Basic Laws Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Ohm’s Law (resistors) Nodes, Branches, and Loops Kirchhoff’s Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Wye-Delta Transformations Applications

Ohm’s Law Resistance R is represented by 1  = 1 V/A Cross-section area A Meterial resistivity  R v + _ i

Resistors Short circuit Open circuit R = 0 v = 0 + _ i R =  v + _ Variable resistor Potentiometer (pot)

Nonlinear Resistors Examples: lightbulb, diodes v Slope = R v i Slope = R(i) or R(v) Examples: lightbulb, diodes All resistors exhibit nonlinear behavior.

Conductance and Power Dissipation Conductance G is represented by 1 S = 1 = 1 A/V  siemens mho A positive R results in power absorption. A negative R results in power generation.

Nodes, Branches, & Loops a c b Brach: a single element (R, C, L, v, i) + _ a c b Brach: a single element (R, C, L, v, i) Node: a point of connection between braches (a, b, c) Loop: a closed path in a circuit (abca, bcb, etc) A independent loop contains at least one branch which is not included in other indep. loops. Independent loops result in independent sets of equations. redrawn + _ c b a

Continued Elements in series Elements in parallel Elements in series (10V, 5) Elements in parallel (2, 3, 2A) Neither ((5/10V), (2/3/2A)) 10V 5 2 3 2A + _

Kirchhoff’s Laws Introduced in 1847 by German physicist G. R. Kirchhoff (1824-1887). Combined with Ohm’s law, we have a powerful set of tools for analyzing circuits. Two laws included, Kirchhoff’s current law (KCL) and Kirchhoff’s votage law (KVL)

Kirchhoff’s Current Law (KCL) Assumptions The law of conservation of charge The algebraic sum of charges within a system cannot change. Statement The algebraic sum of currents entering a node (or a closed boundary) is zero. i1 i2 in

Proof of KCL

Example 1 i1 i3 i2 i4 i5

Example 2 IT I1 I2 I3 IT

Case with A Closed Boundary Treat the surface as a node

Kirchhoff’s Voltage Law (KVL) Statement The algebraic sum of all voltages around a closed path (or loop) is zero. v1 + _ v2 vm

Example 1 Sum of voltage drops = Sum of voltage rises v4 v1 v5 v2 v3 + _ v2 v3

Example 2 V3 V2 V1 Vab + _ a b Vab + _ a b

Example 3 Q: Find v1 and v2. v1 + _ v2 20V 2 3 i Sol:

Example 4 Q: Find currents and voltages. Sol: v1 + _ 30V 8 3 i1 6 Loop 1 Loop 2 a v2 b

Series Resistors v1 + _ v R1 i v2 R2 a b v + _ i Req a b

Voltage Division v1 + _ v R1 i v2 R2 a b v + _ i Req a b

Continued v1 + _ v R1 i v2 R2 a b vN RN v + _ i Req a b

Parallel Resistors i a b R1 + _ R2 v i1 i2 i a b Req or Geq + _ v

Current Division i a b R1 + _ R2 v i1 i2 i a b Req or Geq + _ v

Continued i a b R1 + _ R2 v i1 i2 RN iN i a b Req or Geq + _ v

Brief Summary i1 i2 iN v1 + _ v R1 i v2 R2 a b vN RN i a b R1 + _ R2 v

Example Req 6 3 5 8 2 4 1 Req 2 6 8 4 Req 2.4 8 4 Req 14.4

How to solve the bridge network? Resistors are neither in series nor in parallel. Can be simplified by using 3-terminal equivalent networks. R1 + _ vS R2 R3 R4 R5 R6

Wye (Y)-Delta () Transformations 1 2 3 4 R3 R1 R2 3 4 1 2 Y T Rb Rc 1 2 3 4 Ra Rb Rc 1 2 3 4 Ra  

 to Y Conversion R3 R1 R2 3 4 1 2 Y Rb Rc 1 2 3 4 Ra 

Y to  Conversion R3 R1 R2 3 4 1 2 Y Rb Rc 1 2 3 4 Ra 

Example Rab 12.5 15 5 10 30 20 a b Rab 12.5 15 17.5 70 30 a b 35 Rab 7.292 10.5 21 a b Rab 9.632 a b