MEASUREMENT OF PHASE-MODULATION INDICES AT A RATIONAL FREQUENCY RATIO

V. Ya. Supyan

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

In the study of frequency converters, it is necessary to measure the phase-modulation index in the output signal, whose frequency is neither equal to nor a multiple of the frequency of the reference signal. A necessary condition in measurement is the in-phase relation of the output voltage of the converter with the voltage of the reference signal, or that the frequencies of these voltages be related as integers. The literature [1] describes a method for measuring small phase-modulation indices by means of a cathode-ray tube for the case of the frequency ratio

\[ \frac{m}{n}=\frac{1}{n}. \]

In the present work, the possibility is shown of measuring small phase-modulation indices by means of a cathode-ray tube for any integer ratio of frequencies, and an estimate of the measurement accuracy is given.

The essence of the method is as follows. Let a sinusoidal voltage be applied to the plates of the oscillograph that deflect the beam in the vertical direction,

\[ U_1=U_{m_1}\sin\omega_1 t, \]

and to the plates of horizontal deflection, a voltage

\[ U_2=U_{m_2}\sin[\omega_2 t+\Theta\sin(\Omega t+\Phi)+\varphi_0]. \]

The parametric equations of the figure on the screen of the cathode-ray tube have the form:

\[ x=A\sin n\omega_0 t \tag{1} \]

\[ y=B\sin[m\omega_0 t+\Theta\sin(\Omega t+\Phi)+\varphi_0], \tag{2} \]

where \(A=m_xU_{m_1}\), \(B=m_yU_{m_2}\), and

\[ \frac{\omega_2}{\omega_1}=\frac{m}{n}. \]

If \(\varphi_0=0\) (which can always be ensured by means of an auxiliary phase shifter), then on the screen of the oscillograph we observe the multiple figure of Fig. 1 and

oscillograms in Fig. 2. If \(\Theta=0\) and \(\varphi_0=0\), then the figures are symmetric with respect to the \(X\) and \(Y\) axes [2] and are closed, if the voltages \(u_1\) and \(u_2\) are in phase (Fig. 3). From equation (2), for \(\varphi_0=0\) and \(y=0\),

\[ \omega_0 t=\frac{q\pi-\Theta\sin(\Omega t+\Phi)}{m}, \]

where \(q=0,1,2\ldots\)

The abscissas of the figure lying on the \(X\) axis are determined from (1) as

\[ x_q=A\sin\left[\frac{nq\pi}{m}-\frac{\Theta n}{m}\sin(\Omega t+\Phi)\right]. \tag{3} \]

Fig. 1. Interference pattern for
\[ \varphi_0=0,\quad \frac{m}{n}=\frac{2}{3}. \]

In the absence of phase modulation, \(\Theta=0,\ q=0,\ m,\ 2m\ldots,\ x_{0,m}=0\); the point at the center of symmetry of the figure is determined uniquely. If \(\Theta\ne0\), from (3) we obtain

\[ x_{0,m}=\pm A\sin\left[\frac{n}{m}\Theta\sin(\Omega t+\Phi)\right]. \tag{4} \]

It follows from expression (4) that, with phase modulation, the figure has the greatest blurring at the center of the figure along the \(X\) axis, symmetric about the \(Y\) axis (Fig. 2), when \(\sin(\Omega t+\Phi)=1\).

From the width of the blurring \(cc'_{\max}=2x_{0,m}\) (Fig. 1), the phase modulation index can be determined by the formula

\[ \Theta=\frac{m}{n}\arcsin\frac{cc'_{\max}}{2A}. \tag{5} \]

The phase modulation index can also be determined from the greatest blurring along the \(Y\) axis at the center of the figure, symmetric about the \(X\) axis (Fig. 1).

