Phys 2212L LAB 4 Direct Current Circuits Purpose In this laboratory, we will set up the three basic types of electric circuits: a series, a parallel and a combination circuit. We will use Ohm s Law and Kirchhoff s rules to determine the currents through the circuit elements. For the series and parallel circuits we will determine the equivalent resistance of each circuit. Principles A direct current (DC) circuit is an electric circuit in which the input voltage is constant. Since the voltage is constant, so is the current. This is in contrast to alternating current circuits, in which the voltage and current vary with time. The current and voltage in a DC circuit can be analyzed using Ohm s Law and Kirchhoff s Rules: Ohm s Law: V = IR This gives the voltage drop across a resistor or a combination of resistors in a circuit. Kirchhoff s Rules: 1. Loop rule: The sum of the voltage gains or drops around any current loop equals zero. This is a consequence of conservation of energy. 2. Junction rule: The sum of the currents entering a circuit junction equals the sum of the currents leaving it. This is a consequence of conservation of charge. Series Circuits A series circuit is one in which the resistors or other circuit components are connected in series (see Diagram 1). Since there is only one path for the current to take, the current must be the same in all parts of the circuit. Jan 05 51
Phys 2212L LAB 4 Direct Current Circuits R 1 R 2 Diagram 1: Resistors in series. Ohm s Law tells us the voltage drop across each resistor: V = IR. For different resistors the voltage drops will differ, but the sum of all voltage drops around the circuit must equal the source voltage (the loop rule). Consequently, any series of resistors are equivalent to a single resistor R eq given by: (1) = R + R... R eq 1 2 + We will measure the voltages across two resistors in series and the current through them and confirm these observations. We will also replace the two resistors with a single resistor of value R eq and measure the resulting current. Parallel Circuits Diagram 2 illustrates resistors connected in parallel. In this case, the current has two paths through the circuit, so the current will not be the same through each resistor. However, the voltage drop across the two resistors must be the same, since both resistors connect to points at the same potential. Ohm s Law determines the current through each resistor: I = V/R, where V refers to the voltage across the resistor and R is its resistance. Since the voltage is the same across both resistors but their resistances differ, the current through these two branches of the circuit must differ. However, by the junction rule, the sum of these two currents must equal the current in the unbranched part of the circuit. Jan 05 52
Phys 2212L LAB 4 Direct Current Circuits R 1 R 2 Diagram 2: Resistors in parallel The parallel arrangement of resistors is equivalent to a single resistance given by: (2) 1 1 1 = + R eq R R 1 2 We will set up this circuit and measure the currents and voltages in each branch. We will also replace the resistor combination with R eq and measure the resulting current. Combination Circuit Diagram 3 illustrates a circuit with resistors both in series and in parallel. This circuit has two voltage sources and three currents. Directions for the currents have been arbitrarily indicated; the actual currents may not have these directions. We will use Ohm s Law and measured values for the voltages across the resistors to calculate the current through each resistor. We will compare the calculated with the measured values. As a post-lab assignment, we can predict the current in each branch of the circuit using Kirchhoff s rules: Summing the voltage gains and drops around each of the two current loops and setting the sum equal to zero leads to two equations in the three unknowns I 1, I 2, and I 3 the currents through resistors R 1, R 2 and R 3, respectively. (The knowns are the measured values of the voltage sources, V a and V b, and the three resistances. The junction rule gives us a third equation in I 1, I 2 and I 3. Jan 05 53
Phys 2212L LAB 4 Direct Current Circuits By solving these three equations simultaneously we can determine the currents in terms of the known parameters. We can then compare these with the measured values. R 1 I 1 R 2 I 2 V a V b R 3 I 3 Diagram 3: Combination circuit with three resistors and two voltage sources. Directions for the currents have been arbitrarily assigned; the actual currents may not have these directions. Jan 05 54
Procedures Before setting up the circuits, we will measure the resistances of the resistors to be used and compare the measured values to their color-coded values (Part 1). For the series and parallel circuits (Parts 2 and 3) we will measure the voltage drops across and currents through each resistor and compare the results to V=IR. We will also replace the resistors by a single, equivalent resistance and remeasure the current in the circuit. For the combination circuit (Part 4), we will measure the voltage and current for each resistor and compare the results to V=IR. In the analysis, we will derive an equation for each current in the circuit using the loop and junction rules and compare the calculated currents to their measured values. Equipment Low voltage power supply (2) Digital multimeter (DMM) Resistors (3 ) Decade resistance box Banana wires Alligator clips or other connectors Voltage probes The following is a suggested set of voltage and resistor values to be used in this experiment. : R 1 560 Ω V 0 5.00 V R 2 330 Ω V a 9.00 V R 3 270 Ω V b 5.00 V 1. Resistors Read the resistance of each resistor using the color bands on the resistor. Refer to the section on resistor color codes in the introduction. Record these nominal values for R 1, R 2, and R 3. Also read and record the tolerance for each resistor. Measure the three resistors using the DMM and record these measured values. 55
Procedures Determine the percent difference from the nominal value for each resistor. 56
