DC Circuits and Kirchhoff?s Laws Theory: Gustav Kirchholf formulated rules to make it possible to find the current in each part of a direct current (DC) circuit, no mater how complicated, if we are given the emf?s (voltages) of the sources of potential diference and the resistances of the various circuit elements. These rules apply to junctions where thre or more wires come together and loops that are closed conduction paths that are part of the circuit. Circuit 1 shows a simple example of such a circuit. Circuit 1: Simple DC circuit Kirchhoff?s first rule follows from the conservation of electrical charge and states that the sum of the curents flowing into a junction is equal to the sum of currents flowing out of a junction. For example, in Circuit 1 this gives I 1 = I 2 + I 3 at either junction . The second rule is a consequence of conservation of energy. The work done on the charge around a closed path mut be the same as the work done by the charge on the same path. This gives the rule that the sum of the potential diferences across al elements around any closed loop must be zero. For Circuit 1 this gives for the left loop (moving around the loop in a counterclockwise fashion): ! " 1 #I 1 R 1 #I 2 R 2 =0 For the right loop (moving around the loop in a counterclockwise fashion): ! " 2 +I 2 R 2 #I 3 R 3 =0 The thre equations can be solved for the currents, I 1 , I 2 , and I 3 using either basic substitution or matrix algebra. Aparatus: The equipment wil be the same as the previous two experiments, namely, power supplies, resistors, a breadboard, and multimeters. For al uncertainty values for the multimeter measurements please refer to the lab: Resistors and Simple Circuits. Experiment: In the previous lab you examined a circuit containing thre resistors, two in paralel and one in series with the paralel resistors. You learned from your previous results that the current values for the circuit can be determined from Ohm?s Law but we can now determine if Kirchhof?s Rules also can be applied to find the same results. If so, one may choose how to solve a circuit if sufficient information is available. 1. In this experiment you wil use various resistors. As always the values of the resistors should be measured before the circuits are wired. Please collect the resistor values shown in Table 1 and measure their resistances. Be sure to mark in some way the resistor values as the measured values wil likely be diferent. Nominal Resistor Values (k?) Measured Resistor Value (k?) 0.100 ± 0.100 ± 0.220 ± 0.220 1.0 ± 2.2 ± 2. Now, asemble the resistors and power supply as in the Circuit 2 diagram. Notice this circuit is similar to the paralel and series circuit constructed in the last lab. Set the power supply voltage V1 to 10 Volts. Circuit 2: Kirchhoff?s Laws circuit 1: R 1 =R 4 = 100 ?, R 5 = 1k?, and R 2 =R 5 = 220 ? 3. Using the multimeter as a voltmeter, measure the voltage across each resistor and the power supply making sure to include measurement uncertainties for each value. 4. Using the multimeter as an ameter, measure the current in each leg of the circuit: main leg, R 3 /R 4 and R 5 . Make sure to include asociated measurement uncertainties. 5. Now using Kirchhoff?s current rule and loop rule, write out thre equations that represent Circuit 2. Make sure to use the exact values of the resistances and the power supply and that you show how you defined the direction for each current in the circuit and the loops used. 6. Using the thre equations from #5, solve for the curents in each leg of the circuit using substitution or matrix algebra. If using substitution, make sure to include your handwriten work in your lab report. If using matrix algebra, make sure to show the setup of the matrix and vector and state if you used your calculator or Matlab, etc. to solve. You may ignore uncertainty propagation in this step. Be sure to include discussion of the measured currents obtained in step #4 and the calculated values found in step #6. 7. Circuit 2 only has one power supply, what would you anticipate to happen to the flow of current if a power supply was added as in Circuit 3? Discuss this with your lab partners and include your answer in your lab report. 8. Now, asemble the circuit 3. 9. Measure the voltages and the currents for the circuit, what has changed because of the introduction of the second power supply? Does this follow from your reasoning in #7. If not, what might you have forgotten to take into acount when answering the question in #7? 10. Finaly, write Kirchhof equations for the circuit 3. Would it mater if you defined the current in the wrong direction in any leg of the circuit than what is found in #8? Try this and demonstrate that it doesn?t mater, that the solution simply provides the negative of the current in question. Be sure to include discussion of the measured currents obtained in step #9 and the calculated values found in step #10. Circuit 3: Power Supply 1 = 10 Volts; Power Supply 2 = 5 Volts mary ewell Microsoft Word - DC Circuits and Kirchhoff?s Laws.doc
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