ON A CERTAIN RING CIRCUIT

T. Yu. Mogilevskaya, G. E. Pukhov

In [1] the question of the ring connection of interacting oscillatory circuits is considered (Chapter IV, § 3).

After passing to the equivalent circuit of the network, the problem is solved by the equations:

\[ \left[\frac{U_{N+1}}{I_{N+1}}\right] = M_1^N \left[\frac{U_1}{I_1}\right] = \left[\frac{U_1}{I_1}\right], \tag{1} \]

where \(N+1\) is the number of sections, and

\[ M= \begin{bmatrix} \cos \varphi & j Z_0 \sin \varphi\\ j Y_0 \sin \varphi & \cos \varphi \end{bmatrix}. \tag{2} \]

The method of calculation contains certain inaccuracies. First, the equations given are fully valid only for a through four-terminal network. For the circuit under consideration, however, they are applicable only in the case when it is necessary to know the relation between the input and output quantities, while the remaining quantities (currents in the branches, total voltage) are not of interest. Below the possibility of such an assumption, not noted in [1], is shown. Second, in forming the matrix of a section an error was made which affected the results.

Let us present the exact calculation of the circuit, dividing it into sections as shown in Fig. 1. The equivalent circuit of one section is given in Fig. 2.

We carry out the calculation according to equation [2]:

\[ \begin{bmatrix} U_1\\ I_a\\ U_c \end{bmatrix} = A \begin{bmatrix} U_2\\ -I_b\\ I_0 \end{bmatrix}, \tag{3} \]

where \(I_0=-(I_a+I_c)\).

We represent the section as a series connection of three four-terminal networks, whose chain matrices are equal to:

\[ a_1=a_3= \begin{bmatrix} 1 & 0 & 0\\ \dfrac{1}{2jM\omega} & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}, \tag{4} \]

$$ a_2 = \begin{vmatrix} 1 & j\omega(L-2M)-\dfrac{j}{C\omega} & \dfrac{j}{C\omega} \\ 0 & 1 & 0 \\ 0 & -\dfrac{j}{C\omega} & \dfrac{j}{C\omega} \end{vmatrix} \tag{4.} $$

Fig. 1

Schematic labels: \(M\), \(L\), \(M\), \(L\), \(M\), \(L\), \(M\); \(C\), \(C\), \(C\); I link, II link, III link, and so on.

Fig. 2

Schematic labels: \(I_a\), \(I_b\), \(I_c\); \(u_1\), \(u_2\), \(u_c\); \(2M\), \(L-2M\), \(2M\), \(C\).

Passing to extended chain matrices,

$$ A_1=A_3= \begin{vmatrix} 1 & 0 & 0 & 0 \\ \dfrac{1}{j\,2M\omega} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{vmatrix}, $$

$$ A_2= \begin{vmatrix} 1 & j\omega(L-2M)-\dfrac{j}{C\omega} & 0 & \dfrac{j}{C\omega} \\ 0 & 1 & 0 & 0 \\ 0 & -\dfrac{j}{C\omega} & 1 & \dfrac{j}{C\omega} \\ 0 & 0 & 0 & 1 \end{vmatrix}, \tag{5} $$

by the formula

$$ A_{\text{link}}=A_1A_2A_3 \tag{6} $$

we obtain the extended matrix of the section

\[ A_{\mathrm{zv}}= \left[ \begin{array}{c|c|c|c} 1+\dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}} & j^{\omega}(L-2M)-\dfrac{j}{C\omega} & 0 & \dfrac{j}{C\omega} \\ \hline \dfrac{1}{j2M\omega}\left(2+\dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}}\right) & \dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}}+1 & 0 & \dfrac{1}{2MC\omega^{2}} \\ \hline -\dfrac{1}{2MC\omega^{2}} & -\dfrac{j}{C\omega} & 1 & \dfrac{j}{C\omega} \\ \hline 0 & 0 & 0 & 1 \end{array} \right]. \tag{7} \]

