PARAMETERS OF DC MACHINES IN THE TRANSITION FROM CONVENTIONAL COMMUTATION TO VALVE-MECHANICAL COMMUTATION

O. A. Bratkovsky, V. V. Ivashin, I. A. Miloradov

(Submitted by the scientific seminar of the Departments of Electrical Machines
and General Electrical Engineering)

Let us consider the change in the parameters of a direct-current machine in the transition from the conventional commutation circuit (Fig. 1) to one of the promising circuits of valve-mechanical commutation (VMC) (Fig. 2), with valves included in the commutated section for the entire period of its short-circuiting by the subdivided brush and with the valves loaded by the full current of the brush yoke [1]. It is assumed that the basic parameters (power, voltage, rotational speed, number of poles, geometry and principal dimensions of the active parts, commutator diameter) remain unchanged. In what follows, the parameters of the machine with conventional commutation are denoted by the subscript “1,” and those of the machine with VMC by the subscript “2.”

Initial assumptions

Let us introduce the following notation: \(b\) is the brush width, \(\beta\) is the number of commutator plates covered by the brush, \(t\) is the commutator pitch, \(Z\) is the number of armature slots, \(U_p\) is the number of commutator plates per slot.

In machines with conventional commutation,

\[ \frac{b_1}{t_1} = \beta_1 > 1, \tag{1} \]

In the VMC circuit (Fig. 2), the commutator is doubled and consists of alternating working and idle plates of identical size. In this case

\[ \frac{b_2}{t_2} = \beta_2 = 1, \tag{2} \]

Fig. 1. Conventional commutation circuit:
1 — brush, 2 — commutator, 3 — armature section

It is known [2] that each armature diameter corresponds to its own optimum number of slots. Therefore, if \(Z_1\) has been chosen correctly, then, in the transition to VMC, it is necessary that

\[ Z_2 \approx Z_1 . \tag{3} \]

Taking into account that usually the ratio \(\dfrac{u_{\mathrm{p}1}}{\beta_1}\) is sufficiently close to unity, with allowance for (1)—(3) one may write

\[ \frac{b_2}{b_1} \approx \frac{u_{\mathrm{p}1}\cdot z_1\cdot \beta_2} {u_{\mathrm{p}2}\cdot z_2\cdot \beta_1} \approx \frac{1}{u_{\mathrm{p}2}} . \tag{4} \]

It follows from (4) that, under the adopted conditions, \(b_2/b_1\) cannot be greater than unity. On the basis of the lowest current density in the brush contact, from (4) we obtain the initial relationships for the transition to the VMC circuit (Fig. 2):

\[ b_2 \approx b_1,\quad U_{\mathrm{p}2}=1 . \tag{5} \]

In individual cases deviations from (5) are possible; however, the greatest interest is in analyzing the change of parameters under the condition

\[ b_2=b_1, \tag{6} \]

\[ U_{\mathrm{p}2}=1. \tag{7} \]

Fig. 2. Circuit of valve-mechanical commutation:
1 — subdivided brush,
2 — doubled collector,
3 — armature section,
4 — valve

Number of commutator bars \((K)\)

Taking into account (2), (6), we obtain the number of working commutator bars on the doubled collector (Fig. 2)

\[ K_2=\frac{K_1}{\beta_1}. \tag{8} \]

\(K_2\) must be rounded to the nearest even number. In doing so, we obtain \(b_2 \ne b_1\). However, because of the smallness of this change, it may be neglected in the subsequent analysis.

