- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 1 (83.19 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 2 (26.36 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 3 (26.36 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 4 (79.37 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 5 (79.37 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 6 (80.54 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 7 (73.95 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 8 (84.43 KiB) Viewed 18 times
- 15 Physics Lab Kirchhoff S Rules 470 5 Data Table R 1 680 5 180 B R Percent Error Good V 6v V 12v I 1 39 9 (44.42 KiB) Viewed 18 times
(1000 (2) (6800) R₁ V₂- (4702) (12V) Figure 32.4 Multiloop circuit. Diagram for experimental two-loop circuit. ww R₂ (2200) (330 (2) Ps (1002). V₂. (12 V) (680 (2) (150) Figure 32.5 Multiloop circuit. Diagram for experimental three-loop circuit.
EXPERIMENT 32 Multiloop Circuits: Kirchhoff's Rules I. INTRODUCTION AND OBJECTIVES The analysis of electrical circuits is the first step toward understanding their operation. By "analysis" is meant the process of calculating how the electrical currents in a cir- cuit depend on the values of the voltage sources (or vice versa). For our discussion, we will refer to any voltage source as a battery with a terminal or "operating" voltage V Many electrical circuits can be analyzed by using noth- ing more than Ohm's law (Experiment 25). The simplest situation consists of one battery (V) and one resistor (R) connected in a single closed loop (see Fig. 25.1). In this case, I V/R. Several batteries may appear in series, or several resistors (Experiment 31) may appear in series. In these cases, a combination can be represented by a single equivalent element, and then Ohm's law can be used. Situations that look more complicated include the poten- tiometer (see Fig. 26.3) and the Wheatstone bridge (see Fig. 28.3). These circuits contain more than one loop. However, the loops are independent; that is, there is no current flow be- tween the loops when the bridge is balanced. So even these circuits can be analyzed with Ohm's law in this condition. II. EQUIPMENT NEEDED • Ammeter (0 to 10/100/1000 mA) Voltmeter (0 to 5/25 V) • Two batteries or voltage supplies (6 V and 12 V) Two single-pole, single-throw (SPST) switches III. THEORY The simple multiloop circuit shown in Fig. 32.1 will be used to illustrate the principles of Kirchhoff's rules and the terminology involved. Terminology definitions vary among textbooks, even though the principles remain the same. Therefore, it is important to define terms carefully as they will be used here. A junction is a point in a circuit at which three or more connecting wires are joined together, or a point where the current divides or comes together in a circuit. For example, in Fig. 32.1a points B and D are junctions. A branch is a path connecting two junctions, and it may contain one element or two or more elements. In Fig. 32.1a there are three branches connecting junctions B and D. These are the left branch BAD, the center branch BCD, and the right branch BD with Ry. 375 The more general electrical circuit contains several loops, with batteries and currents shared among the loops. Such a general circuit cannot be analyzed directly by using Ohm's law; however, it can be analyzed using Kirchhoff's rules, or laws as they are sometimes called, named after Gustav Kirchhoff (1824-1887), the German physicist who developed them. In this experiment, we will investigate, use, and ver- ify Kirchhoff's rules in analyzing multiloop circuits. After performing this experiment and analyzing the data, you should be able to: 1. Clearly distinguish between circuit branches and junctions. 2. Apply Kirchhoff's rules to multiloop circuits. 3. Explain how Kirchhoff's rules are related to the con- servation of charge and energy. Contributed in part by Professor L L. Facher, Bergen Community College, New Jersey. Composition resistors, 2-W rating (100 , 150 0. 220 1, 330 1, 470 2, 680 1, 1000 2) .Connecting wires Note: Items may be varied to apply to available equipment. A loop is a closed path of two or more branches. There are three loops in the circuit in Fig. 32.1, as shown in Fig. 32.lb-two inside loops (loops 1 and 2) and one outside loop (loop 3). Notice that each loop in this case is a closed path of two branches. Kirchhoff's Rules These rules do not represent any new physical principles. They embody two fundamental conservation laws: conser- vation of electrical charge and conservation of (electrical) energy. A current flows in each branch of a circuit. In Fig. 32.1a these are labeled I. 1, and I. At a junction, by the conservation of electrical charge, the current (or currents) into a junction equal the current(s) leaving the junction. For example, in Fig. 32.1a.
