EQUATIONS OF A SYMMETRIC HOMOGENEOUS CHAIN CIRCUIT IN THE PRESENCE OF FEEDBACKS

G. E. PUKHOV

As is known, a chain circuit is a very common element of an electric circuit. However, its properties have been well studied only for the case corresponding to a transmission circuit, i.e., a circuit in which feedbacks are absent. Chain circuits of a more general type are almost never encountered in the literature, although circuits of this kind occur in practice, and rather often. Below we consider some questions in the theory of a four-terminal chain circuit for whose sections the following assumptions are made:

1. The sections are symmetric either with respect to their geometric axis, or with respect to both the transverse and the longitudinal axes. In the case of transverse symmetry, interchanging the input and output terminals (Fig. 1a), both for individual sections and for the entire chain as a whole, will not lead to any change in the external-circuit mode. If, in addition, the sections are symmetric with respect to the longitudinal axis, then the external-circuit mode will not change under pairwise crossing of the input and output terminals (Fig. 1b).

Fig. 1

1. All sections of the circuit are passive and are made in exactly the same way.

Equations of a Section Symmetric with Respect to the Transverse Geometric Axis

Since the sections of the chain are connected in series, the equations of a section, whose ordinal number we denote by \(k\), in the general case (in the presence of asymmetries) must have the form [2; 3]:

\[ \left\| \begin{array}{c} u_a^k \\ i_a^k \\ u_c^k \end{array} \right\| = \left\| \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k \\ -i_b^k \\ i_o \end{array} \right\| = a \left\| \begin{array}{c} u_2^k \\ -i_b^k \\ i_o \end{array} \right\| , \tag{1} \]

where \(i_0=-(i_b+i_c)=-(i_b^k+i_c^k)\) (Fig. 2) is the equalizing current caused by the feedbacks. By virtue of the assumed passivity of the section, between the coefficients \(A_{ik}\) of the chain matrix \(a\) there are the relations

\[ \begin{gathered} A_{11}A_{22}-A_{12}A_{21}=1,\\ A_{12}A_{23}-A_{13}A_{22}=A_{32},\\ A_{11}A_{32}-A_{12}A_{31}=-A_{13}, \end{gathered} \tag{2} \]

which follow from the well-known reciprocity theorem. It is likewise not difficult to prove that, in the presence of transverse symmetry alone, among the components \(A_{ik}\) there exist the relations:

\[ A_{23}=A_{13}\frac{A_{11}-1}{A_{12}} \quad\text{and}\quad A_{31}=A_{23}\frac{A_{11}-1}{A_{12}}. \tag{3} \]

Using conditions (2–3), we arrive at the following equations for a section with transverse symmetry:

\[ \left\| \begin{array}{c} u_a^k\\ i_a^k\\ u_c^k \end{array} \right\| = \left\| \begin{array}{ccc} A_{11}, & A_{12}, & A_{13}\\ \dfrac{A_{11}^2-1}{A_{12}}, & A_{11}, & A_{13}\dfrac{A_{11}-1}{A_{12}}\\ - A_{13}\dfrac{A_{11}-1}{A_{12}}, & -A_{13}, & A_{33} \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k\\ - i_b^k\\ i_0 \end{array} \right\|. \tag{4} \]

Fig. 2

In the theory of the ordinary chain circuit, an important role is played by the characteristic resistance \(Z_\lambda\) and the propagation constant \(\gamma\) of a section, determined from the expressions

\[ Z_\lambda=\sqrt{\frac{A_{12}}{A_{21}}} \quad\text{and}\quad A_{11}+\sqrt{A_{12}A_{21}}=e^\gamma, \tag{5} \]

whence

\[ \operatorname{ch}\gamma=A_{11} \quad\text{and}\quad \operatorname{sh}\gamma=\sqrt{A_{12}A_{21}}. \tag{6} \]

In the case under consideration it is convenient to express also the quantities \(A_{13}\) and \(A_{33}\) through hyperbolic functions:

\[ A_{13}=Z_s\,\operatorname{sh}\gamma,\quad A_{33}=A_0-\frac{A_{13}^2}{A_{12}} = A_0-\frac{Z_s^2}{Z_\lambda}\operatorname{sh}\gamma, \tag{7} \]

where \(Z_s\) and \(Z_\lambda\) are certain constants determined by the expressions

\[ Z_s=\frac{A_{13}}{\sqrt{A_{12}A_{21}}}, \]

\[ A_0=A_{33}+\frac{A_{13}^2}{A_{12}}. \tag{8} \]

The convenience of introducing the quantities \(Z_s\) and \(A_0\) will be clear from what follows.

