WIDEBAND FOUR-PHASE OSCILLATOR

V. I. SEDOV, M. S. ROITMAN, A. I. KRAMNYUK

(Presented by the scientific seminar of the Department of Radio Engineering)

The methods considered in [1] for creating two-phase oscillators can ensure a further increase in the accuracy of setting the quadrature of two voltages over a frequency range only when highly stable and high-precision phase-shifter elements are used, as well as matching units with very small phase shifts. This approach requires the use of nonreactive high-precision resistors and capacitors with small loss angles. The creation of amplifiers, phase inverters, and transformers with small phase shifts over a frequency range encounters substantial difficulties. In addition, there arises the need for rigid stabilization of the operating frequency. All of the above makes it extremely difficult to obtain the quadrature of two voltages with an error of no more than \(0.5 \div 0.1^\circ\) in the frequency range \(20\ \text{Hz} \div 100\ \text{kHz}\) and leads to a sharp complication of the apparatus.

An effective way to reduce the error in setting the quadrature of two voltages is to use built-in automatic checking systems (BACS). In this case, the requirements imposed on all the units of the multiphase oscillator, without exception, are sharply reduced, which makes it possible to use mass-produced components.

The block diagram of a four-phase oscillator using a BACS is shown in Fig. 1. To obtain the quadrature of two voltages, a bridge phase shifter is used, one of whose elements is a photoresistor.

The BACS consists of a sum-difference quadrature phase indicator of the type using photoconverters [2] and a dc amplifier. In the event that the phase shift deviates from \(90^\circ\), the sum and difference of the voltages that are in quadrature will not be equal to one another, and the balance of the bridge will be disturbed. The unbalance voltage, after corresponding amplification in the dc amplifier, changes the illumination of the lamp \(L_3\) of the photoconverter \(FP_3\), which is part of the active resistance of the phase shifter; as a result, the phase shift of the phase shifter changes in such a way that the deviation of the phase angle from \(90^\circ\) between the output voltages of the oscillator is reduced.

Let us determine the total error in setting the quadrature; for this purpose we shall open the automatic regulation system at points \(I—I'\) and set an arbitrary value \(\dfrac{d\varphi}{\varphi}\) at the oscillator output,

where \(\varphi\) is the magnitude of the deviation of the phase shift between the output voltages from \(90^\circ\).

Fig. 1

In this case, the voltages acting on lamps \(L_1\) and \(L_2\) will be:

\[ U_1 = U_{\sim}\sqrt{2}\sqrt{1+\sin\varphi} \quad \text{and} \quad U_2 = U_{\sim}\sqrt{2}\sqrt{1-\sin\varphi} \]

and

\[ \frac{dU_1}{U_1} = \frac{\varphi\cos\varphi}{2(1+\sin\varphi)} \cdot \frac{d\varphi}{\varphi} \qquad \frac{dU_2}{U_2} = - \frac{\varphi\cos\varphi}{2(1-\sin\varphi)} \cdot \frac{d\varphi}{\varphi}. \tag{1} \]

Traversing the system around the loop, we obtain:

\[ \frac{di_{\ell 1}}{i_{\ell 1}} = \frac{d_1}{d_2} \cdot \frac{dU_1}{U_1}; \qquad \frac{di_{\ell 2}}{i_{\ell 2}} = \frac{d_3}{d_4} \cdot \frac{dU_2}{U_2}; \tag{2} \]

\[ \frac{dR_{\phi 1}}{R_{\phi 1}} = - k_{i1} \frac{di_{\ell 1}}{i_{\ell 1}}; \qquad \frac{dR_{\phi 2}}{R_{\phi 2}} = - k_{i2} \frac{di_{\ell 2}}{i_{\ell 2}}; \tag{3} \]

\[ \frac{dU_0}{U_0} = \frac{U_M}{U_0} \cdot \kappa \frac{ \dfrac{dR_{\phi 1}}{R_{\phi 1}} - \dfrac{dR_{\phi 2}}{R_{\phi 2}} }{1+a}; \tag{4} \]

\[ \frac{di_{\ell 3}}{i_{\ell 3}} = \frac{d_5}{d_6} \cdot \frac{dU_0}{U_0}; \tag{5} \]

\[ \frac{dR_{\phi 3}}{R_{\phi 3}} = - k_{i3} \frac{di_{\ell 3}}{i_{\ell 3}}; \tag{6} \]

\[ \frac{dR}{R} = \frac{d_8}{d_7} \cdot \frac{dR_{\phi 3}}{R_{\phi 3}}; \tag{7} \]

\[ \frac{d\varphi}{\varphi}=-\frac{3}{2}\frac{\cos^2\varphi}{\varphi}\cdot\frac{dR}{d}. \tag{8} \]

