SYMMETRIC CALIBRATED ATTENUATORS

V. N. STUDENIKIN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In measurement technology, it is often necessary to estimate with sufficiently high accuracy the frequency error of the division ratio of four-terminal networks in the form of symmetric attenuation boxes. To accomplish this task it is necessary to have calibrated sections whose division ratio would not depend on frequency.

Of the known methods considered for verifying the frequency error of attenuation boxes, where inductive and capacitive dividers and four-terminal networks on active resistances are used as reference sections, the most perfect are dividers on resistances. In practice such frequency-independent dividers can be obtained through their structural design.

Let us consider a calibrated section in the form of a symmetric circuit on active resistances (of the BLG type) with a specified transmission-ratio relationship.

If at low frequencies the circuit elements have a purely active character and practically do not depend on frequency, then at elevated frequencies it is necessary to take into account various parasitic inductances and capacitances, whose resistance changes, and the additional frequency error increases.

The equivalent circuit of a four-terminal network, taking reactive admittances into account, is shown in Fig. 1. Here \(L\) represents the inductances in the resistances of the circuit elements; the capacitors correspond to the capacitances between the terminals of the conductors and with respect to other circuit elements. It is not difficult to take such admittances into account by measuring the geometric dimensions of the conductors and the distances between them; their values can be calculated with the aid of formulas [5].

To reduce electromagnetic coupling between the input and output circuits of four-terminal networks, electrostatic shielding is provided. The equivalent circuit in this case has the form shown in Fig. 2, where \(C_1\) is the self-capacitance of the conductor, \(C_2\) is the capacitance of the resistance with respect to the shield, \(C_3\) is the capacitance of the horizontal conductor with respect to the shield, \(C_4\) is the capacitance of the input and output of the horizontal conductor, and \(L_1\) and \(L_2\) are the inductances in the longitudinal and transverse resistance branches.

The through capacitance \(C_4\) is reduced to zero, and such reactances as the mutual inductance between elements and the capacitance between conductors with respect to the shield are neglected because of their small order of magnitude.

The calculation of the transmission coefficient of sections for various attenuations, taking into account the conductivities of the elements, was carried out at the upper limiting frequency \(f = 600\) kHz. Thus, for example, the frequency error of the transmission coefficient for 3 nepers at this frequency was no more than 0.05%.

Fig. 1

Fig. 2

It is very difficult, however, to estimate experimentally the frequency error of shielded dividers with such high accuracy, because of the absence of standard instruments.

It is known from the literature [4] that for all minimum-phase networks there is a one-to-one relation between the frequency and phase characteristics, and, by monitoring the phase of the transmission coefficient, one can judge with very high sensitivity the nonuniformity of the frequency characteristic.

Let us investigate what network the sections of the dividers constructed belong to and determine the nature of the relation between the frequency and phase characteristics.

To simplify the analysis, by virtue of symmetry we shall use one T-shaped section, taking into account the through capacitance \(C_1\). The longitudinal arm of the section, together with the reactances, represents a minimally reactive two-terminal network, and its total resistance is reduced to the equivalent internal resistance of the generator. As shown in Fig. 3, the linear system is composed of three loops with lumped

Fig. 3

parameters. Using the loop-current method, we obtain the following system of equations:

\[ \begin{aligned} Z_{11} I_1 - Z_{12} I_2 - Z_{13} I_3 &= U_{\mathrm{in}},\\ Z_{22} I_2 - Z_{21} I_1 - Z_{23} I_3 &= 0,\\ Z_{33} I_3 - Z_{31} I_1 - Z_{32} I_2 &= 0; \end{aligned} \tag{1} \]

where

\[ Z_{11}=r+\mathrm{i}\omega L+\frac{1}{\mathrm{i}\omega C_2}+Z_H;\qquad Z_{12}=Z_{21}=\frac{1}{\mathrm{i}\omega C_2}; \]

\[ Z_{13}=Z_{23}=Z_{31}=Z_{32}=r+\mathrm{i}\omega L;\qquad Z_H=\frac{R_H}{1+\mathrm{i}\omega C_2 R_H}. \]

