Experiment #5, Series and Parallel Circuits, Kirchhoff s Laws


 Bryan Jacobs
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1 Physics 182 Summer 2013 Experiment #5 1 Experiment #5, Series and Parallel Circuits, Kirchhoff s Laws 1 Purpose Our purpose is to explore and validate Kirchhoff s laws as a way to better understanding simple DC circuits 2 Introduction As we learned from the Ohm s law experiment, an electrical circuit is any continuous path or array of paths along which current may flow. A circuit usually contains a battery or other sources of EMF (electromotive force) to create the current. Without a source of energy to drive the circuit no current (electric charge) will flow. Between the terminals of our power source can be any combination of elements through which the electrons may pass; anything from a single wire to a complicated collection of wires, diodes, transistors, and other circuit elements as resistors and capacitors. Whatever the elements that make up the circuit there are some simple rules that must be obeyed. Two of these rules are Kirchhoff s laws regarding current (flow of electric charge) and voltage (electrical potential difference). Since, as far as we know, charge can be neither created nor destroyed, if we pick a single point in a circuit all of the charge that flows into that point must also flow out of it in a steady state, or unchanging situation. Put in terms of currents Kirchhoff s law derived from the conservation of charge states that at a given node (point in the circuit) the sum of the currents flowing into and out of that node will be zero. This is shown schematically in figure 1 where the sum of the three current flowing into and out of the node, signified by the black dot, adds to zero: I 1 I 2 I 3 0. Note in this case two of the currents are negative, that is to say, they flow out of the node, where the node is the low potential point. The negative () black node dot is the low connection reference point and the three positive (+) dots are the high connection reference points for the voltage measurements in part of this lab. It should be noted that this node is our arbitrary reference point for making measurements. Figure 1. Kirchhoff s law for voltages in a circuit is a consequence of the fact that the electric field exerts a force which is conservative. Thought of in terms of potential energies, the potential energy of a charge at a particular point in space does not depend on the path the charge took to get to that point. Since the potential energy of a charge in a circuit is its charge, Q, times the voltage at the point in the circuit, V, the sum of the voltages measured around any loop in a circuit must add to zero. Otherwise, if the charge went
2 Physics 182 Summer 2013 Experiment #5 2 around that loop its energy would be changed when it returned to its starting point. This is shown schematically in Figure 2. Here the sum of the voltages will be zero, V ab + V bc + V cd + V da = 0. Assuming that the bottom left hand node a is at the higher potential V 1, V 2 and V 3 would be positive and V da would be negative as the voltages are measured counterclockwise. In this part of the lab, V emf = 10.00V will be the power supply that is driving the circuit. This yields: V 1 + V 2 + V V = 0, or V 1 + V 2 + V 3 = 10.00V = V emf IR 1 + IR 2 + IR 3 = 10.00V = V emf (Current I is a constant.) Figure 2 Kirchhoff s laws are important because, if we take all the loops and all the nodes in a complicated circuit and apply Kirchhoff s laws to them then we end up with a series of algebraic equations which can be solved to uniquely determine all of the voltages and currents in our problem. 3 Experimental Apparatus and Procedure 3.1 Apparatus 1. DC power supply. 2. Agilent 34405A digital voltohmmeter (to be used for voltage and resistance measurements). 3. Fluke 75 digital multimeter (to be used for all current measurements). 4. Carbon resistors and connecting cables. If you are curious, read the manuals concerning the operation of the power supply and the digital meters or read the short description in the appendix. Direct Resistance Measurements with a Digital Ohmmeter Resistance in Parallel and Series. Measurements go in the report data section. 1. Measure and record in lab report data section the resistance of the carbon resistors R 1, R 2, R 3 directly using the digital voltmeter (Agilent 34405A) set to measure resistance in ohms (Ohms function switch setting). The correct range will determine the amount of digits after the decimal point. 2. Connect the resistors in parallel and series and measure and record the equivalent parallel and series
