MEASUREMENT OF SMALL PHASE MODULATION INDICES USING INTERFERENCE PATTERNS

K. M. SHULZHENKO, V. Ya. SUPYAN

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

In the study of various four-terminal networks it is often necessary to detect and measure phase modulation with small indices. With the aid of Lissajous figures it is possible to measure not only phase shifts [1], but also to determine the phase-modulation index. If a phase-modulated oscillation is applied to one pair of oscilloscope plates,

\[ u_1 = U_{m1}\sin\left[\omega_0 t + \Theta \sin(\Omega t + \Phi) + \varphi_0\right], \]

and to the other pair an oscillation with a frequency \(n\) times higher than the carrier frequency is applied, \(u_2 = U_{m2}\sin n\omega_0 t\), where \(n = 1, 2, 3\ldots\) is the frequency-multiplication factor, then on the oscilloscope screen, for \(\varphi_0 = 0\) (which can always be ensured by means of an additional phase shifter), we observe the multiple Lissajous figure shown in Fig. 1. The ordinates of the points of this curve are determined by the equation \(y = m_y U_{m2}\sin n\omega_0 t\), and the abscissae by

\[ x = m_x U_{m1}\sin\left[\omega_0 t + \Theta \sin(\Omega t + \Phi)\right], \tag{1} \]

where \(m_x\) and \(m_y\) are the oscilloscope deflection sensitivities along the \(X\) and \(Y\) axes, respectively. Let \(m_x U_{m1} = A\) and \(m_y U_{m2} = B\). The points of intersection of the curve described by the beam on the tube screen with the abscissa axis have \(y=0\), i.e., occur when

\[ n\omega_0 t = k\pi, \tag{2} \]

where \(k = 0, 1, 2, 3\ldots\). Substituting the value \(\omega_0 t = \dfrac{k\pi}{n}\) from (2) into equation (1), we obtain the abscissae of the points of intersection of the curve with the \(X\)-axis

\[ x_k = A\sin\left[\frac{k\pi}{n} + \Theta \sin(\Omega t + \Phi)\right]. \]

The distance between two symmetric—

Fig. 1a. Interference pattern in the presence of phase modulation \(\varphi_0 = 0\); \(n\) even.

…with respect to the \(Y\) axis, the points of intersection of the curve with the \(X\) axis is equal to

\[ x_k-x_{k+n} = A\sin\left[-\frac{k\pi}{n}+\Theta\sin(\Omega t+\Phi)\right] - A\sin\left[\frac{k\pi}{n}+\Theta\sin(\Omega t+\Phi)\right] = 2A\sin\left[\frac{k\pi}{n}+\Theta\sin(\Omega t+\Phi)\right]. \tag{3} \]

Fig. 1b. Oscillogram of the interference pattern in the presence of phase modulation for \(\zeta_0=0\); \(n=8\); \(\Theta=3^\circ\).

For \(k=0\), equation (3) takes the form

\[ x_0-x_n=2A\sin[\Theta\sin(\Omega t+\Phi)]. \tag{4} \]

If \(\Theta=0\), then equation (3) takes the form

\[ x'_k-x'_{k+n}=2A\sin\frac{k\pi}{n}. \tag{5} \]

This corresponds to the interference pattern whose oscillogram is shown in Fig. 2a. All points of intersection of the curve for \(\zeta_0=0\) and \(\Theta=0\) lie on the \(X\) axis. This makes it easy to determine the position of the \(X\) axis in measurements. Dividing (4) by (5), we obtain

\[ \sin[\Theta\sin(\Omega t+\Phi)] = \frac{x_0-x_n}{x'_k-x'_{k+n}}\, \sin\frac{k\pi}{n}. \tag{6} \]