Taking into account the identical nature of the resistances in the various loops and denoting by

\[ Z_1=Z_{12}=Z_{21};\qquad Z_2=Z_{13}=Z_{31}=\ldots;\qquad Z_3=\frac{1}{\mathrm{i}\omega C_1}, \]

we have:

\[ Z_{11}=Z_1+Z_2;\qquad Z_{22}=Z_1+Z_2+Z_H;\qquad Z_{33}=2Z_2+Z_3. \]

The expression for the transmission coefficient can be written in the form (2)

\[ \dot K=\frac{(-1)^{\kappa+1} M_{1\kappa} Z_\kappa}{\Delta}. \tag{2} \]

Substituting the value of the determinant and the minor into expression (2),

\[ \dot K= \frac{\left[Z_1 Z_3+(2Z_1-Z_2)\right]\cdot Z_H} {Z_2\left[(Z_2+Z_H)(2Z_1+Z_3)+Z_2(2Z_1+Z_H)+Z_1Z_3(2Z_3+Z_H)\right]}; \]

express \(Z_1, Z_2\), and \(Z_3\) in terms of inductances, capacitances, and resistances, and replace \(i\omega\) by \(P\). Then expression (2) takes the following form

\[ K(P)= \]

\[ = \frac{ P^4L^2C_1C_2+2P^3LC_1C_2r+P^2(C_1C_2r^2-2LC_1)-2PC_1r-1 }{ P^4L^2C_1C_2+ P^3L\left[2C_1C_2r+\frac{L}{Z_H}(4C_1+C_2)\right]+ P^2\left[L\left(\frac{2r}{Z_H}+1\right)\right. } \times \]

\[ \times \left. (2C_1+C_2)+C_1r\left(C_2r+\frac{2L}{Z_H}\right) \right] +2P\frac{L}{Z_H} + \left(\frac{2r}{Z_H}+1\right). \tag{3} \]

Let us examine the characteristic roots of this function for a minimum-phase circuit. To do this we set \(K(P)=0\) and use the Hurwitz method, according to whose theorem we seek the signs of the determinants (3).

The numerator of expression (3) satisfies an equation of the general form:

\[ b_0P^m+b_1P^{m-1}+b_2P^{m-2}+\ldots+b_{m-1}P+b_m=0. \]

In our case we have a fourth-degree equation:

\[ b_0P^4+b_1P^3+b_2P^2+b_3P+b_4=0, \tag{4} \]

where

\[ b_0=L^2C_1C_2;\quad b_1=2LC_1C_2r;\quad b_2=C_1C_2r^2-2LC_1; \]

\[ b_3=2C_1r;\quad b_4=-1. \]

For equation (4) the determinants, respectively, will be:

\[ \Delta_1=b_1=2LC_1C_2r;\quad \Delta_2=2LC_1^2C_2r^2(\tau_{C_2}-\tau_L); \]

\[ \Delta_3=4LC_1^2C_2r^2\left[L(C_1+C_2)-C_1C_2r^2\right]; \]

\[ \Delta_4=4LC_1^3C_2r^3(\tau_{C_2}-2\tau_L). \]

From the Hurwitz condition it follows that if at least one of the determinants turns out to be less than zero, then the circuit belongs to non-minimum-phase networks.

In calibrated shielded sections, the intrinsic longitudinal capacitance of the conductors is an order of magnitude smaller than the capacitance of the resistor relative to the shield, i.e. \(\tau_{C_2}\geqslant 5\tau_L\); consequently, the determinants \(\Delta_1\), \(\Delta_2\), \(\Delta_3\), and \(\Delta_4\) have a positive sign, and the section considered represents a minimum-phase circuit.

