ON THE DISTRIBUTION OF BRIGHTNESS ON THE SCREEN OF A CATHODE-RAY TUBE UNDER CIRCULAR SWEEP

E. I. FIALKO

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

As is known, there are various methods for measuring phase differences of two oscillations.

Some of these methods are associated with producing a circular sweep, for which one of the voltages under consideration is used:

\[ u = U_m \cdot \sin(\omega t + \varphi). \tag{1} \]

For this purpose the voltage (1) is shifted in phase by \(\frac{\pi}{2}\):

\[ u' = U_m \cdot \cos(\omega t + \varphi), \]

and then the voltages \(u\) and \(u'\) are applied, respectively, to the horizontal and vertical deflection plates of the cathode-ray tube.

The question arises: according to what law will the brightness be distributed on the screen of the cathode-ray tube in the case when the voltage (1) is a fluctuating voltage or contains a harmonic signal and fluctuations?

Basic Relations

The brightness at a certain point of the screen \((a)\) is determined by the relative time of “coverage” of this point by the spot—the base of the beam (Fig. 1):

\[ B = B_0 \cdot \frac{\Delta t}{T}, \tag{2} \]

where

\(B_0\) — brightness of the stationary spot (brightness in the beam);
\(B\) — apparent brightness at the given point during motion of the spot;
\(T\) — observation time;
\(\Delta t\) — time during which point \(a\) is “covered” by the spot, i.e., the time during which the center of the spot is inside a circle of radius \(\frac{\delta}{2}\) with center at point \(a\); \(\delta\) — diameter of the spot.

In doing so, we neglect the nonuniformity of the brightness in the spot.

The ratio \(\dfrac{\Delta t}{T}\) is equal to the probability that the center of the spot falls inside a circle of radius \(\dfrac{\delta}{2}\) with center at the point \(a\):

\[ P=\frac{\Delta t}{T}. \]

Knowing the differential distribution law of the system of two random variables

\[ p_1(x,y) \]

or

\[ p_2(r,l), \]

where \(x, y, r\), and \(l\) are random deviations of the beam (the center of the spot), the meaning of which is clear from Fig. 2, we find

\[ P=p_1(x,y)\cdot ds \]

or

\[ P=p_2(r,l)\cdot ds, \]

where

\[ ds=\frac{\pi\delta^2}{4}. \]

Fig. 1. \(a\)—an arbitrary point on the screen of the cathode-ray tube;
\(\delta\)—spot diameter

Fig. 2. \(r\)—radial deviation of the spot center;
\(l\)—deviation of the spot center from the \(y\)-axis, measured along an arc of radius \(r\);
\(x,y\)—Cartesian coordinates of the spot center

It is evident that, if the differential distribution laws of the random variables \(r\) and \(l\) are described respectively by the functions \(p_r(r)\) and \(p_l(l)\), then

\[ p_2(r,l)=p_r(r)\cdot p_l(l). \]

We find

\[ P=p_r(r)\cdot p_l(l)\cdot \frac{\pi\delta^2}{4}, \tag{3} \]

for which we shall decipher the functions \(p_r(r)\) and \(p_l(l)\).

Distribution laws \(p_r(r)\) and \(p_l(l)\).

If voltages

\[ u=U_m\cdot \sin(\omega t+\varphi) \]

and

\[ u'=U_m\cdot \cos(\omega t+\varphi), \]

are applied to the horizontally and vertically deflecting plates, where \(U_m\) is the voltage envelope, and \(\varphi\) is the initial phase of the high-frequency filling, then the axial deflections of the beam will be

\[ x=\kappa\cdot U_m\cdot \sin(\omega t+\varphi), \]

\[ y=\kappa\cdot U_m\cdot \cos(\omega t+\varphi), \]

where \(\kappa\) is the sensitivity coefficient of the cathode-ray tube (we assume \(\kappa_x=\kappa_y\)). The radial deflection of the spot

\[ r=\sqrt{x^2+y^2} \]

is equal to

\[ r=U_m\cdot \kappa . \]

