EENG 2610: Circuits Analysis Class 2: Kirchhoff’s Laws, Single-Loop Circuits, Single- Node Pair Circuits Oluwayomi Adamo Department of Electrical Engineering College of Engineering, University of North Texas

Some Important Concepts Lumped-Parameter Circuit  Wires in circuits are assumed perfect conductor  Interconnections in circuits have zero resistance  Wire doesn’t consume energy; energy in circuits is lumped in each circuit element. Node  A point of connection of two or more circuit elements.  A node is one end of a circuit element together with all the perfect conductor that are attached to it. Loop  A loop is any closed path through the circuit in which no node is encountered more than once. Branch  A branch is a portion of a circuit containing only a single element and the nodes at each end of the element.

Kirchhoff’s Current Law (KCL) KCL  The algebraic sum of the currents ENTERING any node is zero: Example 2.5: Write all KCL equations Our sign convention for KCL: - The algebraic sign of the current is ‘plus’ if the current is entering the node - The algebraic sign of the current is ‘minus’ if the current is leaving the node

Closed Surface as Super-Node Super-Node  If some set of elements are completely contained within a surface that is interconnected, the surface is called super-node. Generalized KCL for Super-Node  The algebraic sum of the currents entering any closed surface (or super-node) is zero. Write KCL equations for super-nodes

Kirchhoff’s Voltage Law (KVL) KVL  The algebraic sum of the voltages around any loop is zero:  Voltage is defined as the difference in energy level of a unit positive charge located at each of the two points. KVL is based on the conservation of energy: the work required to move a unit charge around any loop is zero. Our Sign Convention for KVL  As moving around a loop, the algebraic sign of voltage is positive in KVL equation if encounter the plus sign first, and the algebraic sign of voltage is negative in KVL equation if encounter the minus sign first.

Example 2.9: Using KVL equation to find = 18 V = 12 V

Convention for Voltage Notation Double-subscript notation  Vab = Va – Vb + and – notation Arrow notation  Use an arrow between two points, pointing from negative node to positive node.  KVL can be applied to a closed path even if part of the closed path is the arrow notation. V = Vab a b Vab

Example 2.11: Use KVL to find V ae and V ec

Single-Loop Circuits Single-loop circuits  Elements are connected in series.  All elements carry same current. Voltage Divider  Source v(t) is divided between two resistors in proportion to their resistance

Multiple Voltage Sources in Single-Loop Circuit + _ + _ Equivalent Transform

Multiple Resistors in Single-Loop Circuit Equivalent Transform

Single-Node Pair Circuits  Elements are connected in parallel  Elements have the same voltage across them Current division

Multiple Current Sources in Single-Node Pair Circuits Equivalent Transform

Multiple Resistors in Single-Node Pair Circuits General current divider: + _ Equivalent Transform

Equivalent circuit Example 2.19: Find the current I L