On One Type of Nonlinear Phase Error Arising During Frequency Conversion

G. I. Levashkin

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

The importance of frequency conversion in modern radio engineering is difficult to overestimate. Converting devices are especially widely used in various areas of instrument engineering. Depending on the purpose of the apparatus in which frequency converters are used, requirements are specified for the spectrum of the converted signal, for the stability of the phase (or frequency), etc. Thus, in creating special frequency standards and phase-measuring apparatus, primary attention is given to eliminating chaotic phase fluctuations and various kinds of nonlinear phase errors arising in the frequency-converter unit.

In a number of works \([1, 2, 3]\) it has been shown that, when converting the frequency of oscillations with an integral ratio of the mixed frequencies, a dependence appears of the amplitude and phase of the difference frequency on the type of nonlinearity and on the amplitudes of the mixed signals. It can be shown that such a dependence is valid not only for the difference frequency, but also for all the other spectral components, including the sum frequency.

If the characteristic of the nonlinear element is described by a power polynomial
\(i = i_0 + i_1 U_0 + i_2 U_0^2 + i_3 U_0^3 + i_4 U_0^4 + i_5 U_0^5 + \ldots\), and the control voltage acting on the nonlinear element is
\(U_0 = U_1 \cos \omega_1 t + U_2 \cos(\omega_2 t + \varphi)\), then we obtain a current whose spectrum can be determined with the aid of a Fourier series¹

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

where

\[ J_0 = \frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} i(U_0)\,dt, \qquad a_p = \frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} i(U_0)\cos p\omega_0 t\,dt, \]

\[ b_p=\frac{2}{T_0}\int\limits_{-T_0/2}^{T_0/2} i(U_0)\sin p\omega_0 t\,dt. \]

The quantity \(T_0\) is determined from the relation of the mixed frequencies. Analytical expressions for the amplitudes and phases of the difference and sum oscillations for small-integer ratios of the mixed frequencies, when the characteristic of the nonlinear element is approximated by a fifth-degree polynomial, are given in Tables 1 and 2.

In Tables 1 and 2 the following notation is used:

\[ i_{\omega}=U_1\left[ i_1+\frac{3}{4}i_3\left(U_1^2+2U_2^2\right) +\frac{5}{8}i_5\left(U_1^4+3U_2^4+6U_1^2U_2^2\right) \right]; \]

\[ i_{\Omega}=U_2\left[ i_1+\frac{3}{4}i_3\left(U_2^2+2U_1^2\right) +\frac{5}{8}i_5\left(U_2^4+3U_1^4+6U_1^2U_2^2\right) \right]; \]

\[ i_{\omega\Omega}=U_1U_2\left[ i_2+\frac{3}{2}i_4\left(U_1^2+U_2^2\right) \right]; \qquad i_{2\omega}=\frac{U_1^2}{2}\left[ i_2+i_4\left(U_1^2+3U_2^2\right) \right]; \]

\[ i_{2\Omega}=\frac{U_2^2}{2}\left[ i_2+i_4\left(U_2^2+3U_1^2\right) \right]; \qquad i_{3\omega}=\frac{U_1^3}{4}\left[ i_3+\frac{5}{4}i_5\left(U_1^2+4U_2^2\right) \right]; \]

\[ i_{3\Omega}=\frac{U_2^3}{4}\left[ i_3+\frac{5}{4}i_5\left(U_2^2+4U_1^2\right) \right]; \qquad i_{4\omega}=\frac{U_1^4}{8}i_4; \]

\[ i_{4\Omega}=\frac{U_2^4}{8}i_4; \qquad i_{5\omega}=\frac{U_1^5}{16}i_5; \qquad i_{5\Omega}=\frac{U_2^5}{16}i_5; \qquad i_{2(\omega\Omega)}=\frac{3}{4}U_1^2U_2^2i_4; \]

\[ i_{2\omega,\Omega}= \frac{U_1^2U_2}{4} \left[ 3i_3+\frac{5}{2}i_5\left(2U_1^2+3U_2^2\right) \right]; \qquad i_{3\omega,\Omega}= \frac{U_1^3U_2}{2}i_5; \]

\[ i_{\omega,3\Omega}= \frac{U_1U_2^3}{2}i_5; \]

\[ i_{\omega,2\Omega}= \frac{U_1U_2^2}{4} \left[ 3i_3+\frac{5}{2}i_5\left(2U_2^2+3U_1^2\right) \right]; \qquad i_{4\omega,\Omega}= \frac{5}{16}U_1^4U_2i_5; \]

\[ i_{\omega,4\Omega}= \frac{5}{16}U_1U_2^4i_5; \]

\[ i_{3\omega,2\Omega}= \frac{5}{8}U_1^3U_2^2i_5; \qquad i_{2\omega,3\Omega}= \frac{5}{8}U_1^2U_2^3i_5. \]

The dependence of the amplitudes and phases of the difference and sum components of the spectrum for mixed frequencies with a large integer ratio will also hold, but it will be determined by the coefficients of the power series at high degrees, i.e., at a greater curvature of the working section of the characteristic of the nonlinear element. From the formulas of Tables 1 and 2, the magnitudes of the phase ad-

For the difference frequency

Table 1

For the sum frequency
Table 3

errors of the intermediate frequency as a function of the phase difference of the signals being mixed, for various ratios of frequencies and amplitudes:

\[ \eta = \varphi_{\mathrm{if}} - \varphi . \]

Here, \(\varphi_{\mathrm{if}}\) is the phase of the intermediate frequency, and \(\varphi\) is the phase difference of the signals being mixed.

