Choosing the Optimal Width of the Active Conductor of a Disk Printed Armature
Abstract Generated abstract
The paper addresses the selection of the active conductor width in disk printed armatures for DC motors, where conductor geometry affects both dynamic response and thermal losses. It develops analytical expressions for the relative conductor width that minimizes the electromechanical time constant and, separately, the width that minimizes rotor electrical losses, including load and eddy-current components. Because these optima generally do not coincide, the authors derive approximate relations for the increase in time constant and rotor losses caused by deviations from each optimum. The resulting procedure supports a practical design choice after the main disk dimensions have been fixed, with inertia usually having the dominant influence.
Full Text
Choosing the Optimal Width of the Active Conductor of a Disk Printed Armature
A. M. Shor, V. M. Kazanskii, L. D. Osnovich
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
In very recent years, DC motors with a printed winding on the armature have gained recognition [1, 2, 4, 5]; their design is based on the need to ensure the best intrinsic speed of response.
The choice of the optimal geometry of a disk rotor from the condition of minimum mechanical inertia is given in [3]. The optimal value of the mean disk diameter (Fig. 1) is limited by the width of the conductor in the end portions, selected as a function of the permissible current density. The width of the active ring of the armature is equal to the active length of the conductor
\[ \frac{1}{2}(D'_e - D'_i)=l_a \tag{1} \]
and is determined by the magnitude of the voltage, the induction in the gap, and the speed of the motor. The degree to which the active ring is filled with copper may vary depending on the width of the active conductor, whose minimum value is determined by the width of the printed conductor in the end portions.
With the same thickness of the printed layer of the active and end portions of the winding, increasing the width of the printed conductor has a contradictory effect on the inertia and thermal loading of the motor.
On the one hand, increasing the width of the active conductor leads to a reduction in the Joule losses from the armature current in the active layer; on the other hand, it leads to an increase in eddy-current losses proportional to the cube of the width of the active conductor.
On the one hand, decreasing the width of the active conductor promotes an increase in the electromechanical time constant of the motor owing to an increase in the armature resistance \(R_a\); on the other hand, it decreases the same constant owing to a reduction in the mass of the rotating armature.
Both contradictions require finding the optimal values of the width of the active conductor.
The electromechanical time constant of the armature is
\[ T_{\mathrm{m}}=\frac{GD^2 R_a}{375\, C_e C_m}. \tag{2} \]
The quantities in expression (2) that depend on the width of the active conductor \(b_a\) enter only into the numerator; therefore the optimum condition will be
\[ \frac{d}{db_a}\left(GD^2R_a\right)=0. \tag{3} \]
Fig. 1.
Introducing into the analysis the relative width of the active conductor
\[ \beta=\frac{b_a}{b_{\mathrm{п}}} \tag{4} \]
in accordance with Fig. 1 we obtain
\[ R_a=\frac{\rho N}{(2a)^2 h_{\mathrm{п}} b_{\mathrm{п}}} \left(l_{\mathrm{п}}+\frac{l_a}{\beta}\right), \tag{5} \]
where \(l_{\ell}=l_e+l_i\) is the total length of the end portions of a winding half-turn;
\(h_{\mathrm{p}}\) is the thickness of the printed conductor;
\(b_{\mathrm{p}}\) is the width of the conductor of the end portions;
\(N\) is the number of conductors in the winding.
The variable component of the armature flywheel moment depending on \(\beta\) is the flywheel moment of the active copper ring \(GD_\beta^2\), which can be represented as the sum of \(N\) flywheel moments of one conductor, a rectangular parallelepiped with sides \(l_a\), \(b_a\), and \(h_{\mathrm{p}}\), rotating about an axis located at a distance \(0.5D_{\mathrm{cp}}\) from its center of gravity.
Then
\[ CD_\beta^2 = N d l_a h_{\mathrm{p}} b_a \left[ \frac{l_a^2+b_a^2}{3}+D_{\mathrm{cp}}^2 \right]. \tag{6} \]
Since \(b_a^2 \ll l_a^2\), then
\[ GD_\beta^2=\beta\left(\frac{l_a^2}{3}+D_{\mathrm{cp}}^2\right)N d l_a h_{\mathrm{p}} b_{\mathrm{p}}, \tag{7} \]
where \(d\) is the specific weight of copper.
The part of the flywheel moment independent of \(\beta\) may be represented by two components, one of which, \(GD_{\ell\mathrm{m}}^2\), is the flywheel moment of the outer and inner copper rings of the end portions, and the second, \(GD_0^2\), is the flywheel moment of all the remaining rotating parts of the armature (the fastening sleeve, the supporting disk base, the shaft collector, etc.).
It is convenient to represent the armature flywheel moment in the following form:
\[ GD^2 = GD_{\ell\mathrm{m}}^2(k_0+1)+GD_\beta^2, \tag{8} \]
where
\[ k_0=\frac{GD_0^2}{GD_{\ell\mathrm{m}}^2} \]
is determined after selecting the main dimensions of the rotor disk in accordance with [3].
