9: Capacitors and Inductors
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- Regina Lynch
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1 E1.1 Analysis of Circuits ( ) and : 9 1 / 12
2 A capacitor is formed from two conducting plates separated by a thin insulating layer. If a currentiflows, positive change,q, will accumulate on the upper plate. To preserve charge neutrality, a balancing negative charge will be present on the lower plate. E1.1 Analysis of Circuits ( ) and : 9 2 / 12
3 A capacitor is formed from two conducting plates separated by a thin insulating layer. If a currentiflows, positive change,q, will accumulate on the upper plate. To preserve charge neutrality, a balancing negative charge will be present on the lower plate. There will be a potential energy difference (or voltage v) between the plates proportional to q. q whereais the area of the plates,dis their separation andǫis v = d Aǫ the permittivity of the insulating layer (ǫ 0 = 8.85 pf /m for a vacuum). E1.1 Analysis of Circuits ( ) and : 9 2 / 12
4 A capacitor is formed from two conducting plates separated by a thin insulating layer. If a currentiflows, positive change,q, will accumulate on the upper plate. To preserve charge neutrality, a balancing negative charge will be present on the lower plate. There will be a potential energy difference (or voltage v) between the plates proportional to q. q whereais the area of the plates,dis their separation andǫis v = d Aǫ the permittivity of the insulating layer (ǫ 0 = 8.85 pf /m for a vacuum). The quantityc = Aǫ hence q = Cv. d is the capacitance and is measured in Farads (F), E1.1 Analysis of Circuits ( ) and : 9 2 / 12
5 A capacitor is formed from two conducting plates separated by a thin insulating layer. If a currentiflows, positive change,q, will accumulate on the upper plate. To preserve charge neutrality, a balancing negative charge will be present on the lower plate. There will be a potential energy difference (or voltage v) between the plates proportional to q. q whereais the area of the plates,dis their separation andǫis v = d Aǫ the permittivity of the insulating layer (ǫ 0 = 8.85 pf /m for a vacuum). The quantityc = Aǫ hence q = Cv. d is the capacitance and is measured in Farads (F), The current,i, is the rate of charge on the plate, hence the capacitor equation: i = dq = C dv. E1.1 Analysis of Circuits ( ) and : 9 2 / 12
6 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. E1.1 Analysis of Circuits ( ) and : 9 3 / 12
7 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. Polystyrene: Two sheets of foil separated by a thin plastic film and rolled up to save space. Values: 10pF to1nf. E1.1 Analysis of Circuits ( ) and : 9 3 / 12
8 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. Polystyrene: Two sheets of foil separated by a thin plastic film and rolled up to save space. Values: 10pF to1nf. Ceramic: Alternate layers of metal and ceramic (a fewµm thick). Values: 1nF to1µf. E1.1 Analysis of Circuits ( ) and : 9 3 / 12
9 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. Polystyrene: Two sheets of foil separated by a thin plastic film and rolled up to save space. Values: 10pF to1nf. Ceramic: Alternate layers of metal and ceramic (a fewµm thick). Values: 1nF to1µf. Electrolytic: Two sheets of aluminium foil separated by paper soaked in conducting electrolyte. The insulator is a thin oxide layer on one of the foils. Values: 1µF to10mf. E1.1 Analysis of Circuits ( ) and : 9 3 / 12
10 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. Polystyrene: Two sheets of foil separated by a thin plastic film and rolled up to save space. Values: 10pF to1nf. Ceramic: Alternate layers of metal and ceramic (a fewµm thick). Values: 1nF to1µf. Electrolytic: Two sheets of aluminium foil separated by paper soaked in conducting electrolyte. The insulator is a thin oxide layer on one of the foils. Values: 1µF to10mf. Electrolytic capacitors are polarised: the foil with the oxide layer must always be at a positive voltage relative to the other (else explosion). Negative terminal indicated by a curved plate in symbol E1.1 Analysis of Circuits ( ) and : 9 3 / 12