Point \(b'\) (Fig. 2b) has intersection number \(k=1\), while point \(b\) corresponds to \(k=1+n\), and the distance \(x'_1-x'_{1+n}=bb'\). The points \(x_{k=0}\) and \(x_{k=n}\) coincide with the origin when \(\Theta=0\). If \(\Theta\ne0\), then these points are displaced relative to the origin. In this case the distance between them is \(x_0-x_n=cc'\). From the lengths of the segments \(bb'\) and \(cc'\), it is easy to calculate the phase modulation index. Substituting into (6) the values \(x'_k-x'_{k+n}=bb'\) for \(k=1\) and \(x_0-x_n=cc'\), we obtain,

\[ \sin [\Theta \sin (\Omega t+\Phi)] = \frac{cc'}{bb'} \sin \frac{\pi}{n}. \tag{7} \]

Taking into account that (in the case \(\Theta < \frac{\pi}{2}\)) the maximum blurring \(cc'_{\max}\) will occur when \(\sin(\Omega t+\Phi)=1\), from (7) we obtain

\[ \Theta = \arcsin \frac{cc'_{\max}}{bb'} \sin \frac{\pi}{n}. \tag{8} \]

Fig. 2a. Oscillogram of the interference pattern in the absence of modulation.

When measuring small modulation indices \(\Theta \ll 1\), one may use the approximate formula

\[ \Theta \simeq \frac{cc'_{\max}}{bb'} \sin \frac{\pi}{n}. \tag{8a} \]

The maximum blurring of the interference pattern \(cc'_{\max}\) will be at the center of the figure (Fig. 1a for even \(n\), and Fig. 3 for odd \(n\)). An oscillogram of the interference pattern for \(n=3\) is shown in Fig. 3a.

Fig. 2b. Interference pattern in the absence of phase modulation, \(\varphi_0=0\); \(n\) even.

In the case of equal frequencies, the distance \(bb'\) should be determined from (5) for \(k=1/2\), i.e., by using the projection of the figure on the \(X\)-axis, equal to \(2A\) (Fig. 3b). The calculation formula for \(n=1\) and \(\varphi_0=0\) takes the form

\[ \Theta = \arcsin \frac{cc'_{\max}}{2A}, \tag{8b} \]

and for \(\Theta \ll 1\),

\[ \Theta \simeq \frac{cc'_{\max}}{2A}. \]

The same method can be used when measuring the phase-modulation index of a signal with a higher frequency

\[ u_2=U_{m_2}\sin [n\omega_0 t+\Theta_2\sin(\Omega t+\Phi)] . \]

However, the sensitivity of measuring the modulation index is then reduced by a factor of \(n\). The calculation formula for this case has the form

\[ \Theta_2=n\arcsin \frac{CC'_{\max}}{bb'}\sin \frac{\pi}{n}. \tag{8в} \]

Fig. 3. Interference pattern in the presence of phase modulation \(\varphi_0=0\); \(n\)—odd.

With amplitude modulation of one of the signals, the interference pattern will have the form shown in the oscillograms of Fig. 4 (modulation of the signal with the lower frequency) and Fig. 5 (modulation of the signal with the higher frequency). As is seen from these figures, in the presence of amplitude modulation of one of the signals, no blurring is observed at the center of the figure along the \(X\) axis. This is not difficult to show mathematically.

Fig. 3a. Oscillogram of the interference pattern for \(\varphi=0\),
\(n=3\) and \(\Theta=4^\circ30'\).

If the signal with the lower frequency has amplitude modulation (Fig. 4), then the equations describing the figure have the form:

\[ x=A[1+M\sin(\Omega t+\Phi)]\sin(\omega_0 t+\varphi_0), \]

\[ y=B\sin n\omega_0 t. \]

The points of intersection with the abscissa axis will occur for \(n\omega_0 t=k\pi\), i.e., when

\[ \omega_0 t=\frac{k\pi}{n}. \]

The abscissas of these points

\[ x=A[1+M\sin(\Omega t+\Phi)]\sin\left(\frac{k\pi}{n}+\varphi_0\right). \]

The distance between two points of intersection with the \(X\) axis that are symmetric with respect to the \(Y\) axis, for \(\varphi_0=0\), is equal to

\[ x_k-x_{k+n}=2A[1+M\sin(\Omega t+\Phi)]\sin\frac{k\pi}{n}. \]

For \(k=0\) and \(k=n\), the points of intersection will be at the origin. The distance \(x_0-x_n=0\), i.e., at the origin there is no blurring (Fig. 4). The blurring along the \(X\) axis will be maximal at the edges of the figure.

