DC CIRCUITS:DC CIRCUITS:DC CIRCUITS:DC CIRCUITS:

CHAPTER 4CHAPTER 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

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.

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.

A capacitor with an applied voltage

Plates – aluminum foil

Dielectric – air/ceramic/paper/mica

Circuit symbols for capacitors

(a) Fixed capacitor

(b) Variable capacitor

Circuit parameters• 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.,

d

AC

= permeability of dielectric

in Wb/Am

A = surface area of plates in m2

d = distance btw the plates m

(1)

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,

dt

dVCi

t

t o

t

o

tVdtiC

dtiC

V

)(1

1

(2)

(3)

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)

2

2

1CVw (6)

C

qw

2

2

or

dt

dVCVVip

t

t

tt tCVdVVCdt

dt

dVVCdtpw

2

2

1

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.

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)

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)

Practice problem 6.3• The current through a 100-F

capacitor is i(t) = 50sin120t mA. Calculate the voltage across it at t = 1 ms and t = 5 ms Take v(0) = 0. (Ans: 93.137V, 1.736V)

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)

Practice problem 6.5• Under dc conditions, find the

energy stored in the capacitors in Fig. 6.13. (Ans: 405J, 90 J)

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.

Parallel capacitances• The equivalent capacitance of N

parallel-connected capacitors is the sum of the individual capacitances.

Neq CCCCC 321

Parallel N-connected capacitors

Equivalent circuit

(7)

Series capacitances• The equivalent capacitance of series-connected

capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

1113

12

11

Neq CCCCC (8)

Series N-connected capacitors

Equivalent circuit

Practice problem 6.6• Find the equivalent capacitance seen at the

terminals of the circuit in Fig. 6.17. (Ans: 40F)

50 F

20 F

60 F

70 F 120 FCeq

Practice problem 6.7• Find the voltage across each of the capacitors

in Fig. 6.20 (Ans: 30V, 30V, 10V, 20V)

40 F

30 F20 F

60 F

60 V

+ V1 - + V3 -+ V

2 -

+ V

4 -

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.

Typical form of an inductor

Circuit symbols for inductors

Air-core iron-coreVariable

iron-core

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),

dt

diLV (9)

t

t o

t

o

tidttVL

dttVL

i

)()(1

)(1

(10)

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

(12)t

t

tt tLidiiLdti

dt

diLdtpw

2

2

1

idt

diLVip

(11)

2

2

1Liw (13)

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.

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)

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)

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)

3 1

0.25 H

F4 A

+ V

C -

iL

Series inductances• The equivalent inductance of N series-

connected inductors is the sum of the individual inductances.

Neq LLLLL 321

Series N-connected inductors

Equivalent circuit

(14)

Parallel inductances• The equivalent inductance of series-connected

inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

1113

12

11

Neq LLLLL (15)

Parallel N-connected inductors

Equivalent circuit

Practice problem 6.11• Calculate the equivalent inductance

for the inductive ladder network in Figure below. (Ans: 25 mH)

20 mH 100 mH

50 mH

40 mH

40 mH 30 mH 20 mHLeq

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). 3 H

6 H8 H

+ V1 -

+ V2 -

i1

i2

+

V

-

i