Lecture 5 Characteristics of Energy-Storing Elements PDF

EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements Capacitors and Inductors In this lecture, we shall introduce two new and important passive linear circuit elements: the capacitor and the inductor. Unlike resistors, which dissipate energy, capacitors and indu...

EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements Capacitors and Inductors In this lecture, we shall introduce two new and important passive linear circuit elements: the capacitor and the inductor. Unlike resistors, which dissipate energy, capacitors and inductors do not dissipate but store energy, which can be retrieved at a later time. For this reason, capacitors and inductors are called storage elements. Capacitors A capacitor is a passive element designed to store energy in its electric field. Capacitors are used extensively in electronics, communications, computers, and power systems. For example, they are used in the tuning circuits of radio receivers and as dynamic memory elements in computer systems. A capacitor is typically constructed as depicted in fig. 6.1. A capacitor consists of two conducting plates separated by an insulator (or dielectric). In many practical applications, the plates may be aluminium foil while the dielectric may be air, ceramic, paper, or mica. When a voltage source v is connected to the capacitor, as in fig 6.2, the source deposits a positive charge q on one plate and a negative charge –q on the other. The capacitor is said to store the electric charge. The amount of charge stored, represented by q, is directly proportional to the applied voltage v so that 𝑞 = 𝐶𝑣 (6.1) where C, the constant of proportionality, is known as the capacitance of the capacitor. The unit of capacitance is the farad (F), in honor of the English physicist Michael Faraday. From equation 6.1, we may derive the following definition. Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F). Note that from eq.1, that 1 farad = 1 coulomb/volt. Although the capacitance C of a capacitor is the ratio of the charge q per plate to the applied voltage v, it does not depend on q or v. It depends on the physical dimensions of the capacitor. For example, for the parallel-plate capacitor shown in fig. 6.1, the capacitance is given by ∈𝐴 𝐶= (6.2) 𝑑 where A is the surface area of each plate, d is the distance between the plates, and ∈ is the permittivity of the dielectric material between the plates. Although eq. 6.2 applies only to parallel- Page 1 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements plate capacitors, we may infer from it that, in general, three factors determine the value of the capacitance: 1. The surface area of the plates – the larger the area, the greater the capacitance. 2. The spacing between the plates – the smaller the spacing, the greater the capacitance. 3. The permittivity of the material – the higher the permittivity, the greater the capacitance. Capacitors are commercially available in different values and types. Typically, capacitors have values in the picofarad (pF) to microfarad (µF) range. They are described by the dielectric material they are made of and by whether they are fixed or variable. Fig. 6.3 shows the circuit symbols for fixed and variable capacitors. Fig. 6.4 shows common types of fixed-value capacitors. Fig. 6.5 shows the most common types of variable capacitors. The current-voltage relationship for a capacitor is shown in eq. 6.4. 𝑑𝑣 𝑖=𝐶 (6.4) 𝑑𝑡 The relationship is illustrated in fig. 6.6 for a capacitor whose capacitance is independent of voltage. Capacitors that satisfy eq. 6.4 are said to be linear. For a nonlinear capacitor, the plot of the current-voltage relationship is not a straight line. Although some capacitors are nonlinear, most are linear. The voltage-current relationship of the capacitor can be obtained by integrating both sides of eq. 6.4. We get 1 𝑡 𝑣= ∫ 𝑖𝑑𝑡 + 𝑣 (𝑡0 ) (6.6) 𝐶 𝑡0 Page 2 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements where (𝑡0 ) = 𝑞 (𝑡0 )⁄𝐶 , is the voltage across the capacitor at time t 0. Eq. 6.6 shows that capacitor voltage depends on the past history of the capacitor current. The energy stored in the capacitor is 1 2 𝑤= 𝐶𝑣 (6.9) 2 or 𝑞2 𝑤= (6.10) 2𝐶 Eq. 6.9 or 6.10 represents the energy stored in the electric field that exists between the plates of the capacitor. This energy can be retrieved, since an ideal capacitor cannot dissipate energy. In fact, the word capacitor is derived from this element’s capacity to store energy in an electric field. We should note the following important properties of a capacitor: 1. A capacitor is an open circuit to dc. However, if a battery (dc voltage) is connected across a capacitor, the capacitor charges. 2. The voltage on a capacitor cannot change abruptly. The voltage on the capacitor must be continuous. 3. The ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit. Sample Problems: 1. (a) Calculate the charge stored on a 3 pF capacitor with 20 V across it. (b) Find the energy stored in the capacitor. 