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9: and • Capacitors • Types of • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/ Continuity • Average Current/Voltage • • Power and • Summary 9: Capacitors and Inductors

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 1 / 12 Capacitors

9: Capacitors and Inductors A capacitor is formed from two conducting plates separated by a thin • Capacitors • Types of Capacitor insulating layer. • Inductors • Passive Components • Series and Parallel If a current i flows, positive change, q, will Inductors • Series and Parallel accumulate on the upper plate. To preserve Capacitors charge neutrality, a balancing negative charge • Current/Voltage Continuity • Average Current/Voltage will be present on the lower plate. • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12 Capacitors

9: Capacitors and Inductors A capacitor is formed from two conducting plates separated by a thin • Capacitors • Types of Capacitor insulating layer. • Inductors • Passive Components • Series and Parallel If a current i flows, positive change, q, will Inductors • Series and Parallel accumulate on the upper plate. To preserve Capacitors charge neutrality, a balancing negative charge • Current/Voltage Continuity • Average Current/Voltage will be present on the lower plate. • Buck Converter • Power and Energy • Summary There will be a difference (or voltage v) between the plates proportional to q. d v = Aǫ q where A is the area of the plates, d is their separation and ǫ is pF the of the insulating layer (ǫ0 = 8.85 /m for a ).

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12 Capacitors

9: Capacitors and Inductors A capacitor is formed from two conducting plates separated by a thin • Capacitors • Types of Capacitor insulating layer. • Inductors • Passive Components • Series and Parallel If a current i flows, positive change, q, will Inductors • Series and Parallel accumulate on the upper plate. To preserve Capacitors charge neutrality, a balancing negative charge • Current/Voltage Continuity • Average Current/Voltage will be present on the lower plate. • Buck Converter • Power and Energy • Summary There will be a potential energy difference (or voltage v) between the plates proportional to q. d v = Aǫ q where A is the area of the plates, d is their separation and ǫ is pF the permittivity of the insulating layer (ǫ0 = 8.85 /m for a vacuum). Aǫ The quantity C = d is the and is measured in (F), hence q = Cv.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12 Capacitors

9: Capacitors and Inductors A capacitor is formed from two conducting plates separated by a thin • Capacitors • Types of Capacitor insulating layer. • Inductors • Passive Components • Series and Parallel If a current i flows, positive change, q, will Inductors • Series and Parallel accumulate on the upper plate. To preserve Capacitors charge neutrality, a balancing negative charge • Current/Voltage Continuity • Average Current/Voltage will be present on the lower plate. • Buck Converter • Power and Energy • Summary There will be a potential energy difference (or voltage v) between the plates proportional to q. d v = Aǫ q where A is the area of the plates, d is their separation and ǫ is pF the permittivity of the insulating layer (ǫ0 = 8.85 /m for a vacuum). Aǫ The quantity C = d is the capacitance and is measured in Farads (F), hence q = Cv. The current, i, is the rate of charge on the plate, hence the dq dv capacitor equation: i = dt = C dt .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. are distinguished by • Inductors • Passive Components the material used as the . • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. Capacitor types are distinguished by • Inductors • Passive Components the material used as the insulator. • Series and Parallel Inductors • Series and Parallel Capacitors : Two sheets of foil separated by a • Current/Voltage Continuity • Average Current/Voltage thin film and rolled up to save space. • Buck Converter Values: to . • Power and Energy 10 pF 1 nF • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. Capacitor types are distinguished by • Inductors • Passive Components the material used as the insulator. • Series and Parallel Inductors • Series and Parallel Capacitors Polystyrene: Two sheets of foil separated by a • Current/Voltage Continuity • Average Current/Voltage thin plastic film and rolled up to save space. • Buck Converter Values: to . • Power and Energy 10 pF 1 nF • Summary : Alternate layers of metal and ceramic (a few µm thick). Values: 1 nF to 1 µF.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. Capacitor types are distinguished by • Inductors • Passive Components the material used as the insulator. • Series and Parallel Inductors • Series and Parallel Capacitors Polystyrene: Two sheets of foil separated by a • Current/Voltage Continuity • Average Current/Voltage thin plastic film and rolled up to save space. • Buck Converter Values: to . • Power and Energy 10 pF 1 nF • Summary Ceramic: Alternate layers of metal and ceramic (a few µm thick). Values: 1 nF to 1 µF.

