1 Chapter 6 Capacitors and Inductors 電路學 ( 一 )

2 Capacitors and Inductors Chapter 6 6.1Capacitors 6.2Series and Parallel Capacitors 6.3Inductors 6.4Series and Parallel Inductors

3 6.1 Capacitors (1) 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).

4 6.1 Capacitors (2) Capacitance C 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). 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 and F (10 –6 )

5 6.1 Capacitors (3) 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 and

6 6.1 Capacitors (4) The energy, w, stored in the capacitor is A capacitor is –an open circuit to dc (dv/dt = 0). –its voltage cannot change abruptly. –The ideal capacitor does not dissipate energy.

7 6.1 Capacitors (5) Example 1 (a)Calculate the charge stored on a 3-pF capacitor with 20 V across it. (b)Find the energy stored in the capacitor.

8 6.1 Capacitors (6) Example 2 The current through a 100-F capacitor is i(t) = 50 sin(120 t) mA. Calculate the voltage across it at t =1 ms and t = 5 ms. Take v(0) =0. Answer: v(1ms) = 93.14mV v(5ms) = V

9 6.1 Capacitors (7) Example 3 An initially uncharged 1-mF 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

Capacitors (8) Example 4 Obtain the energy stored in each capacitor in the figure under dc conditions.

Capacitors (9) Example 5 Under dc condition, find the energy stored in the capacitors in the figure.

Series and Parallel Capacitors (1) The equivalent capacitance of N parallel- connected capacitors is the sum of the individual capacitances.

Series and Parallel Capacitors (2) The equivalent capacitance of N series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

Series and Parallel Capacitors (3) Example 6 Find the equivalent capacitance seen at the terminals of the circuit in the circuit shown below: Answer: C eq = 40F

Series and Parallel Capacitors (4) Example 7 Find the voltage across each of the capacitors in the circuit shown below: Answer: v 1 = 15 V v 2 = 10 V v 3 = 5 V

Series and Parallel Capacitors (5) Example 8 Find the voltage across each of the capacitors in the circuit shown below:

Inductors (1) An inductor is a passive element designed to store energy in its magnetic field. An inductor consists of a coil of conducting wire.

Inductors (2) Inductance 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 ). and

Inductors (3) The current-voltage relationship of an inductor: The power stored by an inductor: An inductor acts like a short circuit to dc (di/dt = 0) The current through an inductor cannot change abruptly.

Inductors (4) Example 9 The current through a 0.1-H inductor is i(t) = 10te -5t A Find the voltage across the inductor and the energy stored in it.

Inductors (5) Example 10 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.

Inductors (6) Example 11 Consider the circuit, under dc conditions, find: (a) i, v C, and i L (b) the energy stored in the capacitor and inductor. Answer: i = 2 A v C = 10 V w L = 4 J w C = 50 J

Inductors (7) Example 12 Determine v c, i L, and the energy stored in the capacitor and inductor in the circuit of circuit shown below under dc conditions.

Series and Parallel Inductors (1) The equivalent inductance of series-connected inductors is the sum of the individual inductances.

Series and Parallel Inductors (2) The equivalent capacitance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

Series and Parallel Inductors (3) Example 13 Calculate the equivalent inductance for the inductive ladder network in the circuit shown below: Answer: L eq = 25mH

Series and Parallel Inductors (4) Example 14 For the circuit, i= 4(2 – e -10t ) mA. If i 2 (0) = -1 mA, find: (a) i 1 (0); (b) v(t), v 1 (t), and v 2 (t); (c) i 1 (t) and i 2 (t)

Series and Parallel Inductors (5) Current and voltage relationship for R, L, C + + +

29 at t = 0 −, inductance acts as a short circuit at t = 0 +, i L cannot change instantaneously. +v2−+v2− +vL−+vL−

30 at t = 0 −, capacitor acts as a open circuit at t = 0 +, v c cannot change instantaneously. icic i2i2

31 at t = 0 −, i L (0 − ) = 10/5 = 2 A v c (0 − ) = i L (0 − ) 3 = 6 V at t = 0 +, i L (0 + ) = i L (0 − ) = 2 A v c (0 + ) = v c (0 − ) = 6 V i c (0 − ) = 0 v L (0 − ) = 0 i c (0 + ) = −i L (0 + ) = −2 A v L (0 + ) = v c (0 + ) − i L (0 + ) 3 = 0 icic +vL−+vL−

32 Example Find i L (0 + ), v c (0 + ), dv c (0 + )/dt, and di L (0 + )/dt i L (0 + ) = i L (0 − ) = 0 v c (0 + ) = v c (0 − ) = −2 V

Applications (1) Integrator An op amp circuit whose output is proportional to the integral of the input signal

34 Example 15 If v 1 = 10 cos 2t mV and v 2 = 0.5t mV, find v o in the op amp circuit. Assume the voltage across the capacitor is initially zero. 6.5 Applications (2)

Applications (3) Differentiator An op amp circuit whose output is proportional to the rate of change of the input signal

36 Example 16 Sketch the output voltage for the circuit, given the input voltage in the figure. Take v o = 0 at t = Applications (4) v i (V)

Applications (5) Analog Computer

Applications (6)

Applications (7)

Applications (8)