11 Types of Capacitor Capacitor symbol represents the two separated plates. Capacitor types are distinguished by the material used as the insulator. Polystyrene: Two sheets of foil separated by a thin plastic film and rolled up to save space. Values: 10pF to1nf. Ceramic: Alternate layers of metal and ceramic (a fewµm thick). Values: 1nF to1µf. Electrolytic: Two sheets of aluminium foil separated by paper soaked in conducting electrolyte. The insulator is a thin oxide layer on one of the foils. Values: 1µF to10mf. Electrolytic capacitors are polarised: the foil with the oxide layer must always be at a positive voltage relative to the other (else explosion). Negative terminal indicated by a curved plate in symbol or -. E1.1 Analysis of Circuits ( ) and : 9 3 / 12
12 are formed from coils of wire, often around a steel or ferrite core. E1.1 Analysis of Circuits ( ) and : 9 4 / 12
13 are formed from coils of wire, often around a steel or ferrite core. The magnetic flux within the coil isφ = µna l i wheren is the number of turns,ais the cross-sectional area of the coil andlis the length of the coil (around the toroid). µ is a property of the material that the core is made from and is called its permeability. For free space (or air): µ 0 = 4π 10 7 = 1.26 µh /m, for steel,µ 4000µ 0 = 5 mh /m. E1.1 Analysis of Circuits ( ) and : 9 4 / 12
14 are formed from coils of wire, often around a steel or ferrite core. The magnetic flux within the coil isφ = µna l i wheren is the number of turns,ais the cross-sectional area of the coil andlis the length of the coil (around the toroid). µ is a property of the material that the core is made from and is called its permeability. For free space (or air): µ 0 = 4π 10 7 = 1.26 µh /m, for steel,µ 4000µ 0 = 5 mh /m. From Faraday s law: v = N dφ = µn2 A l di = Ldi. E1.1 Analysis of Circuits ( ) and : 9 4 / 12
15 are formed from coils of wire, often around a steel or ferrite core. The magnetic flux within the coil isφ = µna l i wheren is the number of turns,ais the cross-sectional area of the coil andlis the length of the coil (around the toroid). µ is a property of the material that the core is made from and is called its permeability. For free space (or air): µ 0 = 4π 10 7 = 1.26 µh /m, for steel,µ 4000µ 0 = 5 mh /m. From Faraday s law: v = N dφ = µn2 A l di = Ldi. We measure the inductance, L = µn2 A l, in Henrys (H). E1.1 Analysis of Circuits ( ) and : 9 4 / 12
16 Passive Components We can describe all three types of passive component by the relationship betweenv andi using, in each case, the passive sign convention. E1.1 Analysis of Circuits ( ) and : 9 5 / 12
17 Passive Components We can describe all three types of passive component by the relationship betweenv andi using, in each case, the passive sign convention. Resistor: v = Ri E1.1 Analysis of Circuits ( ) and : 9 5 / 12
18 Passive Components We can describe all three types of passive component by the relationship betweenv andi using, in each case, the passive sign convention. Resistor: v = Ri Inductor: v = L di E1.1 Analysis of Circuits ( ) and : 9 5 / 12
19 Passive Components We can describe all three types of passive component by the relationship betweenv andi using, in each case, the passive sign convention. Resistor: v = Ri Inductor: v = L di Capacitor: i = C dv E1.1 Analysis of Circuits ( ) and : 9 5 / 12
20 Passive Components We can describe all three types of passive component by the relationship betweenv andi using, in each case, the passive sign convention. Resistor: v = Ri Inductor: v = L di Capacitor: i = C dv Notes: (1) There are no minus signs anywhere whatever you were taught at school. (2) We use lower case, v, for time-varying voltages. E1.1 Analysis of Circuits ( ) and : 9 5 / 12
21 Series and Parallel v = v 1 +v 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
22 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di E1.1 Analysis of Circuits ( ) and : 9 6 / 12
23 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di E1.1 Analysis of Circuits ( ) and : 9 6 / 12
24 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
25 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) E1.1 Analysis of Circuits ( ) and : 9 6 / 12
