Circuit Theory Chapter 6 Capacitors and Inductors Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Capacitors and Inductors Chapter 6 6.1 6.2 6.3 6.4 Capacitors Series and Parallel Capacitors
Inductors Series and Parallel Inductors 2 6.1 Capacitors (1) A capacitor is a passive element designed store energy in its electric field. to A capacitor consists of two conducting plates separated by an insulator (or dielectric). 3 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). q C v and A C d 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. F (106) Unit: F, pF (1012), nF (109), and 4
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 dv i C dt and 1
v C t i d t v(t ) t0 0 5 6.1 Capacitors (4) The energy, w, stored in the capacitor is 1
2 w Cv 2 A capacitor is an open circuit to dc (dv/dt = 0). its voltage cannot change abruptly. 6 6.1 Capacitors (5) Example 1 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) = 1.7361V 7 6.1 Capacitors (6) Example 2 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 8
6.2 Series and Parallel Capacitors (1) The equivalent capacitance of N parallelconnected capacitors is the sum of the individual capacitances. C eq C1 C 2 ... C N Current Divider Rule C1 i1 is C1 C2 C2 i2 is
C1 C2 9 6.2 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. 1 1 1 1 ... C eq C1 C 2 CN
Voltage Divider Rule C2 v1 vs C1 C2 C1 v2 vs C1 C2 10 6.2 Series and Parallel Capacitors (3) Example 3 Find the equivalent capacitance seen at the
terminals of the circuit in the circuit shown below: Answer: Ceq = 40F 11 6.2 Series and Parallel Capacitors (4) Example 4 Find the voltage across each of the capacitors in the circuit shown below: Answer: v1 = 30V v2 = 30V v3 = 10V v4 = 20V
12 6.3 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. 13 6.3 Inductors (2) Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H). di v L dt
and N2 A L l is the inductor core permeability The unit of inductors is Henry (H), mH (103) and H (106). 14 6.3 Inductors (3) The current-voltage relationship of an inductor: 1 i
L t v ( t ) d t i ( t ) 0
t0 The energy stored by an inductor: 1 w L i2 2 An inductor acts like a short circuit to dc (di/dt = 0) and its current cannot change abruptly. 15 6.3 Inductors (4) Example 5 The terminal voltage of a 2-H is v = 10(1-t) V
inductor 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 16 6.3 Inductors (5) Example 6 Determine vc, iL, and the energy stored in the capacitor and inductor in the circuit shown below under dc conditions.
Answer: iL = 3A vC = 3V wL = 1.125J wC = 9J 17 6.4 Series and Parallel Inductors (1) The equivalent inductance of series-connected inductors is the sum of the individual inductances. Leq L1 L2 ... LN Voltage Divider Rule L1 v1 vs
L1 L2 L2 v2 vs L1 L2 18 6.4 Series and Parallel Inductors (2) The equivalent capacitance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances. Current Divider Rule L2 i1
is L1 L2 1 1 1 1 ... Leq L1 L2 LN L1 i2 is L1 L2 19
6.4 Series and Parallel Inductor (3) Example 7 Calculate the equivalent inductance for the inductive ladder network in the circuit shown below: Answer: Leq = 25mH HW6 Ch6: 9, 13, 25, 59, 61, 65, 73, 79 20 6.4 Current and voltage relationship Current and voltage relationship for R, L, C
+ + + 21