The ordinates of the curve lying on the \(Y\) axis are determined from equation (2) for \(x=0,\ \omega_0 t=\dfrac{k\pi}{n}\), where \(k=0,1,2,\ldots\),

\[ y_k=B\sin\left[\frac{m}{n}\pi k+\Theta\sin(\Omega t+\Phi)\right]. \tag{6} \]

If \(k=0,\ n,\ 2n\ldots\), from (6) we obtain

\[ y_{0,n}=\pm B\sin[\Theta\sin(\Omega t+\Phi)]. \tag{7} \]

When \(\sin(\Omega t+\Phi)=1,\ 2y_{0,n}=aa'_{\max}\), the phase modulation index is determined by the formula

\[ \Theta=\arcsin\frac{aa'_{\max}}{2B}. \tag{8} \]

Fig. 3. Oscillograms of interference patterns in the absence of phase modulation:

a) \(z_0 = 0,\ \Theta = 0,\ \dfrac{m}{n} = \dfrac{2}{3}\);

b) \(z_0 = 0,\ \Theta = 0,\ \dfrac{m}{n} = \dfrac{3}{4}\);

c) \(z_0 = 0,\ \Theta = 0,\ \dfrac{m}{n} = \dfrac{3}{8}\).

Fig. 2. Oscillograms of interference patterns under phase modulation at the lower frequency:

a) \(\varphi_0 = 0,\quad \dfrac{m}{n} = \dfrac{2}{3},\quad \Theta = 3^\circ 50'\);

b) \(\varphi_0 = 0,\quad \dfrac{m}{n} = \dfrac{3}{4},\quad \Theta = 9^\circ\);

c) \(\varphi_0 = 0,\quad \dfrac{m}{n} = \dfrac{3}{8},\quad \Theta = 4^\circ 50'\).

It is not difficult to show mathematically that, for any integer frequency ratio and \(\varphi_0=0\), the blurring of the figure will be greatest at the center (segments \(cc'_{\max}\) and \(aa'_{\max}\), Fig. 1). The ordinate of the curve located at the smallest distance from the center is determined from equation (6), for \(\Theta=0\) and \(k=1\) or \(k=n+1\),

\[ y'_{1+n}=-B\sin\frac{m\pi}{n}. \tag{9} \]

If \(\Theta\ne 0\), the ordinate of the curve located at the smallest distance from the center is not determined uniquely, or has a blurring

\[ y_{1+n}=-B\sin\left[\frac{m}{n}\pi+\Theta\sin(\Omega t+\Phi)\right]. \tag{10} \]

One half of this blurring is determined as the difference between formulas (9) and (10):

\[ y_{1+n}-y'_{1+n} = -B\left\{ \sin\frac{m\pi}{n} \left[ 1-\cos\left\{\Theta\sin(\Omega t+\Phi)\right\} \right] - \cos\frac{m\pi}{n}\, \sin\left[\Theta\sin(\Omega t+\Phi)\right] \right\}. \tag{11} \]

In expression (11), for small \(\Theta\),

\[ \left[1-\cos\{\Theta\sin(\Omega t+\Phi)\}\right]\approx 0, \]

then

\[ y_{1+n}-y'_{1+n} \approx -B\cos\frac{m\pi}{n}\, \sin\left[\Theta\sin(\Omega t+\Phi)\right]. \tag{12} \]

From a comparison of expressions (12) and (7) it is seen that, under phase modulation, the blurring along the \(Y\)-axis at the ordinate nearest to the center is \(\cos\dfrac{m\pi}{n}\) times smaller than the blurring at the center of the figure. Analogously, it can be shown that the blurring along the \(X\)-axis at the center of the figure is \(1/\cos\dfrac{m\pi}{n}\) times wider than at the abscissa nearest to the center.

