Multiloop DC Circuits (Kirchhoff s Rules) In analyzing circuits, it is generally the current that is of interest. You have seen how Ohm s Law can be used to analyze very simple circuits consisting of an EMF and a single resistance. This method can be extended to include circuits with more than one resistance, provided the resistances are in series or parallel arrangements that can be replaced with an equivalent resistance. However, there are more complex circuits in which Ohm s Law is inadequate for a full analysis. The resistances may not be in simple series or parallel arrangements, for example. Or, EMF s and currents may be shared between different loops in the circuit. This is where Kirchhoff s Rules are applied. Theory First, some terminology; reference the circuit below. A i 1 R 1 R 4 V 2 V 3 V 1 R 3 R 2 i 3 R 5 B i 2 junction A point at which three or more wires connect; i.e., where three or more currents meet. Points A and B are junctions. branch A single path connecting two junctions. There may be any number of circuit elements in the branch. There are three branches connecting junctions A and B - the left branch containing R 1, V 1, and R 2 ; the center branch containing V 2 and R 3 ; and the right branch containing R 5, V 3, and R 4. loop A closed path of two or more branches. There are three here - the left loop containing R 1, V 1, R 2, R 3, and V 2 ; the right loop containing R 5, V 3, R 4, V 2, and R 3 ; and the outside loop containing V 1, R 1, R 4, V 3, R 5, and R 2. Kirchhoff s two rules are: junction rule The algebraic sum of the currents at any junction is zero. This is a consequence of the conservation of charge. Sign convention is that any current into the junction is positive and 1
any current out of the junction is negative. You assign a direction to the current in each branch when you start - it is completely arbitrary. In the circuit above, the junction rule at B yields i 1 + i 2 i 3 = 0. loop rule The algebraic sum of the changes in potential around any closed loop is zero. This is a consequence of the conservation of energy. Sign convention depends on the direction of the current in a branch as well as the direction taken around the loop (they do not have to be the same). Any time you traverse an EMF from negative to positive, the change in potential is taken as positive; if going from positive to negative, it is negative. For resistances, when you traverse one in the direction of the current, the change in potential is negative; opposite the direction of the current is positive. In the circuit above, the loop rule on the left loop (starting and ending at A and proceeding clockwise) yields V 2 R 3 i 2 + R 2 i 1 + V 1 + R 1 i 1 = 0. These two rules are used to write equations involving the unknown currents in the circuit (there is a different current in each branch); you need the same number of independent equations as you have currents. The equations are then solved simultaneously for the currents. After solving the equations, a negative sign on a current means that the actual direction of the current in that branch is opposite of the direction you chose (the magnitude is the same). Apparatus Power supply, Pasco AC/DC Electronics Laboratory, Patch cords, DMM. Procedure Verifying the Junction and Loop Rules The schematic below shows a bridge circuit. C R 2 R 1 B R 5 A R 4 D R 3 V 1. Assemble a bridge circuit with the five resistors provided. A more realistic diagram for this circuit is shown in Figure 1. 2. Measure the changes in potential around loop BCDB - some of these will be positive and some negative. In order to get consistent signs, keep the DMM in the same orientation as you move around the loop. For example, measure V 2 with the black lead at B and the red lead at C; then get V 5 with black at C at red at D; finally get V 4 with black at D at red at B. Record the values in Table 1 and sum them. Repeat for loop BCADB. 2
Figure 1: Bridge circuit on the board. 3. Measure all currents at junction B. Again, for sign consistency keep the black lead of the DMM at B as you measure each current. Repeat for junction C. Table 1: Verifying the Junction and Loop Rules Loop BCDB Loop BCADB V 2 (V) V 5 (V) V 4 (V) Sum (V) V 2 (V) V 1 (V) V 3 (V) V 4 (V) Sum (V) Junction B Junction C i T (ma) i 2 (ma) i 4 (ma) Sum (ma) i 2 (ma) i 5 (ma) i 1 (ma) Sum (ma) Solving for Currents 1. Assemble the circuit shown schematically below; Figure 2 shows a more realistic representation. 2. Measure the resistance of each resistor in the circuit, recording these values at the corresponding resistor in the schematic diagram. 3
3. Turn on the power supply and measure the potential offered by V 1 and V 2 (adjust V 1 to between 8-10V). Record these values by the corresponding potential in the schematic. When done, turn off the power supply and DMM. R 1 R 2 A V 1 R 5 V 2 R 3 B R 4 Figure 2: Kirchhoff circuit on the board. 4. In the schematic, indicate the direction of the current in each branch and assign each a unique name (i 1, i 2,...). Using the resistances and potentials already recorded, write some Kirchhoff equations below (one junction and two loops) to solve for these currents. 5. Solve your equations for the theoretical currents and place these values in Table 2. 4
Table 2: Currents Theoretical Measured Percent Current Current Error (ma) (ma) 6. Turn on the power supply and measure the current in each branch for comparison. When done, turn off the power supply and DMM. Analysis Verifying the Junction and Loop Rules 1. Do your sums in Table 1 verify the loop rule? Why or why not? 2. Do your sums in Table 1 verify the junction rule? Why or why not? 5
Solving for Currents 1. What was the average percent error between the theoretical and measured currents? 2. Based on your answer to Question 1, did Kirchhoff s rules work? Why or why not? 3. What could account for the difference between the theoretical and measured currents? 6
Pre-Lab: Multiloop DC Circuits (Kirchhoff s Rules) Name Section Answer the questions at the bottom of this sheet, below the line - continue on the back if you need more room. Any calculations should be shown in full. 1. What is a junction? 2. What is a branch? 3. What is a loop? 4. How many different currents are in the circuit shown in the Theory section? 5. Does the direction you assign to each current in a circuit matter when you are setting up the Kirchhoff equations? Why or why not? 6. Solve the following system of equations. You will find this much easier (and the lab calculations will go much faster), if you learn how to solve systems of equations such as these with your TI calculator. x + y z = 0 5.08x 9.96y = 5.31 9.96y 14.72z = 10.37 7