Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law EXERCISE OBJECTIVE When you have completed this exercise, you will be able to calculate the equivalent resistance of multiple resistors in series circuits. You will be introduced to Kirchhoff s voltage law and be able to apply this law to electrical circuits. You will know what a voltage divider is, and how to implement one using either two resistors, a rheostat and a resistor, or a potentiometer. DISCUSSION OUTLINE The Discussion of this exercise covers the following points: Calculating the equivalent resistance in series circuits Kirchhoff s voltage law Voltage dividers Voltage divider consisting of two resistors. Voltage divider consisting of a rheostat and a resistor. Voltage divider consisting of a potentiometer. DISCUSSION Calculating the equivalent resistance in series circuits In Exercise 3, you saw that in series circuits, there is only one path for electrical current to flow. As a consequence, the current flowing in each component of the circuit is equal. In Exercise 5, you learned how to calculate the current flowing in a circuit using Ohm s law. To do so, it is necessary to know the voltage of the power source, as well as the resistance of the circuit. When there is only one load in a circuit, the resistance of the circuit is equal to the load resistance. When there is more than one load in the circuit, however, the equivalent resistance of the circuit is equal to the combined resistance of all loads. It is possible to calculate the combined resistance of all loads in a series circuit using the following equation: (9) where is the equivalent resistance of all resistors in the series circuit, expressed in ohms ( ) is the resistance of each resistor in the circuit, expressed in ohms ( ) As the equation shows, the equivalent resistance of any resistors connected in series is simply equal to the sum of all their resistance values. Consider, for example, the series circuit containing three resistors shown in Figure 81a. To calculate the current flowing in this circuit, it is necessary to calculate the equivalent resistance of the circuit, as shown in Figure 81b. This is calculated using the following equation: Festo Didactic 89688-00 99
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Discussion 15 40 80 25 (a) Series circuit with three resistors (b) Circuit with a resistance equivalent to the resistance of,, and in series Figure 81. Circuits showing how to calculate the equivalent resistance of series resistors. Knowing the equivalent resistance of the circuit, it is now possible to calculate the current flowing in the circuit using Ohm s law: Therefore, a current of 0.30 A flows in both circuits in Figure 81, since both have the same equivalent resistance of 80. Kirchhoff s voltage law Kirchhoff s voltage law is a law described by German physicist Gustav Kirchhoff that is of use in the study of series circuits. It states that the sum of all voltages in a closed circuit loop is equal to 0 V. In practice, this means that, in a closed circuit, the sum of the voltages across the loads in the circuit is equal to the voltage of the power source. In order to understand Kirchhoff s voltage law, it is necessary to know that the voltage measured across passive elements in a circuit (i.e., all elements except power sources) is a voltage drop, and the polarity of this voltage value is inverted with respect to the polarity of the power source voltage. Therefore, if the voltage across a power source is positive, the voltage across all loads connected to it will be negative. This is denoted by the fact that the polarity of the power source opposes that of the loads (+ against +, - against -). Consider, for example, the circuit containing a single resistor shown in Figure 82. In this circuit, we know that the power source voltage is equal to and that the voltage measured across the resistor is equal to. However, both voltage values are of opposite polarities (as shown by the fact that the + signs are opposed in Figure 82). Therefore, the voltage across the resistor is not equal to, but to - when seen with respect to the polarity of the power source. 100 Festo Didactic 89688-00
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Discussion According to Kirchhoff s voltage law, the sum of the voltages across the circuit is equal to 0 V. We can see that this is true, as demonstrated by the following equations: + + - - Figure 82. Kirchhoff s voltage law applied to a circuit containing a single resistor. Now consider the series circuit containing three resistors shown in Figure 83. + 8 - + + 24 - - + 16 - Figure 83. Kirchhoff s voltage law applied to a series circuit containing three resistors. Festo Didactic 89688-00 101