The simultaneous solution of (1)—(8) makes it possible to obtain the stabilization coefficient of the phase shift

\[ n_{\mathrm{st}}=\frac{3}{2}\varkappa_{i1}\cdot\varkappa_{i3}\frac{d_1d_5d_8}{d_2d_6d_7}\cdot K\cdot\cos\varphi\frac{U_m}{U_0}\cdot\frac{1}{1+a}, \tag{9} \]

where

\[ d_1=1+\frac{R_1}{R_{n1}};\qquad d_2=n_1+\frac{R_1}{R_{n1}};\qquad d_3=1+\frac{R_2}{R_{n2}}; \]

\[ d_4=n_2+\frac{R_2}{R_{n2}};\qquad d_5=1+\frac{R_3}{R_{n3}};\qquad d_6=n_3+\frac{R_3}{R_{n3}}; \]

\[ d_7=1+\frac{R_{\phi3}}{R_4};\qquad d_8=\frac{\dfrac{R_{\phi3}}{R_5}}{1+\dfrac{R_{\phi3}}{R_5}}; \]

\(d_1 \div d_6\) are constant coefficients characterizing the operating modes of the photoconverters;

\(d_7, d_8\) are constant coefficients characterizing the degree of switching-on of the photoresistance \(R_{\phi3}\);

\(K_{i1}, K_{i2}, K_{i3}\) are the relative current transfer coefficients of \(FT_1\), \(FT_2\), \(FT_3\), respectively;

\(K\) is the gain coefficient of the DC amplifier.

\[ a=\frac{R_{\phi1}}{R_{n1}}=\frac{R_{\phi2}}{R_{n2}} \]

for

\[ d_1=d_3=d_5=3\quad\text{and}\quad d_2=d_4=d_6=4,\quad a=1, \]

\[ d_7=6,\quad d_8=\frac{6}{7},\quad K=50,\quad \frac{U_m}{U_0}=1,\quad K_{i1}=K_{i2}=K_{i3}=5, \]

\[ n_{\mathrm{st}}=\frac{3}{2}\cdot5\cdot5\cdot\frac{3\cdot3\cdot6}{4\cdot4\cdot6\cdot7}\cdot50\cdot1\cdot0.5=75, \]

i.e., if the quadrature error with the open-loop system is of the order of \(1^\circ\), the error with the WSAP switched on will be of the order of \(0.01^\circ \div 0.02^\circ\).

What has been said is valid provided that the phase indicator of the WSAP system does not introduce a noticeable error. An analysis of possible implementations of phase indicators [2, 3] showed that the highest accuracy is attainable in systems that compare the sum and difference of the vectors of the input voltages by their rms value. In this case the main component of the phase-indicator error, namely the error due to the influence of higher harmonic components, turns out to be negligibly small. Therefore, when rms-value phase indicators are used, the quadrature-setting error of two voltages will be determined by the errors of the circuits that form the sum and difference of the voltage vectors and by the influence of the spread of the input resistances of the indicator.

The equivalent electrical circuit of the network forming the sum and difference of the voltages is shown in Fig. 2. In this circuit \(E_1, E_2, E_3, E_4, r_1, r_2, r_3, r_4\) are the output voltages and output resistances of each output of the four-phase generator; \(R_1\) and \(R_2\) are the resistances of the bridge—

...of the bulbs of the phase-indicator photoconverters; \(R_3\) and \(R_4\) are the resistances of their equivalents.

