PHY 132 - Kirchhoff's Laws manual

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Arizona State University *

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132

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Electrical Engineering

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Apr 3, 2024

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1 KIRCHHOFF'S LAWS SUMMARY In this experiment, we will demonstrate Kirchhoff’s junction and loop rules which, unlike Ohm’s Law, come from fundamental conservation laws. EQUIPMENT PASCO® EM-8622 Circuit Experiment Board, 3 resistors, 2 batteries, wires, multimeter. INTRODUCTION Most circuits are too complicated to be reduced to simple series or parallel combinations of resistors and emf sources. These circuits can be analyzed by using Kirchhoff’s laws. To explain what these laws are, we must first define two terms: junction and loop. A junction is a point in a circuit where at least three conductors meet. For example, in Figure 1, points b and e are junctions. Notice that the corners a, c, d, and f are not junctions, because only two conductors meet at those points. A loop is any part of a circuit where conductors form a closed path. In Figure 1, there are three loops marked with blue labels and rounded rectangles. The direction of any given loop is arbitrary. Kirchhoff’s Current Law (KCL): The sum of all currents into and out of a junction must be zero. In other words, the amount of current that flows into a junction must be equal to the amount that flows out. � 𝐼𝐼 = 0 (1) This follows from the conservation of electric charge. KCL (often called the junction rule) will hold true under these conditions: (1) charge is not created or destroyed, (2) charge does not accumulate inside the junction, and (3) the circuit does not gain or lose charge to the outside environment (i.e. charge doesn’t dissipate into the air). Incoming currents are taken to be (+) and outgoing to be (-) in Eq. (1). For example, applying Eq. (1) to junction b in Figure 1 tells us: 𝐼𝐼 1 + 𝐼𝐼 2 = 𝐼𝐼 3 . (2) For junction e, the equation is 𝐼𝐼 3 = 𝐼𝐼 1 + 𝐼𝐼 2 , which is equivalent to Eq. (2). Figure 1: A circuit that cannot be reduced to a simple series/parallel combination.

2 In general, current directions are not given and you must choose these directions. To be clear, your choice does not alter the behavior of the circuit, but it provides a mathematical basis from which to solve the system; this idea is akin to defining your coordinate system in a projectile motion problem, for example. Do not hesitate to choose the current directions; if you chose wrong, you would get a negative current at the end of the calculation. If this occurs, it simply means the current flows in opposition to your initial guess. In this case, you should make sure to flip the arrow in your drawing and change your junction rules accordingly. In Figure 1, current directions are already chosen in a way that guaranties all three currents to be positive. This is usually not possible, but the circuit in Figure 1 is designed this way to simplify the experiment. In the pre-lab, however, you have to choose the directions yourself and do a full solution. Kirchhoff’s Voltage Law (KVL): The sum of the potential differences around any loop must be zero. � 𝑉𝑉 = 0 (3) This follows from conservation of energy, since electrostatic force is a conservative force. To use KVL (often called the loop rule), simply trace any loop, either clockwise or counterclockwise, add all the potential rises and drops, then set the sum to zero. As you go around a loop, you will encounter different circuit elements. Depending on the direction you travel across them and the direction the current flows through them, the electric potential either rises or drops as you pass these elements. If the electric potential rises, you should add the voltage of that element and, conversely, if the electric potential drops, you should subtract the voltage. As you go around a loop and encounter an emf, if the traveling direction is from (–) pole to (+) pole, add + 𝜀𝜀 (see Figure 2-a); if the traveling direction is from (+) pole to (-) pole, add −𝜀𝜀 (see Figure 2-b) in Eq. (3). As you go around a loop and encounter a resistor, if the traveling direction is the same as the current direction, add −𝐼𝐼𝐼𝐼 (see Figure 3-a); if the traveling direction is opposite to the current direction, add + 𝐼𝐼𝐼𝐼 (see Figure 3-b); in Eq. (3). The starting point of the loop does not matter. When the loop is complete and you are back at the starting point, set the sum to zero. In Figure 1, the loop travel directions are already chosen. The loop travel direction is completely arbitrary, there is no right or wrong direction. Figure 2: emf sign conventions Figure 3: Resistor sign conventions

3 For example, for loop 1 following the path e → d → c → b → a → f → e in Figure 1, KVL tells us: + 𝜀𝜀 2 − 𝐼𝐼 2 𝐼𝐼 2 + 𝐼𝐼 1 𝐼𝐼 1 − 𝜀𝜀 1 = 0 (4) Choosing a different starting point will only change the order of the terms in the sum and choosing the other travel direction will multiply the whole equation by − 1 . Either way, the result will be equivalent. KCL (junction rule) and KVL (loop rule) can be used to solve for whichever quantities are unknown. In general, the three steps to solving a circuit using Kirchhoff’s Laws are: 1. Apply Kirchhoff’s current law (KCL) and and Kirchhoff’s voltage law (KVL) to obtain an equal number of unique equations as there are unknown variables. (Note: this often means you will need to apply KVL more than once). 2. Use algebra to un-tangle the equations and solve for the unknown variables. 3. Plug in the numbers and check the units make sense.

