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CPE220 Electric Circuit Analysis

Chapter 5: Capacitors and Inductors. CPE220 Electric Circuit Analysis. Chapter 5.

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CPE220 Electric Circuit Analysis

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  1. Chapter 5: Capacitors and Inductors CPE220 Electric Circuit Analysis

  2. Chapter 5 Previously we discussed only one kind of passive elements which is a resistor. A resistor will absorb the energy and dissipate it into heat. In this chapter we will discuss two other passive elements which are capacitor and inductor. Both the capacitor and inductor are physical devices that are intentionally designed to store and delivery finite amount of energy. Capacitor stores energy in term of an electric field while inductor stores energy in term of a magnetic field. Capacitors and inductors can delivery finite amount of energy. However, they are not active elements. In chapter 2, the definitions of active and passive are still slightly fuzzy. We only state that active

  3. element supplies the energy while the passive element absorbs the energy. Although both capacitors and inductors can supply the energy, they cannot deliver an average power that is greater than zero over an infinitetime interval. Hence, we now define an active element as an element that is capable of furnishing an average power greater than zero to external device, where the average is taken over an infinite time interval. Capacitors 5.1 A capacitor is a passive element designed to store energy in its electrostatic field. Capacitors are used for many purposes. Capacitors are used, for example, as part of the circuits that tune in the stations on your radio or television. Another example: In a

  4. Figure 5.1 A typical capacitor. portablebattery-operated photoflash unit, a capacitor accumulates charge slowly during the charging process, building up an electric field as it does so. It holds this field and its energy until the energy is rapidly released during the flash. 5.1.1 Ideal Capacitor Model A capacitor consists of two conducting plates separated by an insulator as shown in Fig. 5.1. Conductor (aluminum) Insulator (ceramic or other dielectric material) Conductor (aluminum) In many practical applications, the conductor in Fig. 5.1 may be aluminum foil

  5. C = eA d while insulator may be ceramic, plastic,paper, mica, or air. These are dielectric materials, that is, materials that contain a large number of electric dipoles, which become polarized in the presence of an electric field. Air, however, yields capacitors with very low values of capacitance. Let A be the surface area of each plate, d be the distance between the plates, and e is the permitivity of the dielectric material between the plates. The capacitance C measured in farad (F) of capacitor is defined as: (5.1) When a voltage source v is connected to the capacitor, then positive charges will be transferred to one plate and negative charges in the other. The capacitor is said

  6. to store the electric charge. The amount of charge stored which is represented by q is proportional to the voltage across it such that q = Cv (5.2) where q = the charge in coulombs (C) v = the voltage across capacitors in volts (V) C = the capacitance in farads (F). From Eq. 5.2, we can notice that; 1 farad = 1 coulomb/volt 5.1.2 Capacitor Voltage and Capacitor Current Relationship The capacitance of capacitor, C, is a

  7. = C dq dv i = dt dt measure of how much charge must be put on the plates to produce a certain voltage. From Eq. 5.2, we can see that the greater the capacitance, the more charge is required. If we take the derivative of the both sides of the Eq. 5.2, we will obtain the relationship between current passing through, i, and voltage across, v, the capacitor as follows: (5.3) Hence, even though no DC current is flowing through the capacitor, a time- varying voltage ( an electric field) causes charge to vary with time which is analogous to a current. A current induced by an electric field is called "displacement current". Although the displacement

  8. + + + + - - - - Figure 5.2 A conduction current and a displacement current. current flows internally between the capacitor plates, the amount of it is exactly equal to the conduction current flowing in the capacitor leads. Figure 5.2 illustrates the relationship between these two types of currents. i = dq/dt = conduction current (moving charge) + v_ i = dψ/dt = displacement current (changing stored charge) i = dq/dt = conduction current (moving charge) The voltage across the capacitor can be expressed by integrating Eq. 5.3 as follows:

  9. q(t0) C the capacitor voltage at time t0 in volts (V). = (5.4) where v(t0) = Hence, the capacitor voltage is proportional to the charge, which is the integral of the current, and depends on the past history of the capacitor current. That is capacitors have memory. Eq. 5.4 also implies that the capacitor voltage vc(t) cannot change instantaneously since it will require an infinite current that is physically impossible. But, the capacitor current ic(t) can change instantaneously.