Replace \(P\) by \(j\omega\) and rewrite the expression for the transmission coefficient (3) in the form

\[ \dot K(\omega)= \frac{ \left\{\omega^2\left[LC_1(\omega^2LC_2+2)-C_1C_2r^2\right]-1\right\} -j2\omega C_1r(\omega LC_2-1) }{ \omega^2\left\{\omega^2L^2C_1C_2- \left[L\left(\frac{2r}{Z_H}+1\right)(2C_1+C_2) +C_1r\left(C_2r+\frac{2L}{Z_H}\right)\right]\right\} + } \times \]

\[ \times \frac{1}{ \left(\frac{2r}{Z_H}+1\right) -j\omega L\left\{ \omega^2\left[2C_1C_2r+\frac{L}{Z_H}(4C_1+C_2)\right] +\frac{2}{Z_H} \right\} }; \]

denote

\[ \dot a=C_1(2L-C_2r^2);\quad b=\left[ L(2C_1+C_2)+C_1r\left(\frac{2L+C_2rZ_H}{2r+Z_H}\right) \right]. \]

\[ d=2C_1C_2r+\frac{L}{Z_H}(4C_1+C_2); \]

if quantities of small order are neglected, we obtain the final expression for the transmission coefficient and its argument:

\[ \dot K(\omega)= \frac{ \left(\frac{1}{\omega}-a\omega\right)-2jC_1r }{ \left(\frac{2r}{Z_H}+1\right)\left(\omega b-\frac{1}{\omega}\right) +j\omega L\left(\omega d+\frac{2}{\omega Z_H}\right) }; \tag{5} \]

\[ \varphi=-\operatorname{arctg} \frac{ 2C_1r\left(\frac{2r}{Z_H}+1\right)\left(\omega b-\frac{1}{\omega}\right) +\omega L\left(\omega d+\frac{2}{\omega Z_H}\right)\left(\frac{1}{\omega}-a\omega\right) }{ \left(\frac{2r}{Z_H}+1\right)\left(\frac{1}{\omega}-a\omega\right)\left(\omega b-\frac{1}{\omega}\right) -2\omega LC_1r\left(\omega d+\frac{2}{\omega Z_H}\right) }. \tag{6} \]

The phase characteristics of sections with attenuations of 1 and 3 nepers were investigated experimentally.

The measurements were carried out on a phase meter in the frequency range above 1 MHz, where a significant phase shift was observed for the transmission coefficient of the four-terminal network. Figure 4 shows the curves obtained experimentally for the phase characteristic of sections loaded by a resistance \(R_H=600\ \Omega\). The calculated characteristic for the phase angle according to expression (6) is also shown here by the dashed line.

Fig. 4

As can be seen from the graph, the experimental and theoretical phase characteristics have the same behavior.

Consequently, when estimating the frequency error of calibrated attenuators, the following conclusions can be drawn:

1. In symmetric shielded voltage dividers it is theoretically easy to calculate the values of the reactive quantities that affect the frequency properties of the transmission coefficient.

2. A calibrated resistance section (of the BLP type) with electrostatic shielding has a small frequency error, and such sections can be used as reference dividers in comparison circuits.

3. The sections of the constructed model represent a minimum-phase circuit in the specified frequency range. The error of the division coefficient can be estimated by using phase relationships.

4. The theoretical and experimental phase characteristics obey one and the same law. The phase shift of the calculated characteristic exceeds the value of the angle obtained experimentally. Consequently, the reactive conductances were overestimated in the calculations.

REFERENCES

1. F. Terman, J. M. Pettit. Electronic Measurements. IL, 1955.

2. I. S. Gonorovskii. Radio-Engineering Circuits and Signals. Sovetskoe Radio Publishing House, 1963.

3. G. V. Voishvillo. Low-Frequency Vacuum-Tube Amplifiers. Svyazizdat, 1959.

4. A. D. Artym. Electrical Correcting Circuits and Amplifiers. Energiya Publishing House, 1965.

5. L. R. Neiman, N. A. Kalantarov. Theoretical Foundations of Electrical Engineering, Part II. Gosenergoizdat, 1959.