Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 116　　　　　　　　　　　　　　　　　　　　　　　　　　　　1962

On Phase Relationships in Frequency Conversion

G. I. Levashkin

(Presented by the scientific and technical conference of the Radio Engineering Faculty, dedicated to the 60th anniversary of the Institute)

It is known [1, 2] that the error introduced by the pre-indicator device of a phase-measuring instrument is determined mainly by distortions caused by nonlinear processes that transform the signal waveform: frequency conversion, multiplication, division of frequencies, and the like. As experience in the operation of phase-measuring equipment has shown [1], nonlinear phase errors arising in the frequency-conversion unit may considerably exceed all other phase errors. However, all the causes of the occurrence of these errors and the possibilities for reducing them have received little treatment in the literature.

In the present article an attempt is made to analyze the causes of the appearance of nonlinear phase errors and to determine a possible method for their quantitative evaluation.

Types of nonlinear phase errors in frequency conversion

Earlier investigations of phase relationships in frequency conversion [3, 4, 5] (as also in other nonlinear processes [9, 10, 11]) under conditions of ideal signal selection showed that a dependence of the phase of the output signal may be recorded on changes in the amplitudes of the input voltages, on inaccuracies in tuning the load oscillatory circuit, on drift of the signal or heterodyne frequency, on changes in the parameters of the circuit with changes in temperature, and on similar causes.

The indicated phase errors can be reduced by various methods [1, 2, 9, 10, 11] in the design of the circuit and adjustment of the unit. However, as was noted in [1] and [2], with the usual selectivity of the device phase distortions of another character, which have a time dependence, are also recorded. They appear even when all the above-mentioned causes are eliminated. This type of phase error has been termed parasitic phase modulation and can be fully explained by the presence in the spectrum of the output voltage of interfering combination frequencies, since phase variations with time can be recorded in the case where the signal spectrum consists of a certain number of harmonic oscillations having asymmetry.

—to the amplitudes or phases relative to the fundamental oscillation. The degree of asymmetry is proportional to the magnitude of the phase modulation being measured. In connection with this, it becomes necessary to analyze the spectrum of the signal obtained as a result of frequency conversion.

Nature of the Spectrum of a Frequency-Converted Signal as a Function of the “Frequency Regime” of the Mixer

It is known [6, 7] that the very appearance of combination frequencies is determined exclusively by the relationship among the frequencies participating in the conversion—the “frequency regime.” In addition (since we are interested in the stationary process during conversion), it should be noted that stationary processes in physical systems are in general incommensurable; however, we can always, with any desired degree of accuracy, represent the process as consisting of an integral series of oscillations with commensurable frequencies. These general considerations allow us to assume, without loss of generality, that the “frequency regime” of the mixer is determined by \(N\) oscillations with frequencies \(\omega_1, \omega_2, \ldots, \omega_N\), where \(\omega_1 : \omega_2 : \ldots : \omega_N = x : p : \ldots : z\), where \(x, p, \ldots, z\) are simple integers having no common divisor.

In the anode circuit of the converter there is a nonperiodic sequence of current pulses, which can be represented as a sum of simple harmonic oscillations whose periods \(T_p\) depend linearly on the fundamental repetition period of the current pulses \(T_o\). The latter, in turn, is wholly determined by the periods of oscillation of the input signals \(T_N\) and depends linearly on them. Thus,

\[ T_p = \frac{T_o}{p}, \tag{1} \]

and

\[ T_o = xT_1 = pT_2 = \ldots zT_N. \tag{2} \]

From (1) and (2) it follows: 1) for large integer frequency ratios \(T_o \to \infty\), and the spectrum of the output signal becomes continuous; 2) when mixing the frequencies \(\omega_1; \omega_2; 2\omega_2; \ldots n\omega_2\), the same period of oscillations \(T_o\) is obtained as when mixing the frequencies \(\omega_1\) and \(\omega_2\); 3) in contrast to the commonly used formula for determining combination frequencies

\[ \left|\, n\omega_1 \pm m\omega_2 \pm \ldots \pm k\omega_N \,\right| = \omega_{np} \tag{3} \]

relations (1) and (2) give a unique solution in determining the possible combination frequencies¹.

The amplitudes and phases of the oscillations of the combination frequencies depend on the characteristic of the nonlinear element, the amplitudes of the input signals, the converter circuit, and its tuning. Thus, in order to determine the spectrum of the output voltage it is necessary to find the influence of the above-mentioned factors on the appearance of the various combination frequencies determined on the basis of the “frequency regime” of operation of the mixer.

