Capacitors and Inductors Mustafa Kemal Uygurolu Eastern Mediterranean

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Capacitors and Inductors Mustafa Kemal Uyguroğlu Eastern Mediterranean University 1

Capacitors and Inductors Mustafa Kemal Uyguroğlu Eastern Mediterranean University 1

Chap. 6, Capacitors and Inductors « Introduction « Capacitors « Series and Parallel Capacitors

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

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

Michael Faraday (1971 -1867) Eastern Mediterranean University 4

6. 2 Capacitors « A capacitor consists of two conducting plates separated by an

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

« 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

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. 5 Variable capacitors Eastern Mediterranean University 8

Fig 6. 3 Eastern Mediterranean University 9

Fig 6. 3 Eastern Mediterranean University 9

Fig 6. 2 Eastern Mediterranean University 10

Fig 6. 2 Eastern Mediterranean University 10

Charge in Capacitors « The relation between the charge in plates and the voltage

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 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

I-V Relation of Capacitor + v i C - Eastern Mediterranean University 13

Physical Meaning + v i C - • when v is a constant voltage,

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

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

+ 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

Energy Storing in Capacitor + v i C - Eastern Mediterranean University 17

Model of Practical Capacitor Eastern Mediterranean University 18

Model of Practical Capacitor Eastern Mediterranean University 18

Example 6. 1 (a) Calculate the charge stored on a 3 -p. F capacitor

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

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

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

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

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

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,

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. 4 Eastern Mediterranean University 26

Example 6. 5 « Obtain the energy stored in each capacitor in Fig. 6.

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

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

Fig 6. 14 Eastern Mediterranean University 29

6. 3 Series and Parallel Capacitors « The equivalent capacitance of N parallel-connected capacitors

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

Fig 6. 15 Eastern Mediterranean University 31

Series Capacitors « The equivalent capacitance of series-connected capacitors is the reciprocal of the

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

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

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. 6 Solution: Eastern Mediterranean University 35

Example 6. 7 « For the circuit in Fig 6. 18, find the voltage

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 Eastern Mediterranean University 37

Example 6. 7 Solution: « Two parallel capacitors: « Total charge « This is

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

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

Joseph Henry (1979 -1878) Eastern Mediterranean University 40

6. 4 Inductors « An inductor is made of a coil of conducting wire

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. 22 Eastern Mediterranean University 42

Fig 6. 23 (a) air-core (b) iron-core (c) variable iron-core Eastern Mediterranean University 43

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

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

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 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

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. «

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

+ v L - Eastern Mediterranean University 49

Energy Stored in an Inductor + v L - « The energy stored in

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

Model of a Practical Inductor Eastern Mediterranean University 51

Example 6. 8 « The current through a 0. 1 -H inductor is i(t)

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

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. 9 Eastern Mediterranean University 54

Example 6. 10 « Consider the circuit in Fig 6. 27(a). Under dc conditions,

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

Example 6. 10 Solution: Eastern Mediterranean University 56

Inductors in Series Eastern Mediterranean University 57

Inductors in Series Eastern Mediterranean University 57

Inductors in Parallel Eastern Mediterranean University 58

Inductors in Parallel Eastern Mediterranean University 58

6. 5 Series and Parallel Inductors « Applying KVL to the loop, « Substituting

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

Parallel Inductors « Using KCL, « But Eastern Mediterranean University 60

 « The inductor in various connection has the same effect as the resistor.

« 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

Table 6. 1 Eastern Mediterranean University 62

Example 6. 11 « Find the equivalent inductance of the circuit shown in Fig.

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

Example 6. 11 « Solution: Eastern Mediterranean University 64

Practice Problem 6. 11 Eastern Mediterranean University 65

Practice Problem 6. 11 Eastern Mediterranean University 65

Example 6. 12 « Find the circuit in Fig. 6. 33, If find :

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 Solution: Eastern Mediterranean University 67

Example 6. 12 Eastern Mediterranean University 68

Example 6. 12 Eastern Mediterranean University 68