Formula (3) is convenient to use when there is no amplitude modulation in the voltage applied to the vertical-deflection plates of the oscilloscope, and formula (5) in the case where it is absent in the voltage applied to the horizontal-deflection plates. The accuracy of measuring small indices of phase modulation by formula (5), as compared with formula (3), is the higher the smaller the ratio \(\dfrac{m}{n}\). For small \(\Theta\), one may use the formula obtained by multiplying formulas (8) and (5):

\[ \Theta \approx \frac{1}{2} \sqrt{ \frac{aa'_{\max}\cdot cc'_{\max}}{A\cdot B} } \cdot \sqrt{\frac{m}{n}}. \tag{13} \]

The sensitivity of the measurement can be considerably increased if \(m_x\) and \(m_y\) of the oscilloscope are increased (Fig. 4), and the voltages \(u_{m_1}\) and \(u_m\)

measure by means of a vacuum-tube voltmeter. In this case formula (13) takes the form

\[ \Theta \approx \frac{1}{2}\sqrt{\frac{maa'_{\max}-cc'_{\max}}{nm_xm_yu_{m_1}u_{m_2}}}. \tag{14} \]

To determine the phase-modulation index in a six-terminal network by the method described, the block diagram shown in Fig. 5 is suitable. The voltage from generator 1 is applied to two channels. In the first channel, through auxiliary frequency divider 2, the voltage is applied to the six-terminal network 3 under investigation. In the second channel, through

Fig. 4. Oscillogram of the center of the interference pattern at \(\varphi_0=0,\ \dfrac{m}{n}=\dfrac{2}{3},\ \Theta=6.5^\circ\).

Fig. 5. Block diagram for measuring phase-modulation indices in six-terminal networks.
1—measuring generator; 2 and 4—auxiliary frequency dividers; 5—auxiliary phase shifter; 3—the six-terminal network under investigation.

the zero-phase-shift setting phase shifter 5 and auxiliary frequency divider 4, the voltage is applied to the second pair of terminals of the six-terminal network under investigation. The auxiliary dividers are used to synchronize the frequency of the input voltages. The output voltage of the six-terminal network is applied to one pair of deflection plates of the oscillograph; one of the input signals of the six-terminal network is applied to the second pair of plates. It is necessary that the frequency ratio of these voltages be as small as possible; from this point of view, one or another input signal is chosen as the reference. Then the segments \(cc'_{\max}\) and \(2A\) are determined on the oscillograph screen (Fig. 1), or \(aa'_{\max}\) and \(2B\). From formula (5) or (3) the phase-modulation index is determined. With the aid of vacuum-tube voltmeter 6 (Fig. 5), the voltages \(u_{m_1}\) and \(u_{m_2}\) are measured, while the segments \(cc'_{\max}\) and \(aa'_{\max}\) are measured on the oscillograph screen, and \(\Theta\) is calculated from formula (14).

To measure the parasitic phase modulation of auxiliary dividers or of phase shifts introduced by the dividers, (3) provides for switching \(P_1\) and \(P_2\) to position 2 or 3. Accordingly, when measuring phase modulation in four-terminal networks, the measurement block diagram is considerably simplified, since there is no need to use the second auxiliary frequency divider.

In estimating the accuracy of measuring small phase-modulation indices by the proposed method, it is necessary to clarify the question of the influence of parasitic amplitude modulation during the measurements. If amplitude modulation is present in the signal with frequency \(\omega_2\), the parametric equations of the figure on the screen of the cathode-ray tube have the form

\[ x = A \sin n\omega_0 t \]

\[ y = B[1 + M\sin(\Omega t + \Phi)]\sin(m\omega_0 t + \zeta_0). \tag{15} \]

The ordinates of the figure for \(\omega_0 t = \dfrac{k\pi}{n}\), where \(\kappa = 0, 1, 2\ldots\), from (15) are

\[ y_k = B\left[1 + M \sin\left(\frac{\Omega}{\omega_0}\frac{k\pi}{n} + \Phi\right)\right]\sin\left(\frac{m}{n}k\pi + \zeta_0\right). \]

In the case \(\zeta_0 = 0\) and \(k = 0, n, 2n\ldots gn\), i.e., at the center of the figure, there is no blurring along the \(Y\) axis, since \(y_{0,n} = 0\); only the length of the segment \(2B\) changes. If, however, amplitude modulation is present in the signal with the higher frequency \(\omega_1\), then it can similarly be shown that blurring at the center of the figure will be absent, while the length of the segment \(2A\) will change (Fig. 6). Thus, parasitic modulation does not affect the accuracy of measuring the segments \(aa'_{\max}\), \(cc'_{\max}\). The influence of amplitude modulation on the accuracy of measuring the segments \(2A\) or \(2B\) can be avoided by using either formula (5) or formula (8).

