimum and the efficiency optimum coincide [3].
Thus, as a result of the studies carried out ...
1) relations \(c_A\) and \(c_B\) have been obtained that make it possible to take into account the most rational distribution of electromagnetic loads, with allowance for the characteristics of the DC brushless electric-machine amplifier;
2) it has been shown that the distribution of electromagnetic loads between stages has a substantial effect on the power and gain factor of the brushless electric-machine amplifier.
References
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E. N. Polborskii. Development and investigation of a DC brushless electric-machine amplifier. Dissertation, Tomsk, 1968.
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I. M. Postnikov. Design of electrical machines. Gostekhizdat USSR, 1956.
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V. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.