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Discussion To calculate the voltage across each resistor, it is necessary to first calculate the equivalent resistance of the circuit: Using the equivalent resistance of the circuit, it is then possible to calculate the current flowing in the circuit: Since, in a series circuit, the same current flows in all components of the circuit, it is possible to use the source current to calculate the voltage across each resistor in the circuit, as shown below: As these calculations show, the higher the resistance of the component, the higher the voltage across the component and the greater the voltage drop at its terminals. Using these voltage values, it is then possible to verify that Kirchhoff s voltage law is respected in the circuit, as confirmed below: Voltage dividers The principles behind Kirchhoff s voltage law are often used in electronics and electrical engineering in order to produce what is referred to as a voltage divider. As its name indicates, a voltage divider can be used to divide its output voltage to a fraction of the input voltage. For example, if the input voltage of the voltage divider is 48 V, it is possible to obtain an output voltage equal to anything equal to or below 48 V. Voltage divider consisting of two resistors The most common type of voltage divider consists of a series circuit containing two resistors, as shown in Figure 84. As you saw in the previous section, the voltage across each resistor in the circuit depends on the resistance of each resistor in relation with the equivalent resistance of the circuit. For example, if resistance is equal to 45 and resistance is equal to 15, it is possible to calculate that the voltage across resistor is equal to 36 V and the voltage across resistor is equal to 12 V. Since the output terminals of the voltage divider correspond to the terminals of resistor, any component connected to the output terminals will be applied a voltage of 12 V, just as if the component was connected to a 12 V power source. 102 Festo Didactic 89688-00
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Discussion 45 48 V 15 Voltage divider output 12 V Figure 84. Voltage divider circuit consisting of two resistors. By adjusting the resistance of the resistors composing the voltage divider, it is possible to adjust the output voltage as required. For example, replacing resistor with a 30 resistor would yield an output voltage of 16 V, while replacing it with a 15 resistor would yield an output voltage of. This enables a voltage divider to be tailored specifically for a particular situation, simply by carefully selecting the appropriate resistors. Voltage divider consisting of a rheostat and a resistor Another common type of voltage divider is the variable voltage divider. It consists of a series circuit containing a rheostat and a resistor, as shown in Figure 85. 15 to 45 48 V 15 Voltage divider output Figure 85. Variable voltage divider circuit consisting of a rheostat and a resistor. As the figure shows, the only particularity of the variable voltage divider is that the resistance of resistor can now be dynamically varied so that the output voltage is adjusted as required. This means that the same voltage divider can be used for multiple purposes, or be used for a purpose requiring a variable-voltage power source. For example, the output voltage of the variable voltage divider in Festo Didactic 89688-00 103
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Discussion Figure 85 can vary from (when the rheostat is set to 15 ) to 12 V (when the rheostat is set to 45 ). Voltage divider consisting of a potentiometer Finally, it is possible to implement a voltage divider simply by using a potentiometer, as shown in Figure 86. 48 V 0 to 60 Voltage divider output Figure 86. Variable voltage divider circuit consisting of a potentiometer. As the figure shows, the wiper of the potentiometer effectively divides the potentiometer into two resistors whose respective resistance values depend on the position of the wiper. The higher the resistance across the circuit branches connected to the voltage divider outputs, the higher the output voltage. For example, the output voltage of the variable voltage divider in Figure 86 can vary from 48 V (when all of the potentiometer resistance is encompassed between the voltage divider outputs) to 0 V (when no part of the potentiometer resistance is encompassed between the voltage divider outputs). 104 Festo Didactic 89688-00
Outline PROCEDURE OUTLINE The Procedure is divided into the following sections: a Note that, due to the length of the procedure, it is recommended to separate it into more than one lab session. Setup Equivalent resistance and Kirchhoff s voltage law circuit with two resistors Solving the circuit through mathematical calculations. Solving the circuit through circuit measurements. Equivalent resistance and Kirchhoff s voltage law circuit with three resistors Solving the circuit through mathematical calculations. Solving the circuit through circuit measurements. Voltage divider consisting of two resistors Solving the voltage divider through mathematical calculations. Solving the voltage divider through circuit measurements. Light intensity control circuit implemented using a resistor and a selector switch PROCEDURE High voltages are present in this laboratory exercise. Do not make or modify any banana jack connections with the power on unless otherwise specified. Setup In this section, you will install the training system modules in the workstation. 