Current density in the brush contact \((j_{\mathrm{щ}})\) and collector length

Let us consider only the simplest case—linear commutation, when the current density in the brush contact of a machine with VMC is constant and equal to

\[ j_{\mathrm{shch}2}=\frac{I_{\mathrm{shch}2}}{S_{\mathrm{shch}2}}. \]

Taking into account that \(I_{\mathrm{shch}2}=I_{\mathrm{shch}1}\) and \(b_2=b_1\), we obtain

\[ \frac{j_{\mathrm{shch}2}}{j_{\mathrm{shch}1}} = \frac{l_{\mathrm{shch}1}}{l_{\mathrm{shch}2}} \simeq \frac{l_{\mathrm{k}1}}{l_{\mathrm{k}2}}. \tag{9} \]

Here \(I_{\mathrm{shch}}\) is the maximum current through the brush, \(S_{\mathrm{shch}}\) is the contact area of the brush with the commutator, \(l_{\mathrm{shch}}\) is the brush length (Fig. 3), and \(l_{\mathrm{k}}\) is the length of the working surface of one commutator (Fig. 3).

Fig. 3. Arrangement of brushes on the commutator under conventional commutation (a) and under VMC (b)

Number of armature slots \((z)\)

Taking (2) into account, we obtain

\[ z_2=\frac{u_{\mathrm{p}1}}{\beta_1\cdot u_{\mathrm{p}2}}\cdot z_1 = \frac{u_{\mathrm{p}1}}{\beta_1}\cdot z_1. \tag{10} \]

Armature winding

In machines with the VMC circuit (Fig. 2), the use of both lap and wave windings is in principle possible (single-circuit windings are considered). But under VMC the number of winding sections (as well as the number of working commutator segments) must be even. This requirement imposes a certain limitation on the ratio between the number of commutator segments \(K_2\) and the number of pole pairs \(p_2\).

Width of the commutation zone \((b_{\mathrm{kz}})\)

Taking into account that

\[ b_{\mathrm{kz}} = b\frac{D}{D_{\mathrm{K}}} + \left( u_{\mathrm{p}}+\varepsilon_{\mathrm{k}}-\frac{a}{p} \right) t\cdot\frac{D}{D_{\mathrm{K}}}\cdot[2], \]

\[ t\simeq\frac{b}{\beta},\qquad D_2=D_1,\qquad D_{\mathrm{K}2}=D_{\mathrm{K}1},\qquad \beta_2=1, \]

after transformation we obtain

\[ b_{\mathrm{kz}2} = \frac{ 1+\left(u_{\mathrm{p}2}+\varepsilon_{\mathrm{k}2}-\frac{a_2}{p_2}\right) }{ 1+\left(u_{\mathrm{p}1}+\varepsilon_{\mathrm{k}1}-\frac{a_1}{p_1}\right)\frac{1}{\beta_1} } \cdot b_{\mathrm{kz}1}. \tag{11} \]

Here \(a\) is the number of pairs of parallel branches of the armature winding,

\(p\) is the number of pairs of machine poles,

\(D\) is the armature diameter,

\(D_k\) is the commutator diameter,

\(\varepsilon\) is the shortening of the winding pitch in commutator segments.

Table 1 gives values of \(b_{k32}/b_{k31}\), calculated from (11) with account of (2), (6), for the most common ranges of parameter variation: \(U_{n1}=2—4\); \(\beta_1=(0{,}75—1{,}0)U_{n1}\); \(\varepsilon_k \leq U_{\mathrm{g}}\); \(\varepsilon_{k1}=0—(2—4)\); \(\varepsilon_{k2}=0—1\); \(p_2=p_1=1—3\); \(\dfrac{a_1}{p_1}=\dfrac{a_2}{p_2}=1;\ 1/3\).

Table 1

Number of turns in a section \((W_s)\)

Since, in the transition to VMC, the armature emf does not change, while the divisibility of the armature electric loading decreases by a factor of \(k_2/k_1\), then, taking (2) into account,

\[ w_{s2}=\frac{a_2\cdot K_1}{a_1\cdot K_2}\cdot w_{s1} =\frac{a_2}{a_1}\cdot \beta_1\cdot w_{s1}. \tag{12} \]

Active resistance of a section \((r_s)\)

To a first approximation, it may be assumed that, in the transition to VMC, the copper filling factor of the active layer, the mean turn length, and the armature-winding temperature remain unchanged, while the cross section of the effective conductor changes in proportion to \(a_2/a_1\).