Substituting relations (5), (6), and (7) into equations (4), we obtain

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k\\[2mm] u_c^k \end{array} \right\| = \left\| \begin{array}{ccc} \operatorname{ch}\gamma, & Z_\lambda \operatorname{sh}\gamma, & -Z_s\operatorname{sh}\gamma\\[2mm] \dfrac{\operatorname{sh}\gamma}{Z_\lambda}, & \operatorname{ch}\gamma, & -2\dfrac{Z_s}{Z_\lambda}\operatorname{sh}^2\dfrac{\gamma}{2}\\[3mm] 2\dfrac{Z_s}{Z_\lambda}\operatorname{sh}^2\gamma, & Z_s\operatorname{sh}\gamma, & A_0-\dfrac{Z_s^2}{Z_\lambda}\operatorname{sh}\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k\\[2mm] -i_b^k\\[2mm] i_0 \end{array} \right\|. \tag{9} \]

Let us note that in the absence of feedbacks \((i_0=0)\), equations (9) become the ordinary, well-known equations of a transmission four-terminal network:

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k \end{array} \right\| = \left\| \begin{array}{cc} \operatorname{ch}\gamma, & Z_\lambda \operatorname{sh}\gamma\\[2mm] \dfrac{\operatorname{sh}\gamma}{Z_\lambda}, & \operatorname{ch}\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k\\[2mm] -i_b^k \end{array} \right\|. \tag{10} \]

These same equations will also hold in the presence of feedbacks, provided that the section is a three-terminal network.^1)

Equations of a section symmetric with respect to the transverse and longitudinal geometrical axes

If a section of the chain circuit, in addition to transverse symmetry, also possesses longitudinal symmetry, then between the components \(A_{ik}\) of the matrix \(a\) there exists the additional dependence \(A_{12}=-2A_{13}\). Introducing this relation into equation (4), we obtain

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k\\[2mm] u_c^k \end{array} \right\| = \left\| \begin{array}{ccc} A_{11}, & A_{12}, & -\dfrac{A_{12}}{2}\\[3mm] \dfrac{A_{11}^2-1}{A_{12}}, & A_{11}, & -\dfrac{A_{11}-1}{2}\\[3mm] -\dfrac{A_{11}-1}{2}, & -\dfrac{A_{12}}{2}, & A_0-\dfrac{A_{12}}{4} \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k\\[2mm] -i_b^k\\[2mm] i_0 \end{array} \right\|. \tag{11} \]

Now expressing the components \(A_{ik}\) in terms of hyperbolic functions, we arrive at the equations

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k\\[2mm] u_c^k \end{array} \right\| = \left\| \begin{array}{ccc} \operatorname{ch}\gamma, & Z_\lambda\operatorname{sh}\gamma, & -\dfrac{Z_\lambda}{2}\operatorname{sh}\gamma\\[3mm] \dfrac{\operatorname{sh}\gamma}{Z_\lambda}, & \operatorname{ch}\gamma, & -\operatorname{ch}^2\dfrac{\gamma}{2}\\[3mm] \operatorname{sh}^2\dfrac{\gamma}{2}, & -\dfrac{Z_\lambda}{2}\operatorname{sh}\gamma, & A_0-\dfrac{Z_\lambda}{4}\operatorname{sh}\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2^k\\[2mm] -i_b^k\\[2mm] i_0 \end{array} \right\|. \tag{12} \]

^1) This follows from the fact that the state of a three-terminal network is described by a system consisting of two equations.

We note that these equations could have been obtained from the equations of a section possessing only transverse symmetry, i.e., from equation (9), by the simple replacement of the constant \(Z_s\) by the quantity \(\dfrac{Z_\lambda}{2}\).

Thus, we have obtained the equations of a four-terminal section having feedbacks. We now proceed to derive the equations of the chain circuit.

Equations of the Chain Circuit

Let us first obtain the equations of a chain circuit consisting of sections symmetric only with respect to the transverse geometric axis.