  10. In many real problems the initial capacitor voltage v(t0) is hardly to be discerned. In such instances it will be mathematically convenient to set t0 = - ∞ and v(- ∞) = 0, thus Example 5.1 Given i(t) as shown. Determine v(t) assuming that v(0) = 0. i(t) 100 μF 1 A ... ... t 0 1 2 i + v –

  11. v(t) 104 V 0 1 2 t Solution: By using Eq. 5.4, we get Ans. Example 5.2 Given v(t) = 25e-10t. Determine i(t), vR(t) and vS(t) in the following circuit.

  12. d(25e-10t) dt = C dv dt Solution: Since i(t) = 80x10-6 = -0.02e-10t A. vR(t) = 1000i = 1000 (-0.02e-10t) = -20e-10t V. KVL vS(t) = vR(t) + v(t) = -20e-10t + 25e-10t V. = 5e-10t V. Ans. Note: v, i, vR and vS are all in the form of Ae-10t where A = constant.

  13. p = vi = Cv dv dt 5.1.3 Power and Energy in Capacitors The instantaneous power delivered to the capacitor can be evaluated as: (5.5) The energy stored in the capacitor can be evaluated by integrating Eq. 5.5 as follows:

  14. (5.6) where wC(t) = the energy stored by capacitor in joules (J). v(t0) = the voltage at time t0 (the initial capacitor voltage) in volts (V). If we assume that the initial capacitor voltage is zero, then 1 wC(t) = (5.7) Cv2 J. 2 That is, we assume a zero-energy reference at t0.

  15. = C dv dt Example 5.3 Determine the maximum energy stored in the capacitor of the given circuit and the energy dissipated in the resistor over the interval 0 ≤ t ≤ 0.5 sec. Solution: iR = (100sin2pt)/106 = 10-4sin2pt A Since ic(t)

  16. d(100sin2pt) dt = 20x10-6 = 4p10-3cos2pt A. wc(t) = (1/2)cv2 = 0.1sin22pt J. The value of sin2pt is maximum when 2pt = p/2, thus t = 0.25 sec. Then wc(t) = 0.1sin22p0.25 = 0.1 J. = 5e-10t V. Since pR = (10-4sin2pt)2(106) W. Then wR(t) Ans. = 2.5 mJ.

  17. A capacitor is an open circuit to the direct current (d.c.). • The capacitor voltage cannot change instantaneously (The capacitor voltage Important Characteristics of an Ideal Capacitor As we discussed in Section 5.1.2 and 5.1.3, an ideal capacitor is an electric circuit element that resists an abrupt change in the voltage across it (Eq. 5.3). In the other words, a discontinuous change in capacitor voltage requires an infinite current. An ideal capacitor can store a finite amount of electrical energy without dissipating into the heat like a resistor. A summary of the important characteristics of an ideal capacitor can be summarized as follows:

  18. Figure 5.3 A capacitor voltage must be continuous. On the other hand, a capacitor current can change abruptly. must be continuous) as illustrated in Fig. 5.3. vc ic t t • An ideal capacitor does not dissipate energy, but only stores it. • A finite amount of energy can be stored in a capacitor even if the current through the capacitor is zero, such as when the capacitor voltage is constant.

  19. Figure 5.4 A model of physical capacitor which is an ideal capacitor connecting parallel with a leakage resistor. In fact, a physical capacitor consists of a parallel leakage resistor as illustrated in Fig. 5.4. Practically, the very large value of leakage resistor is desirable, so that a practical capacitor can store the energy for a long period of time. 5.1.4 Capacitors in Series and Parallel As discussed in Chapter 2, there are two types of connections in the electric circuit which are series and parallel connections.

  20. vs = + ... vc 1 + vc + vc 3 N + vc 2 In this section we will discuss how to evaluate the equivalent capacitance of the electric circuit containing a combination of capacitors. Capacitors in Series For an electrical circuit containing N capacitors in series as shown in Fig. 5.5(a), the same current flows through each capacitor. Applying KVL to the loop in Fig. 5.5(a), we get (5.8) Substitute Eq. 5.4 into Eq. 5.8, we get

  21. Figure 5.5 (a) A electric circuit consists of a series connection of N capacitors. (b) An equivalent circuit of the electrical circuit in (a). (a) (b) Hence, or where Ceq be the summation of all capacitance appearing in Fig. 5.5(a). That is,

  22. Capacitor in Series (5.9) The equivalent capacitance of any number of capacitors connected in series is the reciprocal of the sum of the reciprocals of the individual capacitance (Cn). Thus, capacitors in series combine similarly to resistors in parallel. Capacitors in Parallel Capacitors are in parallel when a voltage that is applied across their combination results in the same voltage across each capacitor as illustrated in Fig. 5.6. Applying KCL to the circuit of Fig. 5.6(a), we get

