ELEC 103 Unit 4
The purpose of this unit is to teach students who intend to pursue a career in the electronics industry the correct terminology to use when discussing graphs and graphing. Additionally, the purpose of the unit is to teach the student how to plot graphs on various types of graph paper, how to draw graphs using a French Curve, and how to label graphs correctly.
In order to achieve those objectives, several things must be accomplished. The student must purchase a French Curve. French Curves can be purchased at the College Book Store, or at most art supply stores. Your Professor will show you some examples of the French Curve since they come in various shapes and sizes. Graph paper must be made available. The first three graphs will be plotted on linear graph paper. Therefore, three sheets of the linear graph paper found in Appendix A must be reserved and used to submit the first three graphing projects/ Data for the last two graphing assignments will be supplied to you, in this unit, as the graphs are being discussed. The ability to place the data correctly on the graph paper and to draw the curve is the starting point for this unit.
THE FUNCTION CONCEPT:
A function is a mathematical relationship between two or more variables. Using the Ohms Law formula E = I R, the voltage E and the current I are the variables and the resistance R is the constant. When the data for the first graphing assignment was calculated, at the end of Unit Three, the resistance was held constant at some value throughout the series of calculations. The voltage was chosen to be a specific value that changed for each calculation. Then, the current was calculated using the value of voltage that was chosen and the value for the constant.
The same thing held true for the calculations performed for the Sine Wave and the Cosine Wave. The formula, ei = EP SINq was used. EP was made equal to 10 volts throughout the calculation and therefore, was the constant. Values of theta were chosen and values for ei were calculated using specific values for theta and the appropriate value for the constant. Therefore, SINq and ei were the variables.
On graph paper, the horizontal axis, no matter which way the paper is held, is referred to as the ABSCISSA. The vertical axis is referred to as the ordinate. If you refer to Figure 4-1, you will see the horizontal axis, labeled the X-axis. This axis is referred to as the abscissa. The vertical axis, labeled the Y-axis, is the one that is referred to as the ordinate.
FIGURE 4-1
In Figure 4-1 notice the roman numeral one through roman numeral four. These areas, or quadrants, are referred to as quadrant one, quadrant two, etc. The first graphing assignment will be plotted in quadrant one exclusively. The second and third graphing assignments of Unit Four will be plotted in two of the four quadrants, quadrant one and quadrant four. Graphs can be plotted in only one quadrant or they can be plotted in more than one quadrant, depending upon the data.
COORDINATE SYSTEM:
A commonly used Coordinate System is formed by two perpendicular coordinate Axes, an X-axis and a Y-axis. Each axis is scaled in some convenient manner, as is shown in Figure 4-1. Although the coordinate axes shown in the figure are scaled the same, they can be scaled differently.
ORDERED PAIR:
If we assume a value for "X", we can obtain the corresponding value of "Y" from the functional relation, which in this case is E = I x R. Assign the constant, R, a value of 2. Then, substituting an assumed value of "X" = 3 into the equation, will produce the second number which corresponds to the value of "Y". In this instance, the "X" will be the current and the "Y" will be the voltage. The calculated value for the voltage is 6 and this is the value that corresponds to "Y".
The values X = 3 and corresponding value of Y = 6 make what is known as an Ordered Pair. They can be symbolically stated as (3,6) where the first value within the parentheses is a value of "X" which is associated with the X-axis or abscissa. And, the second value within the parentheses is a value of "Y" which is associated with the Y-axis or ordinate. If you look at Figure 4-2, you will see the coordinate point that is represented by the ordered pair (3,6). That coordinate point was determined by using the ordered pair (3,6).
FIGURE 4-2
The origin is the point where the X-axis and the Y-axis cross. All ordered pair measurements start from this point. Move to the right from the origin, along the X-axis, three scale marks or divisions. Now, move up vertically from that position, which is three divisions to the right of the origin, making a dashed line as you move upward.
Next, move upward from the origin, along the Y-axis, six divisions. Now, move horizontally from that position, which is six divisions up the Y-axis from the origin, making a dashed line as you move horizontally. Where the horizontal and vertical dashed lines cross is called the coordinate point. That point, the coordinate point, is represented by the ordered pair (3,6). Therefore, for each ordered pair, one can plot on an appropriately marked piece of graph paper, one point that is called a coordinate point. The first number of the ordered pair is associated with the X-axis. The second number of the ordered pair is associated with the Y-axis.
A SET OF ORDERED PAIRS:
R = 1000WVoltage (V) / Current (mA)
0 V / 0 mA
2 V / 2 mA
4 V / 4 mA
6 V / 6 mA
8 V / 8 mA
10 V / 10 mA
FIGURE 4-3
A set of ordered pairs will be obtained if we continue the procedure of assuming additional values of the variable "X" and obtaining the additional corresponding values of "Y" from the functional relationship (equation). A set of ordered pairs is usually represented by a data table filled with data calculated by using a specific formula. At the end of the last unit, you saw five data tables and, by this time, the data have been calculated. Each line of a data table consists of an ordered pair.
One of the data tables from the last part of Unit Three is illustrated in Figure 4-3. Ohms Law was used and the constant was the resistance, R, with a value of 1000W. One of the variables was the voltage and values for the voltage were assumed. The assumed voltages were to be over a ten volt range in two volt increments. Therefore, the values ranging from zero volts to ten volts were chosen. This variable, the one that has the values chosen for it, is called the INDEPENDENT variable. In this instance, the voltage is the independent variable.
