Engineer-mechanic R. A. Voronov.

Graphical Analysis of Periodic Curves.

In the study of alternating-current circuits, one often has to deal with the expansion of periodic curves into a series of sinusoidal components of various orders. In such an analysis of curves three kinds of methods are possible: analytical, graphical, and mechanical. The first of these, being the most accurate, require the performance of a series of computations, which, when there are a large number of analyses, is slow and tedious. Mechanical methods are very rapid and convenient, but are available to few, since they require special instruments. The most convenient and easily applicable are the graphical methods of analysis, which, while yielding somewhat in accuracy to the analytical ones, give results much more quickly. In these analyses there is either no computation at all, or very little; but they all require the making of exact graphical constructions, on which the results obtained strongly depend.

In the present article a simple graphical method of analysis is briefly set forth, one which does not require the making of exact constructions and in which there are no computations at all. The performance of the analysis consists in transferring the ordinates of the curve under investigation to a special grid, connecting the points obtained by a broken line, and planimetering this figure. The numerical value of the area will be equal to the required amplitude.

The basis of the present method is formed by the equalities*)

\[ A'_n=\frac{2}{m}\sum_{k=1}^{k=m} y_k \cdot \sin\left(n\frac{k}{m}\cdot 2\pi\right) \]

and

\[ A''_n=\frac{2}{m}\sum_{k=1}^{k=m} y_k \cdot \cos\left(n\frac{k}{m}\cdot 2\pi\right)\ldots\ldots\ldots\ldots\ldots\ldots (A). \]

which, with great accuracy, replace the equations of expansion into a Fourier series

\[ A'_n=\frac{1}{\pi}\int_{0}^{2\pi} y_x \cdot \sin(nx)\cdot dx \]

and

\[ A''_n=\frac{1}{\pi}\int_{0}^{2\pi} y_x \cdot \cos(nx)\,dx \]

) The method described is the graphical solution for Perry’s method. The simplified equations adopted will be the sole place of inaccuracy of both methods. On the deviations obtained from the true values and on a comparison of Perry’s method with others, see Akimov’s article in the journal Elektrichestvo*, No. 8 for 1927. More correct results are given by Clifford’s graphical method, analogous to that described, but based not on simplified, but on exact equations, as a consequence of which its greater complexity also appears.

In these equalities \(A'_n\) and \(A''_n\) are the amplitudes of the curve of the \(n\)-th order, \(k\) is the serial number of the corresponding ordinate, and \(m\) is the number of parts into which the period is divided. The total amplitude of the \(n\)-th order and its initial angle are calculated from the equations

\[ A_n=\sqrt{(A'_n)^2+(A''_n)^2} \]

and

\[ \operatorname{tg} z_n=\frac{A''_n}{A'_n} \quad \text{or} \quad \cos z_n=\frac{A'_n}{A_n} \tag{B} \]

Equations (A) may be represented in the form

\[ A'_n=\sum_{k=1}^{k=m} \left[ y_k \cdot \frac{2\cdot \sin\left(n\frac{k}{m}\cdot 2\pi\right)}{m} \right] = \sum_{k=1}^{k=m}(y_k\cdot a_{nk}) \]

and

\[ A''_n=\sum_{k=1}^{k=m} \left[ y_k \cdot \frac{2\cos\left(n\frac{k}{m}\cdot 2\pi\right)}{m} \right] = \sum_{k=1}^{k=m}(y_k\cdot b_{nk}) . \tag{Б} \]

where the coefficients at the ordinates

\[ a_{nk}=\frac{2}{m}\cdot \sin\left(n\frac{k}{m}\cdot 2\pi\right) \]

and

\[ b_{nk}=\frac{2}{m}\cdot \cos\left(n\frac{k}{m}\cdot 2\pi\right) \tag{C} \]

do not depend on the curve under investigation. Equations (B) may be regarded as certain areas composed of rectangles with bases \(a_{nk}\) or \(b_{nk}\), and with heights equal to the corresponding ordinates of the curve (Figs. 1 and 2). Since the values of the coefficients \(a_{nk}\) and \(b_{nk}\) may be either positive or negative, it is necessary, for each component, to find two areas and take their difference.

