1. Chapter 15 Series and Parallel AC Circuits

2. Complex Number Review A= x 2 + y 2 ϕ= tan −1 y x
𝑥=𝑍 cos⁡(𝜙), 𝑦=𝑍 sin 𝜙
−𝑍∠𝜙=𝑍∠𝜙∓180
𝑍 1 = 𝐴 1 ∠ 𝜙 1
𝑍 2 = 𝐴 2 ∠ 𝜙 2
𝑍 1 𝑍 2 = 𝐴 1 𝐴 2 ∠ 𝜙 1 − 𝜙 2
𝑍 1 ∗ 𝑍 2 = 𝐴 1 ∗ 𝐴 2 ∠ 𝜙 1 + 𝜙 2

3. Introduction

4. Example

5. Impedance
The impedance (𝑍) is a measure of how much the element will “impede” the flow of charge through the network.
The total impedance combine the effect of the resistance, inductive reactance, and capacitive reactance

6. Impedance (R)
R does not vary with time

8. Impedance of an inductive element
Inductive Reactance: 𝑋 𝐿 =𝑤𝐿→ 𝑍 𝐿 =𝑋 𝐿 ∠ 90 𝑜 =𝑗 𝑋 𝐿
For a pure inductor, the current lags the voltage

10. Impedance of a capacitive element
Capacitive Reactance: 𝑋 𝐶 = 1 𝑊𝐶 → 𝑍 𝐶 = 𝑋 𝐶 ∠− 90 𝑜 =−𝑗 𝑋 𝐶
For a pure capacitor, the current leads the voltage

12. Impedance

13. Series Impedance

14. Series Impedance

15. Series Circuit

16. Example 𝐸= 141.4 2 ∠0=100∠ 0 𝑜 𝑍 𝑡 =3+𝑗4=5∠ 53.13 𝑜
𝐸= ∠0=100∠ 0 𝑜
𝑍 𝑡 =3+𝑗4=5∠ 𝑜
𝐼= 𝐸 𝑍 𝑡 =20∠−53.13
𝑉 𝐿 =4∠90 ×𝐼=80∠ 𝑜
𝑉 𝑅 =60∠−53.13
𝑖= 2 ×20 sin 𝑤𝑡−53.13
𝑣 𝐿 = 2 ×80 sin 𝑤𝑡+36.87
𝑣 𝑅 = 2 ×60 sin 𝑤𝑡−53.13

17. Example (cont.)

18. Example 𝐸= 70.7 2 ∠0=50∠ 0 𝑜 𝑍 𝑡 =3+𝑗7−𝑗3=5∠ 53.13 𝑜
𝐸= ∠0=50∠ 0 𝑜
𝑍 𝑡 =3+𝑗7−𝑗3=5∠ 𝑜
𝐼= 𝐸 𝑍 𝑡 =10∠−53.13
𝑉 𝐿 =7∠90 ×𝐼=70∠ 𝑜
𝑉 𝐶 =3∠−90 ×𝐼=30∠−143.13
𝑉 𝑅 =30∠−53.13
𝑖= 2 ×10 sin 𝑤𝑡−53.13
𝑣 𝐿 = 2 ×70 sin 𝑤𝑡+36.87
𝑣 𝐶 = 2 ×30 sin 𝑤𝑡−143.13
𝑣 𝑅 = 2 ×30 sin 𝑤𝑡−53.13

19. Example

20. Voltage Divider

21. Example 𝐸= 70.7 2 ∠0=50∠ 0 𝑜 𝑍 𝑡 =3+𝑗7−𝑗3=5∠ 53.13 𝑜
𝐸= ∠0=50∠ 0 𝑜
𝑍 𝑡 =3+𝑗7−𝑗3=5∠ 𝑜
𝑉 𝐿 =50∠0× 7∠90 5∠ 𝑜 =70∠ 𝑜
𝑣 𝐿 = 2 ×70 sin 𝑤𝑡+36.87

22. Parallel Circuit

23. Parallel Circuit

24. Parallel Circuit Conductance: 𝑌 𝑅 = 1 𝑍 𝑅 = 1 𝑅∠0 =𝐺∠0, 𝐺= 1 𝑅
Susceptance:
𝑌 𝐿 = 1 𝑍 𝐿 = 1 𝑋 𝐿 ∠90 = 𝐵 𝐿 ∠−90, 𝐵 𝐿 = 1 𝑋 𝐿
𝑌 𝐶 = 1 𝑍 𝐶 = 1 𝑋 𝐶 ∠−90 = 𝐵 𝐶 ∠90, 𝐵 𝐶 = 1 𝑋 𝐶
Unit is in Siemens (S)

25. Example

26. Example

27. Current Divider Rule

28. Equivalent Circuit

32. Series-Parallel Circuit

33. Example

34. Ladder Network

35. Find 𝑍 𝑇
Calculate 𝐼 𝑆
Calculate 𝑉 𝐿
The average power delivered by the circuit
The power factor