The error in indicating the quadrature of two voltages, for example \(E_1\) and \(E_2\), will be

\[ \varphi = \frac{\delta}{25(2+m)} + \frac{\delta_i}{50(4+m)} + \frac{\delta \cdot K}{2500(2+m)} + \frac{\delta_i \cdot K}{5000(4+m)} + \frac{\alpha_1}{4+m} + \frac{\alpha_2}{2}, \]

where \(\delta\) is the spread of the resistances \(R_1, R_2, R_3, R_4\), in %;
\(\delta_i\) is the spread of the resistances \(r_1, r_2, r_3, r_4\), in %;
\(K\) is the deviation of the voltage amplitudes of one channel \((E_1\) and \(E_3)\) relative to the voltage amplitudes of the other channel \((E_2, E_4)\), in %;
\(\alpha_1\) is the phase error of the transformer of the first channel (between \(E_1\) and \(E_3\)), in radians;
\(\alpha_2\) is the phase error of the transformer of the second channel (between \(E_2\) and \(E_4\)), in radians.

\[ m=\frac{R}{r}. \]

Fig. 2

Let us consider the main components of the error in indicating the quadrature of two voltages (for \(R=470\ \Omega\), \(r=14.36\ \Omega\)).

1. Error due to the spread of the input resistances of the phase indicator

\[ \varepsilon_R=\frac{\delta}{25(2+m)}, \qquad \text{where } \delta=\frac{\Delta}{R}\cdot 100\%. \]

The spread of the elements \(R_1, R_2, R_3, R_4\) in this case is associated with the difference between the static and dynamic resistances of the filament bulb.

As is known [4], the static resistance of a circuit consisting of a linear resistance and a bulb at unchanged current amplitude is determined as

\[ R_{\text{st}} = R_{\text{add}} + R_{\text{l}}, \]

and the dynamic resistance is

\[ R_{\text{dyn}} = R_{\text{add}} + nR_{\text{l}}, \]

where \(n\) is the nonlinearity index of the current–voltage characteristic of the filament bulb.

The equivalents \(R_3\) and \(R_4\) can, with a sufficiently high degree of accuracy, be selected equal to \(R_{\text{st}}\); therefore

\[ \delta=\frac{\Delta}{R}\cdot 100\% = \frac{R_{\text{dyn}}-R_{\text{st}}}{R_{\text{st}}}\cdot 100\% = \frac{n-1}{d_1}\cdot 100\%. \]

Usually \(d_1 \simeq 3\), \(n=2\). Therefore

\[ \delta \simeq \frac{n-1}{d_1}\cdot 100\% = 33\%. \]

and

\[ \varphi_R=\frac{\delta}{25(2+m)}=\frac{33}{25\cdot 34.7}=0.04^{\mathrm{rad}}\simeq 2.3^\circ . \]

1. Error due to the spread of the output resistances of each phase of the generator.

The indicated spread is determined by the nonidentity of the halves of the secondary windings of the transformer and does not exceed 1%. Therefore

\[ \varphi_r=\frac{\delta_i}{50(4+m)}=\frac{1}{5\cdot 36.7}=5.4\cdot 10^{-4}_{\mathrm{rad}}=0.03^\circ . \]

1. Error due to amplitude asymmetry of the channels

\[ \varphi_k=\frac{K}{100}(\varphi_R+\varphi_r) \]

in the presence of periodic calibration of the channel amplitudes can be reduced to a value below \(0.01^\circ\).

1. Error due to phase asymmetry of the halves of the secondary winding of the output transformers

\[ \varphi_a=\frac{\alpha_1}{4+m}+\frac{\alpha_2}{2}. \]

Without major error it may be assumed that

\[ \alpha_1=\alpha_2=\alpha, \]

then

\[ \varphi_a=\frac{6+m}{2(4+m)}\alpha\simeq 0.52\alpha. \]

As the experiments performed show, for wideband transformers \(\alpha\) can reach several degrees. Thus, the principal components of the error in indicating the quadrature of two voltages in the generator according to the circuit of Fig. 1 are the errors of phase inversion in the transformer circuits and the errors due to the influence of the spread of the input resistances of the phase indicator.

The phase error of a transformer phase-inverting circuit is determined mainly by parasitic parameters, which have very large values, especially in transformers intended for operation over a wide frequency range. This error can be significantly reduced by using correcting \(RC\) circuits. Such correction, with selected \(R\) and \(C\), takes place only at one frequency.

Recently, new circuit solutions have appeared that make it possible sharply to reduce the error of any phase-inverting circuits, the operation of which is based on the principle of self-compensation of the error [5]. The operation of the latter can be explained by the block diagram of Fig. 3, \(a\), and by the vector diagram of Fig. 3, \(b\).