Augmenting the chain matrix \([2, 3]\), taken from equations (9), and then raising it to the \(n\)-th power (\(n\) is the number of sections), we find

\[ A_{\substack{\text{symmetric}\\ \text{transverse}}} = \left\| \begin{array}{cccc} ch\gamma, & Z_\lambda sh\gamma, & 0, & -Z_s sh\gamma \\[4pt] \dfrac{sh\gamma}{Z_\lambda}, & ch\gamma, & 0, & -\dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{\gamma}{2} \\[8pt] \dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{\gamma}{2}, & Z_s sh\gamma, & 1, & A_0-\dfrac{Z_s^2}{Z_\lambda}\,sh\gamma \\[8pt] 0, & 0, & 0, & 1 \end{array} \right\|^{n} = \]

\[ = \left\| \begin{array}{cccc} chn\gamma, & Z_\lambda shn\gamma, & 0, & -Z_s shn\gamma \\[4pt] \dfrac{shn\gamma}{Z_\lambda}, & chn\gamma, & 0, & -\dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{n\gamma}{2} \\[8pt] \dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{n\gamma}{2}, & Z_s shn\gamma, & 1, & nA_0-\dfrac{Z_s^2}{Z_\lambda}\,shn\gamma \\[8pt] 0, & 0, & 0, & 1 \end{array} \right\|. \tag{13} \]

Let us note that if the component \(A_{33}\) had not previously been represented in the form

\[ A_{33}=A_0-\dfrac{A_{13}^2}{A_{12}}, \]

then expression (13) would have been considerably less convenient. Deleting from the augmented matrix (13) the fourth row and the third column, we arrive at its normal form

\[ a_{\substack{\text{symmetric}\\ \text{transverse}}} = \left\| \begin{array}{ccc} chn\gamma, & Z_\lambda shn\gamma, & -Z_s shn\gamma \\[6pt] \dfrac{shn\gamma}{Z_\lambda}, & chn\gamma, & -\dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{n\gamma}{2} \\[8pt] \dfrac{2Z_s}{Z_\lambda}\,sh^2\dfrac{n\gamma}{2}, & Z_s shn\gamma, & nA_0-\dfrac{Z_s^2}{Z_\lambda}\,shn\gamma \end{array} \right\|. \tag{14} \]

This matrix corresponds to the following equations of the equivalent four-terminal network replacing the part of the chain circuit under consideration:

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k\\[2mm] u_b^k \end{array} \right\| = \left\| \begin{array}{ccc} \operatorname{ch} k\gamma, & Z_\lambda \operatorname{sh} k\gamma, & -Z_s \operatorname{sh} k\gamma\\[2mm] \dfrac{\operatorname{sh} k\gamma}{Z_\lambda}, & \operatorname{ch} k\gamma, & -\dfrac{2Z_s}{Z_\lambda}\operatorname{sh}^2\dfrac{k\gamma}{2}\\[3mm] \dfrac{2Z_s}{Z_\lambda}\operatorname{sh}^2\dfrac{k\gamma}{2}, & Z_s \operatorname{sh} k\gamma, & kA_0-\dfrac{Z_s^2}{Z_\lambda}\operatorname{sh} k\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2\\[2mm] -i_b\\[2mm] i_1 \end{array} \right\|, \tag{15} \]

where \(k\) denotes the ordinal number of the section, assigned in accordance with Fig. 2.

If the sections of the chain, in addition to transverse symmetry, also possess longitudinal symmetry, then its equations, after replacing \(Z_s\) in (15) by \(\dfrac{Z_\lambda}{2}\) (see above), take the form:

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k\\[2mm] u_b^k \end{array} \right\| = \left\| \begin{array}{ccc} \operatorname{ch} k\gamma, & Z_\lambda \operatorname{sh} k\gamma, & -\dfrac{Z_\lambda}{2}\operatorname{sh} k\gamma\\[3mm] \dfrac{\operatorname{sh} k\gamma}{Z_\lambda}, & \operatorname{ch} k\gamma, & -\operatorname{sh}^2\dfrac{k\gamma}{2}\\[3mm] \operatorname{sh}^2\dfrac{k\gamma}{2}, & -\dfrac{Z_\lambda}{2}\operatorname{sh} k\gamma, & kA_0-\dfrac{Z_\lambda}{4}\operatorname{sh} k\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2\\[2mm] -i_b\\[2mm] i_0 \end{array} \right\|. \tag{16} \]

Equations (15) and (16) are convenient in that they make it possible, in the simplest way, to find the currents and voltages at the point of connection of any two sections; for this it is sufficient to substitute into these equations, in place of \(k\), the number of the section.

In the absence of the equalizing current \(i_0\), which may occur, for example, when the feedbacks are broken, expressions (15) and (16) reduce to the familiar equations of a through chain circuit:

\[ \left\| \begin{array}{c} u_a^k\\[2mm] i_a^k \end{array} \right\| = \left\| \begin{array}{cc} \operatorname{ch} k\gamma, & Z_\lambda \operatorname{sh} k\gamma\\[2mm] \dfrac{\operatorname{sh} k\gamma}{Z_\lambda}, & \operatorname{ch} k\gamma \end{array} \right\| \cdot \left\| \begin{array}{c} u_2\\[2mm] -i_b \end{array} \right\|. \tag{17} \]

In this case (for \(i_0=0\)) one is usually not interested in the additional equations determining the lateral voltage \(U_c^k\).