1. Refer to the Equipment Utilization Chart in Appendix A to obtain the list of equipment required to perform this exercise. Install the equipment required in the workstation. Make sure that all fault switches are set to the O (off) position. Equivalent resistance and Kirchhoff s voltage law circuit with two resistors In this section, you will set up a circuit containing two resistors. In the first part of this section, you will determine the circuit parameters mathematically (equivalent resistance, current, voltage across the resistors). You will then verify that Kirchhoff s voltage law is respected by confirming that the sum of all calculated voltages in the circuit is equal to 0 V. In the second part of this section, you will measure the circuit parameters. You will verify that the measured parameters are equal to the parameters you calculated, and also that Kirchhoff s voltage law is respected by confirming that the sum of all measured voltages in the circuit is equal to 0 V. Festo Didactic 89688-00 105
2. Consider the circuit shown in Figure 87. 50 250 Figure 87. Circuit containing two resistors used for confirming Kirchhoff s voltage law. Solving the circuit through mathematical calculations 3. Knowing that the resistance of resistor is equal to 50 and that the resistance of resistor is equal to 250, calculate the equivalent resistance of the circuit. a The equivalent resistance of series resistors is calculated using the following equation: Calculated equivalent resistance 4. Knowing that the ac power source voltage is equal to and using the equivalent resistance you calculated in the previous step, calculate the current flowing in the circuit. a Ohm s law states that: Calculated current A 5. Using the current you calculated in the previous step and knowing the resistance values of the resistors, calculate the voltage across each resistor in the circuit. a Ohm s law states that: Calculated resistor voltage V Calculated resistor voltage V 106 Festo Didactic 89688-00
6. Using the circuit parameters you know and those you calculated, can you confirm through calculations that Kirchhoff s voltage law is respected? a According to Kirchhoff s voltage law: Solving the circuit through circuit measurements 7. Make sure that the main power switch on the Power Source module is set to the O (off) position, then connect it to an ac power outlet. Set up the circuit shown in Figure 88. Use the 50 and 250 resistors of the Resistors module to implement resistors and, respectively. This circuit allows measurement of the equivalent resistance of the two series resistors. 50 250 Figure 88. Measuring the equivalent resistance in the circuit of Figure 87. 8. Using an ohmmeter, measure the equivalent resistance in the circuit. Record the value below. Measured equivalent resistance Festo Didactic 89688-00 107
9. Is the equivalent resistance you measured in the previous step virtually equal to the value you calculated in step 3? 10. Set up the circuit shown in Figure 89. For the moment, connect only ammeter A1 to the circuit. 50 250 Figure 89. Measuring the voltages and currents in the circuit of Figure 87. To reduce the risk of electrical shock, connect all ground (green) terminals of the modules in series with the ground (green) terminal of the power source. 11. Turn the power source on. Measure the currents,, and flowing in the circuit by successively connecting the ammeter at the locations shown in Figure 89. Record the current values below. b Make sure to turn the power source off before making any change to the circuit connections. Also make sure to turn it back on before taking any measurements. Measured current A Measured current A Measured current A 108 Festo Didactic 89688-00
12. Based on the currents you measured in the previous step, what can you conclude about the current flowing in series circuits? Briefly explain. 13. Are the currents,, and you measured in the previous step virtually equal to the current you calculated in step 4? 14. Measure the source voltage and the voltage across each resistor in the circuit by connecting the voltmeters at the locations shown in Figure 89. Record the voltage values below. Measured source voltage V Measured resistor voltage V Measured resistor voltage V 15. Are the voltages you measured across each resistor in the previous step virtually equal to the values you calculated in step 5? 16. Using the circuit parameters you know and those you measured in this subsection, can you confirm through calculations that Kirchhoff s voltage law is respected? a According to Kirchhoff s voltage law: 17. Turn the power source off. Festo Didactic 89688-00 109
