# 5 multi loop circuits and kirchhoffs rules unlike the

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5 Multi-Loop Circuits and Kirchhoff’s Rules Unlike the previous section, we will now examine circuits that can not simply be analyzed by combining resistors in series or parallel. These circuits consist of several interrelated loops with multiple EMF sources. To properly evaluate these circuits one must use Kirchhoff’s Rules. When analyzing a multi-loop circuit the first step is to determine the number of branches and for each branch define a current I i with a specified positive direction (keep in mind this over all sign is arbitrarily defined, but must be consistent throughout your calculations). These currents are then related by Kirchhoff’s Junction Rule Eq. (1). When employing Kirchhoff’s Loop Rule one must take care to sum the voltage change across each component appropriately. Going through an EMF (battery, power supply, etc.) from ‘-’ to ‘+’ results in a potential increase (conversely ‘+’ to ‘-’ results in a decrease). Additionally, going through a resistor in the direction of the current also results in a potential decrease (the opposite direction results in an increase). 4

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Figure 5: A multi-loop circuit containing two voltage sources. Here the current conventions are defined arbitrarily. Circuit components: V 1 and V 2 = D-Cell batteries R 1 = 100Ω R 2 = 10Ω R 3 = 330Ω Start by writing out Kirchhoff’s Circuit Rules for the circuit in Figure 5. Leave these equations in a general form, i.e. do not substitute in any of the values for your specific circuit. From these three equations (2 loop and 1 junction equation) solve for the three currents ( I 1 , I 2 , I 3 ) in terms of the resistor and EMF values. Once you’ve determined the values for the components you will be using you can predict the currents that will flow through your circuit. Gather the resistors and batteries to be used for the circuit in Figure 5. Although these components are labeled with values, they may not be accurate; thus, using your DMM measure the resistances of the resistors and voltages of the batteries. Record these values so you may calculate predictions for the currents. Do so by using the previously derived equations for I 1 , I 2 and I 3 . Now assemble the circuit on your Pasco Electronics Board. Note there are two battery holders located on the board itself as seen in Figure 1. Hook up the circuit such that the batteries are not fully connected so as not to drain them until you’re ready to take measurements. Configure Data Studio to register the Voltage and Current Probes plugged into the Science Workshop device. Follow the steps of the “Current Probe Calibration guide” found on the website. Wire the Current Probe to be in series with resistor R 1 and completely close the circuit by hooking up the loose ends of the battery. Now current should be flowing through the circuit in a steady state. Record the current flowing through R 1 being sure to make note of the sign and direction designated by the probe. Using the Voltage Probe measure the voltage differences across each of the five components in the circuit being sure to note which side is at higher potential. Record all these values.
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