Chapter 2 Basic laws SJTU

Some Basic Concepts Branch----- A branch represents a single element such as a voltage source or a resistor. {But usually we think a branch as a path flowing the same current. So maybe includes more than one element.} SJTU

Some Basic Concepts Node-------A node is the point of connection between two or more branches Loop-------A loop is any closed path in a circuit. Mesh------A mesh is a loop which does not contain any other loops within it. SJTU

KIRCHHOFF’S LAWS Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering a node (or a closed boundary) is zero. (Based on the law of conservation of charge)   The sum of the currents entering a node = the sum of the currents leaving the node. SJTU

Note: 1) KCL is available to every node at anytime. KCL also applies to a closed boundary. Note: 1) KCL is available to every node at anytime. 2) KCL is related only to the currents instead of the elements. 3) Pay attention to the current direction. SJTU

KIRCHHOFF’S LAWS Kirchhoff’s Voltage Law (KVL): The algebraic sum of all voltages around a closed path (or loop) is zero. (Based on the principle of conservation of energy) To illustrate KVL, consider the circuit : V1-V2-V3+V4-V5=0 or V2+V3+V5=V1+V4 Sum of voltage drops=Sum of voltage rises SJTU

TWO SORTS OF CONSTRAINTS 1.         Topological constraints Determined by the way of connection among the elements. (Such as KCL KVL) 2.      Element constraints Determined by the elements. (VAR) Using two sorts of constraints, we can analysis any lumped circuit (solve out all the voltages and currents). SJTU

That is called 2b analysis. In a circuit with b branches and n nodes, there are 2b variables should be valued. Then: KCL for n nodes: only n-1 equations are independent. KVL for loops: only b-n+1 equations are independent. (only KVL for meshes) VAR for branches: b equations.   So, (n-1)+(b-n+1)+b=2b, 2b equations to value 2b variables. That is called 2b analysis. SJTU

SERIES RESISTORS AND VOLTAGE DIVISION As we know, Series-connected means that the same current flows in them.   The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.  See illustration with 2 resistors: R is the equivalent resistance. It can be applied to any number of resistors SJTU

Concept of Equivalent: Be equivalent to the outside, not the inside. The equivalent power of any number of resistors connected in series is the sum of the individual powers. Concept of Equivalent: Be equivalent to the outside, not the inside. Principle of voltage division: if a voltage divider has N resistors(R1,R2,…RN) in series with the source voltage v, the nth resistor(Rn)will have a voltage drop of SJTU

PARALLEL RESISTORS AND CURRENT DIVISION As we know, Parallel-connected means that the same voltage covers over them.  The equivalent conductance of resistors connected in parallel is the sum of their individual conductances. See illustration with 2 resistors: G G G is the equivalent conductance. It can be applied to any number of resistors SJTU

Principle of current division The equivalent power of any number of resistors connected in parallel is the sum of the individual powers. Principle of current division If a current divider has N conductors (G1,G2…GN) in parallel with the source current i, the nth conductor (Gn) will have current SJTU

MIXED CONNECTION AND ITS EQUIVALENT RESISTANCE Examples (we combine resistors in series and in parallel) SJTU

WYE-DELTA TRANSFORMATIONS Situations often arise in circuit analysis when the resistors are neither in parallel nor in series. e d R12=? SJTU

WYE-DELTA TRANSFORMATIONS SJTU

Can you imagine another way of transformation? So e d 3 1 2 Can you imagine another way of transformation? SJTU