Equivalent resistance and Kirchhoff s voltage law circuit with three resistors In this section, you will set up a circuit containing three resistors. In the first part of this section, you will determine the circuit parameters mathematically (equivalent resistance, current, voltage across the resistors). You will then verify that Kirchhoff s voltage law is respected by confirming that the sum of all calculated voltages in the circuit is equal to 0 V. In the second part of this section, you will measure the circuit parameters. You will verify that the measured parameters are equal to those you calculated, and also that Kirchhoff s voltage law is respected by confirming that the sum of all measured voltages in the circuit is equal to 0 V. 18. Consider the circuit shown in Figure 90. 50 50 250 Figure 90. Circuit containing three resistors used for confirming Kirchhoff s voltage law. Solving the circuit through mathematical calculations 19. Knowing that the resistance of resistors and is equal to 50, and that the resistance of resistor is equal to 250, calculate the equivalent resistance of the circuit. Calculated equivalent resistance 20. Knowing that the power source voltage is equal to and using the equivalent resistance you calculated in the previous step, calculate the current flowing in the circuit. Calculated current A 110 Festo Didactic 89688-00
21. Using the current you calculated in the previous step and knowing the resistance values of the resistors, calculate the voltage across each resistor in the circuit. Calculated resistor voltage V Calculated resistor voltage V Calculated resistor voltage V 22. Using the circuit parameters you know and those you calculated, can you confirm through calculations that Kirchhoff s voltage law is respected? Solving the circuit through circuit measurements 23. Set up the circuit shown in Figure 91. Use the 50 resistors to implement resistors and, and the 250 resistor to implement resistor. This circuit allows measurement of the equivalent resistance of the three series resistors. 50 50 250 Figure 91. Measuring the equivalent resistance in the circuit of Figure 90. 24. Using an ohmmeter, measure the equivalent resistance in the circuit. Record the value below. Measured equivalent resistance Festo Didactic 89688-00 111
25. Is the equivalent resistance you measured in the previous step virtually equal to the value you calculated in step 19? 26. Connect the circuit shown in Figure 92. For the moment, do not connect any of the voltmeters to the circuit. 50 50 250 Figure 92. Measuring the voltages and current in the circuit of Figure 90. 27. Turn the power source on. Measure the current flowing in the circuit. Record the value below. Measured current A 28. Is the current flowing in the circuit you measured in the previous step virtually equal to the value you calculated in step 20? 112 Festo Didactic 89688-00
29. Measure the source voltage and the voltage across each resistor in the circuit by successively connecting a voltmeter at the locations shown in Figure 92. Record the voltage values below. Measured source voltage V Measured resistor voltage V Measured resistor voltage V Measured resistor voltage V 30. Are the voltages you measured across each resistor in the previous step virtually equal to the values you calculated in step 21? 31. Using the circuit parameters you know and those you measured in this subsection, can you confirm through calculations that Kirchhoff s voltage law is respected? 32. Turn the power source off. Festo Didactic 89688-00 113
Voltage divider consisting of two resistors In this section, you will connect a voltage divider consisting of two resistors. You will calculate the equivalent resistance of the voltage divider, the current flowing in the circuit, and the voltage across each resistor. Using these calculated parameters, you will verify that Ohm s law is respected. You will then repeat the above steps using actual circuit measurements, and compare the measured values to the calculated values. 33. Consider the circuit shown in Figure 93. 500 250 Voltage divider output Figure 93. Voltage divider consisting of two resistors. Solving the voltage divider through mathematical calculations 34. Calculate the equivalent resistance of the voltage divider, the current flowing in the circuit, and the voltages and across each resistor. Record the values below. Calculated equivalent resistance Calculated current A Calculated voltage V Calculated voltage V 35. Using the parameters of the voltage divider you calculated in step 34, can you confirm through calculations that Kirchhoff s voltage law is respected? 114 Festo Didactic 89688-00
Solving the voltage divider through circuit measurements 36. Set up the circuit shown in Figure 94. 500 250 Voltage divider output Figure 94. Measuring the equivalent resistance in the circuit of Figure 93. 37. Using an ohmmeter, measure the equivalent resistance of the voltage divider. Record the value below. Measured equivalent resistance 38. Is the equivalent resistance you measured in the previous step virtually equal to the value you calculated in step 34? 39. Set up the circuit shown in Figure 95. For the moment, do not connect any of the voltmeters to the circuit. 50 250 Voltage divider output Figure 95. Measuring the voltages and current in the circuit of Figure 93. Festo Didactic 89688-00 115
