Lab 7 v6-Thevenin and Norton Equivalent Circuits

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Apr 3, 2024

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Cessar Lechuga Gerardo Gamez Moreno LAB 7

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents A. OBJECTIVES  Observe application of Thevenin equivalent principles  Apply the least squares method to fit a linear equation to experimental data B. EQUIPMENT REQUIRED  Bench top Digital multimeter  Breadboard  Triple Channel Power supply  Miscellaneous Cables C. PARTS REQUIRED  Fourteen 1/2 Watt Resistors, each of following values: 51 ohms, 240 ohms, 560 ohms, 2x1.2 kilohms, 1.8 kilohms, 2x2.2 kilohms, 2x2.4 kilohms, 3.6 kilohms, 3.9 kilohms, 13 kilohms, 18 kilohms.  Hook-up wire (4/20 or #22 solid conductor) D. PRIOR TO LAB (1) Thevenin Equivalent Circuits The purpose of this lab is to experimentally demonstrate the concept of Thevenin equivalent circuits, and then apply some basic data analysis techniques to the resulting measurements. The basic concept of a Thevenin equivalent is illustrated in Figure 8-1. Consider a linear circuit 1 with a pair of output terminals, which we will call A and B, as shown at the left. Connected to these terminals is another resistance R, which we call the load. There is a simplified circuit, shown at right, which is equivalent to the circuit at the left. By equivalent, we mean that if any value R of load resistor is connected the circuit at the right, it gives the same voltage V and the same current I as if the load resistor were connected to the circuit at the left. 1 Linear circuits were discussed in the previous lab. If a circuit consists only of ideal sources and resistors, it is safe to assume it is linear. Page 1 of 10

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents Page 2 of 10

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EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents Figure 8-1 Thevenin Equivalent Circuit The circuit at the right delivers the same voltage and current as the circuit at the left, as long as both have the same value of load resistance R connected. You can verify this by doing a few test cases. For example, if R = 1000 ohms, both circuits shown in Figure 8-1 will give V= 0.222 Volts and I = 0.222 mA. If R = 2000 ohms, both circuits will give V = 0.2857 Volts and I = 0.1429 mA. The Thevenin equivalent is always a single voltage source in series with a single resistance. The value of the voltage source is called the Thevenin voltage or the open- circuit voltage. The resistance is called the Thevenin resistance or the equivalent resistance. It turns out that all linear circuits, with a few exceptions, have Thevenin equivalents. (The exceptions are cases where the equivalent resistance turns out to be infinite.) Once the Thevenin equivalent has been found for a circuit, it is much easier to determine the behavior when different loads are connected to the terminals A and B. Although we have shown the load as a resistor, it could be a complete circuit by itself or a device such as a motor or actuator. Many devices (amplifiers, filters, test instruments, etc.) can be characterized by their equivalents. A few fundamental facts that you will need for this lab are as follows: (a) The short circuit current is the current I that a circuit produces when the load R is reduced to zero ohms. (b) The open circuit voltage is the voltage V that a circuit produces when the load R is removed and the terminals A and B are left open, or, in other words, when the value of R increases to infinity. (c) The open circuit voltage divided by the short circuit current equals the Thevenin equivalent resistance. (d) Maximum Power Transfer Theorem the power dissipated in the resistor R changes as the value of R is changed. This theorem states that the power is maximum when the resistance R of the load is equal to the Thevenin equivalent resistance. Page 3 of 10

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents (2) Current versus Voltage for a Circuit and its Thevenin Equivalent As different values of R are connected to a linear circuit, the voltage V and current I as shown in Figure 8-1 will change. If one looks at a generic Thevenin equivalent, as shown in Figure 8-2, one can see that the voltage drop across the resistor R th is IR th . Therefore, the output voltage V is given by V = V th - IR th . or alternatively, the current I is given by ( V th - V ) / R th . If one plots current I versus voltage V, it can be seen that the equation represents a straight line with the following features: (a) the vertical intercept is at V th /R th which is the short circuit current. (b) the horizontal intercept is at V th , the Thevenin equivalent voltage. (c) the slope is -1/R th . Since a circuit and its Thevenin equivalent give the same voltage and current for any load R, these features should hold true for any linear circuit with a pair of output terminals. Plotting measured data for current versus voltage is one way to experimentally determine its Thevenin equivalent. Figure 8- 2 Current versus Voltage for a Thevenin Equivalent E. IN THE LAB (1) Circuit Set-Up (a) Identify the resistors listed in the parts list on page 1. Measure each with the ohmmeter, using the range which gives best resolution, and record the measurements along with the color codes for each resistor. Printed Resistance Measured Resistance Color Codes 51 Ω 50.4 Ohms Green Brown Black 240 Ω 236.43 ohms Red Yellow Brown Page 4 of 10

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents 560 Ω 0.551 kohms Green Blue Brown 1.2k Ω 1.16 kohms Brown Red Red 1.2k Ω 1.17 kohms Brown Red Red 1.8k Ω 1.7690 kohms Brown Green Red 2.2k Ω 2.16 kohms Red Red Red 2. 2k Ω 2.165 kohms Red Red Red 2.4k Ω 2.37 kohms Red Yellow Red 2.4k Ω 2.37 kohms Red Yellow Red 3.6k Ω 3.51 kohms Orange Blue Red 3.9k Ω 3.85 kohms Orange White Red 13k Ω 12.87 kohms Brown Orange Orange 18k Ω 17.69 kohms Brown Gray Orange (b) Set up your power supply so that one of the outputs is putting out 24 Volts and the other is putting out 10 Volts. Measure the voltages for each supply with the voltmeter and make sure that they are correct to within +/- 0.1 Volts. Record the values for future use. Power Supply # 1:24.003 Volts Power Supply # 2: 9.999 Volts (c) Then construct the circuit shown in Figure 8-3. Figure 8-3 Circuit for Part (1) (2) Current vs. Voltage Measurements (a) Set up the meter to measure DC voltage. In the measurements below, you are responsible for determining and using the voltage range, which will give you the most accurate results. (b) There should be eleven resistors from the parts list on page 1 which were not used in constructing the circuit. Now take these resistors and connect them one by one to the circuit above. For each resistor, measure the voltage. Then use the voltage and your measured value for the resistance to compute the current and power in the resistor. Record your results in the form of a table; for example: Page 5 of 10