\[ r_{s2}=\frac{a_2\cdot w_{s2}}{a_1\cdot w_{s1}}\cdot r_{s1} =\left(\frac{a_2}{a_1}\right)^2\cdot \beta_1\cdot r_{s1}. \tag{13} \]

Inductance of a section \((L_s)\)

It is known [3] that

\[ L_s=2w_s^2\cdot l\cdot \lambda, \]

\[ \lambda=\lambda_p+\lambda_{\mathrm{lb}}\cdot \frac{l_{\mathrm{lb}}}{l}, \tag{14} \]

where \(l\) is the length of the armature stack, \(l_{\mathrm{lb}}\) is the length of the end part of the section, \(\lambda\) is the specific magnetic permeance of the section (per unit length of the armature stack) without taking into account mutual inductance with other sections in the slot; \(\lambda_p\), \(\lambda_{\mathrm{lb}}\) are the magnetic permeances of the slot and end parts of the section per unit of their length.

In the transition to VMC, taking (12) into account, we obtain

\[ L_{s2}=\frac{w_{s2}^{2}\lambda_{2}}{w_{s1}^{2}\lambda_{1}}\cdot L_{s1} =\left(\frac{a_{2}}{a_{1}}\cdot \beta_{1}\right)^{2}\cdot \frac{\lambda_{2}}{\lambda_{1}}\,L_{s1}. \]

In the first approximation,

\[ \frac{\lambda_{2}}{\lambda_{1}}=1;\qquad L_{s2}=\left(\frac{a_{2}}{a_{1}}\cdot \beta_{1}\right)^{2}\cdot L_{s1}. \tag{15} \]

Resultant inductance of a section \((L_{sp})\)

We shall carry out the comparison on the basis of the generally accepted principles of straight-line commutation [3].

By analogy with (14) we have

\[ L_{sp}=2\cdot W_s^2\cdot l\cdot \xi, \tag{16} \]

\[ \xi=\lambda'_p+\lambda_{lb}\cdot \frac{l_{lb}}{l} = k'\cdot \lambda_p+\lambda_{lb}\cdot \frac{l_{lb}}{l}. \]

Here

\(\xi\) is the average specific magnetic permeance of the section (per unit length of its slot part), taking mutual induction into account,

\(\lambda'_p\) is the average specific magnetic permeance of the slot part of the section, taking mutual induction into account,

\(k'\) is a coefficient accounting for the change in \(\lambda_p\) due to the presence of mutual induction.

In the transition to VMC, taking (16) into account, we obtain

\[ L_{sp2}= \frac{w_{s2}^{2}\xi_{2}}{w_{s1}^{2}\xi_{1}}\cdot L_{sp1} = \left(\frac{a_{2}}{a_{1}}\beta_{1}\right)^{2}\cdot \frac{\xi_{2}}{\xi_{1}}\cdot L_{sp1}. \]

For analysis in the first approximation one may take

\[ \lambda_{lb2}\cdot \frac{l_{lb2}}{l_{2}} \approx \lambda_{lb1}\cdot \frac{l_{lb1}}{l_{1}} \approx 0; \qquad \lambda_{p2}=\lambda_{p1}. \]

In this case

\[ \frac{\xi_{2}}{\xi_{1}}=\frac{k'_2}{k'_1}; \qquad L_{sp2}=\left(\frac{a_{2}}{a_{1}}\cdot \beta_{1}\right)^{2}\cdot \frac{k'_2}{k'_1}\cdot L_{sp1}. \tag{17} \]

On the basis of the data of [2], Table 2 gives the limits of variation of the ratio \(k'_2/k'_1\) for \(U_{p1}=2—4\); \(\beta_1=(0.75—1.0)U_{p1}\); \(\varepsilon_{k1}=0—(2—4)\); \(\varepsilon_{k2}=0—1\).

Table 2

Commutating emf in a section \((e_k)\)

With straight-line commutation,

\[ e_k=L_{sp}\cdot \frac{2i_a}{T}, \]

where \(i_a\) is the current in a parallel branch of the armature winding,

\(T\) is the commutation period of a section.

