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DC CIRCUITS:

DC CIRCUITS:. CHAPTER 4. Capacitors and Inductors. Introduction Capacitors: terminal behavior in terms of current, voltage, power and energy Series and parallel capacitors Inductors: terminal behavior in terms of current, voltage, power and energy Series and parallel inductors.

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DC CIRCUITS:

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  1. DC CIRCUITS: CHAPTER 4

  2. Capacitors and Inductors • Introduction • Capacitors: terminal behavior in terms of current, voltage, power and energy • Series and parallel capacitors • Inductors: terminal behavior in terms of current, voltage, power and energy • Series and parallel inductors

  3. Introduction • Two more linear, ideal basic passive circuit elements. • Energy storage elements stored in both magnetic and electric fields. • They found continual applications in more practical circuits such as filters, integrators, differentiators, circuit breakers and automobile ignition circuit. • Circuit analysis techniques and theorems applied to purely resistive circuits are equally applicable to circuits with inductors and capacitors.

  4. Capacitors • Electrical component that consists of two conductors separated by an insulator or dielectric material. • Its behavior based on phenomenon associated with electric fields, which the source is voltage. • A time-varying electric fields produce a current flow in the space occupied by the fields. • Capacitance is the circuit parameter which relates the displacement current to the voltage.

  5. A capacitor with an applied voltage Plates – aluminum foil Dielectric – air/ceramic/paper/mica

  6. Circuit symbols for capacitors (a) Fixed capacitor (b) Variable capacitor

  7. Circuit parameters (1) • The amount of charge stored, q = CV. • C is capacitance in Farad, ratio of the charge on one plate to the voltage difference between the plates. But it does not depend on q or V but capacitor’s physical dimensions i.e.,  = permeability of dielectric in Wb/Am A = surface area of plates in m2 d = distance btw the plates m

  8. Current – voltage relationship of a capacitor • To obtain the I-V characteristic of a capacitor, we differentiate both sides of eq.(1) . We obtain, • Integrating both sides of eq.(2) we obtain, (2) (3)

  9. Instantaneous power and energy for capacitors • The instantaneous power delivered to a capacitor is, • The energy stored in the capacitor, • At V(-∞) = 0 (cap. uncharged at t = -∞, hence (4) (5) (6) or

  10. Important properties of a capacitor • A capacitor is an open circuit to dc. • When the voltage across capacitor is not changing with time (constant), current thru it is zero. • The voltage on a capacitor cannot change abruptly. - The voltage across capacitor must be continuous. Conversely, the current thru it can change instantaneously.

  11. Practice problem 6.1 • What is the voltage across a 3-F capacitor if the charge on one plate is 0.12 mC? How much energy is stored? (Ans: 40V, 2.4mJ)

  12. Practice problem 6.2 • If a 10-F capacitor is connected to a voltage source with v(t) = 50sin2000t V • Calculate the current through it. (Ans: cos2000t A)

  13. Practice problem 6.3 • The current through a 100-F capacitor is i(t) = 50sin120t mA. Calculate the voltage across it at t = 1 ms and t = 5 ms Take v(0) = 0. (Ans: 93.137V, 1.736V)

  14. Practice problem 6.4 • An initially uncharged 1-mF capacitor has the current shown in Figure 6.11 across it. Calculate the voltage across it at t = 2 ms and t = 5 ms. (Ans: 100mV, 400mV)

  15. Practice problem 6.5 • Under dc conditions, find the energy stored in the capacitors in Fig. 6.13. (Ans: 405J, 90 J)

  16. Series/parallel capacitances • Series-parallel combination is powerful tool for circuit simplification. • A group of capacitors can be combined to become a single equivalent capacitance using series-parallel rules.

  17. Parallel capacitances • The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances. (7) Equivalent circuit Parallel N-connected capacitors

  18. Series capacitances • The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances. (8) Series N-connected capacitors Equivalent circuit

  19. Practice problem 6.6 • Find the equivalent capacitance seen at the terminals of the circuit in Fig. 6.17. (Ans: 40F)

  20. Practice problem 6.7 • Find the voltage across each of the capacitors in Fig. 6.20 (Ans: 30V, 30V, 10V, 20V)

  21. Inductors • Electrical component that opposes any change in electrical current. • Composed of a coil or wire wound around a non-magnetic core/magnetic core. • Its behavior based on phenomenon associated with magnetic fields, which the source is current. • A time-varying magnetic fields induce voltage in any conductor linked by the fields. • Inductance is the circuit parameter which relates the induced voltage to the current.

  22. Typical form of an inductor

  23. Circuit symbols for inductors Variable iron-core Air-core iron-core

  24. Current – voltage relationship of an inductor • The voltage across an inductor, • L is the constant proportionality called inductance measured in Henry. • To obtain current integrate eq. (7), (9) (10)

  25. Instantaneous power and energy fir inductors • The instantaneous power delivered to a capacitor is, • The energy stored in the capacitor, • At V(-∞) = 0 (ind. uncharged at t = -∞, hence (11) (12) (13)

  26. Important properties of an inductor • An inductor acts like a short circuit to dc. • When the current thru inductor is not changing with time (constant), voltage across it is zero. • The current thru an inductor cannot change instantaneously. - An important property is its opposition to the change in current flowing thru it. However the voltage across it can change abruptly.

  27. Practice problem 6.8 • If the current through a 1-mH inductor is i(t) = 20cos100t mA, find the terminal voltage and the energy stored. (Ans: -2sin100t mV, 0.2cos2100t J)

  28. Practice problem 6.9 • The terminal voltage of a 2-H inductor is V = 10(1 – t) V. find the current flowing thru it at t=4s and the energy stored in it within 0 < t < 4s. Assume i(0)=2A. (Ans: -18 A, 320 J)

  29. Practice problem 6.10 • Determine VC, iL and the energy stored in the capacitor and inductor in the circuit below under dc conditions. (Ans: 3V, 3A, 1.125J)

  30. Series inductances • The equivalent inductance of N series-connected inductors is the sum of the individual inductances. (14) Series N-connected inductors Equivalent circuit

  31. Parallel inductances • The equivalent inductance of series-connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances. (15) Parallel N-connected inductors Equivalent circuit

  32. Practice problem 6.11 • Calculate the equivalent inductance for the inductive ladder network in Figure below. (Ans: 25 mH)

  33. Practice problem 6.12 • In the circuit of Figure below, given i1(t)=0.6e-2t. If i(0)=1.4A, find (a)i2(0); (b) i2(t) and i(t); (c) V1(t), V2(t) and V(t).

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