The flywheel moment of the end copper is determined as the sum of the flywheel moments of the inner and outer end rings
\[ GD_{\ell\mathrm{m}}^2 = 2Nd h_{\mathrm{p}} b_{\mathrm{p}} \left[ l_e\frac{D_e^2+D_e'^2}{4} + l_i\frac{D_i'^2+D_i^2}{4} \right]. \tag{9} \]
Denoting by coefficients
\[ \kappa_{\ell}=0.5l_e(D_e^2+D_e'^2)+0.5l_i(D_i'^2+D_i^2), \]
\[ \kappa_a=l_a\left(\frac{1}{3}l_a^2+D_{\mathrm{cp}}^2\right). \]
and substituting expressions (7) and (9) into (8), we obtain
\[ GD^2 = N d h_{\mathrm{p}} b_{\mathrm{p}}\left[(k_0+1)\kappa_{\ell}+\beta\kappa_a\right]. \tag{10} \]
From (3), the optimum condition for the width of the active conductor will be
\[ \rho d\left(\frac{N}{2a}\right)^2 \frac{d}{d\beta} \left[(k_0+1)\kappa_{\ell}+\kappa_a\beta\right] \left(l_{\ell}+\frac{l_a}{\beta}\right)=0. \]
After simple transformations, we obtain the optimum value of the relative width of the active conductor from the condition of the minimum electromechanical time constant
\[ \beta_{01}= \sqrt{(k_0+1)\frac{\kappa_{\ell}l_a}{\kappa_a l_{\ell}}}. \tag{11} \]
To determine the length of the end parts of a half-turn, for known disk diameters the following approximate formulas may be recommended:
\[ l_e \simeq \frac{1}{2}\sqrt{(D_e-D'_e)^2+\left(\frac{\pi D_e}{2p}\right)^2}, \]
\[ l_i \simeq \frac{1}{2}\sqrt{(D'_i-D_i)^2+\left(\frac{\pi D'_i}{2p}\right)^2}. \]
The second contradiction is resolved by finding the optimal width of the active conductor, based on the minimum of the electrical losses in the rotor
\[ \frac{d}{d\beta}(P_{ml}+P_{ma}+P_v)=0, \tag{12} \]
where \(P_{ml}\) are the Joule losses from the load current in the end parts of the winding;
\(P_{ma}\) are the Joule losses from the load current in the active parts of the winding,
\(P_v\) are the eddy-current losses in the printed armature winding.
\(P_{ml}\) does not depend on \(\beta\) and is expressed by the constant coefficient
\[ P_{ml} \simeq \left(\frac{I_y}{2a}\right)^2 \frac{\rho(l_e+l_i)}{h_p b_l}=k_1. \tag{13} \]
Accordingly,
\[ P_{ma}\simeq \left(\frac{I_y}{2a}\right)^2 \frac{\rho l_a}{h b_p}\frac{1}{\beta} = \frac{k_2}{\beta}. \tag{14} \]
The eddy-current losses were considered in [6]; the expression obtained there can be written in the form
\[ P_v=k_3\beta^3, \tag{15} \]
where
\[ k_3=\frac{p^2 n^2 B_m^2 l_a h_p b_p^3}{2700\rho(1-\alpha_p)}. \tag{16} \]
From condition (12), after the corresponding transformations, we obtain the optimal value of the relative width of the active conductor from the condition of minimum electrical losses in the rotor,
\[ \beta_{02}=\sqrt[4]{1-\alpha_p}\sqrt{\frac{30\rho j_y}{b_p n B_m p}}, \tag{17} \]
where
\[ j_y=\frac{I_y}{2a h_p b_p} \]
is the permissible current density in the printed winding over the end parts.
The optimal values of the width of the active conductor from the conditions of the minimum electromechanical time constant (11) and from the condition of minimum rotor losses (17) do not, in the general case, coincide.
Therefore, for a rational choice of the width of the active conductor it is necessary to evaluate the variation of the electromechanical time constant \(\Delta T_m\) and the rotor losses \(\Delta P_p\) as a function of the deviation from the corresponding optimal values \(\beta_{01}\) and \(\beta_{02}\).
