Kirchhoff’s Laws Laws of Conservation

Kirchhoff’s Current Law Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or closed boundary) is zero. The sum of the currents entering a node is equal to the sum of the currents leaving the node

KCL (cont.) For current sources combined in parallel, the current is the algebraic sum of the current supplied by the individual sources.

Kirchhoff’s Voltage Law Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero Sum of voltage drops = Sum of voltage rises

KVL (cont.) For voltage sources connected in series, the combined voltage is the algebraic sum of the voltages of the individual sources.

Series and Parallel Elements Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current through them. Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.

Series Resistors The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

Voltage Division To determine the voltage across each resistor we use: The voltage is divided among the resistors in direct proportion to their resistances.

Parallel Resistors The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.

Parallel Resistors (cont.) The equivalent resistance of N resistors in parallel is Req is always smaller than the resistance of the smallest resistor in the parallel combination. If the resistances are equal, simply divide by the number of resistors.

Parallel Conductance It is often more convenient to use conductance when dealing with parallel resistors. The equivalent conductance of resistors connected in parallel is the sum of their individual conductances.

Serial Conductance The equivalent conductance of series resistors is obtained in the same manner as the resistance of resistors in parallel.

Current Division For two resistors in parallel, the resistors will have current

Current Division (cont.) The total current i is shared by the resistors in inverse proportion to their resistances. If a current divider has N conductors in parallel, the nth conductor (Gn) will have current

Examples

Examples Find current io and voltage vo in the circuit.

Examples Find v1 and v2 in the circuit.

Examples Find Req by combining the resistors.

Examples Calculate the equivalent resistance Rab in the circuit.

Examples Find the equivalent conductance Geq for the circuit.

Examples Find Req by combining the resistors.

Examples Find the currents and voltages in the circuit.

Examples Calculate the voltages and currents, and find the power dissipated in the 12 Ω and 40 Ω resistors