From this, deleting the third column and the fourth row, we find

\[ a_{\mathrm{zv}}= \left[ \begin{array}{c|c|c} 1+\dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}} & j^{\omega}(L-2M)-\dfrac{j}{C\omega} & \dfrac{j}{C\omega} \\ \hline \dfrac{1}{j2M\omega}\left(2+\dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}}\right) & \dfrac{L-2M}{2M}-\dfrac{1}{2MC\omega^{2}}+1 & \dfrac{1}{2MC\omega^{2}} \\ \hline -\dfrac{1}{2MC\omega^{2}} & -\dfrac{j}{C\omega} & \dfrac{j}{C\omega} \end{array} \right]. \tag{8} \]

In [ref. 1, p. 133], instead of \(2jM\omega\), \(jM\omega\) is erroneously used, and the matrix of the section is obtained in the form:

\[ a'_{\mathrm{zv}}= \left[ \begin{array}{c|c} 1 & 0 \\ \hline \dfrac{1}{jM\omega} & 1 \end{array} \right] \cdot \left[ \begin{array}{c|c} 1 & \dfrac{1}{jC\omega} \\ \hline 0 & 1 \end{array} \right] \cdot \left[ \begin{array}{c|c} 1 & j^{\omega}(L-2M) \\ \hline 0 & 1 \end{array} \right] \left[ \begin{array}{c|c} 1 & 0 \\ \hline \dfrac{1}{jM\omega} & 1 \end{array} \right] = \tag{9} \]

\[ = \left[ \begin{array}{c|c} 1+\dfrac{L-2M}{M}-\dfrac{1}{\omega^{2}MC} & j\omega(L-2M)+\dfrac{1}{jC\omega} \\ \hline \dfrac{1}{jM\omega}\left(2+\dfrac{L-2M}{M}-\dfrac{1}{MC\omega^{2}}\right) & \dfrac{L-2M}{M}+1-\dfrac{1}{MC\omega^{2}} \end{array} \right]. \]

If the indicated error had not been made, the components of the matrix \(a'_{\mathrm{zv}}\) would be identical to the components of matrix (8). This shows the possibility of the assumption made in [ref. 1], but only in calculating input and output quantities.

Continuing the calculation using the data of the author of the book, we obtain:

\[ \begin{gathered} \cos\varphi = 1+\frac{L-2M}{M}-\frac{1}{\omega^{2}MC} =\frac{L-M}{M}-\frac{1}{\omega^{2}MC},\\[6pt] \frac{1}{\omega^{2}}=(L-M)C-MC\cos\varphi=\left(\frac{\lambda}{2\pi C}\right)^{2},\\[6pt] \lambda_{0}=2\pi C\sqrt{LC},\\[6pt] \left(\frac{\lambda}{\lambda_{0}}\right)^{2} =1-\frac{M}{L}-\frac{M}{L}\cos\varphi, \end{gathered} \tag{10} \]

whence

\[ \cos\varphi=\frac{L}{M}\left[1-\left(\frac{\lambda}{\lambda_{0}}\right)^{2}\right]-1, \tag{11} \]

which does not correspond to the curves shown in Fig. IV,6 (p. 134) of the book (see Fig. 3).

Fig. 3

Continuing our calculation, we obtain the following:

\[ \begin{gathered} \cos\varphi=1+\frac{L-2M}{2M}-\frac{1}{2MC\omega^{2}} =\frac{L}{2M}-\frac{1}{2MC\omega^{2}},\\[6pt] \frac{1}{\omega^{2}}=LC-2MC\cos\varphi=\left(\frac{\lambda}{2\pi C}\right)^{2},\\[6pt] \left(\frac{\lambda}{\lambda_{0}}\right)^{2}=1-\frac{2M}{L}\cos\varphi,\\[6pt] \cos\varphi=\left[1-\left(\frac{\lambda}{\lambda_{0}}\right)^{2}\right]\frac{L}{2M}. \end{gathered} \tag{12} \]

The calculation is then carried out in the form of a table, with the ratios \(\frac{L}{M}\) and \(\frac{\lambda}{\lambda_{0}}\) for both curves taken to be the same as in the book.

Table

These results correspond completely to the indicated curves (Fig. 3), which were evidently obtained not from the calculated data of the author of the book, but in some other way, possibly experimentally.

LITERATURE

1. Zusmanovsky S. A. (ed.) Magnetrons of the centimeter range, Part 1. Published by “Soviet Radio,” 1950, pp. 131–134.
2. Pukhov G. E. Theory of the subcircuit method, Electricity, No. 8, 1952.