Basic Laws

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

2. 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

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

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

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

6. 1 S = 1 = 1 A/V  mho siemens Conductance and Power Dissipation • Conductance G is represented by A positiveR results in power absorption. A negativeR results in power generation.

7. a b + + _ _ c a b c 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. redrawn

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

9. 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)

10. i1 i2 in 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.

11. Proof of KCL

12. i5 i1 i4 i2 i3 Example 1

13. IT I1 I2 I3 IT Example 2

14. Case with A Closed Boundary Treat the surface as a node

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

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

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

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

19. Example 4 + + v2 v3 _ _ 8 i1 i3 a _ v1 + i2 + 6 3 30V _ Loop 1 Loop 2 b Q: Find currents and voltages. Sol:

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

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

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

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

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

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

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

27. 4 1 2 Req 5 6 3 8 4 4 2 Req 6 Req Req 2 14.4 2.4 8 8 Example

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

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

30. 1 3 R1 R2  R3 2 4 Rc 1 3 Rb Ra Y 2 4  to Y Conversion

31. Y- Transformations

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

33. Applications: Lighting Systems

34. Applications: DC Meters Parameters: IFS and Rm

35. Continued

36. Continued

37. Voltmeters Single-range Multiple-range

38. Ammeters Single-range Multiple-range