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4 PROCEDURE 0. Measure the voltages of the two batteries 𝜀𝜀 1 and 𝜀𝜀 2 , then measure the resistances of 𝐼𝐼 1 and 𝐼𝐼 2 with the multimeter and record these values in your worksheet. Do not measure the resistance of 𝐼𝐼 3 yet. 1. Build the circuit in Figure 4 on the PASCO® Circuit Experiment Board. Make sure you put 𝐼𝐼 1 , 𝐼𝐼 2 and the unknown resistor 𝐼𝐼 3 in the correct places corresponding to the circuit diagram. Color or length of the wires do not matter. Pay attention to the poles of the batteries. It would behoove you to ask your TA to double check the circuit is set up correctly before continuing. 2. Disconnect one leg of 𝐼𝐼 1 from the board, then use the TENMA 72-7780 multimeter in mA mode to measure the current 𝐼𝐼 1 (with the minigrabbers). 3. To double check the value of 𝐼𝐼 1 , measure the voltage 𝑉𝑉 1 (across 𝐼𝐼 1 ) using the Fluke 179 in DC Voltage mode. Calculate 𝐼𝐼 1 using Ohm’s “Law” and show your work with units. If this value is significantly different than the value you measured in the previous step, get help from your TA before continuing. 4. Apply KVL for loop 1. Refer to Figures 2 and 3 to know whether to add or subtract each term. Write down the (symbolic) equation you obtain. Pro-tip: your equation should be written using only symbols (don’t plug numbers in too early or it will seem more complicated than it is). Re-arrange your symbolic equation to solve for 𝐼𝐼 2 , then insert the relevant measured values to calculate 𝐼𝐼 2 . Show your work. 5. Repeat steps 2 & 3 for current 𝐼𝐼 2 . Once again, if these values are inconsistent, get help from your TA. 6. Write down the (symbolic) equation you get from applying KCL at point b. Use the directly measured values of 𝐼𝐼 1 and 𝐼𝐼 2 to solve the junction rule for 𝐼𝐼 3 . Show your work. 7. Repeat step 2 for 𝐼𝐼 3 and write the measurement in your worksheet. 8. Apply KVL for loop 2 and loop 3 and write down the symbolic equations you obtain. Re-arrange one of the equations to solve for the unknown resistance 𝐼𝐼 3 (it doesn’t matter which one you use), then insert the relevant measured values and units to calculate 𝐼𝐼 3 . Show your work. 9. Remove 𝐼𝐼 3 from the circuit and measure its resistance using the multimeter. Calculate the percent difference between the measured value ( 𝐼𝐼 𝐷𝐷𝐷𝐷𝐷𝐷 ) and the value you obtained from KVL in step 8. Figure 4: Figure 1 copy-pasted for convenience.

5 K I R C H H O F F ’ S L A W S – W O R K S H E E T Name: Partners: TA: DATA & ANALYSIS KVL for Loop 1 • 𝐼𝐼 1 , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ……………….. (unit: _____ ) • 𝑉𝑉 1 , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ………………. (unit: _____ ) • 𝐼𝐼 1 , 𝑐𝑐𝑚𝑚𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑚𝑚 = ………………. (unit: _____ ) (from Ohm’s rule) • 𝐼𝐼 2 , 𝑐𝑐𝑚𝑚𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑚𝑚 = ………………. (unit: _____ ) (from KVL) • 𝐼𝐼 2 , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ……………….. (unit: _____ ) Clean-up/sign-out (3): Data (30): Analysis (40): Post Lab Qs (15): Lab Report Total (88): Step 0 measurements • 𝜀𝜀 1 = ………………. (unit: _____ ) • 𝜀𝜀 2 = ………………. (unit: _____ ) • 𝐼𝐼 1 = ………………. (unit: _____ ) • 𝐼𝐼 2 = ………………. (unit: _____ ) Calculations Symbolic KVL equation:

6 KCL for point b • 𝐼𝐼 3 , 𝑐𝑐𝑚𝑚𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑚𝑚 = ………………. (unit: _____ ) (from KCL) • 𝐼𝐼 3 , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ……………….. (unit: _____ ) Solving for 𝐼𝐼 3 • 𝐼𝐼 3 , 𝑐𝑐𝑚𝑚𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚𝑚𝑚 = ………..…………. (unit: _____ ) (from KVL) • 𝐼𝐼 𝐷𝐷𝐷𝐷𝐷𝐷 = …………..………. (unit: _____ ) • % difference = ………………..…. (unit: _____ ) Calculations Calculation done using KVL for (circle/highlight one): Loop 2 | Loop 3 Symbolic KVL equation (Loop 2): Symbolic KCL equation: Symbolic KVL equation (Loop 3):

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7 POST LAB QUESTIONS 1. What are the three steps to solving a circuit with Kirchhoff’s Laws? ( Hint: Read the fantastic manual ) 2. Two identical lightbulbs with unknown, but equal resistance and two identical batteries with a voltage of 5V are shown in the diagram below. When the switch is closed, how much current will flow through the middle branch? Which direction (up or down) will the current flow in the middle branch if you flip the polarity of the battery on the right? 3. Based on the circuit shown in Figure 1, calculate the currents in each branch for the case 𝜀𝜀 2 = 0 . This is equivalent to removing ε 2 and replacing it with a continuous wire from d to e. Attach a separate page if needed.