Understanding Capacitors and Inductors in Circuits

1. Chapter 6 Capacitors and Inductors

2. Introduction • Capacitors and inductors • Passive elements in a linear circuit. • Both stores energy.

3. Capacitors • Consist of two conducting plates separated by an insulator. • If it is connected to a voltage source, the source deposits: • +ve charge q on one plate. • -ve charge –q on another plate.

4. Capacitors

5. Capacitors • Capacitors store electric charge. • The amount of charge (q) stored is directly proportional to the applied voltage (v). q = Cv C = known as the capacitance of the capacitor, measured in Farad.

6. Capacitors • Passive sign convention: • Current flows: • Into the +ve terminal of the capacitor when it is charging. • Out of the +ve terminal when it is discharging. • Previously: i=dq/dt and q=Cv i=dCv/dt == Cdv/dt • Current-voltage relationship: i =Cdv/dt

7. Capacitors • Voltage-current relationship v = = where v(t0) = q(t0)/C is the voltage across capacitor at time t0 • Instantaneous power: p = vi = Cvdv/dt

8. Capacitors • Energy stored: = = Thus: Note that v(- )=0 because the capacitor was uncharged at t=-

9. Capacitors - Summary • Charge (q) = cv • Current (i) = cdv/dt • Voltage (v) = • Power (p) = vi= cvdv/dt • Energy (w) =

10. Capacitors • Property 1: • A capacitor is an open circuit at dc • If a voltage is constant (not changing with time), the current across it is 0. • A capacitor is an open circuit when the voltage across capacitor is not charging with time constant, current through is zero I = C dv/dt => if dv/dt = 0 (not charging), I = 0 (open circuit). • If a battery (dc voltage) is connected across a capacitor, the capacitor charges.

11. Capacitors • Property 2: • The voltage on one capacitor must be continuous. • No instantaneous change of voltage on capacitor • But current may change instantaneously. allowed not allowed

12. Capacitors • Property 3: • An ideal capacitor does not dissipate energy. • It takes power from the circuit and stores it. • It returns previously stored energy when delivering power.

13. Capacitors • Example 1: • What is the voltage across 3µF capacitor if the charge on one plate is 0.12 mC. • How much energy is stored? • Ans = 40v,2.4mJ

14. Capacitors • Example 2: • If a 10 µF capacitor is connected to a voltage source with v(t)=50 sin 2000t; find the current through the capacitor. Answer = (cos2000t)A.

15. Capacitors • Example 3: • An initially uncharged 1 mF capacitor has the current as below. Calculate the voltage across it at t=2 ms and at t=5ms. ans = 100mV, 400mV.

16. Capacitors • Example 4: • Under dc conditions, find the energy stored in the capacitors. Ans = 90µJ,405µJ.

17. Parallel Capacitors The equivalent capacitance of N-parallel-connected capacitors is the sum of the individual capacitances.

18. Series Capacitors  thus The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

19. Capacitors • Example 5: • Determine the equivalent capacitance seen at the terminals of the circuit. Ans = 40µF.

20. Capacitors • Example 6: • Find the voltages across each of the capacitors. Ans = v1 = 30, v2 = 30, v3 = 10, v4 = 20

21. Inductors

22. Inductors • It is a passive elements and it stores energy in its magnetic field. • Voltage-current relationship: • The voltage across an inductor is directly proportional to the time rate of change of a current flowing through it. L=Constant of proportionality, known as inductance, Measured in Henry.

23. Inductors • Current-voltage relationship: Where i(t0) is the total current for and i( )=0. • Power (p) = vi = or

24. Inductors • Energy stored: Since i( ) = 0

25. Inductors - Summary • Voltage (v) = • Current (i) = • Power (p) = vi= • Energy (w) =

26. Inductors • Property 1: • An inductor acts like a short circuit at dc. • Voltage across inductor is 0 when current is constant.

27. Inductors • Property 2: • The current through an inductor cannot change instantaneously • However, voltage may change instantaneously. allowed not allowed

28. Inductors • Property 3: • An ideal inductor does not dissipate but stores energy. • It takes power from the circuit and stores it. • It returns previously stored energy when delivering power.

29. Inductors • Example 1: • If the current through 1 mH inductor is i(t)=20 cos 100t mA, find the terminal voltage and the energy stored. • Ans = (-2sin100t)mV, (0.2cos 100t)µj 2

30. Inductors • Example 2: • The terminal voltage of 2H inductor is v=10(1-t) v. • Find the current flowing through it at t=4s and energy stored in it within 0<t<4s. • Assume i(0)=2A. • Ans = -18A, 320J.

31. Inductors • Example 3: • Determine VC, iL and the energy stored in the capacitor and inductor in the circuit under dc conditions. ans = 3A, 3V, 9J, 1.125J.

32. Series Inductors  • The equivalent inductance of series connected inductors is the sum of the individual inductances. • Similar to resistors connected in series.

33. Parallel Inductors   • The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances. • Similar to resistors connected in parallel.

34. Inductors • Example 4: • Calculate equivalent inductance for the inductive ladder network. ans = 25mF.

35. Inductors • Example 5: • In the circuit; i1(t)=0.6e-2tA. If i(0)=1.4A; find: • i2(0) • i2 (t) and i(t) • v(t), v1 (t) and v2 (t)