PHYS112 : Labs
Subsections


6 DC Circuits

Introduction

Kirchhoff's Rules

In Lab 5, you explored the behavior of circuits with resistors connected in series and in parallel. You were able to analyze the behavior of these circuits using equations for equivalent resistance. This approach does not generally work for circuits with more than one power source. While equivalent resistance still has some application, in a multi-loop, multi-source circuit like the one you will be investigating, you will need a more general approach to circuit analysis - Kirchhoff's rules.

The first rule, the loop rule states that the total change in potential around a closed loop must be zero. This means that the sum of the potential differences across all of the devices in a closed loop must be zero,

\begin{displaymath} \sum_\mathrm{loop} \Delta V = 0 \end{displaymath} (14)

The second rule, known as the junction rule, applies to currents. This rule states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction,
\begin{displaymath} \sum i_\mathrm{in} = \sum i_\mathrm{out} \end{displaymath} (15)

These rules became evident in your measured potential differences and currents in Lab 5. Here, you will use Kirchhoff's rules to predict the behavior of a more complex circuit.

The Breadboard

Figure 11: A scanned image of a breadboard with rectangles showing conducting connections between the sockets.
\scalebox{0.5}{ \includegraphics{DC1-breadboard.eps} }

A useful tool for constructing temporary circuits for measurement or testing is known as a breadboard. A scan of the small breadboard you will be using is shown in Figure 11. A breadboard allows for easy assembly and modification of a circuit by sliding components and wires into the various sockets. These sockets are linked in an easily recognizable pattern that allows for the components and wires to be connected in a circuit. This pattern is indicated by the rectangles in Figure 11. The horizontal rows at the top and bottom are each connected in one chain. These are often used for power and ground, respectively, but we will not be using them here. The vertical columns are connected in groups of five as shown. These are not connected with the rows on the other side of the central trench.

Experiments

Determining the Internal Resistances of a Battery

Figure 12: Circuit for determining the internal resistance $r$ of a battery.
\includegraphics{DC1-internalr.eps}

You have been supplied with two D-cell batteries in holders, a small ($< 50 \Omega$) resistor, a digital multimeter (DMM), alligator-clip leads for connecting the batteries to the resistors, and a breadboard for connecting the resistors to each other.

  1. Record the resistance of the small ($< 50 \Omega$) resistor indicated by its color code, and measure and record its value with the DMM. (They should be consistent within the tolerance indicated by the color code on the resistor.)

  2. Choose a battery to test. While the battery is not connected to anything else, turn the knob on the DMM to a suitable voltage scale, and measure and record the voltage across the battery to determine the $\mathcal{E}$.

  3. Use the battery and the small ($< 50 \Omega$) resistor to set up the circuit shown in Figure 12.

  4. Turn the knob on the DMM to a suitable voltage scale, and record the voltage across the resistor.

  5. Connect your DMM in series with the resistor. Remember never to connect an ammeter in parallel with anything! Turn the knob on the DMM to a suitable current scale, and record the current $i$ flowing through the circuit.

  6. Using Kirchhoff's loop rule, Ohm's law, and your measurements, determine the value of the internal resistance $r$ of the battery.

Investigating Contact Resistance

Use the DMM to measure the of various series combinations of your alligator-clip leads and your small ($< 50 \Omega$) resistor. Any resistance you measure above the known resistance of your resistor is due to contact resistance. Investigate this phenomenon carefully. Determine its approximate magnitude and whether or not it is reproducible.

A Multi-loop Circuit with Two Power Sources

Figure 13: A multi-loop DC circuit with two power sources and three resistors.

\includegraphics{DC1-circuit-diagram.eps}

\scalebox{0.5}{ \includegraphics{DC1-circuit-image.eps} }

  1. You will be constructing the circuit shown in Figure 13. Based on your work thus far, choose three different resistors, that are large enough that the internal resistances of the batteries and the contact resistances of the alligator-clip leads will not make significant impacts (more than 1%) on the currents and potential differences in your circuit.

  2. Measure and record the resistance of each resistor with the DMM, and check your results with the color codes.

  3. Measure and record the $\mathcal{E}$s of your batteries. It is important not to connect them to anything other than the DMM when you make these measurements. (Why?)

  4. Construct the circuit shown in Figure 13. Make sure that the batteries' orientations are correct and that you record which $\mathcal{E}$ value goes with which battery.

  5. Turn the knob on the DMM to a suitable voltage scale, and record the voltage across each of the resistors and the batteries.

  6. Remember never to connect an ammeter in parallel with anything! Use the DMM to measure magnitudes and directions the three unique currents in the circuit $i$, $i_1$, and $i_2$. This will require you to connect the DMM in series with each separate branch of the circuit. Be careful not to change the way the circuit is arranged as you make these measurements. If you're not sure how, ask for help!

    In order to determine the direction of the current, remember that a positive current flows into the red lead and out of the black lead. The directions you measure may not agree with the arrows in Figure 13, and that's OK! Record what you actually observe.

  7. Check your measurements by using your measured currents and voltages along with Ohm's Law to calculate the resistances of the resistors. Compare them with the actual values measured with the DMM in step 2. Resolve any inconsistencies before you move on!

Analysis

Use Kirchhoff's rules and your measured resistances and emfs to predict the magnitudes and directions of each of the unique currents in the circuit shown in Figure 13.

Group Assignment (Before You Leave Lab)

Discuss with your instructor preliminary answers to the questions below.

Individual Assignment

  1. Why was it important to measure the $\mathcal{E}$s of your batteries while they were not connected to anything other than the DMM?

  2. Give the internal resistance of the battery you tested. Show your calculations.

  3. Describe your investigation of contact resistance. Approximately how large is this effect?

  4. What is the order of magnitude of the resistances you chose for your two-loop circuit? Explain your choice.

  5. Show the calculations you made to predict the currents and potential differences in the circuit shown in Figure 13. Compare your predictions to your measurements, and discuss any significant ($> 5\%$) discrepancies.


Copyright © 2006-2009, L.A. Riley, T. J. Carroll, J.S. Scott Updated Sun Apr 26 23:00:14 2009

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