Electrolytic: Two sheets of foil separated by soaked in conducting . The insulator is a thin layer on one of the foils. Values: 1 µF to 10 mF.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. Capacitor types are distinguished by • Inductors • Passive Components the material used as the insulator. • Series and Parallel Inductors • Series and Parallel Capacitors Polystyrene: Two sheets of foil separated by a • Current/Voltage Continuity • Average Current/Voltage thin plastic film and rolled up to save space. • Buck Converter Values: to . • Power and Energy 10 pF 1 nF • Summary Ceramic: Alternate layers of metal and ceramic (a few µm thick). Values: 1 nF to 1 µF.

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: 1 µF to 10 mF. 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 (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Types of Capacitor

9: Capacitors and Inductors Capacitor symbol represents the two separated • Capacitors • Types of Capacitor plates. Capacitor types are distinguished by • Inductors • Passive Components the material used as the insulator. • Series and Parallel Inductors • Series and Parallel Capacitors Polystyrene: Two sheets of foil separated by a • Current/Voltage Continuity • Average Current/Voltage thin plastic film and rolled up to save space. • Buck Converter Values: to . • Power and Energy 10 pF 1 nF • Summary Ceramic: Alternate layers of metal and ceramic (a few µm thick). Values: 1 nF to 1 µF.

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: 1 µF to 10 mF. 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 (2017-10110) Capacitors and Inductors: 9 – 3 / 12 Inductors

9: Capacitors and Inductors Inductors are formed from coils of , often • Capacitors • Types of Capacitor around a steel or core. • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 4 / 12 Inductors

9: Capacitors and Inductors Inductors are formed from coils of wire, often • Capacitors • Types of Capacitor around a steel or . • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors The magnetic flux within the coil is µNA where is the number of • Current/Voltage Continuity Φ = l i N • Average Current/Voltage turns, A is the cross-sectional area of the coil and l is the length of the coil • Buck Converter • Power and Energy (around the toroid). • Summary µ is a property of the material that the core is made from and is called its −7 µH permeability. For free space (or air): µ0 = 4π × 10 = 1.26 /m, for mH steel, µ ≈ 4000µ0 = 5 /m.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 4 / 12 Inductors

9: Capacitors and Inductors Inductors are formed from coils of wire, often • Capacitors • Types of Capacitor around a steel or ferrite core. • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors The magnetic flux within the coil is µNA where is the number of • Current/Voltage Continuity Φ = l i N • Average Current/Voltage turns, A is the cross-sectional area of the coil and l is the length of the coil • Buck Converter • Power and Energy (around the toroid). • Summary µ is a property of the material that the core is made from and is called its −7 µH permeability. For free space (or air): µ0 = 4π × 10 = 1.26 /m, for mH steel, µ ≈ 4000µ0 = 5 /m. 2 dΦ µN A di di From Faraday’s law: v = N dt = l dt = L dt .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 4 / 12 Inductors

9: Capacitors and Inductors Inductors are formed from coils of wire, often • Capacitors • Types of Capacitor around a steel or ferrite core. • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors The magnetic flux within the coil is µNA where is the number of • Current/Voltage Continuity Φ = l i N • Average Current/Voltage turns, A is the cross-sectional area of the coil and l is the length of the coil • Buck Converter • Power and Energy (around the toroid). • Summary µ is a property of the material that the core is made from and is called its −7 µH permeability. For free space (or air): µ0 = 4π × 10 = 1.26 /m, for mH steel, µ ≈ 4000µ0 = 5 /m. 2 dΦ µN A di di From Faraday’s law: v = N dt = l dt = L dt . 2 µN A We measure the , L = l , in Henrys (H).

