DC Circuit Analysis 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 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. Or, EMF s and currents may be shared between different loops in a circuit. This is where Kirchhoff s Rules are applied. Theory When resistors are connected in series, there is a constant current through all of them. resistors can be replaced with a single equivalent resistance given by Therefore, the R eq = R 1 + R 2 + R 3 +... (1) On the contrary, when resistors are connected in parallel, there is the same potential drop across all of them. Hence the resistors can be replaced with a single equivalent resistance given by 1 R eq = 1 R 1 + 1 R 2 + 1 R 3 +... (2) These equations can be applied multiple times when there are combinations of series and parallel arrangements of resistors in the same circuit. As for Kirchhoff s Rules, first some terminology; reference the circuit below. V 1 R 1 i 1 V 2 R 3 R 2 A i 2 B R 4 i 3 V 3 R 5 junction A point at which three or more wires connect; i.e., where three or more currents meet. Points A and B are junctions. 1
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 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 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) yields i 1 R 1 V 1 i 1 R 2 + i 2 R 3 + V 2 = 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 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 Note: Do not leave the power supply on. Turn it on only when measuring potentials or currents in a circuit - then turn it off. Select three resistors - designated as R 1, R 2, and R 3 - and determine their resistance (consult a color code chart). Series and Parallel Resistances 1. Assemble the circuit shown in Figure 1. Using the resistances determined above, calculate the equivalent resistance R eq for this circuit, using Equations 1 and 2. 2. Measure the potential V T of and the current I T at the source (power supply). Use these values along with Ohm s Law to calculate the equivalent resistance of the circuit. 3. Measure the potential drop across each individual resistor (V 1, V 2, and V 3 ) as well as the current (I 1, I 2, and I 3 ) through it. 4. Repeat the procedure for the circuits shown in Figure 2 and Figure 3. 2
Figure 1: Series Resistances Figure 2: Parallel Resistances 3
Figure 3: Series/Parallel Combination of Resistances Kirchhoff s Rules 1. Assemble the circuit shown in Figure 4, which is a bridge circuit. You will need two more resistors. 2. Measure the current in each of the six branches. In addition, note the direction of the current in each branch. 3. Measure the potential drop across each resistor. In addition, note which side of the resistor is at the higher potential. 4. Measure the potential at the source. Figure 4: Kirchhoff Circuit Analysis Series and Parallel Resistances 1. Compare the equivalent resistances for each circuit. 2. Do you think Equations 1 and 2 were verified? Why or why not? 3. Go back and determine the tolerance of resistors R 1, R 2, and R 3. Take these as the uncertainties in R 1, R 2, and R 3. For the series circuit (Figure 1), calculate the uncertainty in R eq from Equations 1 and 2. Does this alter your answer for Question 2? 4
4. Comment extensively on potential drops and currents in series circuits. Repeat for parallel circuits. Kirchhoff s Rules 1. What is the sum of the currents at node A? This should be calculated using the current magnitudes and directions obtained in the procedure. Repeat for nodes B, C, and D. Is the junction rule verified? Why or why not? 2. Select a loop in the circuit. Designate it by nodes; e.g., loop ABDA. Sum the changes in potential around the loop, using the potentials obtained in the procedure. Repeat for two more loops. Is the loop rule verified? Why or why not? 5
Pre-Lab: DC Circuit Analysis 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 the common physical quantity in two resistors in series? 2. What is the common physical quantity in two resistors in parallel? 3. What is a junction? 4. What is a branch? 5. What is a loop? 6. How many different currents are in the circuit shown in the Theory section? 7. 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