EXPERIMENT 32/Multiloop Circuits: Kirchhoff's Rules (20) (12V) R₂ 1 C (60) (6V) (40) D (a) Loop 3 Loop Loop 2 (b) Figure 32.1 Multiloop circuit. (a) By Kirchhoff's june- tion theorem, the sum of the currents at a junction is zero; that is, the current into the junction equals the current out, or 1,1, 1₂, at junction B. (b) The circuit has three loops, about which the sum of the voltage changes is zero (Kirchhoff': loop theorem). 4-1₂+h (current in current out) By the conservation of electrical charge, this means that charge cannot "pile up" or "vanish" at a junction. This cur- rent equation may be written 4-4-4-0 (32.1) Of course, we do not generally know whether a particular current flows into or out of a junction by looking at a multi- loop circuit diagram. We simply assign labels and assume the directions the branch currents flow at a particular junction. If these assumptions are wrong, we will soon find out from the mathematics, as will be shown in a following ex- ample. Notice that once the branch current directions are assigned at one junction, the currents at a common branch junction are fixed; for example in Fig. 32.1a, at junction D. 2+1,-1, [current(s) in- current out]. Equation 32.1 may be written in mathematical nota- tion as $1,=0 (32.2) 376 which is a mathematical statement of Kirchhoff's first rule or junction theorem: The algebraic sum of the currents at any junction is zero. In a simple single-loop circuit as in Fig. 25.1, it is easy to see that by the conservation of energy the voltage "drop" across the resistor must be equal to the voltage "rise" of the battery,* i.e.. V battery resistor where the voltage drop across the resistor is by Ohm's law equal to IR, i.e., Vesistor = IR. By the conservation of energy, this means that the en- ergy (per charge) delivered by the battery to the circuit is the same as that expended in the resistances. The conserva- tion law holds for any loop in a multiloop circuit, although there may sometimes be more than one battery and more than one resistor in a particular loop. Similar to the summation of the currents in the first rule, we may write for the voltages Kirchhoff's second rule or loop theorem: EV,=0 (32.3) or The algebraic sum of the voltage changes around a closed loop is zero. Since one may go around a circuit loop in either a clock- wise or a counterclockwise direction, it is important to es- tablish a sign convention for voltage changes. For example, if we went around a loop in one direction and crossed a re- sistor, this might be a voltage drop (depending on the cur- rent flow). However, if we went around the loop in the opposite direction, we would have a voltage "rise" in terms of potential. We will use the sign convention illustrated in Fig. 32.2. The voltage change of a battery is taken as positive when the battery is traversed in the direction of the "positive" ter- minal (a voltage "rise"), and negative if the battery is tra- versed in the direction of the "negative" terminal. Note that the assigned branch currents have nothing to do with deter- mining the voltage change of a battery, only the direction one goes around a loop or through a battery. The voltage change across a resistor, on the other hand, involves the direction of the assigned current through the resistor. The voltage change is taken to be negative if the re- sistor is traversed in the direction of the assigned branch current (a voltage "drop") and positive if traversed in the opposite direction. The sign convention allows you to go around a loop ei- ther clockwise or counterclockwise at your choice. Going • Here we take the terminal or "operating" voltage of the battery instead of the emf (6). The terminal or "operating" voltage of the battery is V- 8-Ir, where r is the internal resistance of the battery (not usually known) and Ir is the internal voltage "drop" of the battery (see Experiment 26).