The independent variable is usually plotted along the horizontal, which is referred to as the abscissa.
The variable that has values calculated for it, the current in this instance, is called the dependent variable. The dependent variable is usually plotted along the vertical, which is referred to as the ordinate. As graphs are plotted in this unit, the values for the independent variable will always be plotted along the abscissa and the values for the dependent variable will always be plotted along the ordinate.
In the data table of Figure 4-3, the last line in the data table is 10 volts and 10 milliamperes. Those two pieces of data make up an ordered pair. The current, which is the dependent variable, was calculated by dividing the independent variable by the constant. The dependent variable for each ordered pair is always calculated by using the independent variable and the constant. In order to plot a graph one must have a formula that can be used to calculate values for the dependent variable.
The dependent variable need not be named Y. The dependent variable in the case of Ohms Law was named I, for the current. The important things to remember are, number one, the dependent variable is the variable for which data is calculated and, number two, the dependent variable is plotted along the ordinate.
The independent variable need not be named X. The independent variable in the case of Ohms Law was named V, for the voltage. The important things to remember are, number one, the independent variable is the variable for which data is assumed and, number two, the independent variable is plotted along the abscissa. In the following example, using Figure 4-4, 4-5 and 4-6, a formula is supplied, data assumed, a data table developed and the curve plotted.
Using the formula y = 2x + 6, values for the independent variable, which in this instance is x, are chosen. If you ask the question, "Why where those values for x chosen?". The answer is, the values chosen for x where chosen to make the coordinate points fall on the existing scaled axes. It really is unimportant at this time. What is important at this time is learning how to setup a data table, how to plot the ordered pairs to produce coordinate points and how to draw the line or curve, as the case may be. The values for the dependent variable are calculated by using the chosen values for x, one at a time, in the existing formula. For each value of x there is a calculated value for y. Those values make up the ordered pairs. In this case the ordered pairs are (1,8), (0,6), (-1,4), (-2,2), (-3,0), (-4,-2) and (-5,-4). These values are displayed in the data table of Figure 4-4 and other figures.
FIGURE 4-4
After the data table has been setup as shown in Figure 4-4, the ordered pairs are utilized to plot the coordinate point. Start with the first ordered pair, (-5,-4). Of course, if x = -5, then from the formula, y = -4. Using Figure 4-4, start at the origin and move to the left five divisions. Draw a dashed line from that position on the abscissa, downward so it is parallel to the Y-axis. The dashed line is drawn downward since the second number of the ordered pair is a negative number. If the second number of the ordered pair was a positive number the dashed line would have been drawn upward. You have now taken care of the first number of the ordered pair.
FIGURE 4-5
Next, using the second number of the ordered pair, start at the origin and move downward four divisions. Draw a dashed line from that position on the ordinate, to the left so it is parallel to the X-axis. You have now taken care of the second number of the ordered pair. Where the two dashed lines cross, a coordinate point may be drawn. On some graphs the ordered pair is placed next to the coordinate point. However, the graphs for this unit will not utilize that concept. Do not place ordered pairs in parentheses next to the associated coordinate point in this unit.
The remaining ordered pairs are plotted to obtain their coordinate points as shown in Figure 4-5. Notice that the dashed lines are not drawn to identify each and every coordinate point. The dashed line concept is shown only to illustrate a method of plotting the coordinate point. After all of the coordinate points have been located and plotted from the ordered pairs available in the data table, the last step is to connect the coordinate points together in an appropriate manner.
FIGURE 4-6
In this case, a line can be drawn with a straight edge through all the coordinate points. If the coordinate points have been carefully placed on the graph paper so they are centered at the intersection of the ordered pair, the line will cross directly over the coordinate point. Indeed, the straight line should be drawn directly through the center of each and every coordinate point.
This type of graph, the one put together in Figures 4-4, 4-5, and 4-6 is called a linear or straight line graph.
PLOTTING LINEAR GRAPHS:
You will remember the three data tables displayed earlier, at the end of Unit Three, where the dependent variable was the current. The independent variable was the voltage and the constant was the resistance. The graph paper in Figure 4-7 is setup to plot the data from the data table in Figure 4-8.
The first graphing assignment is in the Learning Objectives for this unit but basically the assignment states that the student will: Compute and tabulate values of current flowing through a 1000W resistor for 2 volt increments over a range from 0 volts to 10 volts. Repeat the calculations for a 2500W resistor and a 10,000W resistor. Plot the data on the same sheet of graph paper using the current in mA as the ordinate and the voltage in Volts as the abscissa.
The first step will be to identify a piece of linear graph paper and then set it up with the voltage along the abscissa and the current along the ordinate. Look at the figure on the next page. That figure, Figure 4-7, illustrates a sheet of rectangular coordinate graph paper.
You will see a piece of linear graph paper setup with the voltage as the abscissa and the ordinate setup for current as requested in the assignment. Notice, the vertical divisions and the horizontal divisions are all the same size. The scale for the abscissa is linear, in that each division is one volt. The ordinate scale is also linear in that each division is two milliamperes. The abscissa and the ordinate need not have the same scale and, indeed, in this example they do not have the same scale. Both abscissa and ordinate can have different scales but each division on the abscissa must be exactly the same number of units. Also, each division on the ordinate must be exactly the same as the unit before it and after it. Therefore, each division on the abscissa is scaled as one volt and each division on the ordinate is scaled so each division is two milliamperes.