Radio Broadcast (May 1928-Apr 1929)

_ RADIO BROADCAST No. 17 Radio Broadcast's Home-Study Sheets PLOTTING CURVES — PART I March, 1929 A LL experimenters should be able to draw curves, graphs, or plots and to interpret what these pictures mean. Also they should be able to interpret what the curves drawn by other experimenters mean. A note book full of curves is a source of concentrated information of infinite variety. In a few pages it may contain a summary of a month's work in a laboratory, or of many week's work with complex mathematical formulas. It is always a visual picture or representation of some physical, electrical, or mechanical phenomenon. This "Home-Study Sheet" is written in the hope that some of the less apparent facts about curve plotting may be brought to light and that it may encourage more experimenters to keep their data in this convenient form. To state that a graph is a visual representation of a mathematical expression may not convey B H N — fi r 1 w — ( Chics E, go 4 E W— S Fig. 1 — A map is a form of graph much to the average experimenter, but such is a fact nevertheless. Every graph or plot or curve may be expressed in the form of a mathematical equation. Some curves, however, are so complex that the expression would be very difficult to figure out. Conversely, every mathematical expression may be plotted in the form of a graph. A graph is a visual statement that two factors are related to each other in some fashion, either simple or complex. Thus, one factor may increase when the other increases, directly or according to a square or a more complicated law, or it may decrease as the other increases. A form of graph with which everyone is familiar is a map. We say that a certain town, "A," is so many miles north and so many miles west of Chicago. Anyone with a map could put his finger on such a place at a moment's notice. A map has the essentials of every graph, namely, two coordinates (axes) or directions, north-south and east-west, an origin, in this case Chicago, and a point which we wish to locate with respect to this origin. Fig. 1 shows how we would locate the town of "A." Some maps have the coordinates marked off as shown at the top and down the left side of Fig. 1 and so "A" on such a map would be defined as existing at (B, 3) . In this case the origin is at the top left-hand corner of the graph. Problem 1. Mark on the map a town, "B," which is at (F, 6). Such a means of locating a point on a map is everyday knowledge. The next problem is a bit more complex. How would you state that a railroad runs north and south and at a distance of 50 miles from Chicago? Here we must locate not a point on a map but a straight line perpendicular to one axis (coordinate) and parallel to another. A simple expression for such a line, representing a railroad, would be (west 50 miles) signifying that the road ran north and south and was 50 miles west of Chicago at its nearest point. Problem 2. A road runs south of Chicago through (D, 6) and straight east and west. Mark it on the map. The next problem would be to describe a road that ran at an angle to the two axes and approached to within 50 miles to Chicago. We could state that it ran through two towns and then give their locations on the map just as we located the point (B, 3) above. Problem 3. A road runs through (B, 2) and (F. 6.5). Place it on the map. How far south of Chicago is the nearest approach? A point on a map is located at the intersection of two lines; a line is defined when two points through v/hich it passes are located. The points are always given in certain distances away from vertical and horizontal axes or coordinates. Other Types of Graphs A graph is no different from a map, even though the axes or coordinates may be called X and Y instead of north-south and east-west. Also such high-sounding words as "ordinates" and "absissa," etc., may be used to express the distance up or down, and right or left, from some point chosen as the origin. In a graph the units of measurements, instead of being miles or feet, may be amperes, dollars, watts, volts, or merely unnamed units. Generally the origin is at the lower left-hand corner of the graph, although there is no reason why it cannot be somewhere else; for example in plotting the plate current of a vacuum tube against the grid voltage, the vertical axis (representing plate current) is usually near the center of the graph instead of at one corner of it so that both positive and negative values of grid voltage may be represented. Wherever the origin is, to plot the position of a point with respect to the origin, we need only move a certain number of units to the right (or left) and erect a perpendicular line; then move a certain number of units up (or down) and make a horizontal line. Where these two lines cross each other is the position or location of the point. For example, on Fig. 2 is plotted the point (X=5, Y=5). We find this position by moving 5 units to the right of the origin (where both X and Y are equal to zero). At this point we erect the perpendicular line which contains all points which are 5 units to the right of X = 0. Then we draw the