Fig. 3

The output voltages of any phase-inverting circuit differ from one another both in magnitude and in phase. If between the outputs

include a precision divider by 2 in the circuit, then the voltage value \(U_0\) between the midpoint of the divider and ground will contain information on the voltage asymmetry both in magnitude and in phase. Having amplified the voltage \(U_0\) by a factor of two and added it to either of the output voltages \((U_1)\), at the output of the device we now obtain two equal antiphase voltages \(U'_1\) and \(U_2\).

In the case of an ideal divider and amplifier, the error of such a phase-inverting device will be zero. In practice, both the gain of the amplifier and the division ratio of the divider will be complex quantities. It is not difficult to show that in this case the error of such a phase-inverting device will be, in magnitude,

\[ \delta_{\mathrm{eqv}} = \delta_1 \delta_2 + \varepsilon \cdot \varphi_{\Sigma} \]

and, in phase,

\[ \varphi_{\mathrm{eqv}} = \delta_2 \varphi - \delta_1 \varphi', \]

where \(\delta_{\mathrm{eqv}}, \varphi_{\mathrm{eqv}}\) are the asymmetry in magnitude and phase of the voltages of the phase-inverting device;

\(\delta_1, \varphi\) are the asymmetry in magnitude and phase of the voltages of the phase-inverting circuit;

\[ \delta_2 = 1 - \frac{K_y K_c}{K_d}; \]

\(K_d, K_y, K_c\) are the moduli of the transfer coefficients of the divider, amplifier, and summing unit;

\(\varphi_{\Sigma} = \varphi_d + \varphi_y + \varphi_c\) is the total phase shift of the divider, amplifier, and summing unit.

Since the errors of the divider and summing unit are considerably smaller than the errors of the amplifier, the resulting error of the device is completely determined by the quality of the latter.

As follows from the formulas given, the phase error of the phase-inverting device depends strongly on the asymmetry in magnitude and, for large values of the latter, may be completely determined by it. But since in devices of this type both the asymmetry in magnitude and that in phase decrease, the phase asymmetry (error) decreases much more strongly than the magnitude asymmetry.

Figure 4 gives the dependence of the phase shift of the transformer on frequency without compensation and with compensation of the phase error. Reduction of the second component of the error in setting the quadrature of two voltages, namely the error due to the spread of the input resistances of the phase indicator, can be achieved by increasing the ratio

\[ m = \frac{R}{r}, \]

where \(R\) is the value of the input resistance of the phase indicator, and \(r\) is the output resistance of each phase of the generator.

An increase in \(m\) is possible both as a result of increasing \(R\) and of decreasing \(r\). A decrease in the output resistance of each phase of the generator can be obtained by inserting power amplifiers after the output transformer, which sharply increases the complexity of the generator and makes it possible to reduce \(r\) by approximately one order of magnitude.

The effort to increase \(m\) by increasing \(R\) leads us to two variants. The first of them consists in inserting matching networks with high input resistance between the outputs of the four-phase generator and the phase indicator. This variant also complicates the circuit; moreover, very high requirements are imposed on the matching networks with respect to their identity in transfer coefficient and

input resistance. This variant is examined in sufficient detail in [3].

Another way of increasing \(m\) consists in using phase-indicator circuits that have high input resistances.

Fig. 4

REFERENCES

1. E. D. Koltik. Measuring two-phase alternating-current generators. Publishing House of the Committee of Standards, Measures, and Measuring Instruments under the Council of Ministers of the USSR. Moscow, 1963.

2. M. S. Roitman. On phase indicators based on comparison of the effective values of voltages. Automatic Control and Methods of Electrical Measurements. Proceedings of the III Conference. Vol. I, Novosibirsk, 1961.

3. M. S. Roitman, V. I. Sedov, A. I. Kramnyuk, V. M. Sergeev. Investigation of ways and methods for creating a universal calibration installation. Report on a research topic. Tomsk, TPI, 1970.

4. M. S. Roitman, E. I. Tsimbalist, A. I. Lysov. A photoelectric converter as a controllable element of an electrical circuit. — Izvestiya Vuzov. Instrument Engineering. Vol. IX, No. 6, Leningrad, 1966.