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DET 101 - Electric Circuit Fundamentals I

DET 101 - Electric Circuit Fundamentals I. CHAPTER 4 INDUCTANCE & CAPACITANCE. Capacitors. A capacitor is a passive element designed to store energy in its electric field. A capacitor consists of two conducting plates separated by an insulator (or dielectric).

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DET 101 - Electric Circuit Fundamentals I

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  1. DET 101 - Electric Circuit Fundamentals I CHAPTER 4 INDUCTANCE & CAPACITANCE

  2. Capacitors • A capacitor is a passive element designed to store energy in its electric field. • A capacitor consists of two conducting plates separated by an insulator (or dielectric).

  3. In many practical application: (a) Plates maybe aluminum foil (b) Dielectric maybe air, ceramic, paper or mica When a voltage source, v is connected to the capacitor, the source deposits a positive charge q on one plate and a negative charge –q on the other

  4. The capacitor is said to store energy and the amount of charge stored, represent by q is directly proportional to the applied voltage, v • CapacitanceC is the ratio of the charge q on one plate of a capacitor to the voltage difference v between the two plates, measured in farads (F). • The unit of capacitance is the farad, in honor of the English physicist Michael Faraday.

  5. 1 farad = 1 coulomb/volt • Where  is the permittivity of the dielectric material between the plates, A is the surface area of each plate, d is the distance between the plates. • Unit: F, pF (10–12), nF (10–9), and F (10–6)

  6. Three factors determine the value of the capacitance: • The surface area of the plate – the larger the area, the greater the capacitance • The spacing between the plates – the smaller the spacing, the greater the capacitance • The permittivity of the material – the higher the permittivity, the greater the capacitance

  7. If i is flowing into the +ve terminal of C • Charging => i is +ve • Discharging => i is –ve • The current-voltage relationship of capacitor according to above convention is V(t0) is the voltage across the capacitor at time t0

  8. The instantaneous power delivered to the capacitor • The energy, w, stored in the capacitor is

  9. A capacitor is • an open circuit to dc (dv/dt = 0). • its voltage cannot change abruptly • (It opposes the change in voltage.

  10. Example 1 Calculate the charge stored on a 3pF capacitor with 20V across it. Find also the energy stored in the capacitor

  11. Example 2 An initially uncharged 1mF capacitor has the current shown below across it. Calculate the voltage across it at t = 2 ms and t = 5 ms. Answer: v(2ms) = 100 mV v(5ms) = 500 mV

  12. Voltage across from t=0 untill t=2ms

  13. Voltage across from t=2ms until t=5ms

  14. Example 3 If a 10μF capacitor is connected to a voltage source with v(t)=50 sin(2000t)V determine the current through the capacitor

  15. The equivalent capacitance of Nparallel-connected capacitors is the sum of the individual capacitances.

  16. The equivalent capacitance of Nseries-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

  17. Example 4 Find the equivalent capacitance seen at the terminals of the circuit in the circuit shown below: Answer: Ceq = 40F

  18. 60μF series with 120μF 40μF parallel with 20μF 60μF series with 120μF 50μF parallel with 70μF

  19. Answer: v1 = 30V v2 = 30V v3 = 10V v4 = 20V Example 4 Find the voltage across each of the capacitors in the circuit shown below:

  20. Find Ceq 60μF and 30μF - series 20μF and 20μF - parallel 40μF and 40μF - series Imagine that charge act like current

  21. 60μF and 30μF - series

  22. EXERCISE 1 • Determine the equivalent capacitance for each of the circuit Ans: (a) 3F (b) 8F (c) 1F

  23. EXERCISE 2 • Determine the equivalent capacitance at terminal a-b Ans: 2.5uF

  24. EXERCISE 3 • Determine (a) the voltage across each capacitor (b) energy stored in each capacitor Ans: v30μF = 90V w30μF = 121.5mJ v60μF = 30V w60μF = 27mJ v14μF = 60V w14μF = 25.2mJ v20μF = 48V w20μF = 23.04mJ v80μF =12V w80μF = 5.76mJ

  25. Inductance • Inductor • Relationship between voltage, current, power and energy • Series-parallel combinations for inductance

  26. Inductors • An inductor is a passive two terminal electrical component designed to store energy in its magnetic field. • A practical inductor is usually formed into a cylindrical coil with many turns of conducting wire

  27. Inductors • Inductance L is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H). • The unit of inductors is Henry (H), mH (10–3) and H (10–6). • Inductance of an inductor depends on its physical dimension and construction.

  28. Inductance • Inductance, L • µ depends on the material of core. The core may be made of iron, steel or plastic L = inductance in henrys (H). N = number of turns µ = core permeability A = cross-sectional area (m2) ℓ = length (m)

  29. Inductance can be increased by • Increasing the number of turns of coil • using material with higher permeability as the core • increasing the cross-sectional area • reducing the length of coil

  30. Relationship between voltage, current, power and energy Inductor symbol Inductor Voltage

  31. Inductor current Where i(t0) is the total current for Power

  32. Assuming that energy is zero at time t= ,then inductor energy is:

  33. An inductor • Act s like a short circuit to dc • The current through an inductor cannot change instantaneously • Voltage across an inductor can change abruptly • An ideal capacitor does not dissipate energy • Practically – non-ideal inductor has significant resistive components so it will dissipate energy

  34. Example 5 The terminal voltage of a 2-H inductor is v = 10(1-t) V Find the current flowing through it at t = 4 s and the energy stored in it within 0 < t < 4 s. Assume i(0) = 2 A. Answer: i(4s) = -18A w(4s) = 320J

  35. v = 10(1-t) V, L=2H 0 < t < 4 s. Assume i(0) = 2 A

  36. v = 10(1-t) V, L=2H 0 < t < 4 s. Assume i(0) = 2 A

  37. Example 6 If the current through a 1mH inductor is i(t)=20 cos 100t mA, find the terminal voltage and energy stored

  38. To find terminal voltage

  39. To find energy stored

  40. Example 7 Determine vc,iL, and the energy stored in the capacitor and inductor in the circuit of circuit shown below under dc conditions. Answer: iL = 3A vC = 3V wL = 1.125J wC = 9J

  41. In dc, inductor act like short circuit and capacitor act like an open circuit Using current divider At circuit

  42. Vc is equals to voltage at resistor 1Ω Energy stored in capacitor Energy stored in inductor

  43. Series and Parallel Inductors • The equivalent inductance of series-connected inductors is the sum of the individual inductances.

  44. Series and Parallel Inductors • The equivalent capacitance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

  45. Example 8 Calculate the equivalent inductance for the inductive ladder network in the circuit shown below: Answer: Leq = 25mH

  46. SUMMARY • Current and voltage relationship for R, L, C + + +

  47. Example 9 In the circuit i1(t)=0.6e-2t A. If i(0)=1.4A find • (a) i2(0) • (b) i2(t) and i(t) • (c) v1(t), v2(t) and v(t)

  48. (a) i2(0) i1(t)=0.6e-2t i(0)=1.4A At circuit,

  49. (b) i2(t) and i(t) i1(t)=0.6e-2t

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