Capacitors and Inductors Mustafa Kemal Uygurolu Eastern Mediterranean
- Slides: 68
Capacitors and Inductors Mustafa Kemal Uyguroğlu Eastern Mediterranean University 1
Chap. 6, Capacitors and Inductors « Introduction « Capacitors « Series and Parallel Capacitors « Inductors « Series and Parallel Inductors Eastern Mediterranean University 2
6. 1 Introduction « Resistor: a passive element which dissipates energy only « Two important passive linear circuit elements: 1) Capacitor 2) Inductor « Capacitor and inductor can store energy only and they can neither generate nor dissipate energy. Eastern Mediterranean University 3
Michael Faraday (1971 -1867) Eastern Mediterranean University 4
6. 2 Capacitors « A capacitor consists of two conducting plates separated by an insulator (or dielectric). Eastern Mediterranean University 5
« Three factors affecting the value of capacitance: 1. Area: the larger the area, the greater the capacitance. 2. Spacing between the plates: the smaller the spacing, the greater the capacitance. 3. Material permittivity: the higher the permittivity, the greater the capacitance. Eastern Mediterranean University 6
Fig 6. 4 (a) Polyester capacitor, (b) Ceramic capacitor, (c) Electrolytic capacitor Eastern Mediterranean University 7
Fig 6. 5 Variable capacitors Eastern Mediterranean University 8
Fig 6. 3 Eastern Mediterranean University 9
Fig 6. 2 Eastern Mediterranean University 10
Charge in Capacitors « The relation between the charge in plates and the voltage across a capacitor is given below. q Linear Nonlinear v Eastern Mediterranean University 11
Voltage Limit on a Capacitor « Since q=Cv, the plate charge increases as the voltage increases. The electric field intensity between two plates increases. If the voltage across the capacitor is so large that the field intensity is large enough to break down the insulation of the dielectric, the capacitor is out of work. Hence, every practical capacitor has a maximum limit on its operating voltage. Eastern Mediterranean University 12
I-V Relation of Capacitor + v i C - Eastern Mediterranean University 13
Physical Meaning + v i C - • when v is a constant voltage, then i=0; a constant voltage across a capacitor creates no current through the capacitor, the capacitor in this case is the same as an open circuit. • If v is abruptly changed, then the current will have an infinite value that is practically impossible. Hence, a capacitor is impossible to have an abrupt change in its voltage except an infinite current is applied. Eastern Mediterranean University 14
Fig 6. 7 « A capacitor is an open circuit to dc. « The voltage on a capacitor cannot change abruptly. Abrupt change Eastern Mediterranean University 15
+ v i C - « The charge on a capacitor is an integration of current through the capacitor. Hence, the memory effect counts. Eastern Mediterranean University 16
Energy Storing in Capacitor + v i C - Eastern Mediterranean University 17
Model of Practical Capacitor Eastern Mediterranean University 18
Example 6. 1 (a) Calculate the charge stored on a 3 -p. F capacitor with 20 V across it. (b) Find the energy stored in the capacitor. Eastern Mediterranean University 19
Example 6. 1 Solution: (a) Since (b) The energy stored is Eastern Mediterranean University 20
Example 6. 2 « The voltage across a 5 - F capacitor is Calculate the current through it. Solution: « By definition, the current is Eastern Mediterranean University 21
Example 6. 3 « Determine the voltage across a 2 - F capacitor if the current through it is Assume that the initial capacitor voltage is zero. Solution: « Since Eastern Mediterranean University 22
Example 6. 4 « Determine the current through a 200 - F capacitor whose voltage is shown in Fig 6. 9. Eastern Mediterranean University 23
Example 6. 4 Solution: « The voltage waveform can be described mathematically as Eastern Mediterranean University 24
Example 6. 4 « Since i = C dv/dt and C = 200 F, we take the derivative of to obtain « Thus the current waveform is shown in Fig. 6. 10. Eastern Mediterranean University 25
Example 6. 4 Eastern Mediterranean University 26
Example 6. 5 « Obtain the energy stored in each capacitor in Fig. 6. 12(a) under dc condition. Eastern Mediterranean University 27
Example 6. 5 Solution: « Under dc condition, we replace each capacitor with an open circuit. By current division, Eastern Mediterranean University 28
Fig 6. 14 Eastern Mediterranean University 29
6. 3 Series and Parallel Capacitors « The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitance. Eastern Mediterranean University 30
Fig 6. 15 Eastern Mediterranean University 31