In the transition to VMC

\[ i_{a2}=i_{a1}\cdot \frac{a_1}{a_2};\qquad T_2=T_1, \]

Taking into account (17), and also the fact that in practice \(e_k\) is increased by \(k_y\) times in order to create accelerated commutation, and \(k_{y2}\) is usually greater than \(k_{y1}\), we obtain

\[ e_{k2}=\frac{a_2}{a_1}\cdot \beta_1^2\cdot \frac{k_{y2}\cdot k'_2}{k_{y1}\cdot k'_1}\cdot e_{k1}, \tag{18} \]

where \(k_y\) is the commutation-acceleration coefficient.

Number of turns of the interpole winding \((W_d)\)

In normally designed machines, the magnetic circuit of the interpoles must be unsaturated, and the drop of magnetomotive force in its steel parts may be neglected. Then

\[ w_d=\frac{F_{ra}+F_{\delta d}}{I_a};\qquad F_{\delta d}=m\cdot B_{\delta d};\qquad B_{\delta d}=\frac{e_k}{2w_s\cdot V\cdot l}. \tag{19} \]

Here

\[ \begin{aligned} F_{ra} &\text{— m.m.f. of armature reaction }(F_{ra2}=F_{ra1}),\\ F_{\delta d} &\text{— m.m.f. of the air gap under the interpole,}\\ I_a &\text{— armature current }(I_{a2}=I_{a1}),\\ B_{\delta d} &\text{— induction in the air gap under the interpole,}\\ m &\text{— coefficient of proportionality between }F_{\delta d}\text{ and }B_{\delta d}\;(m_2=m_1),\\ V &\text{— linear velocity of the armature }(V_2=V_1). \end{aligned} \]

After transformations, taking into account (12), (18), (19), we obtain

\[ B_{\delta d2}=\beta_1\frac{k_{y2}\cdot k'_2}{k_{y1}\cdot k'_1}B_{\delta d1};\qquad w_{d2}= \frac{ 1+\beta_1\cdot \frac{k_{y2}\cdot k'_2}{k_{y1}\cdot k'_1} \cdot \frac{F_{\delta d1}}{F_{ra1}} }{ 1+\frac{F_{\delta d1}}{F_{ra1}} } \cdot w_{d1}. \tag{20} \]

Cross section of the interpole \((S_d)\), the armature back \((S_a)\), and the inductor yoke \((S_i)\)

As indicated above, in the transition to VMC the width of the commutation zone changes (Table 1) and the induction in the gap under the interpole changes (20). It is obvious that, in the transition to VMC, the flux in the magnetic sections of the interpole circuit will change by

\[ \beta_1\cdot \frac{k_{y2}\cdot k'_2 b_{kz2}}{k_{y1}\cdot k'_1\cdot b_{kz1}} = A \text{ times}. \]

If the inductions in the magnetic sections of the interpole circuit in a machine with conventional commutation have been chosen correctly (there is no saturation), then, proceeding from their preservation, we obtain

\[ S_{\mathrm{d}2}=A S_{\mathrm{d}1}; \qquad S_{\mathrm{a}2(j2)}= \frac{1+A\dfrac{\Phi_{\mathrm{d}1}}{\Phi_{\mathrm{g}1}}} {1+\dfrac{\Phi_{\mathrm{d}1}}{\Phi_{\mathrm{g}1}}}\cdot S_{\mathrm{a}1(j1)}, \tag{21} \]

where \(\Phi_{\mathrm{d}}\) and \(\Phi_{\mathrm{g}}\) are the magnetic fluxes of the interpole and the main pole.