3 Physics 182 Summer 2013 Experiment #5 3 resistance of these networks. The correct range will determine the amount of digits after the decimal point. Measurements go into the data section of the lab report. Compare your measured values with the theoretical values calculated from the following formulas: a. Equivalent parallel resistance: 1/R p = 1/R 1 + 1/R 2 + 1/R 3 b. Equivalent series resistance: R s = R 1 + R 2 + R 3 Calculations of R p and R s go into the calculations and analysis section of the lab report. Show both formulas and your calculations Kirchhoff's Voltage Law Assemble the series circuit using three carbon resistors as shown in the figure to the right. Connect the power supply across this network with the milliammeter (Fluke 75) in series. The milliammeter (current meter) measures the current (1 ma = A, where A = ampere). The current is a constant in this series circuit. Set the power supply to V emf = V. The correct range on the voltmeter will determine the amount of digits after the decimal point. Record the current I exp in the circuit and measure the voltage drop V across each resistor using the voltmeter (Agilent 34405A). Results go into the data section of the report Kirchhoff's Current Law. In this section we will be studying Kirchhoff s current law. The circuit we will be examining is shown to the right. The node we are going to explore is the point where the three resistors connect. Because we need to insert the current meter in series with each leg of the circuit of interest there will be a series of measurements with the ammeter in different place in the circuit. All measurement will be made with the same voltage, 5.00 volts, applied to the circuit. First, record the voltage drop across each of the three resistors. All voltage measurements should be made with the node as the low side of the potential. As R 2 and R 3 form a parallel connection, V 2 and V 3 should be the same negative voltage. V 1 is a positive voltage. Next, insert the ammeter between the end of the first resistor and the low side which is the node. To do so, cables will have to be removed when the ammeter is connected. Record the current running through the first resistor. Next, insert the ammeter between the second resistor and the node. Record the current running through the second resistor. Finally,
4 Physics 182 Summer 2013 Experiment #5 4 insert the ammeter between the third resistor and the node. Record the current running through the third resistor. Both I 2 and I 3 should have a negative reading. I 1 is a positive current. Results go into the data section of the report. If I 2 and I 3 were measured relative to the direction of the current flow, these two values would have been positive. They are negative as we used the node as common reference point. Same applies to V 2 and V 3. 4 Calculations and Analysis for Lab Report. Show all formulas and calculations. 4.1 Analysis of Resistance in Parallel and Series 1. Calculate R p. Use percent error to compare the experimentally measured and theoretically calculated value of R p. When doing the % error, substitute the theoretical value for the accepted value. 2. Calculate R s. Use percent error to compare the experimentally measured and theoretically calculated value of R s. 4.2 Analysis of a simple series circuit and Kirchhoff s Voltage Law 1. Use percent error to compare the experimentally measured value of the current I in the circuit with the theoretical value calculated using the equation I = V/R s where R s is the equivalent series resistance of the resistors as directly measured experimentally (not calculated). 2. Use percent error to compare each experimentally measured value of the voltage drop across each resistor with the theoretical value calculated using the formula V = IR (use the experimentally measured value for I and the measured resistance of each resistor for V = IR). 3. Verify Kirchhoff's voltage law, that both the experimental and theoretical algebraic sum of the voltage drops around the circuit in a complete loop is zero. Note that the power supply is a source of EMF so that the current flows through the power supply from negative to positive terminal. Thus, denoting the power supply EMF in volts by V emf, Kirchhoff's voltage law can be written as: Experimental Value: V emf,exp = V 1,exp + V 2,exp + V 3,exp Theoretical Value: V emf,th = I exp R 1 + I exp R 2 + I exp R 3 Where the V's refer to corresponding measured voltage drops across each of the three resistors in the circuit as indicated in Figure 2. Use R measured for the theoretical value. Calculate % error. 4.3 Analysis of a simple circuit and the Kirchhoff Current Law. 1. Use percent error to compare the experimentally measured value of the current, I exp, in the circuit with the theoretical value calculated using the equation I 1,th = V 1 /R 1 wherev 1 is the measured voltage drop and R 1 is the measured resistance of the first resistor. Do the same for the second and third resistor. 2. Verify Kirchhoff s current law, that the algebraic sum of currents in a node is zero. Note that one current flows into the node (positive) and two flow out (negative). Show that: I 1,exp + I 2,exp + I 3,exp = 0 and I 1,th + I 2,th + I 3,th = 0 5 Questions 1. Did you verify the Kirchhoff s voltage law for the simple series circuit which you constructed? Explain how this was shown. If the verification was not exact, how large was the deviation? 2. Did you verify the Kirchhoff s current law for the circuit which you constructed? Explain how this was shown. If the verification was not exact, how large was the deviation?