Series Capacitors « The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances. Eastern Mediterranean University 32
Summary « These results enable us to look the capacitor in this way: 1/C has the equivalent effect as the resistance. The equivalent capacitor of capacitors connected in parallel or series can be obtained via this point of view, so is the Y-△ connection and its transformation Eastern Mediterranean University 33
Example 6. 6 « Find the equivalent capacitance seen between terminals a and b of the circuit in Fig 6. 16. Eastern Mediterranean University 34
Example 6. 6 Solution: Eastern Mediterranean University 35
Example 6. 7 « For the circuit in Fig 6. 18, find the voltage across each capacitor. Eastern Mediterranean University 36
Example 6. 7 Eastern Mediterranean University 37
Example 6. 7 Solution: « Two parallel capacitors: « Total charge « This is the charge on the 20 -m. F and 30 -m. F capacitors, because they are in series with the 30 -v source. ( A crude way to see this is to imagine that charge acts like current, since i = dq/dt) Eastern Mediterranean University 38
Example 6. 7 « Therefore, « Having determined v 1 and v 2, we now use KVL to determine v 3 by « Alternatively, since the 40 -m. F and 20 -m. F capacitors are in parallel, they have the same voltage v 3 and their combined capacitance is 40+20=60 m. F. Eastern Mediterranean University 39
Joseph Henry (1979 -1878) Eastern Mediterranean University 40
6. 4 Inductors « An inductor is made of a coil of conducting wire Eastern Mediterranean University 41
Fig 6. 22 Eastern Mediterranean University 42
Fig 6. 23 (a) air-core (b) iron-core (c) variable iron-core Eastern Mediterranean University 43
Flux in Inductors « The relation between the flux in inductor and the current through the inductor is given below. ψ Linear Nonlinear i Eastern Mediterranean University 44
Energy Storage Form « An inductor is a passive element designed to store energy in the magnetic field while a capacitor stores energy in the electric field. Eastern Mediterranean University 45
I-V Relation of Inductors « An inductor consists of a coil of conducting wire. i + v L - Eastern Mediterranean University 46
Physical Meaning « When the current through an inductor is a constant, then the voltage across the inductor is zero, same as a short circuit. « No abrupt change of the current through an inductor is possible except an infinite voltage across the inductor is applied. « The inductor can be used to generate a high voltage, for example, used as an igniting element. 47 Eastern Mediterranean University
Fig 6. 25 « An inductor are like a short circuit to dc. « The current through an inductor cannot change instantaneously. Eastern Mediterranean University 48
+ v L - Eastern Mediterranean University 49
Energy Stored in an Inductor + v L - « The energy stored in an inductor Eastern Mediterranean University 50
Model of a Practical Inductor Eastern Mediterranean University 51
Example 6. 8 « The current through a 0. 1 -H inductor is i(t) = 10 te-5 t A. Find the voltage across the inductor and the energy stored in it. Solution: Eastern Mediterranean University 52
Example 6. 9 « Find the current through a 5 -H inductor if the voltage across it is Also find the energy stored within 0 < t < 5 s. Assume i(0)=0. Solution: Eastern Mediterranean University 53
Example 6. 9 Eastern Mediterranean University 54
Example 6. 10 « Consider the circuit in Fig 6. 27(a). Under dc conditions, find: (a) i, v. C, and i. L. (b) the energy stored in the capacitor and inductor. Eastern Mediterranean University 55
Example 6. 10 Solution: Eastern Mediterranean University 56
Inductors in Series Eastern Mediterranean University 57
Inductors in Parallel Eastern Mediterranean University 58
6. 5 Series and Parallel Inductors « Applying KVL to the loop, « Substituting vk = Lk di/dt results in Eastern Mediterranean University 59
Parallel Inductors « Using KCL, « But Eastern Mediterranean University 60
« The inductor in various connection has the same effect as the resistor. Hence, the Y-Δ transformation of inductors can be similarly derived. Eastern Mediterranean University 61
Table 6. 1 Eastern Mediterranean University 62
Example 6. 11 « Find the equivalent inductance of the circuit shown in Fig. 6. 31. Eastern Mediterranean University 63
Example 6. 11 « Solution: Eastern Mediterranean University 64
Practice Problem 6. 11 Eastern Mediterranean University 65
Example 6. 12 « Find the circuit in Fig. 6. 33, If find : Eastern Mediterranean University 66
Example 6. 12 Solution: Eastern Mediterranean University 67
Example 6. 12 Eastern Mediterranean University 68
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