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EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents Nominal Value Voltage Measured Resist. Current Power 51 ohms 0.368 Volts 51 ohms 0.722 mA 2.66 mW 240 ohms 1.605 Volts 240 ohms 6.69 mA 10.73 mW etc. etc. etc. etc. etc. Table 7-1 Load resistance’s Voltage and Current Resistance Nominal Value [Ω] Measured Resistance [Ω] copy from p3 Measured Output Voltage [V] Calculated Current[mA] I = V/R Calculated Power[mW] P=VI=RI 2 51 Ω 50.4 Ohms 371.46 mV 7.37 mA 2.73 240 Ω 236.43 ohms 1.616 V 6.30 mA 9.38 560 Ω 0.551 kohms 3.354 V 6.08 mA 20.36mW 1.2k Ω 1.16 kohms 5.833 V 5.02 mA 29.23 1.8k Ω 1.76 kohms 7.53 V 4.28 mA 32.24 2.2k Ω 2.16 kohms 8.42 V 3.90 mA 32.85 2.4k Ω 2.37kohms 8.805 V 3.71 mA 32.62 3.6k Ω 3.51 kohms 10.47 V 2.98 mA 31.17 3.9k Ω 3.85 kohms 10.55 V 2.74 mA 28.90 13k Ω 12.87 kohms 14.70 V 1.14 mA 16.72 18k Ω 17.69 kohms 15.34 V 0.87 mA 13.38 Page 6 of 10

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents Figure 8-4 Voltage Measurements (c) As soon as you are finished, make an Excel plot of current versus output/load voltage based on the data in your table, and make sure that there are no obviously incorrect points. (You will do a more careful plot in the "After the Lab" section.) If there is a bad point, measure again that resistance and retake the voltage, current and power. Use the PC at your workstation to plot a graph of current vs. voltage from Table 7.1. Insert a “scatter Chart” for this purpose. One you have completed the plot, right click on any one of the points, select “Add Trend line”, use Linear plot, Set, and Display equation on chart 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 f(x) = − 2.35 x + 17.39 Current versus output/load voltage Current (mA) Voltage (V) (d) From your graph estimate the open circuit voltage, the short circuit current, and the equivalent resistance and record your numbers. Thevenin Resistance = Open Circuit Voltage / Short Circuit Current =17.39/7.4= 2350 Ω Thevenin Voltage = Open Circuit Voltage = 17.39 Volts Also, look at the results for power. At what resistance does the power (mili Watts) reach a maximum? 2.16 k Ohms. Is this close to the Thevenin resistance? YES NO Page 7 of 10

EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents Do the results seem to agree with the maximum power transfer theorem? Yes, the results seem to agree with the maximum power transfer theorem (3) Direct Measurement of Thevenin Equivalent Circuit. Remove the load resistor (a) Open Circuit Voltage to measure the open circuit voltage, simply measure the voltage from point A to point B (positive reference terminal at with no load resistor present . Voc = 17.34 V (b) Short Circuit Current To measure the short circuit current, set up your meter to measure current and connect it directly to the points A and B, again with the positive reference at A. Isc = 7.544 mA (c) Equivalent Resistance to measure the equivalent resistance, kill both of the voltage sources in your circuit. If you do not remember how to kill a voltage source, read the following. (Hint: Remove the two leads from the power supply positive terminals and connect them to ground, on the breadboard.) Then set up your meter to measure resistance and directly connect the meter to points A and B. R TH = 2.30 k ohms (d) Compare your results to those obtained in part (d) of Part (2) from your sketch of current versus voltage. If any parameter differs by more than 10% there was probably a mistake in your measurements. The percent error is 0.85% (e) What is the maximum power transfer to the load in this particular example? Maximum Power Transfer = 32.17 milliWatts F. AFTER THE LAB Analysis and Calculations (1) Plotting Data Make a neat and accurate plot of your current versus voltage data either by hand or using plotting software. Both axes should be linear. Note: if you use business-oriented software such as Excel and Quattro-Pro, make sure that the scales are linear. On a linear scale each tic mark represents the same amount of increase in value. Take a screenshot of the graph and insert it here. On the graph’s cartesian axes, mark and label the short circuit current I SC , and the open circuit voltage V OC Page 8 of 10

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EECE 2105-2 Spring 2024 LAB 7 v5 Thevenin & Norton Equivalents 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 f(x) = − 2.35 x + 17.39 Current versus output/load voltage Current ISC (mA) Voltage VOC (V) (2) Theoretical Thevenin Equivalent Using circuit analysis techniques, determine the Thevenin equivalent for the circuit constructed in the laboratory; use the measured rather than the nominal values for the voltage sources and the resistors. Compare your answers to your experimental results and comment on any discrepancies. R Thevenin = 2350 Ω V Thevenin 17.39 V Page 9 of 10