Method for analyzing the frequency error of an inductive divider
Abstract Generated abstract
This paper presents a simplified analytical method for estimating the frequency error of inductive voltage dividers, intended to make such calculations more accessible for engineering practice than earlier computer based approaches. The method introduces auxiliary electromotive forces at divider tap points, reduces the coupled winding system to equivalent four terminal network parameters, and derives current balance equations for a decade inductive divider wound with a twisted wire bundle. The resulting formulas give the output voltages at each tap as functions of circuit parameters, including leakage inductance, winding resistance, and dynamic distributed capacitance, enabling calculation of systematic division ratio errors with frequency. The authors report that calculated values agreed with experimental data within 10 percent over the range from 20 Hz to 200 kHz.
Full Text
METHOD FOR ANALYZING THE FREQUENCY ERROR OF AN INDUCTIVE DIVIDER
A. I. KRAMNYUK, M. S. ROITMAN
(Presented by the scientific seminar of the Department of Radio Engineering)
To extend the capabilities and improve the accuracy of various electrical and radio-engineering devices, voltage dividers with a small error in the division ratio are needed. Inductive dividers (ID) are promising in this respect.
In works on the investigation of IDs, carried out both abroad [1, 4] and in our country [3], it has been shown that the division error at low frequencies is no more than \(10^{-6}\). As the frequency increases, the frequency error that appears reduces the accuracy of the division ratio. In [2] an attempt is made to estimate the frequency error. However, the complicated method of analysis using computer facilities reduces clarity and creates certain difficulties for its application in engineering practice.
Below we consider a comparatively simple method for calculating the frequency error of an inductive divider, based on the methods of direct calculation developed by Academician G. E. Pukhov.
Suppose that, in a complex circuit with many mutual couplings, it is necessary to determine the potential difference between two points (for example: points A, B and the neutral conductor. Fig. 1a). Let us denote the potentials of the indicated points relative to the common conductor by \(U_a\) and \(U_b\). Connecting extraneous EMFs to points A and B, whose magnitudes are respectively equal to \(U_a\), \(U_b\), will produce no measurements in the circuit, and the currents through the auxiliary sources will be equal to 0. If the circuits \(a— a'\) and \(b— b'\) are opened, the operating conditions will also not be disturbed, but in this case currents begin to flow through the auxiliary sources, the values of which are determined by the expressions:
\[ \dot{i}_a = f_1(\dot{E}_{\mathrm{in}}, \dot{U}_a, \dot{U}_b), \]
\[ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \tag{1} \]
\[ \dot{i}_{b'} = f_4(\dot{E}_{b x}, \dot{U}_a, \dot{U}_b). \]
Taking into account that the total values of the currents through the auxiliary sources are equal to 0, we have
\[ \begin{aligned} \dot{i}_a + \dot{i}_{a'} &= 0;\\ \dot{i}_b + \dot{i}_{b'} &= 0. \end{aligned} \tag{2} \]
Solving system (2), we determine the values of \(\dot{U}_a\) and \(\dot{U}_b\).
Of greatest interest is the analysis of a decade divider, i.e., one having ten different values of the divider output voltage, as the most acceptable for practical application. Therefore, the analysis is devoted mainly to precisely this type of divider, but the method presented can also be successfully applied to dividers with another attenuation factor.
Fig. 1.
The dividers are wound with a bundle of uniformly twisted wires, the number of which determines the required number of output-voltage graduations. The windings formed by each wire are connected in series and in phase. Since we are interested in the voltage values at the winding-connection points, auxiliary EMFs are connected to these points (Fig. 2).
Fig. 2.
When finding the components of the currents \(\dot I_1 \div \dot I_{10}\), the equivalent circuit of the divider is reduced to that shown in Fig. 3a, and for the currents \(\dot I_{11} \div \dot I_{19}\) to the circuit in Fig. 3b.
Fig. 3.
Representing the long line by a chain circuit and letting the number of sections tend to infinity, we obtain the matrix of the equivalent four-terminal network:
\[ \gamma = \left| \begin{matrix} 1+\dfrac{n\Delta}{2} & a_0\dfrac{n+1}{n} \\[1.2em] \dfrac{\left(1+\dfrac{n}{6}\Delta\right)(n-1)}{b_0} & 1+\dfrac{n\Delta}{2} \end{matrix} \right| = \left| \begin{matrix} A & B\\ C & D \end{matrix} \right|, \tag{3} \]
where
\[ a_0=r_0+j\omega L'_{s0}, \tag{4} \]
\[ b_0=\frac{1}{j\omega C_{0g}},\qquad \Delta=\frac{a_0}{b_0}, \]
\(n\) is the number of windings of the divider,
\(r_0\) is the active resistance of the windings,
\(L'_{s0}\) is the leakage inductance,
\(C_{0g}\) is the distributed capacitance between windings.