In the presence of amplitude modulation in the signal with the higher frequency, the coordinates of the points of the figure (Fig. 5) are determined by the equations:

\[ x=A\sin\omega_0 t, \tag{9} \]

\[ y=B[1+M\sin(\Omega t+\Phi)]\sin(n\omega_0 t+\varphi_0). \tag{10} \]

Fig. 3b. Lissajous figure \(n=1\); \(\varphi_0=0\) \((\Phi=18^\circ)\).

We find the abscissas of the points of intersection of the curve with the \(X\) axis by setting \(y=0\), i.e.,

\[ \text{for } n\omega_0 t+\varphi_0=k\pi. \tag{11} \]

Substituting the values of \(\omega_0, t\), determined from (11), into (9), we obtain

\[ x=A\sin\left(\frac{k\pi}{n}-\frac{\varphi_0}{n}\right). \]

Fig. 4. Oscillogram of the interference pattern under amplitude modulation in the signal with the lower frequency \(\varphi_0=0; n=2\).

The distance between points of intersection of the curve with the \(X\) axis that are symmetric with respect to the \(Y\) axis is

\[ x_k-x_{k+n}=2A\sin\left(\frac{k\pi}{n}-\frac{\varphi_0}{n}\right). \]

Fig. 5. Oscillogram of an interference pattern under amplitude modulation of the signal with the higher frequency \(\varphi_0=0\); \(n\) is even.

For \(k=0\), the distance \(x_0-x_n\) is determined by the phase shift \(\varphi_0\) between the voltages \(u_1\) and \(u_2\). If \(\varphi_0=0\), then \(x_0-x_n=0\). In this case there is also no blurring of the figure along the \(X\) axis (Fig. 5).

When \(\varphi_0=0\), there is no blurring along the \(Y\) axis at the center of the figure either. Thus, for \(x=0\), i.e., for \(\omega_0 t=q\pi\), where \(q=0,1,2,3,\ldots\), from (10) with \(\varphi_0=0\) we obtain

\[ y=B[1+M\sin(\Omega t+\Phi)]\sin nq\pi=0. \]

Fig. 6. Simplified block diagram for measuring small indices of phase modulation. 1—measuring generator; 2—trigger divider; 3—the four-terminal network under investigation; 4—phase shifter.

Thus, amplitude modulation of any of the voltages at the center of the Lissajous figure does not produce blurring along the \(X\) axis and therefore does not introduce an error into the determination of the phase-modulation index. A simplified structural diagram for measuring small indices of phase modulation by the method described above is shown in Fig. 6. One of the signals is fed to the oscillograph through a phase shifter, and the other through a frequency divider and the four-terminal network under investigation, which are separated from one another by cathode followers. By switching \(P\) from position 1 to 2, one can determine the phase shift introduced by the four-terminal network under investigation, and the parasitic

phase modulation of the auxiliary divider. When measuring phase modulation, we determine the segment \(bb'\) (Figs. 1a, 2b, or 3b) and the position of the \(X\) axis on the oscilloscope screen, using the phase shifter to make \(\varphi_0 = 0\). We determine the length of the segment \(cc'_{\max}\) (Figs. 1a, 3, or 3b) from the maximum blurring at the center of the figure. Using the formulas given above (8), (8a), (8b), or (8c), one can calculate the phase-modulation index. The proposed measurement technique is especially convenient for determining parasitic phase modulation arising in frequency converters (during multiplication, addition of frequencies, etc.). The required frequency multiplicity is provided by using dividers based on trigger cells, which possess sufficient stability of phase relationships. By the method described, we carried out measurements of phase-modulation indices on the order of \(0.5^\circ\). Increased measurement sensitivity is achieved by increasing the frequency multiplicity and stretching the figure along the \(X\) axis.

REFERENCES

1. Gyudin A. I. Measurement of small phase shifts using an ELP. Izmeritel’naya tekhnika, No. 5, 1958.