The construction and planimetering of stepped figures is somewhat inconvenient; therefore it is better to replace them by equal-area regions bounded by broken lines. Such a replacement can easily be made on the basis of the following considerations. If one lays off (Fig. 3) the segment \(BK\), equal to the segment \(AB\), and draws the ordinate \(KK'\), then the area of the trapezoid \(AA'K'K\) will be equal to the sum of the areas of the rectangles \(AA'B'B\) and \(BB'K'K\). Then, laying off the segment \(CM\), equal to \(KC\), we find that the area \(KK'M'M\) is equal to the sum of the areas \(KK'C'C\) and \(CC'M'M\). Dividing the whole figure into a series of trapezoids, we see that the resulting area, equal in area to the previous one, will be bounded by a broken line. The ordinates at the points of inflection are equal to the ordinates of the curve, while the distances between them depend only on the coefficients \(a_{nk}\) and \(b_{nk}\), i.e., they are entirely identical for all curves and can be laid off in advance. When the areas for negative and positive coefficients are plotted on one diagram, the required quantity is represented as their difference. In Fig. 4 such a difference of areas is shaded.

Since in the usual cases of alternating current the curves under investigation contain no components of even orders, it is sufficient to carry out the analysis only for one half of the period, doubling the result obtained. In drawings 5–12 grids have been constructed for laying off the ordinates of the curve under investigation for one half-period. For their construction the half-period was divided into 12 parts \((m=24)\), and the coefficients \(a_{nk}\) and \(b_{nk}\) were doubled:

\[ a_{nk}=\frac{\sin k\,\frac{n\pi}{12}}{6} \quad \text{and} \quad b_{nk}=\frac{\cos k\,\frac{n\pi}{12}}{6}. \]

If these are laid off on the scale one unit equals 10 cm, then the area obtained, expressed in square centimeters, will numerically be equal to one tenth of the required amplitude, likewise expressed in centimeters, or directly equal to it, but expressed in millimeters.

When the half-period is divided into 12 parts, analysis is possible up to the 11th order; moreover, into each result of the \(n\)-th order there will also enter components of all those orders for which

\[ n' = pm \pm n \]

where \(p\) is any positive integer. Usually these components are very small, as a result of which the result obtained is regarded as belonging only to the single \(n\)-th order.

On all the appended grids there are lines for laying off ordinates both for positive and for negative areas. Near each line is placed the number of the ordinate which is to be laid off on it; the numbers above the line refer to positive area, and those below the line to negative area. For determining the second components of the 1st, 5th, 7th, and 11th orders, identical grids are obtained, differing only in the numbers of the ordinates, which are given in a separate table (drawing 11). The same grid is also obtained for the 3rd and 9th orders (drawing 12). The arrows indicate the direction of tracing with the planimeter (see also drawings 4, 13, 14, 15, and 16).

If a negative ordinate occurs in the half-period (which happens very rarely), it must be laid off downward from the base line. In constructing the grids it was assumed that the division of the half-period into 12 parts is such that the first and 12th ordinates coincide with the points where the curve crosses zero, i.e. are also equal to zero (drawing 1). If, however, they are not equal to zero, and \(y_0=-y_{12}\), then in the second of equations (A) one term, equal to

\[ \frac{4y_0}{m} \]

will be missing. In order not to alter the grids constructed for this case, one must add algebraically to the result obtained from them the missing quantity. For \(m=24\), this addition is equal to

\[ \frac{y_0}{6}=-\frac{y_{12}}{6}. \]

The zero ordinate does not enter into the equations for determining the first components; that is, the grids always give the complete result.

In drawings 13–16 an example is given of the extraction, by means of the grids, of the components \(A'_1\), \(A''_1\), \(A'_5\), and \(A''_5\) for the curve of drawing 1.

For the analysis of curves containing even and constant components, grids of the same kind must be constructed, but for the whole period. One may also apply addition and subtraction of half-periods, in which all the grids given for odd ...

components. If the two halves of the period (Fig. 17) are superposed on one another, subtracting (algebraically) the ordinates \(y_k\) and \(y_{k+\frac{m}{2}}\), then the resulting new curve will contain only the odd terms, doubled, while the even ones will be equal to zero (Fig. 18). Conversely, when the ordinates are added, only the even terms remain, while the odd ones vanish (Fig. 19). Such an elimination of the even and odd terms follows from the relations

\[ A_n \cdot \sin (nx+\alpha_n)=A_n \cdot \sin [\,n(x+\pi)+\alpha_n\,] \]

for even \(n\), and

\[ A_n \cdot \sin (nx+\alpha_n)=-A_n \cdot \sin [\,n(x+\pi)+\alpha_n\,] \]

for odd \(n\). The first of the curves obtained is analyzed by the usual grids; for the second, the same grids are constructed, but including also the zero ordinate (Figs. 20–23). The result obtained must be reduced by a factor of two. The constant term is obtained by simple planimetry of the curve in Fig. 17 or 19. Instead of drawing the new curves, it is possible directly to set off the difference or the sum of the ordinates on the corresponding lines of the grids.

