Lab #3 Linearity, Proportionality, and Superposition

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1 This lab experiment will focus on three concepts. Those concepts are linearity, proportionality, and superposition. Linearity and proportionality are like twins; they look similar at first glance, but you can find differences on closer inspection. Linearity is a system property that is very, very desirable. Consider an airplane. You would like to think that if it is 6 months old, or 25 years old, that it is pretty much the same condition it was when it left manufacturing plant. You would also like to think that if you were flying 100mph or 1000mph that the airplane would react the same way (effects of drag, etc. scaled of course). That is a linear system. It exhibits the same constant linear properties over any time frame and input. In reality, an airplane is a highly complex grouping of many linear and nonlinear systems. The fact that it can take you from point A to point B is the property of linearity focused upon in this example. A circuit that contains only resistors and sources is a special type of linear circuit. It is a linear algebraic circuit because all the elements can be considered constants. A true linear system can have time varying quantities (speed, mass) as long as the plant (airplane), behaves like a constant. Proportionality is a way to relate two quantities together. In a linear system, this means that if you supply more input, you get more output that is proportional to you input. In other words, when you crank the volume on your stereo to 10, it takes 10x units of input to get that output, but only 1x unit of input to get a volume of 1. Consider the circuit given in Figure 3.1. V in Linear Resistive Circuit R Figure 3.1: Linearity and Proportionality. In the linear circuit shown in Figure 3.1, we can find the output voltage relative to the input voltage as V = k V out Where the k is called the proportionality constant of the circuit. You should note that you could get a different proportionality constant for each set of quantities you would like to relate. This is a fundamental property of a linear circuit. Superposition is another way to solve a linear electrical circuit. Let us first state the superposition theorem. in 31

2 Superposition Theorem: In any linear electrical circuit, any voltage or current value can be obtained by taking the individual contributions to that voltage or current as a result of each source taken alone and summing them together. What this means is that if we have a circuit with two sources and need to find the output voltage, we can first determine the output voltage as a result of source 1 (getting rid of source 2) and then adding it to the output voltage resulting from source 2 (getting rid of source 1). Consider the circuit with two sources shown in Figure 3.2. R 1 R 2 V 1 R 3 V 2 Figure 3.2: Circuit to Illustrate Superposition. If we would like to solve for in Figure 3.2, we can use the superposition theorem to break this circuit into two sub circuits. These are shown in Figures 3.3 and 3.4. R 1 R 2 R 1 R 2 V 1 1 R 3 2 R 3 V 2 Figure 3.3: Superposition Circuit 1. Figure 3.4: Superposition Circuit 2. By the superposition theorem, as shown in Figure 3.2 is the sum of 1 and 2 as shown in Figures 3.3 and 3.4. In mathematical form, V = V V out out1 out 2 It is very important to note how you eliminate sources when applying the superposition theorem. Voltage sources are replaced with short circuits and current sources are simply opened, or left disconnected. 32

3 Instructional Objectives 3.1 Measure circuit parameters to determine if they are linearly related. 3.2 Measure circuit parameters to determine proportionality constants. 3.3 Verify the superposition theorem. Procedure 1. Construct the circuit shown in Figure 3.5 on your breadboard. V in Figure 3.5: Linear Circuit to Study Proportionality. 2. At 5 different voltages of Vin between 0 and 15V, measure the output voltage,. Calculate the proportionality constant that relates the output voltage to the input voltage. Record your data in Table 3.1. Input Voltage V in (V) Output Voltage (V) Proportionality Constant K = /V in Table 3.1: Data for Figure

4 3. We will now verify the superposition circuit. Since we will be constructing three circuits, it is advised that each lab partner builds one circuit and takes the measurements for that circuit. Construct the circuit shown in Figure 3.6 on your breadboard and measure. Record your data in Table V 6.2kΩ 15V Figure 3.6: Circuit to Verify Superposition. Quantity Measured Voltage (V) V 5V V 15V Table 3.2: Data to Verify Superposition. 4. Now remove the 15V source by replacing it with a short circuit. This is shown in Figure 3.7. Measure the voltage across the 6.2kΩ resistor. 5V V 5V 6.2kΩ Figure 3.7: Superposition Circuit with One Source Removed. 5. Reconnect the 15V source and replace the 5V source with a short circuit and measure the voltage across the 6.2kΩ resistor. This circuit is shown in Figure

5 V 15V 6.2kΩ 15V Figure 3.8: Superposition Circuit Missing A Voltage Source. Post Lab Questions 1. Plot the data in Table 3.1 on the same plot along with your data using the proportionality constant obtained in the prelab. Connect your measured data with a bestfit line and be sure to distinguish the actual measured data points. The theoretical data may be plotted with a smooth line. 2. For the circuit of Figure 3.5, how did the measurements made in the lab compare to the predicted values calculated using proportionality? Explain any differences. 3. For each of the three circuits you built for the superposition portion of this lab, how well did the calculated value from the prelab compare to the measured outputs? Explain any differences. 4. If each of the voltage supplies were independently increased in magnitude (not polarity) by 1V, which one would have a bigger effect on the change in the output voltage? Explain your results. 5. If the input in Figure 3.5 were a 5V peak to peak sine wave instead of a constant DC voltage, plot V in and. Is there still a linear relationship between V in and? 35

6 Name: Section: 1. Solve for V load in Figure 4.0. PreLab #4: Thevenin s Theorem 5V V load 15V Figure 4.0: Circuit for Problem Why does using a Thevenin equivalent circuit make circuit analysis easier? 36