Fig. 6. Oscillogram of the interference pattern with AM in the lower-frequency signal, \(\dfrac{m}{n} = \dfrac{2}{3}\), \(\Theta = 0\) and \(\zeta_0 = 0\).

The errors in measuring the phase-modulation index by the method considered are due mainly to the finite dimensions of the luminous spot on the oscilloscope screen, the curvature of the screen of the cathode-ray tube, and the presence of nonlinear distortions of the signals.

The diameter of the luminous spot in oscilloscopes can be set equal to \(0.5 \div 1\) mm. The distance between two points of the oscillogram can be determined with an accuracy of \(\pm d\), where \(d\) is the line thickness, equal to the diameter of the luminous spot. Determining the phase-modulation index by formula (8), for the limiting possible values of \(\Theta_1\) and \(\Theta_2\), taking into account the error in measuring the distances \(aa'_{\max}\) and \(2B\), we obtain

\[ \sin \Theta_1 = \frac{aa'_{\max} + d}{2B - d} \quad \text{and} \quad \sin \Theta_2 = \frac{aa'_{\max} - d}{2B + d}. \tag{16} \]

Taking

\[ \frac{\Theta_1+\Theta_2}{2}=\Theta, \]

and

\[ \frac{\Theta_1-\Theta_2}{2}=\Delta\Theta \]

and neglecting small quantities of the second order, we obtain

\[ \sin \Delta\Theta \simeq \frac{d}{2B}\left[\frac{1+\sin\Theta}{\cos\Theta}\right]. \tag{17} \]

From expression (17) one can find the limits of variation of \(\sin\Delta\Theta\), if it is taken into account that \(0\leq\Theta\leq \dfrac{\pi}{n}\),

\[ \frac{d}{2B}\leq \sin\Delta\Theta \leq \frac{d}{2B}\left[\frac{1+\sin\dfrac{\pi}{n}}{\cos\dfrac{\pi}{n}}\right]. \]

For example, with a tube screen diameter of \(125\) mm one can obtain a segment length \(2B=100\) mm; then, for a spot diameter \(d=1\) mm and \(n=10\), we have \(\Delta\Theta_{\max}\simeq 29'\).

When formula (5) is used, this error is reduced by

\[ \frac{m}{n} \]

times.

The error caused by the nonlinearity of the tube may be determined through a coefficient indicating the maximum deviation of the sensitivity over the working portion of the screen,

\[ \gamma=\frac{m_{\max}-m_{\min}}{m_{\min}}. \]

The experimentally determined nonlinearity coefficient of tube 3037 has values of the order of \((0.05\div 0.06)\). Let us specify a law of sensitivity variation for which the error in determining the ratio of the segments \(a=\dfrac{aa'_{\max}}{2B}\) will be maximal. Then we obtain the following expressions for determining the segments:

\[ aa'_{\max}=m_{\max}\cdot u_1 \]

and

\[ 2B=m_{\max}u_1+m_{\min}(u_2-u_1). \]

The ratio of the segments, taking into account such nonuniformity of sensitivity, is equal to

\[ a'=\frac{a(1-\gamma)}{a\gamma+1}. \tag{18} \]

Accordingly, we obtain values for \(\Theta\) and \(\Theta'\):

\[ \sin\Theta=a \quad \text{and} \quad \sin\Theta'=a' \tag{19} \]

From expression (19), taking (18) into account and after neglecting small quantities of the second order, we obtain

\[ \sin\frac{\Delta\Theta_{\max}}{2}\simeq \gamma \operatorname{tg}\Theta. \]

For small values of the modulation index, \(\Delta\Theta \simeq 2\gamma\Theta\), or \(\Delta\Theta \simeq (0.1 \div 0.12)\Theta\). The maximum measurement error is an order of magnitude smaller than the measured quantity, which is quite permissible.