The total values of the longitudinal and transverse impedances of the equivalent four-terminal network are determined as
\[ \dot Z_1+(n-1)\dot Z_1=n\dot Z_1=\frac{A-1}{C} = \frac{n\Delta b_0}{2\left(1+\dfrac{n}{6}\Delta\right)(n-1)}; \]
\[ \dot Z_2= \frac{b_0}{\left(1+\dfrac{n}{6}\Delta\right)(n-1)}. \tag{5} \]
Introducing the notation
\[ Z'=\frac{b_0}{\left(1+\dfrac{n}{6}\Delta\right)(n-1)}, \tag{6} \]
we obtain
\[ \dot Z_1=\frac{\Delta \dot Z'}{2},\qquad \dot Z_2=\dot Z'. \tag{7} \]
Having the values of the resistances and denoting
\[ 9\Delta \dot Z'=\dot Z,\qquad \frac{\Delta}{4}=\Delta', \tag{8} \]
[Circuit diagram with labels: \(\dot I_a\), \((n-1)\dot Z_1\), \((n-1)\dot Z_1\), \(\dot I_b\), \(E_n\), \(\dot Z_2\), \(\dot I_{cd}\), \(\dot Z_1\), \(\dot Z_1\), \(\dot I_{cd}\).]
Fig. 4.
we obtain the expression for the currents in the following form
\[ \dot i_a=\dot E\,\frac{1+9\Delta'}{\dot Z}=\dot E\cdot \dot Y_a, \]
\[ \dot i_b=\dot E\,\frac{1}{\dot Z(1+9\Delta')}=\dot E\cdot \dot Y_b, \tag{9} \]
\[ \dot i_{cd}=\dot E\,\frac{9\Delta'}{\dot Z(1+10\Delta')}=\dot E\cdot \dot Y_{cd}. \]
Taking into account the symmetry of the divider with respect to the central tap, we may write the relations:
\[ \dot E_9=\dot E_{10}-\dot E_1,\qquad \dot E_8=\dot E_{10}-\dot E_2, \]
\[ \dot E_7=\dot E_{10}-\dot E_3,\qquad \dot E_6=\dot E_{10}-\dot E_4, \tag{10} \]
\[ \dot E_5=\frac{1}{2}\dot E_{10}. \]
On the basis of (2, 10) and (Figs. 2, 3a, b) we find the values of the sums of currents:
\[ \dot I_1+\dot I_{11} = \dot E_1\dot Y_a -\frac{1}{9}\dot Y_{cd}\sum_{k=2}^{10}\dot E_k -\frac{1}{9}\dot Y_{cd}\sum_{k=1}^{9}\dot E_k + \]
\[ +\dot E_1\dot Y_a -\frac{1}{9}\dot Y_{cd}\sum_{k=2}^{9}\dot E_k -\dot E_2\dot Y_b -\frac{1}{9}\dot Y_{cd}\left(\dot E_1+\sum_{k=3}^{10}\dot E_k\right)=0, \tag{11} \]
where
\[ \sum_{\kappa=2}^{10} \dot E_\kappa=\frac{11}{2}\dot E_{10}-\dot E_1,\qquad \sum_{\kappa=1}^{9}\dot E_\kappa=\frac{9}{2}\dot E_{10}; \]
\[ \sum_{\kappa=2}^{9}\dot E_\kappa=\frac{9}{2}\dot E_{10}-\dot E_1,\qquad \dot E_1+\sum_{\kappa=3}^{10}\dot E_\kappa=\frac{11}{2}\dot E_{10}-\dot E_2. \tag{12} \]
Substituting (12) into (11), we obtain the first equation
\[ \left(2\dot Y_a+\frac{2}{9}\dot Y_{cd}\right)\dot E_1 -\left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_2 =\frac{20}{9}\dot Y_{cd}\dot E_{10}. \tag{13} \]
Similarly, we find the remaining equations:
\[ \left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_1 -\left(2\dot Y_a+\frac{2}{9}\dot Y_{cd}\right)\dot E_2 +\left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_3 =\frac{20}{9}\dot Y_{cd}\dot E_{10}; \]
\[ \left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_2 -\left(2\dot Y_a+\frac{2}{9}\dot Y_{cd}\right)\dot E_3 +\left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_4 =\frac{20}{9}\dot Y_{cd}\dot E_{10}; \tag{14} \]