-V--A -V .V..IR (a) Traversing a battery (b) Traversing a resistor Figure 32.2 Sign convention for Kirchhoff's rules. (a) When one goes around a loop and passes through a bat- mery, the voltage change is taken to be positive when the bat- tery is traversed toward the positive terminal and negative when traversed toward the negative terminal. (b) When one passes through a resistor in going around a loop, the volt. age change across the resistor is taken to be negative ("volt- age drop") if it is traversed in the direction of the assigned branch current and positive if traversed in the opposite direction. in opposite directions merely makes all the signs opposite. and Eq. 32.2 is the same. Kirchhoff's rules may be used in circuit analysis in several ways. We will consider two methods. BRANCH (CURRENT) METHOD First, label a current for each branch in the circuit. This is done by a current arrowhead, which also indicates the cur- rent direction, and is most conveniently done at a junction as in Fig. 32.1 at junction B. Kirchhoff's first rule applies at any junction. Remember, the current directions are arbi- strary, but there must be at least one current in and one cur- rent out. (Why?) Then, draw loops so that every branch is in at least one loop. This is shown for the circuit in Fig. 32.1, which has three loops. Again, the direction of a loop is arbitrary be- cause of our sign convention. With this done, current equations are written for each junction according to Kirchhoff's junction theorem (rule 1). In general, this gives a set of equations that includes all branch currents. For the simple circuit in Fig. 32.1, this is one equation, since the sum of the currents at junction Dis the same as that at junction B. Then, Kirchhoff's loop theorem (rule 2) is applied to the circuit loops. This gives additional equations that form a set of Nequations with Nunknowns, which can be solved or the unknowns. There may be more loops than neces- sary. Only the number of loops that include all branches is needed. To illustrate this method, the circuit in Fig. 32.1 is an- malyzed in the following example. (Assume the component values in the circuit to be exact, that is, ignore significant Tigures.) Example 32.1 Apply Kirchhoff's rules and the above sign convention to the circuit shown in Fig. 32.1 and find the value of the current in each branch. EXPERIMENT 32/Multiloop Circuits: Kirchhoff's Rules 377 Solution By rule 1 (junction theorem), 4-42+4 (32.4) with directions as assigned in the figure. Going around loop 1 as indicated in the figure, with Kirchhoff's second rule (loop theorem) and our sign con- vention, we have, starting at battery 1, V₁-1,R₁-V₂-R₂-0 or with known values (units omitted for convenience), 6-1(2)-12-1(4)-0 and 4₁ +21₂--3 (32.5) Similarly, around loop 2, starting at battery 2. V₂-R₂ + R₂0 or 12-1,(6)+1(4)-0 and 31, -21₂-6 (32.6) Equations 32.4-32.5 constitute a set of three equations with three unknowns from which the values of I, I, and I, can be found. Solving these equations for the currents, one obtains 4- HA The negative values of I, and I, indicate that the wrong di- rections were assumed for these currents. In the actual cir- cuit, I would flow into junction B, and I, would flow out of the junction as well as I, as assumed. Hence, if we had guessed correctly, we would have written for junction B. 12=4+4 A- A+#A (current in) (current out) and 21,-0 as required by rule 1. In looking at Fig. 32.1 more carefully, one might have surmised this. Battery 2 (12 V) has twice the voltage of bat- tery 1 (6V) and it would have been a good guess that (con- ventional) current would flow out of battery 2 toward junction B. If battery I were a rechargeable battery, it would be recharging in the circuit. (Why?) Notice that loop 3 was not used to solve the problem. This loop would have provided a redundant equation with the other two loop equations. However, loop 3 could have been used with one of the other loops to solve the problem.