To explain the application of the equations derived, let us determine the impedance of a complex two-terminal network whose circuit is shown in Fig. 3. Let the inductance \(L\) and capacitance \(C\) be related by the condition \(\omega L=\dfrac{1}{\omega C}\) (\(\omega\) is the frequency of the applied voltage). Replacing the elements coupled by mutual induction with \(T\)-equivalent circuits, we arrive at a chain circuit operating in the regime \(U_a=U_2=0\) (Fig. 4). The impedance of the two-terminal network\(^1\) is evidently equal to

\[ Z=-\left(\frac{\dot U_b}{\dot i_0}\right)_{\dot U=\dot U_2=0}. \tag{18} \]

Since the circuit possesses only transverse symmetry, equations (15) should be used to find \(Z\). Setting in the latter \(\dot U_a=\dot U_2=0\), we find

\[ Z=-nA_0=-n\left(A_{33}+\frac{A_{13}^{2}}{A_{12}}\right), \tag{19} \]

\(^1\) In what follows, the steady-state sinusoidal regime of the two-terminal network is meant.

Fig. 3

Visible labels in the circuit: \(L\), \(2L\), \(2L\), \(\ldots\), \(2L\), \(L\); \(C\), \(\dfrac{C}{2}\), \(\dfrac{C}{2}\), \(\ldots\), \(\dfrac{C}{2}\), \(C\); \(\dot I_o\); \(\dot u_\delta\).

Fig. 4

Visible labels in the circuit: \(L-M\), \(L-M\), \(L-M\), \(L-M\), \(\ldots\), \(L-M\), \(L-M\), \(\ldots\), \(L-M\), \(L-M\); \(M\), \(M\), \(M\), \(M\); \(C\), \(C\), \(C\), \(C\), \(\ldots\), \(C\), \(C\), \(\ldots\), \(C\), \(C\); \(\dot I_o\); section labels \(n\), \(n-1\), \(k\), \(1\).

Fig. 5

Visible labels in the circuit: \(\dot I_a^k\), \(\dot u_a^k\), \(L-M\), \(M\), \(L-M\), \(\dot I_b^k\), \(\dot u_2^k\), \(C\), \(C\), \(\dot I_c^k\), \(\dot u_c^k\).

where \(n\) is the number of sections of the chain replacement circuit (Figs. 3–5), and \(A_{ik}\) are the components of the chain matrix of a section. To determine the quantities \(A_{ik}\), we write the equation of one section in the form (4). We have

\[ \left\| \begin{array}{c} U_a^k\\[2mm] i_a^k\\[2mm] U_c^k \end{array} \right\| = \left\| \begin{array}{ccc} 0, & -j\omega M, & -\dfrac{1}{j\omega C}\\[3mm] \dfrac{1}{j\omega M}, & 0, & -\dfrac{1}{\omega^2MC}\\[3mm] -\dfrac{1}{\omega^2MC}, & \dfrac{1}{j\omega C}, & \dfrac{2-\dfrac{1}{\omega^2MC}}{j\omega C} \end{array} \right\| \cdot \left\| \begin{array}{c} U_2^k\\[2mm] -i_b^k\\[2mm] I_0 \end{array} \right\|. \tag{20} \]

Hence

\[ A_{12}=-j\omega M,\quad A_{13}=-\frac{1}{j\omega C} \quad \text{and} \quad A_{33}=\frac{1}{j\omega C}\left(2-\frac{1}{\omega^2MC}\right). \tag{21} \]

Substituting these quantities into formula (19) and denoting by \(k=M/L\) the coupling coefficient of the circuits, we arrive at the following value of the resistance of the complex two-terminal network under consideration:

\[ \dot Z=2nj\omega L\frac{1-k}{k}. \tag{22} \]

Let us note that the calculation of the circuit considered, given in [4], is not correct, since it was carried out on the basis of the ordinary equations (10, 17).

REFERENCES

1. Kalantarov P. L. Theory of Alternating Currents, 1940.
2. Pukhov G. E. Geometric theory of circuits consisting of four-terminal networks. Scientific Notes of LPI, vol. X, Lvov, 1949.
3. Pukhov G. E. Theory of the method of subcircuits. Elektrichestvo, No. 8, 1952.
4. Zusmanovskii S. A. (ed.) Centimeter-Band Magnetrons, part 1, 1950.