40. Turn the power source on. Measure the current flowing in the circuit, as well as the voltages,, and across each resistor. Do so by successively connecting the voltmeter at the locations shown in Figure 95. Record the values below. b Make sure to turn the power source off before making any change to the circuit connections. Also make sure to turn it back on before taking any measurements. Measured current A Measured voltage V Measured voltage V Measured voltage V 41. Are the parameters of the voltage divider you measured in the previous step virtually equal to those you calculated in step 34? 42. Using the parameters of the voltage divider you measured in this subsection, can you confirm through calculations that Kirchhoff s voltage law is respected? 43. Without modifying the source voltage, what should be done to obtain a particular voltage value across resistor? 44. Turn the power source off. 116 Festo Didactic 89688-00
Light intensity control circuit implemented using a resistor and a selector switch In this section, you will connect a circuit that represents a light intensity control implemented using a selector switch and a resistor. The light intensity control has two settings: low and high. You will observe what happens to the light for each setting. You will also make voltage measurements in the circuit and confirm that Kirchhoff s voltage law is respected. 45. Set up the circuit shown in Figure 96. Use the 250 resistor to implement resistor. This circuit represents a light intensity control circuit with two possible intensity settings (low and high), controlled using a toggle switch. Selector switch 250 Figure 96. Light intensity control circuit using a resistor and a light with two intensity settings controlled using a toggle switch. 46. Set the toggle switch to position B. Turn the power source on. What happens to the indicator light as you turned the power source on while the toggle switch is set to position B? Explain why and indicate to which light intensity this setting of the toggle switch corresponds (low or high intensity). Light intensity Festo Didactic 89688-00 117
47. Using a voltmeter, measure the source voltage, the voltage across the resistor, and the voltage across the indicator light. Record the values below. Source voltage V Resistor voltage V Indicator light voltage V 48. Using the voltages you measured in the previous step, can you confirm through calculations that Kirchhoff s voltage law is respected when the toggle switch is set to position B? 49. Set the toggle switch to position A. 50. What happens to the indicator light as you set the toggle switch to position A? Explain why and indicate to which setting of the indicator light this setting of the toggle switch corresponds (low or high intensity). Light intensity setting 51. Using a voltmeter, measure the voltage across the indicator light. Record the value below. Indicator light voltage V 118 Festo Didactic 89688-00
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Conclusion 52. Using the voltage you measured in the previous step, can you confirm through calculations that Kirchhoff s voltage law is respected when the toggle switch is set to position A? 53. Based on you observations, can you conclude that the circuit in Figure 96 effectively represents a light intensity control circuit with two possible intensity settings (high and low)? 54. Turn off the power source and the measuring instruments. Disconnect your circuit. Return the leads to their storage location. CONCLUSION In this exercise, you learned how to calculate the equivalent resistance of multiple resistors in series circuits. You were introduced to Kirchhoff s voltage law and learned how to apply it to electrical circuits. You also learned what a voltage divider is, and how to implement one using resistors. Festo Didactic 89688-00 119
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Review Questions REVIEW QUESTIONS 1. Consider the circuit in Figure 97. Calculate the current flowing in the circuit from the indicated parameters. 25 50 V 70 30 Figure 97. Circuit for review question 1. 2. State the basic principle of Kirchhoff s voltage law. 3. Name three possible ways to implement a voltage divider. 120 Festo Didactic 89688-00
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Review Questions 4. Consider the circuit in Figure 98. Calculate the output voltage of the voltage divider, knowing that the source voltage is equal to 80 V and that the resistances of the resistors and are equal to 20 and 40, respectively. 20 80 V 40 Voltage divider output Figure 98. Circuit for review question 4. Festo Didactic 89688-00 121
Exercise 6 Solving Series Circuits and Kirchhoff s Voltage Law Review Questions 5. Consider the circuit in Figure 99. Calculate the range of the voltage divider output voltage, knowing that the source voltage is equal to, that the rheostat has a resistance range of 10 to 50, and that the resistor has a resistance of 30. 10 to 50 30 Voltage divider output Figure 99. Circuit for review question 5. 122 Festo Didactic 89688-00