Radio Broadcast (May 1928-Apr 1929)

"ZL RADIO BROADCAST No. 18 Radio Broadcast's Home-Study Sheets PLOTTING CURVES — PART II March, 1929 CURVES may bp plotted either from a mathematical formula or equation or from a set of data obtained in a laboratory or from someone who has already done the laboratory work. To plot these curves properly, all one needs is a hard pencil or a ruling pen, some India Ink (Higgins' American India ink), a celluloid triangle or rule, a French curve, and some cross-section paper. The latter may be bought from Keuffel and Esser, Deitzgen, Codex, and several other manufacturers, and it comes in many colors, many rulings, and sizes, some of which are punched for loose-leaf note books. Keuffel and Esser paper No. 359-6 and 355-2R are both convenient and are ruled 10 x 10 to the inch and are punched for standard size note books. Another good paper is Keuffel and Esser No. 359-11 which is ruled 20 x 20 to the inch. Dietzgen No. 340-10 is ruled 10 x 10 and is punched. Codex 2 and 3 cycle logarithmic paper. No. 3135 and 3112, and Keuffel and Esser double logarithmic three cycles, No. 359-120, are useful in plotting frequency characteristics of audio transformers, amplifiers, loud speakers, etc. Vacu um-Tube Character ist ics The characteristics of a vacuum tube are usually represented on a sheet of graph paper and are called the characteristic curves. Because there are three variable factors involved, plate current, grid voltage, and plate voltage, a complete picture of the tube and its action in a circuit cannot be represented on a single sheet of paper, (which has only two dimensions), but two curves are needed, or better still a three dimension model made of plaster of Paris or wax. Some very beautiful models of this sort are used in the course on vacuum tubes given at Cruft Laboratory, Harvard University, and are part of the equipment of any good radio engineering course. We can get a good idea of what a tube will do by making two curves called the Eg-Ip and the Ep-Ip curves. These show what the plate current is at various values of grid and plate voltage. The slopes of these curves are important tube factors. Problem 1. Plot the data in Table 1, making the vertical axis, the current axis (in mA.). Determine the slope and, remembering that the mutual conductance is the change in amperes divided by change in grid volts, calculate the mutual conductance. The slope of the plate-voltage plate-current curve, using amperes and volts of course, gives the reciprocal of the plate resistance of the tube. The slope of the curve must be divided into 1.0 to get the resistance. Calculate the plate resistance at several points on the curve. Plot the mutual conductance and the plate resistance against grid volts, plate volts, and plate current. In each case assume one of the variable as fixed, e.g., when calculating and plotting the plate resistance assume the grid voltage is some constant value for one set of values, and then assume another value for another set of data. Table 1 Grid volts = E 0 —4 -6 —8 plate volts = Ep 60 80 135 2.75 4.5 10.25 .25 1.0 4.75 0 .25 2.75 0 1.25 Correcting Errors of Measurement A curve which is a visual picture of a given laboratory experiment may be very useful in detecting or 4000 5000 6000 FREQUENCY IN CYCLES 10,000 Fig. 1 — Frequency characteristics of transformer plotted on cross-section paper correcting errors in measurement. For example, if we know that the relation between two factors is a straight line, and when we plot the curve, several points seem to be off this line, these points indicate errors in measurement. In calibrating a wavemeter, according to "Home-Study Sheet No. 13," errors may occur, and the only way to tell them is to plot the curve of wavelength squared against capacity, or wavelength against condenser degrees. The first of these curves will be a straight line, and the latter will be a smooth curve. Points off the curve should be considered wrong and must be repeated or disregarded. Problem 2. Plot the data in Table 2. first, showing the relation between wavelength squared and 500 1000 3000 5000 10,000 FREQUENCY IN CYCLES Fig. 2 — Frequency characteristics of transformer plotted on logarithmic paper condenser capacity, and, secondly, the variation of wavelength with condenser degrees. Determine which points are wrong, and indicate what the wavelength should be instead of the values given. If the slope of the straight line, i.e., (wavelength)2 against capacity is divided by 3.54, the inductance of the circuit will result. Determine the inductance. Wavelength meters 197.5 245. 253 300 Table 2 Condenser degrees 15 25 35 55 Condenser capaci ty 100 mmfd. 150 •' 200 " 300 " A mplificat ion-Frequency Character ist ics The frequency characteristics of amplifiers and audio transformers may be plotted directly against frequency. It has now become standard practice to plot amplification against frequency arranged in octaves, so that each change in frequency gets equal attention. For example, the curve of Fig. 1 represents a transformer of the olden days when lowfrequency amplification was unthought of. Note what a long flat portion the curve has. Then look at the curve of Fig. 2 in which the same data is presented on logarithmic paper. Here the low frequencies, i.e., from 100 to 1000 cycles are not all cramped into a very small part of the whole horizontal scale but get the same horizontal space as does the range from 1000 to 10,000 cycles — and both of these spaces represent a 10 to 1 change in frequency. The ear hears according to a logarithmic scale, and so amplifier characteristics are usually plotted against transmission units (db| of loss or gain with some given frequency as standard. That is, the response at all frequencies is plotted with respect to the response of some intermediate frequency as standard. For example, we may measure the power output of an amplifier obtained at 1000 cycles and then compare the power output of other frequencies to the value at 1000 cycles. Or we may simply plot the power output at all frequencies without regard to any given frequency as standard. One curve gives the characteristic, the other tells us the power output. The characteristic may be obtained from the power output curve by noting from it how much more power is obtained at one frequency than another. Characteristics of amplifiers should always be plotted with a logarithmic horizontal frequency scale and preferably with a vertical scale either in logarithmic units (db) or on a logarithmic scale. Problem 3. Transfer the data given in the curve of Fig. 3 to db, first calculating the number of db up or down from 1000 cycles, where the voltage amplification is 850 and secondly plotting the number of db corresponding to the voltage amplification, e.g., a voltage amplification of 100 corresponds to a db of 40. Remembering that the ear can hear with some difficulty changes in power output of 3 db and cannot hear smaller changes than this, plot the data in Table 3 and determine whether or not the amplifier is a good one. Plot in db using the power output at 1000 cycles as standard. Will the loss in response at 100 and 5000 cycles be noticeable to the ear? Table 3 1000 800 700 600 . 500 i 400 i ■ 300 i ) ' 200 100 Three Sta ge Put lie / \id 'es » . VI) IF li fier 50 100 1000 FREQUENCY IN CYCLES 10,000 Frequency Power output cycles milliwatts 60 175 100 350 200 600 400 700 1000 700 2000 700 4000 435 6000 280 8000 105 Fig. 3 — Power output of an amplifier plotted on Log-Log paper Summary A graph is a visual representation of some physical or mathematical law. To plot the curve when the law or equation is known, it is oidy necessary to assume various values for one of the related factors and to calculate what the other values are. Thus we can plot Ohm's Law by assuming values of voltage and calculating what the current will be at a known resistance. Then voltage and current values are plotted against each other. More complicated relations between two factors give curves which are not straight lines and the mathematical equation or formula is seldom known. • march, 1929 page 308 «