On the Calculation of Anode Current Dividers with Parallel Connection of Valves
Abstract Generated abstract
This paper develops a calculation method for anode current dividers used to ensure ignition of ionic valves connected in parallel, especially in ring circuits with two-core dividers. It argues that earlier formulas underestimate the required inductance for more than four valves because they assume equal current rise in all valves during ignition, and instead derives expressions for the minimum anode voltage under two ignition conditions: avalanche propagation and simultaneous ignition. Using a nine-valve example and extending the results to smaller numbers, the authors obtain coefficients for determining divider or reactor inductance from the minimum ignition voltage and load current rise rate. The results indicate that for more than four parallel valves, reactors may be preferable to two-core anode dividers for ensuring ignition with smaller required dimensions.
Full Text
On the Calculation of Anode Current Dividers in Parallel Connection of Valves
V. V. Ivashchin, L. I. Minenko
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The problem of ensuring parallel operation of ionic valves whose anode ignition voltage is considerably greater than the voltage drop across the valve during its operation is typical in many branches of electronic and converter technology. Parallel operation is usually ensured by connecting multi-rod or two-rod anode current dividers \[1, 2\], or reactors \[6\], into the anode circuits of the valves. A number of works have shown that, when the number of valves is large, instead of multi-rod anode dividers or instead of circuits with cascade current division \[3\], it is preferable to use two-rod anode dividers connected in a ring circuit \[4\]. At the same time, no analytical expressions have been obtained for a sufficiently accurate determination of the required inductance of the anode divider. The formulas \[4, 5\] give substantially underestimated values of the inductance of the anode divider when the number of parallel-connected valves is greater than 4, since in their derivation the crude assumption was made that the rate of rise of the current, or the magnitude of the current, is the same in all valves during the period of their ignition.
In the present work, what is considered is not the division of current among parallel-connected valves, which in many cases does not cause great technical difficulties, but rather the conditions for their simultaneous ignition. As an example, a circuit of 9 parallel-connected valves is considered. The results of the investigation for a smaller number of valves are summarized in tables.
In carrying out the calculations, the following assumptions were made:
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The voltage drop across a valve does not depend on the magnitude of the anode current.
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The anode dividers do not affect the rate of change or the magnitude of the current in the load.
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The inductances of the windings of all dividers are identical.
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The voltage drop across the active resistances of the anode divider is equal to zero.
Let us consider the operation of nine valves connected in a ring circuit (Fig. 1).
Until the instant of ignition, the full voltage of the power source is applied to the anodes of the valves. With a sufficiently large inductance of the anode dividers, the application of the clearing voltage to the control electrodes will cause ignition of the valves. Owing to the existing difference in the starting characteristics of the individual valves, their igni-
tion cannot be simultaneous, but occurs over a certain time interval [7]. We shall assume that this interval is less than the time during which saturation of the anode dividers is possible because of unequal current values in the individual valves. As ignition proceeds, the anode voltage on the valves that are not yet conducting current decreases. Ignition of the last valves occurs at anode voltages considerably lower than those of the first. The minimum value of the anode voltage \(U_m\) depends on the number of valves connected in parallel, the inductance of the anode dividers \(L\), the mutual inductance of the coils of the anode divider \(M\), and the rate of change of the current \(\dfrac{di}{dt}\) in the load circuit.
It is of interest to analyze the system for two different operating conditions.
Under the first condition it is assumed that, during the time interval between ignition of the first and the last device, on any of the devices that have not yet entered operation there will at some instant exist a voltage exceeding the minimum permissible ignition voltage.
Under the second condition it is assumed that, during the specified time interval, on any of the non-burning valves there is always a voltage sufficient for ignition of the valve.
The difference between these conditions is that under the first condition avalanche ignition of the valves is ensured, whereas under the second condition simultaneous ignition is ensured.
Condition 1 is characterized by the fact that when any device is ignited, the voltage on the anodes of the neighboring valves, i.e., those connected by anode dividers with the burning valve, will be greater than in all the others. Consequently, these devices ignite earlier than the others and cause an increase in the voltage on the anodes of neighboring valves. Such a process propagates in avalanche fashion, from valve to valve, and ends with ignition of the last one. With a large number of valves connected in parallel, several avalanches may exist. The ignition time of all valves depends on the rate of propagation of the avalanche process, which is determined by the lag of the anode current relative to the anode-voltage pulse. Suppose that 8 valves are burning and that it is necessary to ensure the ignition condition of the last valve, for example valve 5 (Fig. 1).
