Reduction of Phase Shifts in a Phase Detector Transformer
Abstract Generated abstract
This paper examines voltage-dependent phase shifts in low-frequency transformers used with synchronous phase detectors, where transformer nonlinearity causes dynamic zero drift that cannot be removed by compensating a constant phase offset. Using an equivalent circuit with source resistance, winding resistances, leakage inductances, and detector load resistance, the authors derive expressions for the tangent of the phase shift and discuss simplified cases relevant to small leakage and toroidal transformers. The analysis indicates that relative phase variation can be reduced by choosing transformer parameters to minimize the absolute phase shift, by applying negative feedback, and by lowering the source internal resistance. Experimental curves are reported to show that series voltage feedback with a feedback depth of 10 substantially reduces voltage-dependent zero drift with only modest circuit complication.
Full Text
Reduction of Phase Shifts in a Phase-Detector Transformer
L. I. Olomutskii, M. S. Roitman
(Presented at the Tomsk City Scientific and Technical Conference dedicated to the 90th anniversary of the birth of V. I. Lenin)
It is known that low-frequency transformers are nonlinear elements, since the inductance of the primary winding and the loss resistance depend on the voltage applied to the transformer. The phase shift produced by the transformer will likewise vary with the voltage, which in a number of cases is inadmissible.
Fig. 1.
In most cases the transformer is the load of electron tubes; consequently, the internal resistance of the source of e.m.f. feeding the transformer may be regarded as active and equal to \(R_i\). Since the transformer load is a synchronous detector, which may be represented in the form of an active resistance \(R_n\), the equivalent circuit of the transformer takes the form shown in Fig. 1.
To simplify the analysis, we assume that the resistance equivalent to core losses is \(r_c = 0\). Writing Kirchhoff’s equations for the given circuit, we obtain
\[ \dot{E} = \dot{I}_1 \left( R_i + r_1 + j\omega L_{s1} + j\omega L \right) - \dot{I}_2 j\omega L, \]
\[ 0 = -\dot{I}_1 j\omega L + \dot{I}_2 \left( R_n + r'_2 + j\omega L'_{s2} + j\omega L \right). \]
The transmission coefficient of the cascade is equal to
\[
K^* =
\frac{\omega L R_{\mathrm{н}}}
{(R_i+r_1)\,3\omega L+(R_{\mathrm{н}}+r'_2)\,2\omega L+\omega L(R_{\mathrm{н}}+R_i+r_1+r'_2)}
\]
\[
=
\frac{\omega L R_{\mathrm{н}}}
{+\,j\left[\alpha\beta\omega^2L^2+(\alpha+\beta)\omega^2L^2-(R_{\mathrm{н}}+r'_2)(R_i+r_1)\right]},
\]
where
\[ \alpha=\frac{L_{s1}}{L}, \qquad \beta=\frac{L'_{s2}}{L}. \]
The tangent of the phase-shift angle introduced by the cascade is equal to
\[ \operatorname{tg}\zeta= \frac{(R_{\mathrm{н}}+r'_2)(R_i+r_1)-\omega^2L^2(\alpha+\beta+\alpha\beta)} {\omega L\left[(R_i+r_1)(1+\beta)+(R_{\mathrm{н}}+r'_2)(1+\alpha)\right]} . \tag{1} \]
Taking into account that \(\alpha\ll 1\) and \(\beta\ll 1\), and neglecting the active resistance of the transformer windings \(r_1\) and \(r'_2\), we obtain
\[ \operatorname{tg}\zeta= \frac{R_{\mathrm{н}}R_i-\omega L(L_{s1}+L'_{s2})} {\omega L(R_i+R_{\mathrm{н}})} . \tag{2} \]
The presence of transformer nonlinearity leads to the appearance of a dynamic (relative) phase shift that depends on the voltage \(U\) at the transformer input, \(\Delta\varphi=\psi(U)\). If a constant phase shift is easily compensated, the problem is much more complicated with a dynamic phase shift.
To reduce the relative phase shift introduced by the transformer, several methods may be proposed. One of them consists in the fact that, by a rational choice of the transformer parameters, it is possible, as follows from (2), to achieve a reduction of the absolute shift. Then, when \(L\) changes within certain limits owing to a change in the voltage at the transformer input, the relative phase shift will vary little.
Fig. 2.
A second method is the introduction of negative feedback. It is known that, when negative feedback is introduced, the phase relations are improved. For linear systems, the phase shift introduced by the system is reduced by a factor of \(1+K\beta\). At the same time, owing to a reduction in the internal resistance of the source (2), the zero drift of the synchronous detector (1) is reduced.
The dependence of the zero drift \(\Delta\varphi\) on changes in the voltage at the transformer input was recorded in the presence of series voltage feedback with a depth \(1+K\beta=10\) (curve II, Fig. 2) and without it (curve 1, Fig. 2). From comparison of the curves it is clear that good results can be obtained with only a slight complication of the circuit.
In the case where the leakage inductance can be neglected (which is true for toroidal transformers, in which leakage is very small), formula (1) takes the form
\[ \operatorname{tg}\varphi = \frac{(R_H+r'_2)(R_i+r_1)} {\omega L(R_H+R_i+r'_2+r_1)} . \tag{3} \]
It follows from (3) that the smaller the internal resistance of the source, the smaller the relative phase shift of the transformer caused by the influence of the change in the inductance of the primary winding when the voltage at the transformer input changes.
The easiest way to obtain a small value of the internal resistance of the source is to use a cathode follower whose load is the transformer.
LITERATURE
- Roitman M. S., Gofman S. A., Olomutskii L. P., Karmadonov A. N. On the stability of the zero of synchronous detectors using semiconductor diodes and triodes, Izmeritel'naya tekhnika, No. 9, 1961.