378 EXPERIMENT 32/Multiloop Circuits: Kirchhoff's Rules LOOP (CURRENT) METHOD (Optional) This method is similar to the previous branch method, but some consider it to be simpler mathematically. Loops are drawn as before so that every branch is in at least one loop. Then, each loop is labeled as if it were a current, as shown in Fig. 32.3 for the circuit in Fig. 32.1. The "current loop" direction is arbitrary as in the previous method for loop directions. Kirchhoff's loop theorem (rule 2) is then used to write an equation for each loop, applying the sign convention. For the circuit in Fig. 32.3, there are two loop equations for the "loop currents" I, and la. These are, starting at V, and V₂, respectively. (loop A) (loop B) V₁-IAR₁ - V₂ - 1₁R₂ +1₂R₂ = 0 V₂-18₂-18R₂+1/R₂=0 Note that you have to take into account all the "loop cur- rents" that affect each loop. That is, when going around a loop and for a branch resistor through which two "loop cur- rents" flow, the voltage changes for both currents must be considered for the resistor as for R₂ in this case. Traversing the current loops produces two equations for this circuit with two unknowns, I and I. (Note that in the branch method analysis in Example 32.1, the loop equations had three unknowns.) Putting in the known val- ues, the equations may be solved simultaneously to find the value of each unknown: (loop A) 6-1(2)-12-(4)+1(4)-0 12-1(6)-(4)+1(4)-0 (loop B) These reduce to -31 +21=3 Solving for I and Ig, we find B-A The computed values of the "loop currents" may then be utilized in a straightforward way to find the actual branch R, (20) (612) (402) Figure 32.3 Loop method. In this method, the loops are taken to be "current loops," with a particular current as- signed to each loop. 21-518--6 A currents. Kirchhoff's junction theorem is applied to the var- ious junctions in the circuit, and the branch currents are compared with the "loop currents." For example, compare the branch currents at junction B in Fig. 32.1 and the "loop currents" in Fig. 32.3. (The branch currents may be drawn directly on the current-loop diagram.) It should be evident by comparison that: 4-4--A ----A-HA--A 4-4-A the same values obtained for the circuit by the branch method. IV. EXPERIMENTAL PROCEDURE 1. Examine the resistors. The colored bands on composi tion resistors conform to a color code that gives the re- sistance value of the resistor. Look up the color code in Appendix A. Table A5, to identify each resistor. Note that the actual resistance value may vary according to the tolerance indicated by the last band (gold ±5%, silver ±10%, no band ±20%). 2. Connect the two-loop circuit as illustrated in Fig. 32.4. If you are using variable power supplies, adjust each power supply as closely as possible to the values spec- ified in the figure. Leave the switches open until the circuit has been checked by the instructor. Note: Lay out the circuit on your table exactly as shown in the diagram. This will help prevent errors and will facilitate your measurements. 3. After your circuit has been checked, close the switches and measure the "operating" value of each battery (V₁ and V₂) by temporarily connecting the voltmeter across it. Record these operating values in Data Table 1. Caution: To avoid damage to the voltmeter, al- ways start with the meter on its least-sensitive scale. Increase the sensitivity of the meter only as needed for accurate measurement, and remember to return the meter to its least-sensitive scale before proceeding. 4. Temporarily open the switches. Insert the ammeter in series with one of the branches. Close the switches, measure and record the branch current, then open the switches. Caution: Observe the same precautions described under procedure 3 to avoid damage to the ammeter. Also, if the ammeter deflects downscale (below zero). open the switches before reversing the polarity of the meter. 5. Repeat procedure 4 for each of the branches.
1₁ R₂ (1000 (2) (680 (2) R₁ V₂. (4702) (12 V) Figure 32.4 Multiloop circuit. Diagram for experimental two-loop circuit. 6. Compute the theoretical values of each branch current for this circuit. In the analysis, use the measured val- ues of the batteries V, and V₂ and the labeled values of the resistors (procedure 1). Compare the measured val- ues of the branch currents with the computed theoreti- cal values by finding the percent error. EXPERIMENT 32/Multiloop Circuits: Kirchhoff's Rules (2202) R₁ (330) R₁ R₂ (1002). V₂ (1502) (680 (2) (12 V) Figure 32.5 Multiloop circuit. Diagram for experimental three-loop circuit. 7. Connect the three-loop circuit as shown in Fig. 32.5. Repeat procedures 3 through 6 for this circuit. The in- structor may wish to provide you with a different cir- cuit to investigate. (In Fig. 32.5, why is there no current indicated between the two connection points on the bottom wire?) 379