Table 3 gives, as an example, values found with account taken of (1)—(21) and Tables 1 and 2 (ratios of machine parameters for \(U_{\mathrm{n}1}=2—4\); \(\beta_1=(0.75—1.0)U_{\mathrm{n}1}\); \(\varepsilon_{\mathrm{k}}\le U_{\mathrm{n}}\); \(a_2=a_1\); \(b_2=b_1\); \(l_{\mathrm{k}2}=0.5l_{\mathrm{k}1}\); \(U_{\mathrm{n}2}=\beta_2=1\); \(k_{\mathrm{y}1}=1.15\); \(k_{\mathrm{y}2}=1.5—2\); \(\dfrac{\Phi_{\mathrm{d}1}}{\Phi_{\mathrm{g}1}}=0.1—0.2\); \(F_{\mathrm{d}1}/F_{\mathrm{ry}1}=0.05—0.2\). The figures in Table 3 on the left refer to values \(\beta_1=1.5\), and those on the right to \(\beta_1=4\).

Table 3

Conclusions

1. As is seen from Table 3, in the transition from conventional commutation to VMC (Fig. 2), the following changes occur in the machine parameters, depending mainly on \(\beta_1\):

a) the number of commutator bars and the number of sections decreases, i.e., the subdivision of the armature winding decreases, which, naturally, will lead to some increase in the ripple of the emf and the armature current;

b) the current density in the brush contact \(j_{\mathrm{sh}}\) increases, with unchanged commutator dimensions (or its dimensions increase, with unchanged current density). It should be noted that maintaining \(j_{\mathrm{sh}}\) in the transition to VMC is not always obligatory. It is known [4] that this VMC circuit can operate normally at \(j_{\mathrm{sh}}\) up to \(50\ \mathrm{A/cm^2}\), i.e., the double commutator may, in terms of dimensions, be no different from an ordinary one;

c) the number of turns \(W_s\), the active resistance \(r_s\), and the inductance \(L_s\) of the armature sections increase. It is known [4, 5] that an increase in \(W_s\) and \(L_s\) does not cause deterioration of commutation as it does in ordinary machines. An increase in \(r_s\) somewhat worsens commutation when the machine is started in motor mode;

d) in most cases the following increase: the resulting inductance of the sections \(L_{\mathrm{sp}}\), the induction under the interpole \(B_{\mathrm{d}}\), and the number of turns of the interpole \(W_{\mathrm{d}}\). Their increase can be limited

by shortening the winding pitch under VMC, \(\varepsilon_{k2}=1\). With an increase in \(W_d\), the dimensions of the additional-pole winding also increase somewhat;

d) the commutating emf in the section, \(e_k\), increases, which is physically explained by the increase in \(W_s\), \(L_{sp}\), and by the need, under VMC, for more accelerated commutation. The increase in \(e_k\), like that in \(W_s\), does not cause any deterioration of the commutation conditions;

e) the width of the commutation zone \(b_{k3}\) decreases when \(\varepsilon_{k2}=0\), and when \(\varepsilon_{k2}=1\) in individual cases it increases;

zh) the cross sections of the additional-pole core \(S_d\), the armature back \(S_a\), and the yoke of the inductor \(S_i\) may increase or decrease. On average, their possible increase is smaller than that of \(e_k\) or \(B_{\delta d}\).

1. The relations \(b_2 \approx b_1\) and \(U_{n2}=1\) may be used as initial ones in determining the parameters of a machine with a VMC circuit (Fig. 2).

2. In the transition to VMC, there is no noticeable deterioration of the characteristics or substantial increase in the weight of the machine, while its commutation stability is significantly increased [5].

REFERENCES

1. O. A. Bratkovskii, V. V. Ivashin, I. A. Miloradov. Circuits for valve-mechanical commutation of direct-current machines (see the present collection).

2. P. S. Sergeev et al. Design of Electrical Machines, Gosenergoizdat, 1956.

3. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, Part 1. “Energiya,” 1964.

4. Bates I., Sridhar T., Thyristor-assisted sliding contact commutation, Proc. Inst. El. Engrs, 1966, Vol. 113, No. 2.

5. O. A. Bratkovskii, V. V. Ivashin, I. A. Miloradov. Experimental verification of a direct-current machine with valve-mechanical commutation (see the present collection).