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 R v +_+_ i 1  = 1 V/A Cross-section area A Meterial resistivity  ohm

Resistors R = 0 v = 0 +_+_ i R =  v +_+_ i = 0 Variable resistorPotentiometer (pot) Open circuitShort circuit

Nonlinear Resistors i v Slope = R v i Slope = R(i) or R(v) Linear resistorNonlinear resistor 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 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. + _ a c b + _ c ba redrawn

Continued Elements in parallelElements in series –(10V, 5  ) Elements in parallel –(2 , 3 , 2A) Neither –((5  /10V), (2  /3  /2A)) 10V 55 22 33 2A2A + _

Kirchhoff’s Laws Introduced in 1847 by German physicist G. R. Kirchhoff ( ). 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) i1i1 i2i2 inin 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.

Proof of KCL

Example 1 i1i1 i3i3 i2i2 i4i4 i5i5

Example 2 I1I1 I2I2 I3I3 ITIT ITIT

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. v1v1 + _ v2v2 + _ vmvm + _

Example 1 v4v4 v1v1 v5v5 + _ + _ + _ v2v2 + _ v3v3 + _ Sum of voltage drops = Sum of voltage rises

Example 2 V3V3 V2V2 V1V1 V ab + _ + _ + _ + _ a b + _ + _ a b

Example 3 Q: Find v 1 and v 2. Sol: v1v1 + _ v2v2 + _ 20V 22 33 + _ i

Example 4 Q: Find currents and voltages. Sol: v1v1 + _ 30V 88 33 + _ i1i1 66 + _ v3v3 i3i3 i2i2 Loop 1Loop 2 a + _ v2v2 b

Series Resistors v1v1 + _ v R1R1 + _ i v2v2 + _ R2R2 a b v + _ i v + _ R eq a b

Voltage Division v1v1 + _ v R1R1 + _ i v2v2 + _ R2R2 a b v + _ i v + _ R eq a b

Continued v + _ i v + _ R eq a b v1v1 + _ v R1R1 + _ i v2v2 + _ R2R2 a b vNvN + _ RNRN

Parallel Resistors i a b R1R1 + _ R2R2 v i1i1 i2i2 i a b R eq or G eq + _ v v

Current Division i a b R1R1 + _ R2R2 v i1i1 i2i2 i a b R eq or G eq + _ v v

Continued i a b R eq or G eq + _ v v i a b R1R1 + _ R2R2 v i1i1 i2i2 RNRN iNiN

Brief Summary i a b R1R1 +_ R2R2 v i1i1 i2i2 RNRN iNiN v1v1 + _ v R1R1 + _ i v2v2 + _ R2R2 a b vNvN + _ RNRN

Example R eq R eq 14.4

How to solve the bridge network? R1R1 +_ vSvS R2R2 R3R3 R4R4 R5R5 R6R6 Resistors are neither in series nor in parallel. Can be simplified by using 3-terminal equivalent networks.

Wye (Y)-Delta (  ) Transformations R3R3 R1R1 R2R R3R3 R1R1 R2R RbRb RcRc RaRa RbRb RcRc RaRa Y T  

 to Y Conversion R3R3 R1R1 R2R Y RbRb RcRc RaRa 

Y-  Transformations

Example R ab a b R ab a b 35 R ab a b R ab a b

Applications: Lighting Systems

Applications: DC Meters Parameters: I FS and R m

Continued

Voltmeters Single-range Multiple-range

Ammeters Single-range Multiple-range