The simplicity of the method is evident from its description; its accuracy depends chiefly on the planimetry. With careful work it can easily be brought to \(0.1\) sq. cm and even better, which corresponds to an amplitude accuracy of \(0.1\) mm. One series of grids usually serves for a very long time, provided they are drawn on good paper and all constructions are made with a soft pencil. For independent construction of the grids, the distances in millimeters from the extreme left-hand line of the grids in Figs. 5–12 and 20–23 to all the others are given below.

For greater accuracy, they may be increased several times, reducing the result obtained in the same ratio. Such an increase is especially desirable for large ordinates of the curve. The height of the grids must be taken greater than the ordinates being set off.

To the article by Eng. R. A. Voronov: “Graphical Analysis of Periodic Curves.”

Fig. 1.

Visible labels: \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\); \(y_1, y_2, y_3, y_4, y_5, y_6, y_7, y_8, y_9, y_{10}, y_{11}, y_{13}, y_{14}\); \(\Pi\).

Fig. 2.

Visible labels: \(y_1, y_2, y_3, y_4\); \(a_1, a_2, a_3, a_4\).

Fig. 3.

Visible labels: \(A, B, K, C, M, D, N, E\); \(A', B', K', C', M', D', N', E'\); \(B'', C'', D''\).

Fig. 4.

Visible labels: \(A, F, \varepsilon, D, C, B\).

Visible labels on the lower diagram: \(A'_1\); \(1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11\).

Corresponding Member R. N. [[unclear: surname]], “Graphical Analysis of Periodic Curves.”

Fig. 6.
Diagram labeled \(A'_3\). Visible markings: \(1/3,\ 2/6,\ 3/7,\ 7,\ 9,\ 10,\ 11\).

Fig. 7.
Diagram labeled \(A'_5\). Visible markings: \(5/5,\ 2,\ 1,\ 1',\ 1/4,\ 1/8,\ 6,\ 9,\ 10,\ 11,\ 10,\ 2\).

Fig. 8.
Diagram labeled \(A'_7\). Visible markings: \(5/2,\ 4,\ 3,\ 1/6,\ 1,\ 1/6,\ 9,\ 11,\ 10,\ 8,\ 2\).

Fig. 9.
Diagram labeled \(A'_8\). Visible markings: \(1/5,\ 6/8,\ 3/10,\ 9,\ 11/7,\ 11\).

Fig. 11.
Diagram labeled \(A''_{M}\). Visible markings: \(a,\ b,\ c,\ d,\ e,\ e\).

Fig. 10.
Diagram labeled \(A'_{11}\). Visible markings: \(1/2,\ 3,\ 1/5,\ 5,\ 1/6,\ 1/6,\ 7,\ 1/8,\ 9,\ 1/10,\ 11\).

Fig. 12.
Diagram labeled \(A''_{M}\). Visible markings: \(k,\ k,\ l,\ m,\ n,\ n\).

For Fig. 11.

For Fig. 12.

To the article by engineer E. A. Voronov, “Graphical Analysis of Periodic Curves.”

Drawing 13.

Inscriptions in the drawing:

\[ A_1' \]

\[ O = 38.1\ \mathrm{cm}^2 \]

\[ A_1' = 38.1\ \mathrm{mm} \]

Bottom markings: \(1,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 1\).

Drawing 14.

Inscriptions in the drawing:

\[ A_1'' \]

\[ O = -0.6\ \mathrm{cm}^2 \]

\[ A_1'' = -0.6\ \mathrm{mm} \]

Bottom markings: \(\frac{5}{7},\ \frac{4}{8},\ \frac{3}{9},\ \frac{2}{10},\ \frac{1}{1},\ \frac{1}{11}\).

Drawing 15.

Inscriptions in the drawing:

\[ A_5'' \]

\[ O = 2.6\ \mathrm{cm}^2 \]

\[ A_5'' = 2.6\ \mathrm{mm} \]

Bottom markings: \(\frac{1}{11},\ \frac{4}{3},\ \frac{9}{5},\ \frac{10}{2},\ \frac{6}{7},\ \frac{5}{7}\).

Drawing 16.

Inscriptions in the drawing:

\[ A_5' \]

\[ O = -0.9\ \mathrm{cm}^2 \]

\[ A_5' = -0.9\ \mathrm{mm} \]

Bottom markings: \(\frac{5}{3},\ 2,\ 1,\ 1,\ \frac{1}{4},\ [[unclear: fraction]],\ 6,\ [[unclear: number]],\ 1,\ 1,\ 10,\ 2\).

To the article by Eng. R. A. Voronov: “Graphical Analysis of Periodic Curves.”

Fig. 17.

Fig. 18.

Fig. 19.

Fig. 22.

Fig. 20.

Fig. 21.

To Figs. 20, 21, 22.

Fig. 23.

To Fig. 23.