5 Physics 182 Summer 2013 Experiment # In the circuit used to verify Kirchhoff s current law, R 2 and R 3 are connected in parallel. a. Calculate the equivalent parallel resistance R p using the measured values of R 2 and R 3. b. Your value of R p, calculated above in part 3.a, is in series with R 1. Calculate R s using the measured value of R 1 and your calculated value of R p in part 3.a. c. Using the value of R s from part 3.b above, and 5.00 volts, calculate the current. Call this current I cal. d. Use percent error to show the difference between the calculated current in 3.c and the positive sum of I 2,exp and I 3,exp. (For this calculation, simply change both negative values to positive values. Nothing complicated.) Example. If I 2,exp = ma and I 3,exp = ma, than their sum for this question is I 2+3,exp = 1.38 ma. Simple. Took the absolute value. Use the 3.c value as the experimental current value when doing percent error. e. Why should the positive sum of I 2,exp and I 3,exp equal the current in 3.c? 6 Conclusion This section should have a clear statement of the results of the experiment and the extent to which the results are in agreement with the theory being tested. Include percent error for this comparison. To make this comparison meaningful, you should include the impact of the experimental error on your results. In addition, please include a statement of what you have learned, a critique of the experiment, and any suggestions you have which you think could improve the experiment or the lab handout.
6 Physics 182 Summer 2013 Experiment #5 6 Instrumentation 1. The DC Voltmeter measures the voltage difference between two points to which its terminals are connected. The voltmeter is always connected in parallel to the part of the circuit across which the potential difference is to be measured (Figure A1). In this experiment the Agilent 34405A multimeter (multimenter) is used to measure voltage differences and the resistance of the carbon resistor. 2. The DC Milliammeter measures the electric current between any two points to which its terminals are connected. To measure the current in any part of the circuit, the circuit must be broken at that point and the ammeter must be inserted in the gap with loose ends connected to its terminals, i.e., the meter is connected in series with that part of the circuit where the current is to be measured (Figure A2). The Fluke 75 battery powered multimeter is used to measure the current in milliamperes in this experiment. 3. Connecting wire leads. Assumed to be made of a good conducting material, e.g., a metal such as copper, and of a large enough crosssection to have negligible resistance and voltage drop across them. 4. The DC Power supply is a device used to supply a constant source of EMF between its output terminals. The electric current flows internally in the power supply from the minus to the plus terminal. In the external circuit it flows from the plus to the minus terminal (Figure A3). The voltmeter on the actual power supply is not accurate and should not be used to determine the output voltage. The Agilent 34405A multimeter is used to measure voltages on the power supply.
7 Physics 182 Summer 2013 Experiment #5 7 Data Sheet. All measurements go into the data section of the lab report. 1 Individual Resistors. The correct range will determine the amount of digits after the decimal point. Measured using Agilent 34405A Measured value R 1 = Put in correct units for all measurements. Measured value R 2 = Measured value R 3 = 2 Resistor Combinations. The correct range will determine the amount of digits after the decimal point. All calculations and % errors go into the calculation section of the lab report. Parallel Combination. Experimental: R p,exp = Theoretical: R p,th = [1/R 1 + 1/R 2 + 1/R 3 ] 1 = (use measured values of R) % error = Series Combination Experimental: R s,exp = Theoretical: R s,th = R 1 + R 2 + R 3 = (use measured values of R) % error = 3 Study of Simple Circuits: Kirchhoff's Voltage Law A. Applied Voltage V emf = (10 volts) (Enter as a positive value.) B. Current I Measured using the Fluke 75. Experimental: I exp = (1 ma = 1 milliampheres = A = 1x103 A) Theoretical: I th = V emf /R s,exp = (Use experimentally measured R s,exp ) % error = C. Voltage drops across the individual resistor. All values should be positive. The correct range will determine the amount of digits after the decimal point. Measured (Agilent 34405A) Theoretical V = IR (Use Experimental I exp & R Measured) V 1,exp = V 1,th = % error = V 2,exp = V 2,th = % error = V 3,exp = V 3,th = % error =
8 8 D. Verification of Kirchhoff's Law. Sum should equal V emf = 10 Volts V emf,exp = V 1,exp + V 2,exp + V 3,exp V emf,th = V 1,th + V 2,th + V 3,th Using experimental values from part C: V emf,exp = Using theoretical values from part C: V emf,th = % error = 4 Study of Simple Circuits: Kirchhoff's Current Law A. Applied Voltage = (5 volts) B. Individual measured experimental voltages and currents. (Results will be both positive and negative.) V 1 = I 1,exp = V 2 = I 2,exp = V 3 = I 3,exp = C. Predicted theoretical currents (use voltages just measured in circuit and resistances that were measured) and the % error with the experimentally measured currents. V 1 /R 1 = I 1,th = % error = V 2 /R 2 = I 2,th = % error = V 3 /R 3 = I 3,th = % error = D. Verification of Current Law. Sum should equal zero. Using Measured values; I 1,exp + I 2,exp + I 3,exp = Using Predicted Values I 1,th + I 2,th + I 3,th = For lab report, all measured values must go into the data section. All calculations, including % errors, goes into the calculations and analysis section of the lab report. Keep these separate. Keep things neat and well organized.
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