On a Method for Measuring Attenuation at Elevated Frequencies

V. N. Studenikin, V. S. Petrov

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

Accurate measurement of attenuation over a wide frequency range is one of the tasks of modern measurement technology. Existing methods of measuring attenuation are, as a rule, based on measuring the input and output signals or on comparison with a reference device having a known error-distribution law.

The attenuation of any passive four-terminal network, matched at the input and output, is determined by the well-known relation [1]

\[ b=\ln \frac{A_1(t)}{A_2(t)} . \]

For attenuation boxes produced by domestic industry in the frequency range up to 300 kHz, the specified permissible error is determined by the following expression:

\[ \Delta b_{\mathrm{perm}} \leqslant 0.03 + 0.015\sqrt{bx}, \tag{1} \]

where \(bx\) is the measured quantity.

Consequently, when measuring attenuation of several nepers it is necessary to have accurate instruments ensuring an error no worse than \(0.1\%\).

The existing verification procedure [4] provides for measuring attenuation at direct current by determining the actual value of the characteristic resistance \(Z_{cg}\) (for each nominal value \(b_g\)), and at elevated frequencies in the range up to 20 kHz by the method of measuring the ratio of voltages at its input and output, using complex verification equipment for attenuations of not more than 2 nepers.

In assessing the frequency error of attenuation boxes produced by manufacturing plants, this procedure does not satisfy the technical requirements. However, at elevated frequencies attenuation up to 2 nepers can be measured without resorting to determination of the input and output voltages. It is sufficient to know the actual value of the characteristic resistance and the short-circuit resistance within the limits of the permissible error. The attenuation and the characteristic resistance are related through the open-circuit and short-circuit parameters by the following expressions:

\[ \operatorname{th} b=\sqrt{\frac{Z_{\mathrm{sc}}}{Z_{\mathrm{oc}}}};\qquad Z_c=\sqrt{Z_{\mathrm{sc}}\cdot Z_{\mathrm{oc}}}. \]

After the corresponding transformations we have

\[ \operatorname{th} b=\frac{Z_{\mathrm{sc}}}{Z_c}. \]

Let us introduce notation for the errors of the measured parameters:

\[ \Delta_c=Z_c-Z_{cg}; \qquad \Delta b=b-b_g; \qquad \Delta_{\mathrm{sc}}=Z_{\mathrm{sc}}-Z_{\mathrm{sc}g}; \]

where \(\Delta_c\), \(\Delta_{\mathrm{sc}}\), and \(\Delta b\) are measurement errors. To establish the relation between them, we find the general expression of the form

\[ \Delta Y=\frac{\partial f}{\partial X_1}\Delta X_1+ \frac{\partial f}{\partial X_2}\Delta X_2+\cdots+ \frac{\partial f}{\partial X_n}\Delta X_n. \tag{2} \]

For the particular case, taking into account the above notation, the maximum total error will be equal to:

\[ \Delta b_{\mathrm{meas}}= \frac{1}{Z_c^2-Z_{\mathrm{sc}}^2} \left(\Delta_{\mathrm{sc}}Z_c+\Delta_c Z_{\mathrm{sc}}\right). \tag{3} \]

Taking expression (1) into account, we have

\[ 0.015\sqrt{bx}+0.03 \leq \frac{1}{Z_c^2-Z_{\mathrm{sc}}^2} \left(\Delta_{\mathrm{sc}}Z_c+\Delta_c Z_{\mathrm{sc}}\right). \]

After simplification we obtain the final expression for the value of the attenuation being measured:

\[ \left[ \frac{\Delta_{\mathrm{sc}}Z_c+\Delta_c Z_{\mathrm{sc}}} {0.015\left(Z_c^2-Z_{\mathrm{sc}}^2\right)} -2 \right]^2 \leq bx. \tag{4} \]

Thus, knowing the permissible error of the attenuation box and having measured the actual values \(Z_{cg}\) and \(Z_{\mathrm{sc}g}\), one can establish limiting conditions for each individual decade of the instrument being tested. The nominal value for \(Z_{\mathrm{sc}}\) and \(Z_{\mathrm{oc}}\) is calculated from the following formulas:

\[ Z_{\mathrm{sc}}=Z_c\operatorname{th} bx; \qquad Z_{\mathrm{oc}}=\frac{Z_c}{\operatorname{th} bx}. \tag{5} \]

Usually the characteristic resistance is active in nature and, for various types of instruments, is equal to:

\[ Z_c=(135\ \Omega,\quad 300\ \Omega,\quad 600\ \Omega) \]

As the analysis has shown, in order to satisfy inequality (4), when measuring attenuations in the range from \(0.01\) to \(0.5\) neper it is sufficient to have a measuring device whose measurement error would not exceed \(\leq 0.5\%\). For attenuations up to 2 nepers, the error of the reference instrument must not exceed \(\gamma \leq 0.1\%\).

The most promising circuit for calibrating attenuation boxes is an instrument in the form of a measuring bridge with inductively coupled arms, developed at the National Research Institute of Metrology [3]. The error in measuring active resistances with this instrument at frequencies up to 1.5 MHz does not exceed \(\gamma \leq 0.05\%\). By using such an instrument in calibration practice, it is possible to calibrate nonsymmetrical attenuation boxes produced by industry at elevated frequencies up to 1 MHz.

As an example, a calculated table of errors is given for attenuation boxes with \(Z_c=600\ \Omega\).

As can be seen from the results given in Table 1, when calibrating an attenuation box by means of a bridge setup, its measuring po-

error does not exceed the permissible value. Consequently, the instrument guarantees verification of four-terminal networks made according to an asymmetrical circuit over the specified frequency range.

Table 1

References

1. N. N. Solov'ev. Fundamentals of Measuring Technology for Wire Communication, part III, GEI, 1959.
2. L. A. Ostrovskii. Fundamentals of the General Theory of Electrical Measuring Devices, GEI, 1965.
3. Measurement Technology, No. 8, 1964.
4. Methodological Instruction No. 161, published by the Committee of Standards, Measures, and Measuring Instruments, 1959.