Since [1, 2]

\[ p_m(U_m)=\frac{U_m}{\sigma^2}\cdot e^{-\frac{U_m^2+U_c^2}{2\sigma^2}}\cdot J_0\left(j\frac{U_c\cdot U_m}{\sigma^2}\right), \]

where \(\sigma\) is the effective voltage of the fluctuations applied to the deflecting plates of the tube,
\(U_c\) is the amplitude of the harmonic signal applied to the deflecting plates of the tube, then

\[ p_r(r)=\frac{r}{\sigma^2\cdot \kappa^2}\cdot e^{-\frac{\left(\frac{r}{\kappa}\right)^2+U_c^2}{2\sigma^2}}\cdot J_0\left(j_0\frac{U_c\cdot r}{\sigma^2\cdot \kappa^2}\right) \tag{4} \]

and, in the case of absence of a signal,

\[ p_r(r)=\frac{r}{\sigma^2\cdot \kappa^2}\cdot e^{-\frac{r^2}{2\sigma^2\cdot \kappa^2}} . \]

It is not difficult to show that

\[ p_l(l)=\frac{1}{2\pi r}. \tag{5} \]

Knowing \(p_r(r)\) and \(p_l(l)\), we find \(P\), and consequently also \(B(r,l)\).

Brightness Distribution on the Screen of a Cathode-Ray Tube

From (3), (4), and (5):

\[ P=\frac{\delta^2}{8\cdot\sigma^2\cdot\kappa^2}\cdot e^{ -\frac{\left(\frac{r}{\kappa}\right)^2+U_c^2}{2\cdot\sigma^2} } \cdot J_0\left(j\frac{U_c\cdot r}{\sigma^2\cdot\kappa^2}\right), \]

and the apparent brightness (2) is equal to

\[ B=B_0\cdot \frac{\delta^2}{8\cdot\sigma^2\cdot\kappa^2}\cdot e^{ -\frac{\left(\frac{r}{\kappa}\right)^2+U_c^2}{2\cdot\sigma^2} } \cdot J_0\left(j\frac{U_c\cdot r}{\sigma^2\cdot\kappa^2}\right). \]

For \(r=0\) (at the center of the screen):

\[ B=B_{\max}. \]

\[ B_{\max}=B_0\cdot\frac{\delta^2}{8\cdot\sigma^2\cdot\kappa^2}\cdot e^{ -\frac{U_c^2}{2\cdot\kappa^2\cdot\sigma^2} }. \]

The relative brightness is

\[ \frac{B}{B_{\max}}= e^{-\frac{r^2}{2\cdot\sigma^2\cdot\kappa^2}} \cdot J_0\left(j\frac{U_c\cdot r}{\sigma^2\cdot\kappa^2}\right). \]

In the absence of a signal,

\[ \frac{B}{B_{\max}}=e^{-\frac{r^2}{2\cdot\kappa^2\sigma^2}}. \]

Denoting \(\kappa^2\cdot\sigma^2=\sigma_r^2\), we obtain

\[ \frac{B}{B_{\max}}=e^{-\frac{r^2}{2\cdot\sigma_r^2}} \]

and, in the presence of a signal,

\[ \frac{B}{B_{\max}}= e^{-\frac{r^2}{2\cdot\sigma_r^2}} \cdot J_0\left(j\frac{U_c\cdot r}{\sigma_r^2}\right). \tag{6} \]

Conclusions

In the case where a fluctuation voltage is applied to the horizontal- and vertical-deflection plates of a cathode-ray tube, the distribution of the screen’s luminous brightness along the radius obeys the normal law.

In the case of the presence of a signal and fluctuations, the brightness distribution is expressed by formula (6).

References

1. Bunimovich V. I. Fluctuation Processes in Radio Receiving Devices, Svyazizdat, 1951.

2. Rice S. O. Theory of Fluctuation Noise. Theory of the Transmission of Electrical Signals in the Presence of Interference (collection of articles), I. L., 1953.