The corresponding relationships are
\[ \Delta T_m=f_1\left(\frac{\Delta\beta}{\beta_{01}}\right) \quad \text{and} \quad \Delta P_p=f_2\left(\frac{\Delta\beta}{\beta_{02}}\right). \]
can be obtained by expanding the dependences of the electromechanical time constant and the losses in the rotor on the relative width of the active conductor \(\beta\) in Taylor series in neighborhoods of the corresponding points \(\beta=\beta_{01}\) and \(\beta=\beta_{02}\)
\[ T_{\mathrm{m}}=T_{\mathrm{m}0}+T'_{\mathrm{m}0}(\beta-\beta_{01})+ \frac{T''_{\mathrm{m}0}}{2!}(\beta-\beta_{01})^2+ \frac{T'''_{\mathrm{m}0}}{3!}(\beta-\beta_{01})^3+\ldots , \tag{18} \]
where \(T_{\mathrm{m}0}\) is the optimal value of the electromechanical time constant at \(\beta=\beta_{01}\),
\[ T_{\mathrm{m}0}= \frac{\rho d\left(\dfrac{N}{2a}\right)^2}{375c_ec_m} \left[(\kappa_0+1)\kappa_l+\beta_{01}\kappa_a\right] \left(l_l+\frac{l_a}{\beta_{01}}\right) = \]
\[ = C\left[\kappa_a l_a+(\kappa_0+1)\kappa_l l_l+ \frac{2}{\beta_{01}}\kappa_l l_a(\kappa_0+1)\right], \tag{19} \]
\(T'_{\mathrm{m}0}\), \(T''_{\mathrm{m}0}\), \(T'''_{\mathrm{m}0}\), etc. are the values of the derivatives of the electromechanical time constant with respect to \(\beta\) at the optimal conductor width \(\beta=\beta_{01}\)
\[ \left. \begin{aligned} T'_{\mathrm{m}0}&=0,\\[4pt] T''_{\mathrm{m}0}&=C2!(\kappa_0+1)\kappa_l l_a\frac{1}{\beta_{01}^3},\\[4pt] T'''_{\mathrm{m}0}&=-C3!(\kappa_0+1)\kappa_l l_a\frac{1}{\beta_{01}^4}. \end{aligned} \right\} \tag{20} \]
Then the dependence of the deviation of the electromechanical constant as a function of the deviation \(\Delta\beta=\beta-\beta_{01}\) will have the form
\[ \frac{T_{\mathrm{m}}-T_{\mathrm{m}0}}{T_{\mathrm{m}0}} = \frac{l_a/\beta_{01}l_l}{\left(l_a/\beta_{01}l_l+1\right)^2} \left[ \left(\frac{\Delta\beta}{\beta_{01}}\right)^2 - \left(\frac{\Delta\beta}{\beta_{01}}\right)^3 +\ldots \right]. \tag{21} \]
Restricting ourselves, for \(\Delta\beta<\beta_{01}\), to the first term of the expansion, we obtain
\[ \frac{\Delta T_{\mathrm{m}}}{T_{\mathrm{m}0}} = \frac{l_a/\beta_{01}l_l}{\left(\dfrac{l_a}{\beta_{01}l_l}+1\right)^2} \left(\frac{\Delta\beta}{\beta_{01}}\right)^2 . \tag{22} \]
By an analogous expansion in a Taylor series of the function of active losses, after simple transformations we obtain
\[ \frac{\Delta P_p}{P_{p0}} \simeq \frac{1.5}{1+\dfrac{3}{4}\dfrac{\kappa_1}{\kappa_2}\beta_{02}} \left(\frac{\Delta\beta}{\beta_{02}}\right)^2 . \tag{23} \]
The ratio \(\dfrac{\kappa_1}{\kappa_2}\) represents the ratio of the active losses in the end and active parts for the same conductor width. Then, taking (13) and (14) into account, expression (23) will have the form
\[ \frac{\Delta P_p}{P_{p0}} \simeq \frac{1.5}{1+0.75\dfrac{l_l}{l_a}\beta_{02}} \left(\frac{\Delta\beta}{\beta_{02}}\right)^2 . \tag{24} \]
Thus, in designing electric motors with a printed disk armature, after selecting the principal dimensions of the disk in accord-
accordance with [3], the optimal width of the active conductor is determined from (11) and (17). Then, using (22) and (24), the degree of influence of the deviation of the width of the active conductor from the optimum on the inertial and thermal properties of the electric motor is determined, and, on the basis of their comparison, the final decision is made.
In most cases the optimum with respect to inertia has the predominant influence.
REFERENCES
-
V. A. Ratmirov. Electric motors with windings made by the printed method. Elektropromyshlennost i priborostroenie, No. 19, 1960.
-
V. M. Kazanskii. Some prospects for the design of dc power motors with printed windings. Proceedings of NEI, vol. III, 1961.
-
L. D. Osnovich. Selection of the optimal dimensions of a disk in dc machines with printed windings. Elektromekhanika, No. 6, 1961.
-
L. D. Osnovich. Elements of calculation of a dc motor with a printed winding on a disk armature. Dissertation for the degree of Candidate of Technical Sciences, 1962.
-
V. M. Kazanskii, L. D. Osnovich, Z. A. Osnovich, A. M. Shor, N. V. Kinsht. A dc commutator motor with a printed winding on the armature. Elektropromyshlennost, No. 10, 1962.
-
L. D. Osnovich, V. M. Kazanskii. Eddy-current losses in the rotor winding of a dc machine with a printed winding. Elektromekhanika, No. 4, 1963.