26 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
27 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 ( 1 = v L 1 + v L 2 = v L L 2 ) E1.1 Analysis of Circuits ( ) and : 9 6 / 12
28 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 ( 1 = v L 1 + v L 2 = v v = 1 1 L L 2 di L L 2 ) E1.1 Analysis of Circuits ( ) and : 9 6 / 12
29 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 ( 1 = v L 1 + v L 2 = v v = 1 1 L L 2 di L L 2 ) Same as a single inductor of value 1 1 L L 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
30 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 ( 1 = v L 1 + v L 2 = v v = 1 1 L L 2 di L L 2 ) Same as a single inductor of value 1 1 = L 1L 2 L + 1 L 1 L 1 +L 2 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
31 Series and Parallel v = v 1 +v 2 = L 1 di +L 2 di = (L 1 +L 2 ) di Same equation as a single inductor of valuel 1 +L 2 di = d(i 1+i 2 ) = di 1 + di 2 ( 1 = v L 1 + v L 2 = v v = 1 1 L L 2 di L L 2 ) Same as a single inductor of value combine just like resistors. 1 1 = L 1L 2 L + 1 L 1 L 1 +L 2 2 E1.1 Analysis of Circuits ( ) and : 9 6 / 12
32 Series and Parallel i = i 1 +i 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
33 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv E1.1 Analysis of Circuits ( ) and : 9 7 / 12
34 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv E1.1 Analysis of Circuits ( ) and : 9 7 / 12
35 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
36 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) E1.1 Analysis of Circuits ( ) and : 9 7 / 12
37 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
38 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 = i C 1 + i C 2 = i ( ) 1 C C 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
39 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 = i C 1 + i C 2 = i ( ) 1 C C 2 1 dv i = 1 C C 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
40 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 = i C 1 + i C 2 = i ( ) 1 C C 2 1 dv i = 1 C C 2 Same as a single capacitor of value 1 1 C C 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
41 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 = i C 1 + i C 2 = i ( ) 1 C C 2 1 dv i = 1 C C 2 Same as a single capacitor of value 1 1 = C 1C 2 C + 1 C 1 C 1 +C 2 2 E1.1 Analysis of Circuits ( ) and : 9 7 / 12
42 Series and Parallel dv i = i 1 +i 2 = C 1 +C 2 dv = (C 1 +C 2 ) dv Same equation as a single capacitor of value C 1 +C 2 dv = d(v 1+v 2 ) = dv 1 + dv 2 = i C 1 + i C 2 = i ( ) 1 C C 2 1 dv i = 1 C C 2 Same as a single capacitor of value 1 1 = C 1C 2 C + 1 C 1 C 1 +C 2 2 combine just like conductances (i.e. parallel capacitors add). E1.1 Analysis of Circuits ( ) and : 9 7 / 12
43 Current/Voltage Continuity Capacitor: i = C dv E1.1 Analysis of Circuits ( ) and : 9 8 / 12
44 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
45 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... E1.1 Analysis of Circuits ( ) and : 9 8 / 12
46 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
47 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
48 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. Inductor: v = L di E1.1 Analysis of Circuits ( ) and : 9 8 / 12
49 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. Inductor: v = L di For the current to change abruptly di = v =. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
50 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. Inductor: v = L di For the current to change abruptly di = v =. This never happens so... E1.1 Analysis of Circuits ( ) and : 9 8 / 12
51 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. Inductor: v = L di For the current to change abruptly di = v =. This never happens so... The current through an inductor never changes instantaneously. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
52 Current/Voltage Continuity Capacitor: i = C dv For the voltage to change abruptly dv = i =. This never happens so... The voltage across a capacitor never changes instantaneously. Informal version: A capacitor tries to keep its voltage constant. Inductor: v = L di For the current to change abruptly di = v =. This never happens so... The current through an inductor never changes instantaneously. Informal version: An inductor tries to keep its current constant. E1.1 Analysis of Circuits ( ) and : 9 8 / 12
