1. CHAPTER I APPLICATION OF CIRCUIT LAWS

2. 2 Introduction Generally, we require 3 steps to analyze AC Circuit Transform the circuit to the phasor / frequency domain Solve the problem using any technique Transform the resulting phasor / frequency domain to the time domain expression.

3. 3 Time domain to Phasor

4. 4 Impedance in Frequency Domain ELEMENTIMPEDANCEFORMULA PHASOR FORM RECTANGULAR FORM RZRZR R R  0 R + j0 LZLZL jωL X L  90 0 + jX L CZCZC 1/jωC X C  -90 0 - jX C

5. 5 Methods of solving Kirchhoff’s Voltage Law (KVL) Kirchhoff’s Current Law (KCL) Voltage Divider Rule (VDR) Current Divider Rule (CDR) Star / Delta Transformation (  /  )

6. 6 Kirchhoff’s Voltage Law (KVL) Algebraic sum of voltage drops around closed loop is zero  V = 0 Voltage drop = voltage rise V= V 1 + V 2 + … + V N = I (Z 1 + Z 2 + … + Z N )

7. 7 Kirchhoff’s Current Law (KCL) Algebraic sum of current at any node is zero  I = 0 Current In = Current Out I = I 1 + I 2 + … + I N = V (1 / Z 1 + 1 / Z 2 + … + 1 / Z N )

8. 8 Voltage Divider Rule (VDR)

9. 9 Current Divider Rule (CDR)

10. 10 Star / Delta (  /  ) Transformation  -  Conversion

11. 11 Delta / Star (  / Y ) Transformation  - Y Conversion

12. 12 Example 1 Find the input impedance, Z in of the circuit (ω=50rad/s)

13. 13 Solution Example 1 Let Z 1 = Impedance of the 2mF capacitor Z 2 = Impedance of the 3  resistor in series with the 10mF capacitor Z 3 = Impedance of the 0.2H inductor in series with 8  resistor Then

14. 14 Solution Example 1 The input impedance is

15. 15 Example 2 Find the input impedance, Z in of the circuit (ω=10rad/s)

16. 16 Solution Example 2 Let Z 1 = Impedance of the 2mF capacitor in series with 20  resistor Z 2 = Impedance of the 4mF capacitor Z 3 = Impedance of the 2H inductor in series with 50  resistor Then

17. 17 Solution Example 2 The input impedance is

18. 18 Example 3 Determine V o (t)

19. 19 Solution example 3 Step 1: Transfer the circuit into frequency domain

20. 20 Solution Example 3 Step 2: Solve the circuit using any method Let Z 1 = Impedance of the 60  resistor Z 2 = Impedance of the parallel combination of the 10mF capacitor and 5H inductor Then

21. 21 Solution Example 3 By using voltage divider rule Step 3: Convert the result to the time domain

22. 22 Example 4 Determine V o (t)

23. 23 Solution Example 4 Step 1: Transfer to the frequency domain Voltage source  0.5H inductor  (1/20)F capacitor 

24. 24 Solution Example 4 Step 2: Solve the circuit using any method Let Z 1 = Impedance of the 0.5H inductor in parallel with the 10  resistor Z 2 = Impedance of the (1/20)F capacitor Then

25. 25 Solution Example 4 By using voltage divider rule Step 3: Convert the result to the time domain

26. 26 Example 5 Find current I

27. 27 Solution Example 5 Step 1: Transform the circuit from delta to star connection (  to Y )

28. 28 Solution Example 5 Calculate new impedances after the transformation

29. 29 Solution Example 5 The total input impedance is

30. 30 Solution Example 5 The desire current is

31. 31 Example 6 Find current I

32. 32 Solution Example 6 Step 1: Transform the circuit from delta to star connection (  to Y )

33. 33 Solution Example 6 Calculate new impedances after the transformation

34. 34 Solution Example 6 The total input impedance is

35. 35 Solution Example 6 The desire current is