Procedures 2. Series Circuit R 1 R 2 A 1 A 2 V 0 V 1 V 2 A 0 Diagram 4: Series circuit, with locations for current (ammeter) and voltage (voltmeter) measurements indicated. Connect R 1 and R 2 in series with the DC power supply, as indicated in Diagram 4. Don t insert the meters in the circuit until you are ready to take measurements. The meter symbols in the diagram indicate where in the circuit the measurements are to be taken. Set the current knob on the power supply to one half turn. Set its output voltage to 5.00 VDC. Use the DMM to measure the power supply s output. Call this voltage V 0. Measure V 1 and V 2, the voltages across R 1 and R 2 and record these values. Measure the current in the circuit at A 0, A 1 and A 2, as indicated in Diagram 4. Record these measured values for I 0, I 1, and I 2 in the data table. Remove R 1 and R 2 from the circuit but leave them connected to each other. With the DMM, measure the resistance of the R 1 + R 2 combination and record this value. Record V 0 as the voltage across the combination of R 1 and R 2. Since the wire leads have essentially zero resistance, the voltage across the combination will be the same as the power supply output. Measure this directly to confirm if you like. Calculate R eq using equation (1) and record this. Replace the two resistors with the decade resistance box set to R eq. Adjust the power supply output, if necessary, to V 0. Measure I eq, the current through R eq and record this value. Analysis: Using Ohm s Law, calculate the currents through R 1, R 2, the series combination R 1 + R 2, and the resistance box, R eq. 57
Procedures Calculate the percent difference between the calculated and measured values. Subtract: V 0 -V 1 - V 2. What does this confirm? How do I 0, I 1 and I 2 relate to one another? What does this mean about the current in the circuit? Take the percent difference between I eq and I 0. 3. Parallel Circuit R 1 A 1 A 2 V 0 R 2 A 0 Diagram 5: Parallel circuit, with ammeters inserted for measuring currents in each of the branches. Connect R 1 and R 2 in parallel across power supply, as indicated in Diagram 5. Again, don t insert the meters until you are ready to take measurements. Set the current knob on the power supply to one half turn. Set the power supply output voltage to 5.00 VDC. Use the DMM to set the output and call this V 0. Measure V 1 and V 2, the voltages across R 1 and R 2. Record these values. Measure and record the current in the circuit at A 0, A 1 and A 2, as indicated in Diagram 5. Remove R 1 and R 2 from the circuit but leave them connected in parallel. Measure and record the resistance of the parallel combination. Record V 0 as the voltage across the combination of R 1 and R 2, since the combination connects directly to the power supply at V 0. Calculate R eq for the parallel combination using equation (2). Replace the two resistors with the decade resistance box set to R eq. Adjust the power supply output, if necessary, to V 0. Measure I eq, the current through R eq and record this value. 58
Procedures Analysis Calculate I 1 and I 2 using Ohm s Law and your measured values for the voltages across and the resistances of the two resistors. Calculate I 0 using Ohm s Law and your measured values of V 0 and R 1 + R 2 in parallel. Calculate I eq using Ohm s Law and V 0 and R eq. Take the percent difference between your measured and calculated values for the currents. Compare your measured values of V 1 and V 2 to V 0. Subtract I 0 - I 1 - I 2. What does the result show? Compare I eq to I 0 and take the percent difference. 4. Combination Circuit Connect R 1, R 2 and R 3 with two power supplies as in Diagram 6. Set the current knob on the power supplies to one half turn. Set V a to 9.00 VDC and V b to 5.00 VDC, as indicated in the diagram. Measure and record V 1, V 2, and V 3. Measure the currents through each resistor (I 1, I 2 and I 3, as indicated in Diagram 6). Record these values, including the sign. Analysis Calculate I 1, I 2, and I 3 from Ohm s Law, using your measured values of the voltages and resistors. Take the percent difference between the calculated and measured values of the three currents in the circuit. Add I 2 and I 3. Compare with I 1 : take the percent difference. 59
Procedures R 1 I 1 R 2 I 2 V a 9V R 3 I 3 V b 5V Diagram 6: Combination circuit with three resistors and two voltage sources. Directions for the currents have been arbitrarily assigned. Write down an equation for each loop in the combination circuit, using the loop rule and Ohm s Law. Write down an equation for the currents using the junction rule. Solve these circuit equations algebraically and derive expressions for I 1, I 2, and I 3 in terms of the resistances and power supply voltages. Calculate the currents using these expressions and the given voltage and resistance values. Compare these predictions with your measured values for the current. 60
Data 1. Resistors Element Resistance: Nominal value Tolerance Resistance: Measured value Percent difference R 1 R 2 R 3 Series Circuit V 0 : Element Resistance V Current Measured Calculated Resistor 1 R 1 V 1 I 1 I 1 % Difference Resistor 2 R 2 V 2 I 2 I 2 Series combination R 1 + R 2 V 0 I 0 I 0 Resistance Box R eq V 0 I eq I eq V 0 V 1 V 2 = This confirms: 61
Data I 1 I 2 = I 1 I 0 = This means: I 0 = I eq = % difference Parallel Circuit V 0 : Current Element Measured Resistance V Measured Calculated Resistor 1 R 1 V 1 I 1 I 1 % Difference Resistor 2 R 2 V 2 I 2 I 2 Parallel combination R 1 + R 2 V 0 I 0 I 0 Resistance Box R eq V 0 I eq I eq V 1 V 0 = V 2 V 0 = This confirms: I 0 - I 1 - I 2 = This confirms: I 0 = I eq = % difference 62
Data 3. Combination Circuit V a V b Current Element Measured Resistance V Measured Calculated % difference (from V=IR) Resistor 1 R 1 V 1 I 1 I 1 Resistor 2 R 2 V 2 I 2 I 2 Resistor 3 R 3 V 3 I 3 I 3 I 2 + I 3 = I 2 + I 3 I 1 = Current Equations & Predicted Currents (Solve for each current and evaluate the resulting expressions.) % error I 1 = I 2 = I 3 = 63
Data 64