Fig. 1. Phase error of the difference frequency for

\[ \frac{\omega_1}{\omega_2} = \frac{1}{2}. \]

1. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV};\)
2. \(U_n = U_v = 50\,\mathrm{mV};\)
3. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}.\)

Fig. 2. Phase error of the difference frequency for

\[ \frac{\omega_1}{\omega_2} = \frac{1}{3}. \]

1. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV};\)
2. \(U_n = U_v = 50\,\mathrm{mV};\)
3. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}.\)

Fig. 3. Phase error of the difference frequency for

\[ \frac{\omega_1}{\omega_2} = \frac{1}{4}. \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV};\)
2. \(U_n = U_v = 50\,\mathrm{mV};\)
3. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}.\)

Fig. 4. Phase error of the difference frequency:

for

\[ \frac{\omega_1}{\omega_2} = \frac{1}{4} \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV};\)

for

\[ \frac{\omega_1}{\omega_2} = \frac{2}{3} \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV};\)

for

\[ \frac{\omega_1}{\omega_2} = \frac{3}{4} \]

\[ \begin{cases} 3 - U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV};\\ 4 - U_n = U_v = 50\,\mathrm{mV}. \end{cases} \]

Fig. 5. Phase error of the difference frequency:

for

\[ \frac{\omega_1}{\omega_2}=\frac{2}{3} \]

1. \(U_n = U_v = 50\,\mathrm{mV}\);
2. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\);

for

\[ \frac{\omega_1}{\omega_2}=\frac{1}{4} \]

1. \(U_n = U_v = 50\,\mathrm{mV}\);
2. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\).

Fig. 6. Phase error of the sum frequency for

\[ \frac{\omega_1}{\omega_2}=\frac{1}{2}. \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}\);
2. \(U_n = U_v = 50\,\mathrm{mV}\);
3. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\);

\(\times\) — values of the phase error verified experimentally.

Fig. 7. Phase error of the sum frequency for

\[ \frac{\omega_1}{\omega_2}=\frac{1}{3}. \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}\);
2. \(U_n = U_v = 50\,\mathrm{mV}\);
3. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\);

\(\times\) — values of the phase error verified experimentally.

Fig. 8. Phase error of the sum frequency for

\[ \frac{\omega_1}{\omega_2}=\frac{1}{4}. \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}\);
2. \(U_n = U_v = 50\,\mathrm{mV}\);
3. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\);

\(\times\) — values of the phase error verified experimentally.

Fig. 9. Phase error of the sum frequency for

\[ \frac{\omega_1}{\omega_2}=\frac{2}{3}. \]

1. \(U_n = 10\,\mathrm{mV},\ U_v = 50\,\mathrm{mV}\);
2. \(U_n = U_v = 50\,\mathrm{mV}\);
3. \(U_n = 50\,\mathrm{mV},\ U_v = 10\,\mathrm{mV}\);

\(\times\) — values of the phase error verified experimentally.

For the calculation, data were taken from an approximation of the nonlinear characteristic of a P-14 semiconductor triode. The experiment was carried out with a single-grid mixer assembled using the indicated triode. To form the required integer ratios, frequencies of 5 kHz, 10 kHz, 20 kHz, 30 kHz, and 40 kHz were taken from a coherent-frequency grid generator.

Measurement and monitoring of the phase of the intermediate frequency were performed using Lissajous figures; as the sweep voltage, a voltage of higher frequency (2–5 times higher) from the same frequency-grid generator was used. The phase difference of the mixed signals was set by means of the simplest phase shifter for the phase of the higher frequency. The results of the calculation and experiment are presented in Figs. 1–9. They show that the manifestation of the dependence of the phase of the combination signal on the form of the characteristic and on the amplitudes of the input signals will be more noticeable and greater if

a) in order to increase the transfer coefficient of the mixer, the operating point is chosen on a section of the characteristic of the nonlinear element with greater curvature (i.e., the operating section is approximated by a polynomial of higher degree);

b) the ratio of the mixed signals is of the same order as the degree of nonlinearity of the operating section of the characteristic \((\alpha+\beta <\) the greatest exponent \(n)\);

c) the amplitude of the signal voltage (the higher frequency) is less than or equal to the amplitude of the heterodyne voltage (the lower frequency);

d) the mixed signals are incoherent \((\varepsilon \ne 0)\).

In the general case, the influence of the amplitudes of the mixed signals and of the form of the characteristic on the phase of the converted oscillation is stronger for the difference frequency than for the sum frequency.

References

1. L. I. Mandelstam and N. D. Papaleksi. Author’s certificate, class 21-a, No. 43685.

2. B. I. Yuzvitskii. On certain methods of receiving radio waves while preserving the constancy of phase relationships. ZhTF, Vol. XI, Nos. 1–2, 1941.

3. V. G. Vlasov. Phase relationships in frequency conversion of oscillations with an integer frequency ratio. Scientific Notes of the Leningrad Higher Naval Engineering School named after Admiral Makarov, Issue II, 1956.

1. Finding the spectrum of the mixer current with the aid of (1) can also be carried out, for very large integral ratios of the mixed frequencies, for irrational ratios as well, with one or another degree of approximation. ↩