2. The voltage across a 5 µF capacitor is v(t) = 10 cos 6000t V. Calculate the current through it. 3. Determine the voltage across a 2 µF capacitor if the current through it is i(t) = 6e -3000t mA. Assume that the initial capacitor voltage is zero. Page 3 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements 4. Obtain the energy stored in each capacitor in fig. 6.12(a) under dc conditions. Series and Parallel Capacitors The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances. The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the individual capacitances. For N = 2, i.e. two capacitors in series, eq. 6.16 becomes Sample Problems: 5. Find the equivalent capacitance seen between terminals a and b of the circuit below. Page 4 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements 6. For the circuit in fig. 6.18, find the voltage across each capacitor. Inductors An inductor is a passive element designed to store energy in its magnetic field. Inductors find numerous applications in electronic and power systems. They are used in power supplies, transformers, radios, TVs, radars, and electric motors. Any conductor of electric current has inductive properties and may be regarded as an inductor. But in order to enhance the inductive effect, a practical inductor is usually formed into a cylindrical coil with many turns of conducting wire, as shown in fig. 6.21. An inductor consists of a coil of conducting wire. If current is allowed to pass through an inductor, it is found that the voltage across the inductor is directly proportional to the time rate of change of the current. Therefore, where L is the constant of proportionality called the inductance of the inductor. The unit of inductance is the henry (H), named in honor of the American inventor Joseph Henry. It is clear from eq. 6.18 that 1 henry equals 1 volt-second per ampere. Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H). The inductance of an inductor depends on its physical dimension and construction. For the inductor (solenoid) shown in fig. 6.21, where N is the number of turns, ℓ is the length, A is the cross-sectional area, and µ is the permeability of the core. We can see from eq. 6.19 that inductance can be increased by increasing the number of turns of coil, using material with higher permeability as the core, increasing the cross-sectional area, or reducing the length of the coil. Page 5 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements Like capacitors, commercially available inductors come in different values and types. Typical practical inductors have inductance values ranging from a few micro-henrys (µH), as in communication systems, to tens of henrys (H) as in power systems. Inductors may be fixed or variable. The core may be made of iron, steel, plastic or air. The terms coil and choke are also used for inductors. Common inductors are shown in fig. 6.22. The circuit symbols for inductors are shown in fig. 6.23. Eq. 6.18 is the voltage-current relationship for an inductor. Fig. 6.24 shows this relationship graphically for an inductor whose inductance is independent of current. Such an inductor is known as a linear inductor. The current-voltage relationship is obtained as where i(t0) is the total current for -∞ < t < t0 and i(-∞) = 0. The inductor is designed to store energy in its magnetic field. The energy stores in the inductor is We should note the following important properties of an inductor. 1. An inductor acts like a short circuit to dc. Page 6 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements 2. The current through an inductor cannot change instantaneously. An important property of the inductor is its opposition to the change in current flowing through it. 3. Like the ideal capacitor, the ideal inductor does not dissipate energy. The energy stored in it can be retrieved at a later time. The inductor takes power from the circuit when storing energy and delivers power to the circuit when returning previously stored energy. Sample Problems: 7. 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. 8. Find the current through a 5 H inductor if the voltage across it is Also, find the energy stored at t = 5s. Assume i(v) > 0. 9. Consider the circuit in fig. 6.27(a). Under dc conditions, find: (a) i, vc, and iL. (b) the energy stored in the capacitor and inductor. Series and Parallel Inductors The equivalent inductance of series-connected inductors is the sum of the individual inductances. The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocal of the sum of the reciprocals of the individual inductances. Page 7 of 8 EE 401 Electrical Circuits I Lecture 5 Characteristics of Energy-storing Elements For two inductors in parallel, As long as all the elements are of the same type, the Δ – Y transformations for resistors discussed in previous topics can be extended to capacitors and inductors. Sample Problems: 10. Find the equivalent inductance of the circuit shown in fig. 6.31. 11. For the circuit in fig. 6.33, i(t) = 4(2 – e-10t) mA. If i2(0) = -1 mA, find: (a) i1(0); (b) v(t), v1(t), and v2(t); (c) i1(t) and i2(t). Page 8 of 8