DC CIRCUITS CHAPTER 4 Capacitors and Inductors Introduction

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DC CIRCUITS: CHAPTER 4

DC CIRCUITS: CHAPTER 4

Capacitors and Inductors • Introduction • Capacitors: terminal behavior in terms of current, voltage,

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

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

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

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 symbols for capacitors (a) Fixed capacitor (b) Variable capacitor

Circuit parameters • The amount of charge stored, q = CV. (1) • C

Circuit parameters • The amount of charge stored, q = CV. (1) • 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 m 2 d = distance btw the plates m

Current – voltage relationship of a capacitor • To obtain the I-V characteristic of

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

Instantaneous power and energy for capacitors • The instantaneous power delivered to a capacitor

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

Important properties of a capacitor • A capacitor is an open circuit to dc.

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

Practice problem 6. 1 • What is the voltage across a 3 - F capacitor if the charge on one plate is 0. 12 m. C? How much energy is stored? (Ans: 40 V, 2. 4 m. J)

Practice problem 6. 2 • If a 10 - F capacitor is connected to

Practice problem 6. 2 • If a 10 - F capacitor is connected to a voltage source with v(t) = 50 sin 2000 t V • Calculate the current through it. (Ans: cos 2000 t A)

Practice problem 6. 3 • The current through a 100 - F capacitor is

Practice problem 6. 3 • The current through a 100 - F capacitor is i(t) = 50 sin 120 t m. A. Calculate the voltage across it at t = 1 ms and t = 5 ms Take v(0) = 0. (Ans: 93. 137 V, 1. 736 V)

Practice problem 6. 4 • An initially uncharged 1 -m. F capacitor has the

Practice problem 6. 4 • An initially uncharged 1 -m. F 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: 100 m. V, 400 m. V)

Practice problem 6. 5 • Under dc conditions, find the energy stored in the

Practice problem 6. 5 • Under dc conditions, find the energy stored in the capacitors in Fig. 6. 13. (Ans: 405 J, 90 J)

Series/parallel capacitances • Series-parallel combination is powerful tool for circuit simplification. • A group

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 parallelconnected capacitors is the sum of

Parallel capacitances • The equivalent capacitance of N parallelconnected capacitors is the sum of the individual capacitances. (7) Parallel Nconnected capacitors Equivalent circuit

Series capacitances • The equivalent capacitance of series-connected capacitors is the reciprocal of the

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

Practice problem 6. 6 • Find the equivalent capacitance seen at the terminals of

Practice problem 6. 6 • Find the equivalent capacitance seen at the terminals of the circuit in Fig. 6. 17. (Ans: 40 F)

Practice problem 6. 7 • Find the voltage across each of the capacitors in

Practice problem 6. 7 • Find the voltage across each of the capacitors in Fig. 6. 20 (Ans: 30 V, 10 V, 20 V)

Inductors • Electrical component that opposes any change in electrical current. • Composed of

Inductors • Electrical component that opposes any change in electrical current. • Composed of a coil or wire wound around a nonmagnetic 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

Typical form of an inductor

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

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

Current – voltage relationship of an inductor • The voltage across an inductor, •

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)

Instantaneous power and energy fir inductors • The instantaneous power delivered to a capacitor

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

Important properties of an inductor • An inductor acts like a short circuit to

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 -m. H inductor

Practice problem 6. 8 • If the current through a 1 -m. H inductor is i(t) = 20 cos 100 t m. A, find the terminal voltage and the energy stored. (Ans: -2 sin 100 t m. V, 0. 2 cos 2100 t J)

Practice problem 6. 9 • The terminal voltage of a 2 -H inductor is

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=4 s and the energy stored in it within 0 < t < 4 s. Assume i(0)=2 A. (Ans: -18 A, 320 J)

Practice problem 6. 10 • Determine VC, i. L and the energy stored in

Practice problem 6. 10 • Determine VC, i. L and the energy stored in the capacitor and inductor in the circuit below under dc conditions. (Ans: 3 V, 3 A, 1. 125 J)

Series inductances • The equivalent inductance of N seriesconnected inductors is the sum of

Series inductances • The equivalent inductance of N seriesconnected inductors is the sum of the individual inductances. (14) Series Nconnected inductors Equivalent circuit

Parallel inductances • The equivalent inductance of series-connected inductors is the reciprocal of the

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

Practice problem 6. 11 • Calculate the equivalent inductance for the inductive ladder network

Practice problem 6. 11 • Calculate the equivalent inductance for the inductive ladder network in Figure below. (Ans: 25 m. H)

Practice problem 6. 12 • In the circuit of Figure below, given i 1(t)=0.

Practice problem 6. 12 • In the circuit of Figure below, given i 1(t)=0. 6 e-2 t. If i(0)=1. 4 A, find (a)i 2(0); (b) i 2(t) and i(t); (c) V 1(t), V 2(t) and V (t).