Indeed, let \(i = f(U_o)\) be the characteristic of the nonlinear element; then, on the basis of knowledge of the period of oscillation of the current pulses from (1) and (2), one can write that

\[ i = J_0 + \sum_{p=1}^{\infty} \sqrt{a_p^2 + b_p^2}\cos\left(p\omega_0 t + \operatorname{arctg}\frac{b_p}{a_p}\right), \tag{4} \]

where

\[ J_0 = \frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} f(U_d)\,dt, \qquad a_p = \frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} f(U_d)\cos p\omega_0 t\,dt, \]

\[ b_p = \frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} f(U_d)\sin p\omega_0 t\,dt. \]

From (4) it is seen that the amplitude and phase of the \(p\)-th combination frequency are determined uniquely, in contrast to the previously used methods for determining the amplitudes and phases of combination products \([8]^1\).

Fig. 1. Phase distribution in the spectrum
of the mixer output current
at \(\dfrac{\omega_1}{\omega_2}=1\).

For convenience in solving the problem, let us represent the voltage acting on the nonlinear element, \(U_d\), and the characteristic of the nonlinear

Fig. 2. Phase distribution in the spectrum of the mixer output current
at \(\dfrac{\omega_1}{\omega_2}=\dfrac{19}{20}\).

\(^1\) For example, in the method set forth in [8], determination of the amplitude for the frequency \(\omega_{np_i}\) was reduced to determining the amplitude and phase of the oscillation of the combination \(\pm n_i\omega_1 \pm m_i\omega_2 \pm \ldots \pm k_i\omega_N\). But, as was indicated above, there may be an infinite set of such combinations; therefore, for an unambiguous determination of the amplitude of oscillations with frequency \(\omega_{np_i}\), it is necessary to find all combinations \(n_i, m_i,\ldots,k\) from (3), determine the amplitudes and phases of all these combinations, and then perform vector addition.

Fig. 3. Phase distribution in the spectrum of the mixer output current for
\[ \frac{\omega_1}{\omega_2}=\frac{5}{6}. \]

Fig. 4. Phase distribution in the spectrum of the mixer output current for
\[ \frac{\omega_1}{\omega_2}=\frac{11}{20}. \]

Fig. 5.

element in the form of sums of even and odd functions; then the quantities \(l_0\), \(a_p\), \(b_p\) can be found from the expressions:

\[ \left. \begin{aligned} l_0&=\frac{1}{\pi}\int_{-\pi}^{\pi}\left[f_{\mathrm{e}}(U_0)+f_{\mathrm{o}}(U_{\mathrm{o}})\right]\,dx;\\[4pt] a_p&=\frac{1}{\pi}\int_{-\pi}^{\pi}\left[f_{\mathrm{e}}(U_0)+f_{\mathrm{o}}(U_{\mathrm{o}})\right]\cos px\,dx;\\[4pt] b_p&=\frac{1}{\pi}\int_{-\pi}^{\pi}f_{\mathrm{o}}(U_{\mathrm{o}})\sin px\,dx, \end{aligned} \right\} \tag{4} \]

where the subscript \(\mathrm{e}\) denotes the even part, and \(\mathrm{o}\) the odd part.

For \(U_{\mathrm{o}}=0\), the coefficients \(b_p=0\), i.e., the distribution of phases over the frequencies of the spectrum will be linear. It can be seen that solution (4) makes it possible to trace the dependence of the phase on changes in the amplitudes of the input signals, on instability of the signal and heterodyne frequencies, and on other causes. As is evident from Figs. 1, 2, 3, 4, the dependence of the phase of the spectral component \((p\omega_0)\) on the amplitudes and phases of the mixed signals will be more noticeable if the ratio of the mixed frequencies approaches \(1/2\), and the highest degree of nonlinearity is comparable with the value determined by the sum of the numbers expressing the ratio of the mixed frequencies \((\alpha+\beta)\). To find the spectrum of the output voltage it is then necessary to consider the action of the obtained spectrum on the nonlinear selective system.

Determination of Parasitic Phase Modulation of a Signal with Converted Frequency in the Presence in the Spectrum of Two Interfering Combination Frequencies

Suppose that at the output of the mixer we have a signal spectrum consisting of three frequencies (Fig. 5): the useful frequency \(\Omega=p\omega_0\) and the interfering combination frequencies \(\omega_1=(p-1)\omega_0\) and \(\omega_2=(p+1)\omega_0\). The amplitudes and phases of these oscillations, determined according to (4), are respectively equal to \(l\), \(a\), \(b\) and \(\varphi_0\); \(\varphi_1=\varphi_0-\Delta\varphi_1\); \(\varphi_2=\varphi_0+\Delta\varphi_2\). Then

\[ \varphi=\operatorname{arctg} \frac{ \sin\left(\arcsin\frac{\sin 2\omega_0 t}{\sqrt{1+k^2+2k\cos 2\omega_0 t}} -\omega_0 t+\frac{\Delta\psi}{2}\right) }{ \frac{r}{\sqrt{1+k^2+2k\cos 2\omega_0 t}} +\cos\left(\arcsin\frac{\sin 2\omega_0 t}{\sqrt{1+k^2+2k\cos 2\omega_0 t}} -\omega_0 t+\frac{\Delta\psi}{2}\right) } \ldots \]

\[ \ldots -\omega_0 t+\frac{\Delta\psi}{2}\right), \]

and

\[ 2\varphi_{\max}=\operatorname{arctg}\left\{ \frac{2}{r^2-2(1+k^2)} \cdot \sqrt{ (1+k)^2\left[ r^2\sin^2\frac{\Delta\psi}{2}-(1-k)^2 \right] +r^2\cos^2\frac{\Delta\psi}{2}(1-k)^2 } \right\}. \tag{5} \]

i.e.