line Y = 5 and let them cross. Equation of a straight line A point is represented as follows, (X = 5, Y = 5). A straight line is a bit more complex because it goes through two points whose locations must be given. We can get around this complexity by knowing one point through which it goes and the slope of the line, that is the change in its Y units that are caused by a change along i ts X axis. In general a fine is represented by an equation of this form, Y = MX + B. where M is the slope of the line, and B is the point where it crosses the Y axis. Thus the line Y = 2X — 4 crosses the Y axis 4 units below the X axis and has a slope of 2. A line parallel to the Y axis is expressed as X = so-and-so; X = 5, for example, because it represents all points 5 units to the right of Y. Similarly a line parallel to the X axis and so many units above it is described in the same manner. For example, a line parallel to the X axis and 5 units above it is represented as Y = 5. A line going through the origin, such as OA of Fig. 3, crosses the Y axis at Y = OandsoBin theequation above equals zero. The formula then becomes simpler, Y = MX where M is the slope and is actually equal to Y/X. In Fig. 3 M is equal to J or 0.5 and so0the equation of this line becomes Y = 0.5 X. A line going through points B and C, Fig. 3, crosses the Y axis at Y = 10 and has a slope equal to — Y/X (because it points in the opposite direction to OA) or — 'sa and so the line becomes Y = — JX Problem 4. Locate on Fig. 2 a point (X = 3, Y =2). Describe in mathematical language the position of the point P on Fig. 2. Problem 5. Mark off several units in both X and Y directions on a sheet of cross-section paper. Draw on it the following lines, (a). Y = 3, (b) X = —4, (c) Y = 4 x + 3, (d) Y = 3 X + 4, (e) Y = X — 3, (f) Y = 2X — 3. Ohm's Law The equation representing Ohm's law reads, I = E/R, may be written I = (1/R) E which looks axis 6 5 (X = 5\ }Y = 5/ > ' ^ Y= 5 4 3 2 ®p X=5 1 X -t ,+ 1 I 1 "1 \ 1 +" 2 3 4 5 6 *-} -Y (X = 0) VY-0/ X axis like the general expression for a straight line through the origin, Y = MX in which M 1/R. Now the reciprocal of the resistance of a circuit, is called its "conductance" and the lower the resistance the greater the conductance. We may write Ohm's law as I = K E in which K is the conductance and is always equal to 1 — R. K (or 1/R) is the slope of the line which expresses the relation between the current and voltage in a circuit. Problem 6. Assume the resistance of a circuit is 1 ohm, and plot the relation between E and I, making E the X axis and I the Y axis. (Assume various values for E, calculate I when R = 1, and plot). Then assume several other values to R and plot all on the same sheet of graph paper. Suppose, however, we have a current of 4 amperes flowing in a circuit having a resistance of 2 ohms. If we add another battery and vary its voltage in 1 r M = y X a 6 c Fig. 2 — This drawing illustrates the location of a point on a graph 01234 56 789 10 X Fig. 3 — Drawing shows method of plotting a line on a graph the current in the circuit will change. How can we express the relation between the total current flowing and the variations in the additional voltages? Let I be equal to the current flowing. Then I=| + 4or-^E + 4orI=±E + 4 The above formula looks like our general expression, Y = MX + B. In this case a current of 4 amperes is always flowing; and so when E = O, I = 4 and the line crosses the vertical or current axis at 1=4. When E = 4 volts, I =4/2+4 =6 amperes. And so on. Problem 7. Assume several values for E in the above case and plot the current on cross-section paper. Draw a line through them. Then assume another value of R and replot. Then assume a negative value of E. calculate I and plot. This is equivalent to reversing the battery so that it bucks the battery which is producing the steady current of 4 amperes. Units The appearance of a curve may be changed somewhat by changing the units into which the vertical and horizontal axes are divided. As an example, plot the following data which give the d.c. output voltage of a cx-380 rectifier tube as the load current is changed, and as various a.c. voltages are put on the plate of the tube. There will be three curves for the three plate voltages applied to the tube. First make the vertical divisions, 100, 150, 200, 250 volts, etc. Then make the same divisions, 100, 200, 300, etc. and note how much flatter the curves seem. The slope of these curves is an indication of the "regulation" of the rectifier, that is, how many volts drop is caused by increasing the output current. Problem 8. The ratio between the voltage and the current at any point on these curves gives the d.c. resistance of the rectifier. The slope of the line, that is the change in voltage divided by the change in current is the a.c. resistance of the rectifier. Calculate the d.c. resistance at each value of output current and plot against current. Data for Example Volts per plat* Volts output of Rectifier Current output 300 260 220 Milliamperes 20 375 330 280 40 350 300 250 60 330 280 230 80 310 260 210 100 290 240 190 120 280 230 180 • march, 1929 page 307 •