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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