  23. i = + ... Figure 5.6 (a) N capacitors in parallel having the same voltage across each capacitor. (b) An equivalent circuit of the electrical circuit in (a). ic 1 + ic + ic 3 N + ic 2 (a) (b) (5.10) Substitute Eq. 5.3 into Eq. 5.10 and since all capacitor have the same voltage v, Eq. 5.10 can be simplified into the form of or

  24. where Ceq be the summation of all capacitance appearing in Fig. 5.6(a). That is, Capacitor in Parallel (5.11) The equivalent capacitance of any number of capacitors connected in parallel is the the sum of the individual capacitance (Cn). Thus, capacitors in parallel combine similarly to resistors in series. Example 5.4 Determine C1eq , C2eq, V1 and V2 in the following circuit.

  25. C2eq = C1eq = (18 series 36 series 24)parallel with 8 (20 series 20)parallel with 8 and 6 Solution: = 24 mF. = 16 mF. Hence, the total equivalent capacitance seeing by the voltage source 90V is Ceq = (16 series 16) = 8 mF. The total charge is q = CeqV = 8 x 90 = 720 mC.

  26. which is the charge on the 16 mF and C2eq. Let treat the charge like a current since i = dq/dt. Thus, V2 = q/C2eq = 720/16 = 45 V. Similarly V1 = 15 V. Ans. Inductors 5.2 An inductor is a passive element designed to store energy in its magnetic field. Inductors are used in many electronic devices and power system, such as, power supply, transformer, radio, TV, radar and electric motor.

  27. Figure 5.7 A typical inductor. 5.2.1 Ideal Inductor Model Inductor is commonly built by winding a coil of wire around a core, which can be an insulator or a ferromagnetic material (plastic or iron) as shown in Fig. 5.7. i + magnetic flux lines Electrical System v N turns - insulator or ferromagnetic material inductor In an ideal inductor, the resistance of the wire is zero, so that a constant current

  28. dΦ v = N dt Φ N L = i through the inductor will flow without no voltage drop. When the current is allowed to flow through an inductor, it causes magnetic flux to link (pass through) each turn of the coil. Faraday's Law yields the voltage across the inductor is directly proportional to the time rate of change of the average magnetic flux Φ. That is, (5.12) where Φ = the average magnetic flux Linking a turn, N = the number of turns. The inductance L is defined as: (5.13) which is the property where by an inductor

  29. di v = L dt exhibits opposition to the change of current flowing through it. It is measured in henry (H). From Eq. 5.12 and Eq. 5.13, we can rewrite the relationship between the inductor voltage and current as follows: (5.14) Thus 1 H equals 1 volt-sec/ampere. Generally, the inductance of an inductor depends on its physical dimension and construction. Formulas for calculating the inductance of inductors of different shapes are derived from electromagnetic theory and can be found in standard electrical engineering handbooks. For the inductor shown in Fig. 5.7 (solenoid), the inductance can be evaluated as:

  30. N2mA (5.15) L = l where N = the number of turns, A = the area of a core, l = the length of a core, m = the permeability of the core. The inductor current can be obtained by integrating Eq. 5.14 as follows: (5.16) In the present of a DC source, from Eq. 5.14, the inductor voltage will be zero.

  31. Figure 5.8 A inductor current must be continuous. On the other hand, a inductor voltage can. Hence, the inductor acts as a short circuit to DC. Eq. 5.16 implies that the inductor current cannot change instantaneously since it will require an infinite inductor voltage that is physically impossible. But, the capacitor voltage can change instantaneously(see Fig. 5.8). iL vL t t

  32. Example 5.5 Given i1 (t) = 0.6e-2t and i(0) = 1.4A. Find all unknown voltages and currents. Solution: di1(t) v1(t) = 6 = -7.2e-2t V. dt i2(0) = i(0) - i1(0) = 1.4 - 0.6 = 0.8 A. Since i2(t) = -0.4 + 1.2e-2t A.

  33. p = vi = Li di dt Hence i(t) = i1(t) + i2(t) = 0.6e-2t -0.4 + 1.2e-2t A. = -0.4 + 1.8e-2t A. di(t) v2(t) = 8 = -28.8 e-2t V. dt v(t) = v1(t) + v2(t) = -7.2e-2t - 28.8e-2t V. = -36e-2t V. Ans. 5.2.2 Power and Energy in Inductors The instantaneous power delivered to the inductor can be evaluated as: (5.17)

  34. The energy stored in the inductor can be evaluated by integrating Eq. 5.17 as follows: (5.18) where wL(t) = the energy stored by inductor in joules (J). i(t0) = the current at time t0 (the initial inductor current) in amperes (A).