Nonlinear distortions of the voltages \(u_1\) and \(u_2\) applied to the deflecting plates of the tube lead to an error in determining the segments \(aa'_{\max}\) and \(2B\), \(cc'_{\max}\) and \(2A\). Representing the voltage under investigation as a sum of harmonic components, we obtain the equations describing the curve on the oscilloscope screen in the form

\[ x = A\sin n\omega_0 t, \]

\[ y = \sum_{k=1}^{\infty} B_k \sin k\left[m\omega_0 t + \Theta \sin(\Omega t+\Phi)\right]. \tag{20} \]

Putting \(x = 0\), i.e. \(\omega_0 t = \dfrac{q\pi}{n}\), we find from (20) the ordinates of the points of intersection with the \(Y\)-axis:

\[ y_q = \sum_{k=1}^{\infty} B_k \sin k\left[\frac{mq\pi}{n} + \Theta \sin(\Omega t+\Phi)\right]. \tag{21} \]

From (21) we determine the width of the blur along the \(Y\)-axis, symmetric with respect to the \(X\)-axis, by putting \(q = 0, n, 2n,\ldots,pn\):

\[ 2y_{0,n} = 2\sum_{k=1}^{\infty} B_k \sin k\left[m\pi + \Theta \sin(\Omega t+\Phi)\right]. \tag{22} \]

From (22) we determine the relative error

\[ \frac{\Delta\Theta}{\Theta} = \frac{B_2\sin 2\Theta}{B\sin\Theta} + \frac{B_3\sin 3\Theta}{B\sin\Theta} +\ldots \]

In the case of small \(\Theta\), we obtain

\[ \frac{\Delta\Theta}{\Theta} \simeq 2\gamma_2 + 3\gamma_3 + \ldots, \]

where \(\gamma_2\) and \(\gamma_3\) are the nonlinearity coefficients for the harmonics.

Thus, we see that small nonlinear distortions of the signal under investigation do not lead to a substantial error in measurements of the phase-modulation index.

Taking into account that the errors considered are independent random quantities, it may be assumed that the total relative error will be small. It decreases with increasing \(n\) and decreasing \(m\). A possible inaccuracy in setting \(\varphi_0 = 0\) does not lead to a substantial error in determining the segments \(cc'_{\max}\) and \(aa'_{\max}\), since these segments are enclosed between parallel lines of the figure.

Conclusions

1. In phase modulation of one of the signals forming an interference pattern on the oscilloscope screen, the phase-modulation index can be determined from two linear segments of the figure for any integer frequency ratio.

1. The accuracy of measuring the modulation index increases when the ratio \(\frac{m}{n}\) decreases, if the phase modulation is present in the signal with the lower frequency.

2. Amplitude modulation of one of the signals forming the interference pattern does not affect the measurement results.

3. Small nonlinear distortions of the signal under study do not lead to a substantial error in measurement.

4. The proposed method for measuring small indices of phase modulation does not require preliminary calibration and is distinguished by the simplicity of the equipment used.

REFERENCES

1. Shulzhenko K. M. and Supyan V. Ya. Measurement of small phase-modulation indices by means of an electron-beam tube. Izvestiya TPI, vol. 105, 1960.

2. Shchegolev E. I. On the phase difference of periodic processes with a rational frequency ratio and on its measurement by means of a cathode oscilloscope, ZhTF, vol. IV, 1934.

3. Moss N. Method of Lissajous figures for oscillographic comparison of frequencies and phases. Sovremennyi katodnyi ostsillograf, part 1, 1951.