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ECE 170 Lab #3 Linearity, Proportionality, and Superposition Lab #3 Linearity, Proportionality, and Superposition 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 non-linear 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. + Vin Linear Resistive Circuit R Vout - 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 Vout k Vin 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. 3-1 ECE 170 Lab #3 Linearity, Proportionality, and Superposition 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. R1 R2 + V1 Vout R3 V2 Figure 3.2: Circuit to Illustrate Superposition. If we would like to solve for Vout 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. R1 R2 R1 + V1 Vout1 R2 + R3 Vout2 R3 V2 - - Figure 3.4: Superposition Circuit 2. Figure 3.3: Superposition Circuit 1. By the superposition theorem, Vout as shown in Figure 3.2 is the sum of Vout1 and Vout2 as shown in Figures 3.3 and 3.4. In mathematical form, Vout Vout1 Vout 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. 3-2 ECE 170 Lab #3 Linearity, Proportionality, and Superposition Instructional Objectives 3.1 3.2 3.3 Measure circuit parameters to determine if they are linearly related. Measure circuit parameters to determine proportionality constants. Verify the superposition theorem. Procedure 1. Construct the circuit shown in Figure 3.5 on your breadboard. 10k 10k + Vin 10k 10k 10k Vout - Figure 3.5: Linear Circuit to Study Proportionality. 2. At 5 different voltages of Vin between 0 and 15V, measure the output voltage, Vout. Calculate the proportionality constant that relates the output voltage to the input voltage. Record your data in Table 3.1. Input Voltage Vin (V) Output Voltage Vout (V) Table 3.1: Data for Figure 3.5. 3-3 Proportionality Constant K = Vout/Vin ECE 170 3. Lab #3 Linearity, Proportionality, and Superposition 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 Vout. Record your data in Table 3.2. 4k 5k + 5V Vout 6.2k 15V Figure 3.6: Circuit to Verify Superposition. Quantity Measured Voltage (V) Vout V5V V15V 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 6.2k resistor. 4k 5k + 5V V5V 6.2k 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 6.2k resistor. This circuit is shown in Figure 3.8. 3-4 ECE 170 Lab #3 Linearity, Proportionality, and Superposition 4k 5k + V15V 6.2k 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 pre-lab. Connect your measured data with a best-fit 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 pre-lab 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 Vin and Vout. Is there still a linear relationship between Vin and Vout? 3-5 ECE 170 Lab #3 Linearity, Proportionality, and Superposition Name: ____________________ Section: ____________________ Pre-Lab #4: Thevenin’s Theorem 1. Solve for Vload in Figure 4.0. 5k 4k + Vload 10k - 5V 15V 10k Figure 4.0: Circuit for Problem 1. 2. Why does using a Thevenin equivalent circuit make circuit analysis easier? 3-6