The voltage on valve 5 is determined by the following system of equations:
\[ U_5 = U_1 + 2L_1 \frac{di_1}{dt} - M \frac{di_2}{dt} - M \frac{di_9}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
\[ U_5 = U_2 + 2L_1 \frac{di_2}{dt} - M \frac{di_1}{dt} - M \frac{di_3}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
\[ U_5 = U_3 + 2L_1 \frac{di_3}{dt} - M \frac{di_2}{dt} - M \frac{di_4}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
\[ U_5 = U_4 + 2L_1 \frac{di_4}{dt} - M \frac{di_3}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
\[ U_5 = U_6 + 2L_1 \frac{di_6}{dt} - M \frac{di_7}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \tag{1} \]
\[ U_5 = U_7 + 2L_1 \frac{di_7}{dt} - M \frac{di_6}{dt} - M \frac{di_8}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
Fig. 1. Circuit for parallel connection of valves with two-core anode dividers
\[ U_5 = U_8 + 2L_1 \frac{di_8}{dt} - M \frac{di_7}{dt} - M \frac{di_9}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
\[ U_5 = U_9 + 2L_1 \frac{di_9}{dt} - M \frac{di_8}{dt} - M \frac{di_1}{dt} + M \frac{di_4}{dt} + M \frac{di_6}{dt}, \]
where \(U_1, U_2 \ldots, U_9\) are the voltage drops across the corresponding valve. Taking into account that, in the ring circuit, the rate of change of the current in valves arranged symmetrically with respect to the fifth can be regarded as identical, that is,
\[ \frac{di_1}{dt} = \frac{di_9}{dt}; \qquad \frac{di_2}{dt} = \frac{di_8}{dt}; \qquad \frac{di_3}{dt} = \frac{di_7}{dt}; \qquad \frac{di_4}{dt} = \frac{di_6}{dt}, \tag{2} \]
and that \(U_1 = U_2 = U_3 = U_4 = U\), the system of eight equations can be reduced to four.
\[ U_M = (2L_1 - M)\frac{di_1}{dt} - M\frac{di_2}{dt} + 0 + 2M\frac{di_4}{dt}, \]
\[ U_M = - M\frac{di_1}{dt} + 2L_1\frac{di_2}{dt} - M\frac{di_3}{dt} + 2M\frac{di_4}{dt}, \]
\[ U_M = + 0 - 2M\frac{di_2}{dt} + 2L_1\frac{di_3}{dt} + M\frac{di_4}{dt}, \tag{3} \]
\[ U_M = + 0 + 0 - M\frac{di_3}{dt} + 2(L_1 + M)\frac{di_4}{dt}, \]
where \(U_M = U_5 - U\).
The derivative of the current in the unbranched circuit:
\[ \frac{di}{dt} = \frac{di_1}{dt} + \frac{di_2}{dt} + \frac{di_3}{dt} + \frac{di_4}{dt} + \frac{di_6}{dt} + \frac{di_7}{dt} +\cdots \]
\[ + \frac{di_8}{dt} + \frac{di_9}{dt}. \tag{5} \]
Taking (2) into account, equation (5) can be written as
\[ \frac{di}{dt} = 2\left( \frac{di_1}{dt} + \frac{di_2}{dt} + \frac{di_3}{dt} + \frac{di_4}{dt} \right). \tag{6} \]
The terms of equation (6) are found by solving the system of equations (4) by Cramer’s rule [8].
\[ \frac{di}{dt} = 2\left( \frac{\Delta_1+\Delta_2+\Delta_3+\Delta_4}{\Delta} \right), \tag{7} \]
where \(\Delta_1,\Delta_2,\Delta_3,\Delta_4,\Delta\) are the determinants of the system of equations (3). Solution (7) makes it possible to find
\[ \frac{di}{dt} = \frac{ 4U_{\mathrm{m}}\left(16L_1^{3}+6L_1^{2}M-6L_1M^{2}-M^{3}\right) }{ 16L_1^{4}+8L_1^{3}M-12L_1^{2}M^{2}-4L_1M^{3}+M^{4} }. \tag{8} \]
It is evident from equation (8) that, for specified values of the rate of change of current in the load \(\frac{di}{dt}\) and of the minimum required voltage \(U_{\mathrm{m}}\), the value of the inductance of one coil of the anode divider depends on the coefficient of mutual inductance \(M\). The value of the inductance can be represented in the form
\[ L_1=\eta_1\frac{U_{\mathrm{m}}}{\left(\dfrac{di}{dt}\right)}, \tag{9} \]
where \(\eta_1=f(M)\).
