Theory All DC circuit analysis (the determining of currents, voltages and resistances throughout a circuit) can be done with the use of three rules. These rules are given below. 1. Ohm's law. This law states that the current in a circuit is directly proportional to the potential difference across the circuit and inversely proportional to the resistance in the circuit. Mathematically, this can be expressed as I = V R . (1)
Ohm's law can be applied to an entire circuit or to individual parts of the circuit. 2. Kirchoff's node rule. This rule states that the algebraic sum of all currents at a node (junction point) is zero. Currents coming into a node are considered negative and currents leaving a node are considered positive. For the situation in figure 1, we have -I1 + I2 + I3 = 0 or I 1 = I 2 + I3
Figure 1 This is a statement of the law of conservation of charge. Since no charge may be stored at a node and since charge cannot be created or destroyed at the node, the total current entering a node must equal the total current out of the node. 3. Kirchoff's loop rule. This rule states that the algebraic sum of all the changes in potential (voltages) around a loop must equal zero. A potential difference is considered negative if the potential is getting smaller in the direction of the current flow. For the situation in figure 2, we have or V1 = V2 + V3 +V1 - V2 - V3 = 0
Figure 2 This is a statement of the law of conservation of energy. Since potential differences correspond to energy changes and since energy cannot be created or destroyed in ordinary electrical interactions, the energy dissipated by the current as it passes through the circuit (V2 + V3) must equal the energy given to it by the power supply ( V1 ). To illustrate the application of these rules, some common electrical circuits are analyzed below.
The Simple Series Circuit Consider the circuit shown in figure 3. It consists of a single loop. There is only one path through which a current can flow. Such a circuit is called a series circuit. Let V1 , V2 and V3 be respectively the potential differences across the resistances R1, R2 and R3; and let I1, I2 and I3 be respectively the currents flowing through those same resistances. Let VT be the electromotive force supplied by the power supply and IT and RT be respectively the total current flowing in the circuit and the total resistance in the circuit.
Figure 3 Writing Kirchoff's loop rule for this circuit yields VT = V 1 + V 2 + V 3 Using Ohm's law, equation (2) can be rewritten as . (2)
IT RT = I1R1 + I2R2 + I3R3
But Kirchoff's node rule tells us that since charge can not pile up in any part of the circuit and since there is only one path for the current to follow, the current in any one part of the circuit must be equal to that in any other part of the circuit. In other words, IT = I1 = I2 = I3. Therefore, equation (3) can be simplified to IT RT = IT R1 + R2 + R3 . (4)
Equation (4) tells us that the total resistance is equal to the sum of the individual resistances. Equations (2) and (4), although specifically describing the circuit in figure 3, can easily be generalized for a series circuit containing n elements. Therefore, application of our three rules leads to these general relationships for series circuits: IT = I1 + I2 + . . . + In VT = V 1 + V 2 + . . . + V n RT = R1 + R2 + . . . + Rn . . . (5a) (5b) (5c)
The Simple Parallel Circuit Consider the circuit shown in figure 4. It consists of three loops. There is more than one path for the current to follow in going from the power supply through the circuit and back to the power supply. Such a circuit is called a parallel circuit. Let V1, V2 and V3 be respectively the potential differences across the resistances R1, R2 and R3; and let I1, I2 and I3 be respectively the currents flowing through those same resistances. Let VT be the electromotive force supplied by the power supply and IT and RT be…