\[ r=\frac{1}{b}; \qquad \frac{\Delta \psi}{2}=\frac{\Delta \varphi_2}{2}-\frac{\Delta \varphi_1}{2} \quad \text{and} \quad k=\frac{b}{a}\quad \text{for } b<a. \]

For \(\varphi_1=-\varphi_2,\ \Delta\psi=0\) (there is no phase asymmetry),

\[ 2\varphi_{\max}=\operatorname{arctg} \frac{2(1-k)}{r^2-2(1+k^2)} \sqrt{r^2-(1+k)^2}. \]

If \(k\ll 1\), then

\[ 2\varphi_{\max}=\operatorname{arctg}\frac{2r}{r^2-2}. \]

The presence of phase asymmetry alone \((k=1)\) gives

\[ 2\varphi_{\max}=\operatorname{arctg} \frac{4r\sin \dfrac{\Delta\psi}{2}}{r^2-4}. \]

With four interfering frequencies the expressions for \(\varphi\) and \(\varphi_{\max}\) become considerably more complicated, but for small amplitudes of the interfering combination frequencies (when \(a/l\) and \(b/l<1\)) it may be assumed that

\[ \varphi_{\max.\,\text{total}} \simeq \sum_1^N \varphi_{\max_i}, \]

where \(i\) is the number of an interfering pair of combination frequencies symmetrically spaced from the selected frequency.

Conclusion

Despite the difficulties in solving equations (4′), the given method for analyzing nonlinear phase errors makes it possible to obtain an unambiguous solution in determining the spectrum of the output signal for the case of frequency conversion, to obtain the distribution of amplitudes and phases in the spectrum of the mixer output voltage, to analyze various methods of frequency mixing, to make an optimal choice of circuit on the basis of a specified “frequency regime,” and to estimate the error in transmitting the phase difference of the signals being mixed.

Using (4′) and (5), one can calculate the possible phase modulation at the mixer output in the presence of two interfering frequencies.

LITERATURE

1. L. N. Duve. Phase meters of medium accuracy. Transactions of the Scientific Research Institute of the State Committee of Standards, Measures, and Measuring Instruments of the USSR, issue 1, 1955.

2. A. M. Bonch-Bruevich, V. I. Shirokov. Some questions of phase measurements. ZhTF, vol. XXV, issue 10, 1955.

3. V. I. Yuzvinskii. On one variant of an interference-instrument circuit for determining the phase distribution of an electromagnetic field. ZhTF, vol. XI, issue 1—2, 1941.

4. V. I. Yuzvinskii. On certain methods of receiving radio waves while preserving the constancy of phase relationships. ZhTF, vol. XI, issue 1—2, 1941.

5. B. G. Vlasov. Phase relationships in frequency conversion. Scientific Notes of the Admiral Makarov LVIMU, issue 2, 1956.

6. B. A. Khanov. On the spectrum of a signal subjected to frequency conversion. Elektrosvyaz’, No. 10, 1960.

7. I. M. Guk. On the question of choosing initial frequencies in interpolation converters. Proceedings of Higher Educational Institutions of the USSR, Radio Engineering, No. 3, 1960.

8. L. T. Kim. Determination of the amplitudes of combination current components arising when a nonlinear resistance is acted upon by a sum of sinusoidal voltages. Collection of Scientific Works of TsNIIS, issue 1, 1959.

1. Yu. A. Sverdlov. A method for calculating amplitude and phase modulation in a multistage frequency multiplier. Radio Engineering and Electronics, vol. 4, no. 6, 1959.

2. K. M. Shulzhenko. On phase relationships in frequency multiplication. Proceedings of the Tomsk City Scientific-Technical Conference on the Automation of Production Processes, Tomsk, 1959.

3. V. G. Efremov. Reduction of phase modulation by means of a trigger cell. Proceedings of the jubilee scientific-technical conference dedicated to the 100th anniversary of the birth of A. S. Popov, Tomsk, 1959.

1. Indeed, it is possible to find an infinitely large number of values of \(n, m, \ldots, k\) for which (3) will be satisfied. Consequently, (3) gives an ambiguous determination of the frequency spectrum of the output signal, whereas from (1) and (2) the value of each \(\omega_{np}\) is obtained only once. ↩