Under the second condition, it is necessary to ensure the possibility of a simultaneous, rather than avalanche, process of ignition of all valves. From consideration of the circuit in Fig. 1 it follows that the worst ignition conditions will occur in the case when only three valves are not conducting, while the remaining ones are conducting. In this case the smallest anode voltage will be at the “middle” valve, into whose anode circuit no additional voltages are transformed.
Naturally, this assertion is valid only in the case when the total number of valves connected for parallel operation is sufficiently large.
Suppose that, of 9 valves, only six are conducting \((1,2,3,7,8,9)\) (Fig. 1).
It is necessary to ensure ignition of valve 5.
Taking into account that \(i_4=i_5=i_6=0\),
\[ \frac{di_4}{dt} = \frac{di_5}{dt} = \frac{di_6}{dt} = 0, \]
\[ i_1=i_9,\qquad i_2=i_8,\qquad i_3=i_7, \]
from (1), after transformations, we obtain
\[ U_{\mathrm{m}} = (2L_2-M)\frac{di_1}{dt} + M\frac{di_2}{dt}, \]
\[ U_{\mathrm{m}} = M\frac{di_1}{dt} + 2L_2\frac{di_2}{dt} - M\frac{di_3}{dt}, \tag{10} \]
\[ U_{\mathrm{m}} = 0 - M\frac{di_2}{dt} + 2L_2\frac{di_3}{dt}. \]
Taking into account that
\[ \frac{di}{dt} = 2\left( \frac{di_1}{dt} + \frac{di_2}{dt} + \frac{di_3}{dt} \right), \]
in solving (10) we obtain:
\[ \frac{di}{dt} = 4U_{\mathrm{m}}\, \frac{6L_2^{\,2}+2L_2M-M^2} {4L_2^{\,2}(2L_2-M)-2L_2M^2(2L_2-M)M_1^{\,2}} . \tag{11} \]
From (11), for any particular value of \(M\), one can determine \(L_2\):
\[ L_2=\eta_2\,\frac{U_{\mathrm{m}}}{\left(\dfrac{di}{dt}\right)}, \tag{12} \]
where \(\eta_2\) is a coefficient that uniquely characterizes the magnitude of the inductance of one winding of the anode divider. The minimum anode ignition voltage and the rate of change of the current in the load at the instant of valve ignition are usually known; therefore the problem of determining the inductance \(L_1\) or \(L_2\) from (9) and (12) reduces to determining the coefficients \(\eta_1\) and \(\eta_2\).
From (8) and (11) one can calculate the coefficients for any values of \(M\), but the cases of greatest interest are those in which \(M=L\) \((\eta_{1a}, \eta_{2a})\) and \(M\approx 0\) \((\eta_{1p}, \eta_{2p})\). Actual designs of devices that ensure ignition of valves when they are connected in parallel satisfy one of these conditions. The case \(M=L\) corresponds to the parallel operation of valves when anode dividers connected in a ring circuit, with a coupling coefficient between the windings equal to unity, are included in their anode circuits. The dimensions of the magnetic core of the anode divider must be determined taking into account that it has two windings, the inductance of each of which is determined by formulas (9) or (12). In calculations, the inductances of the two windings can be reduced to one equivalent inductance
\[ L_{\mathrm{ea}}=4L, \tag{13} \]
where \(L\) is determined by the indicated formulas.
For the coefficients \(\eta\) corresponding to these equivalent inductances, we obtain
\[ \eta_{\mathrm{ea}}=4\eta_a, \tag{14} \]
where \(\eta_a\) is determined from (8) or (11) for \(M=L\). For \(M\approx 0\), the ring circuit of Fig. 1 is transformed into a circuit in which the parallel operation of the valves is ensured by reactors having no inductive coupling with one another. It should be taken into account that, in the anode circuits according to Fig. 1, there are then two inductances each. Therefore the inductance that determines the dimensions of the reactor is
\[ L_{\mathrm{er}}=2L. \tag{15} \]
For the coefficients \(\eta\) corresponding to the inductance \(L_{\mathrm{er}}\), we obtain
\[ \eta_{\mathrm{er}}=2\eta_p, \tag{16} \]
where \(\eta_p\) is determined from (8) or (11) for \(M=0\).
Tables 1 and 2 give formulas for calculating the coefficients \(\eta_1, \eta_2\) for the cases \(M\ne L\), and their numerical values for \(M=0\) and \(M=L\), as well as the values of the equivalent coefficients \((\eta_{1\mathrm{er}}, \eta_{1\mathrm{ea}}, \eta_{2\mathrm{er}}, \eta_{2\mathrm{ea}})\), which characterize the dimensions of the anode dividers and anode reactors that ensure ignition of valves connected in parallel.