  35. If we assume that the initial inductor current is zero, then 1 wL(t) = (5.19) Li2 J. 2 That is, we assume a zero-energy reference at t0. Important Characteristics of an Ideal Inductor Like an ideal capacitor, there are some important characteristics of an ideal inductor that we should note. These important characteristics will be summarized as follows:

  36. An inductor is an short circuit to the direct current (d.c.). • The inductor current cannot change instantaneously (The inductor current must be continuous). • An ideal inductor does not dissipate energy, but only stores it. • A finite amount of energy can be stored in an inductor even if the voltage across the inductor is zero, such as when the inductor current is constant. In fact, a physical inductor consists of a series leakage resistor as illustrated in Fig. 5.9. Practically, the very small value of leakage resistor is desirable, so that a

  37. Leakage Resistor L Figure 5.9 A model of physical inductor which is an ideal inductor connecting in series with a leakage resistor. practical inductor can store the energy for a long period of time. Example 5.6 Determine the maximum energy stored in the inductor and calculate how much energy is dissipated in the resistor in the time during which the energy is being stored in and then recovered from the inductor.

  38. Solution: Hence, the energy stored in the inductor is wL(t) = (1/2)Li2 = 216sin2(p/6)t J. The value of sin(p/6)t is maximum when (p/6)t = p/2, thus t = 3 sec. That is, the maximum energy stored in the inductor is 216 J. at t = 3 sec. The power dissipated in the resistor can be evaluated as: pR(t) = i2R = 14.4sin2(p/6)t W

  39. It will take 6-sec. Interval for the energy to convert into heat, hence = 43.2 J Ans. 5.2.3 Inductors in Series and Parallel A combination of inductors in series or in parallel yields the equivalent inductance in the same manner of a combination of resistors in series or in parallel which will discussed in this section.

  40. vs = vL + vL + vL + ... + vL 1 2 3 N Inductors in Series For an electrical circuit containing N inductors in series as shown in Fig. 5.10(a), the same current flows through each inductor. Applying KVL to the loop in Fig. 5.10(a), we get (5.20) Substitute Eq. 5.14 into Eq. 5.20, we get or where Leq be the summation of all inductance appearing in Fig. 5.10(a). That is,

  41. Figure 5.10 (a) A electric circuit consists of a series connection of N inductors. (b) An equivalent circuit of the electrical circuit in (a). (a) (b) Inductor in Series (5.21) The equivalent inductance of any number of inductors connected in series is the the sum of the individual capacitance (Ln).

  42. iL + iL + iL + ... + iL i = 1 2 3 N Inductors in Parallel For an electrical circuit containing N inductors in parallel as shown in Fig. 5.11(a), the same voltage applied across each inductor. Applying KCL to the circuit of Fig. 5.11(a), we get (5.22) Substitute Eq. 5.16 into Eq. 5.22 and since all capacitor have the same voltage v, Eq. 5.22 can be simplified into the form of

  43. Figure 5.11 (a) A electric circuit consists of a parallel connection of N inductors. (b) An equivalent circuit of the electrical circuit in (a). (a) (b) Hence, or where Leq be the summation of all inductance appearing in Fig. 5.11(a). That is,

  44. Inductors in Parallel (5.23) The equivalent inductance of any number of inductors connected in parallel is the reciprocal of the sum of the reciprocals of the individual inductance (Ln). Summary 5.3 Capacitors and inductors have a number of extremely important characteristics. At this stage, it is appropriate to summarize these characteristics before we are going to consider the electrical circuits containing all three passive elements: resistor, capacitor and inductor.

  45. All important characteristics of capacitor and inductor discussed in this chapter are summarized in Table 5.1 as follows: Table5.1 Important characteristics of capacitors and inductors. Capacitor (C) Inductor (L) Open circuit to d.c. Short circuit to d.c. vC(t) cannot change instantaneously (no jump). iL(t) cannot change instantaneously (no jump). Ideal capacitors and inductors only store energy. However, Physical capacitor normally has a large resistance in parallel. Physical inductor normally has a small resistance in series.

  46. From Table 5.1, we can notice that capacitors and inductors have a dual relationship. That is, their mathematical form listed in Table 5.1 will be identical if we interchange "conductance, C" with "inductance, L" and "current, i" with " voltage, v" and vice versa. Because of this relationship, they share many of complementary features. For example, the voltage across the capacitor must be continuous whereas the current passing through the inductor must be continuous.

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