\[ -\left(\dot Y_b-\frac{1}{9}\dot Y_{cd}\right)\dot E_3 +\left(2\dot Y_a+\frac{2}{9}\dot Y_{cd}\right)\dot E_4 =\left(\frac{39}{18}\dot Y_{cd}+\frac{\dot Y_b}{2}\right)\dot E_{10}. \]
Introducing the notation
\[ \dot Y_1=2\dot Y_a+\frac{2}{9}\dot Y_{cd}, \]
\[ \dot Y_2=\dot Y_b-\frac{1}{9}\dot Y_{cd}, \tag{15} \]
we have
\[ \dot Y_1\dot E_1-\dot Y_2\dot E_2+0+0 =\frac{20}{9}\dot Y_{cd}\dot E_{10}; \]
\[ -\dot Y_2\dot E_1-\dot Y_1\dot E_2-\dot Y_2\dot E_3+0 =\frac{20}{9}\dot Y_{cd}\dot E_{10}; \]
\[ 0-\dot Y_2\dot E_2+\dot Y_1\dot E_3-\dot Y_2\dot E_4 =\frac{20}{9}\dot Y_{cd}\dot E_{10}; \tag{16} \]
\[ 0+0-\dot Y_2\dot E_3+\dot Y_1\dot E_4 =\left(\frac{39}{18}\dot Y_{cd}+\frac{\dot Y_b}{2}\right)\dot E_{10}. \]
Solving (16) with allowance for (9), we find the values of the output voltages:
\[ \dot E_1=\dot E_{10}\frac{1}{10}(1+59.75\Delta); \]
\[ \dot E_2=\dot E_{10}\frac{2}{10}(1+39.75\Delta); \]
\[ \dot E_3=\dot E_{10}\frac{3}{10}(1+23.25\Delta); \]
\[ \dot E_4=\dot E_{10}\frac{4}{10}(1+10\Delta); \]
\[ \dot E_5=\dot E_{10}\frac{5}{10}; \tag{17} \]
\[ \dot E_6=\dot E_{10}\frac{6}{10}(1-6.66\Delta); \]
\[ \dot E_7=\dot E_{10}\frac{7}{10}(1-9.95\Delta); \]
\[ \dot E_8=\dot E_{10}\frac{8}{10}(1-10\Delta); \]
\[ \dot E_9=\dot E_{10}\frac{9}{10}(1-6.66\Delta). \]
When calculating the frequency error of an inductive divider by formulas (17), the following circumstance must be taken into account. In (4), \(C_{0g}\) is understood to mean the value of the dynamic distributed capacitance. Its magnitude can be determined from the relation
\[ C_{0g}=\frac{2C_{\mathrm{st}}}{n}, \tag{18} \]
where \(C_{\mathrm{st}}\) is the static value of the distributed capacitance measured between any two wires in the harness. For the case \(n=10\),
\[ C_{0g}=\frac{C_{\mathrm{st}}}{5}. \tag{19} \]
Because of its complexity, the proof of relation (18) is not given here. It will be presented in subsequent works.
Conclusions
The fairly simple expressions obtained for the values of the output voltage make it possible to determine the frequency error of the division ratio and, since it is systematic in character, to take it into account when using inductive dividers.
In conclusion it should be noted that the investigations carried out showed good agreement between the calculated and experimental data. In the range from 20 Hz to 200 kHz the discrepancy was no more than 10%.
References
- “Proceedings of the Institution of Electrical Engineers,” No. 5, Vol. 115, 1968, USA.
- I. Eiskop, H. Sillamaa. Frequency characteristic of an inductive voltage divider. Problems of Radio Electronics, series “Radio-Measurement Technology,” issue 5, 1968.
- V. M. Baikov. Analysis of the errors of transformer voltage dividers. Second All-Union Scientific-Technical Conference on Measuring Amplifier Technology. Leningrad, 1967.
- F. L. Hermach. Low-frequency electrical calibrations of the National Bureau of Standards. “Instrum. Soc. Amer. Preprint,” 1967, N-MES-TIND-5/1, USA.