Figure 2 shows curves of the equivalent coefficients obtained from conditions I and II. It follows from these curves that:
- In the case of using anode dividers, simultaneous ignition of the valves will be ensured if the value of the inductance is determined from curve 2 when the number of valves is \(N\le 5\), and from curve 4 when \(N\ge 5\).
Table 1
| Number of parallel-connected valves \(N\) | General expression for \(\eta_1\) | Coefficient \(\eta_1\), \(\eta_1^{p}\) \((M=0)\) | Coefficient \(\eta_1\), \(\eta_1^{a}\) \((M=L_1)\) | Equivalent values \(\eta_1^{\mathrm{eq}}\) \((M=0)\) | Equivalent values \(\eta_1^{a}\) \((M=L_1)\) |
|---|---|---|---|---|---|
| 3 | \(\displaystyle \frac{2L_1}{2L_1+M}\) | 1 | 0,67 | 2 | 2,67 |
| 4 | \(\displaystyle \frac{(3L_1+2M)L_1}{2L_1(L_1+M)}\) | 1,5 | 1,25 | 3 | 5 |
| 5 | \(\displaystyle \frac{2(4L_1+M)L_1}{4L_1^{2}+2L_1M-M^{2}}\) | 2 | 2 | 4 | 8 |
| 6 | \(\displaystyle \frac{(20L_1^{2}+6L_1M-M^{2})L_1}{2(4L_1^{3}+4L_1^{2}M-L_1M^{2}-M^{3})}\) | 2,5 | 2,92 | 5 | 11,6 |
| 7 | \(\displaystyle \frac{4(6L_1^{2}+2L_1M-M^{2})L_1}{8L_1^{3}+4L_1^{2}M-4L_1M-M^{3}}\) | 3 | 4 | 6 | 16 |
| 8 | \(\displaystyle \frac{(28L_1^{3}+21L_1^{2}M-6L_1M^{2}-M^{3})L_1}{4(2L_1^{4}+2L_1^{3}M-L_1^{2}M^{2}-L_1M^{3})}\) | 3,5 | 5,25 | 7 | 21 |
| 9 | \(\displaystyle \frac{4(16L_1^{3}+6L_1^{2}M-6L_1M^{2}-M^{3})L_1}{16L_1^{4}+8L_1^{3}M-12L_1^{2}M^{2}-4L_1M^{3}+M^{4}}\) | 4 | 6,67 | 8 | 26,7 |
Table 2
| Number of valves connected in parallel \(N\) | General expressions for \(\eta_2\) | Coefficient \(\eta_2\): \(\eta_{2p}\) \((M=0)\) | Coefficient \(\eta_2\): \(\eta_{2a}\) \((M=L_2)\) | Equivalent values \(\eta_{2e}\): \(\eta_{2p e}\) \((M=0)\) | Equivalent values \(\eta_{2e}\): \(\eta_{2a e}\) \((M=L_2)\) |
|---|---|---|---|---|---|
| 4 | \(\dfrac{1}{2}\) | 0.5 | 0.5 | 1 | 2 |
| 5 | \(\dfrac{2}{2L_2-M}\cdot L_2\) | 1 | 2 | 2 | 8 |
| 6 | \(\dfrac{3L_2+2M}{2L_2^2-M^2}\cdot L_2\) | 1.5 | 5 | 3 | 20 |
| 7 | \(\dfrac{2(4L_2^2+M)}{4L_2^2-2L_2M-M^2}\cdot L_2\) | 2 | 10 | 4 | 40 |
| 8 | \(\dfrac{20L_2^2+16L_2M-M^2}{2(4L_2^3-3L_2M^2)}\cdot L_2\) | 2.5 | 17.5 | 5 | 70 |
| 9 | \(\dfrac{4(6L_2^3+2L_2M-M^2)\cdot L_2}{4L_2^2(2L_2-M)-2L_2M^2(2L_2-M)}\) | 3 | 28 | 6 | 112 |
Fig. 2. Dependence of the inductance of anode dividers on the number of valves connected in parallel under various conditions: 1 — coefficient \(\eta_{1er}\) of the reactor under condition I; 2 — coefficient \(\eta_{1ea}\) of the anode divider under condition I; 3 — coefficient \(\eta_{2er}\) of the reactor under condition II; 4 — coefficient \(\eta_{2ea}\) of the anode divider under condition II.
- To ensure simultaneous ignition of valves when their number is more than four, it is advisable to use not two-core anode dividers, but reactors. In this case the dimensions of the reactors are smaller than those of the anode dividers. The reactors should be calculated with allowance for condition I.
LITERATURE
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