53 Average Current/Voltage For a capacitori = C dv. E1.1 Analysis of Circuits ( ) and : 9 9 / 12
54 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv E1.1 Analysis of Circuits ( ) and : 9 9 / 12
55 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv E1.1 Analysis of Circuits ( ) and : 9 9 / 12
56 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) E1.1 Analysis of Circuits ( ) and : 9 9 / 12
57 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) E1.1 Analysis of Circuits ( ) and : 9 9 / 12
58 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. E1.1 Analysis of Circuits ( ) and : 9 9 / 12
59 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. (2) Ifv is bounded then the average current 0 as(t 2 t 1 ). E1.1 Analysis of Circuits ( ) and : 9 9 / 12
60 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. (2) Ifv is bounded then the average current 0 as(t 2 t 1 ). The average current through a capacitor is zero E1.1 Analysis of Circuits ( ) and : 9 9 / 12
61 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. (2) Ifv is bounded then the average current 0 as(t 2 t 1 ). The average current through a capacitor is zero and, likewise, the average voltage across an inductor is zero. E1.1 Analysis of Circuits ( ) and : 9 9 / 12
62 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. (2) Ifv is bounded then the average current 0 as(t 2 t 1 ). The average current through a capacitor is zero and, likewise, the average voltage across an inductor is zero. The circuit symbols remind you of this. E1.1 Analysis of Circuits ( ) and : 9 9 / 12
63 Average Current/Voltage For a capacitori = C dv. Take the average of both sides: 1 t 2 t 1 t2 t 1 i = 1 t 2 t 1 t2 t 1 C dv = C t 2 t 1 v(t2 ) v(t 1 ) dv = C t 2 t 1 [v] v(t 2) v(t 1 ) = C t 2 t 1 (v(t 2 ) v(t 1 )) (1) Ifv(t 1 ) = v(t 2 ) then the average current exactly equals zero. (2) Ifv is bounded then the average current 0 as(t 2 t 1 ). The average current through a capacitor is zero and, likewise, the average voltage across an inductor is zero. The circuit symbols remind you of this. Average can either be over an exact number of periods of a repetitive waveform or else the long-term average (provided v and i remain bounded). v is bounded means v always stays less than a predefined maximum value. E1.1 Analysis of Circuits ( ) and : 9 9 / 12
64 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 E1.1 Analysis of Circuits ( ) and : 9 10 / 12
65 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. E1.1 Analysis of Circuits ( ) and : 9 10 / 12
66 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. WhenS opens, the currenti L cannot change instantly and so it must flow through the diode (we assume the diode is ideal). E1.1 Analysis of Circuits ( ) and : 9 10 / 12
67 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. WhenS opens, the currenti L cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value ofxisav the average value ofy must also beav. E1.1 Analysis of Circuits ( ) and : 9 10 / 12
68 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. WhenS opens, the currenti L cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value ofxisav the average value ofy must also beav. The average current throughris av R so, since the average current through C must be zero, the average currenti L must also be av R. E1.1 Analysis of Circuits ( ) and : 9 10 / 12
69 Buck Converter A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. WhenS opens, the currenti L cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value ofxisav the average value ofy must also beav. The average current throughris av R so, since the average current through C must be zero, the average currenti L must also be av R. C dy = i L i R ifc is large, then the variations iny will be very small. E1.1 Analysis of Circuits ( ) and : 9 10 / 12
70 Buck Converter [Do not memorize this circuit] A buck converter converts a high voltage,v,into a lower one,y. The switch,s, closes for a fractiona of the time. a is the duty cycle and in this example. is 1 3 WhenS is closed,x = v, and a currenti L flows. WhenS opens, the currenti L cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value ofxisav the average value ofy must also beav. The average current throughris av R so, since the average current through C must be zero, the average currenti L must also be av R. C dy = i L i R ifc is large, then the variations iny will be very small. E1.1 Analysis of Circuits ( ) and : 9 10 / 12
71 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). E1.1 Analysis of Circuits ( ) and : 9 11 / 12
72 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. E1.1 Analysis of Circuits ( ) and : 9 11 / 12
73 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv E1.1 Analysis of Circuits ( ) and : 9 11 / 12
74 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv E1.1 Analysis of Circuits ( ) and : 9 11 / 12
75 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) E1.1 Analysis of Circuits ( ) and : 9 11 / 12
76 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) = 1 2 C( v 2 (t 2 ) v 2 (t 1 ) ) E1.1 Analysis of Circuits ( ) and : 9 11 / 12
77 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) = 1 2 C( v 2 (t 2 ) v 2 (t 1 ) ) Ifv(t 1 ) = v(t 2 ) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls. E1.1 Analysis of Circuits ( ) and : 9 11 / 12
78 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) = 1 2 C( v 2 (t 2 ) v 2 (t 1 ) ) Ifv(t 1 ) = v(t 2 ) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls. The energy stored in a capacitor is 1 2 Cv2 E1.1 Analysis of Circuits ( ) and : 9 11 / 12
79 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) = 1 2 C( v 2 (t 2 ) v 2 (t 1 ) ) Ifv(t 1 ) = v(t 2 ) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls. The energy stored in a capacitor is 1 2 Cv2 and likewise in an inductor 1 2 Li2. E1.1 Analysis of Circuits ( ) and : 9 11 / 12
80 Power and Energy Electrical power absorbed by any component at the instanttisv(t) i(t). So total energy absorbed between timest 1 an 2 isw = t 2 t=t 1 vi. For a capacitori = C dv, so W = C t 2 t=t 1 v dv = C v(t 2 ) v=v(t 1 ) vdv = C [ 1 2 v2] v(t 2 ) v(t 1 ) = 1 2 C( v 2 (t 2 ) v 2 (t 1 ) ) Ifv(t 1 ) = v(t 2 ) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls. The energy stored in a capacitor is 1 2 Cv2 and likewise in an inductor 1 2 Li2. Ifv andiremain bounded, then the average power absorbed by a capacitor or inductor is always zero. E1.1 Analysis of Circuits ( ) and : 9 11 / 12
81 Summary Capacitor: i = C dv E1.1 Analysis of Circuits ( ) and : 9 12 / 12
82 Summary Capacitor: i = C dv parallel capacitors add in value E1.1 Analysis of Circuits ( ) and : 9 12 / 12
83 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. E1.1 Analysis of Circuits ( ) and : 9 12 / 12
84 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero E1.1 Analysis of Circuits ( ) and : 9 12 / 12
85 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero Inductor: v = L di E1.1 Analysis of Circuits ( ) and : 9 12 / 12
86 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero Inductor: v = L di series inductors add in value (like resistors) E1.1 Analysis of Circuits ( ) and : 9 12 / 12
87 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero Inductor: v = L di series inductors add in value (like resistors) average v is zero, i never changes instantaneously. E1.1 Analysis of Circuits ( ) and : 9 12 / 12
88 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero Inductor: v = L di series inductors add in value (like resistors) average v is zero, i never changes instantaneously. average power absorbed is zero E1.1 Analysis of Circuits ( ) and : 9 12 / 12
89 Summary Capacitor: i = C dv parallel capacitors add in value average i is zero, v never changes instantaneously. average power absorbed is zero Inductor: v = L di series inductors add in value (like resistors) average v is zero, i never changes instantaneously. average power absorbed is zero For further details see Hayt et al. Chapter 7. E1.1 Analysis of Circuits ( ) and : 9 12 / 12
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