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 4 / 12 Passive Components

9: Capacitors and Inductors We can describe all three types of passive component by the relationship • Capacitors • Types of Capacitor between V and I using, in each case, the . • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 5 / 12 Passive Components

9: Capacitors and Inductors We can describe all three types of passive component by the relationship • Capacitors • Types of Capacitor between V and I using, in each case, the passive sign convention. • Inductors • Passive Components • Series and Parallel : v = Ri Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 5 / 12 Passive Components

9: Capacitors and Inductors We can describe all three types of passive component by the relationship • Capacitors • Types of Capacitor between V and I using, in each case, the passive sign convention. • Inductors • Passive Components • Series and Parallel Resistor: v = Ri Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy : di • Summary v = L dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 5 / 12 Passive Components

9: Capacitors and Inductors We can describe all three types of passive component by the relationship • Capacitors • Types of Capacitor between V and I using, in each case, the passive sign convention. • Inductors • Passive Components • Series and Parallel Resistor: v = Ri Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy Inductor: di • Summary v = L dt

dv Capacitor: i = C dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 5 / 12 Passive Components

9: Capacitors and Inductors We can describe all three types of passive component by the relationship • Capacitors • Types of Capacitor between V and I using, in each case, the passive sign convention. • Inductors • Passive Components • Series and Parallel Resistor: v = Ri Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy Inductor: di • Summary v = L dt

dv Capacitor: i = C dt

Notes: (1) There are no minus signs anywhere whatever you were taught at school. (2) We use lower case, v, for time-varying .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 5 / 12 Series and Parallel Inductors

9: Capacitors and Inductors • Capacitors v = v1 + v2 • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) dt = dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt = v + v = v 1 + 1 L1 L2  L1 L2 

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt = v + v = v 1 + 1 L1 L2  L1 L2  1 di v = 1 1 dt L1 + L2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt = v + v = v 1 + 1 L1 L2  L1 L2  1 di v = 1 1 dt L1 + L2

1 Same as a single inductor of value 1 1 L1 + L2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt = v + v = v 1 + 1 L1 L2  L1 L2  1 di v = 1 1 dt L1 + L2

1 L1L2 Same as a single inductor of value 1 1 = L1+L2 L1 + L2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Inductors

9: Capacitors and Inductors di di • Capacitors v = v1 + v2= L1 dt + L2 dt • Types of Capacitor di • Inductors =(L1 + L2) dt • Passive Components • Series and Parallel Same equation as a single inductor of value L1 + L2 Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary di d(i1+i2) di1 di2 dt = dt = dt + dt = v + v = v 1 + 1 L1 L2  L1 L2  1 di v = 1 1 dt L1 + L2

1 L1L2 Same as a single inductor of value 1 1 = L1+L2 L1 + L2 Inductors combine just like .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 6 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors • Capacitors i = i1 + i2 • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dt = dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt = i + i = i 1 + 1 C1 C2  C1 C2 

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt = i + i = i 1 + 1 C1 C2  C1 C2  1 dv i = 1 1 dt C1 + C2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt = i + i = i 1 + 1 C1 C2  C1 C2  1 dv i = 1 1 dt C1 + C2 1 Same as a single capacitor of value 1 1 C1 + C2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt = i + i = i 1 + 1 C1 C2  C1 C2  1 dv i = 1 1 dt C1 + C2

1 C1C2 Same as a single capacitor of value 1 1 = C1+C2 C1 + C2

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Series and Parallel Capacitors

9: Capacitors and Inductors dv dv • Capacitors i = i1 + i2= C1 dt + C2 dt • Types of Capacitor • Inductors dv =(C1 + C2) • Passive Components dt • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage Same equation as a single capacitor of value • Buck Converter C1 + C2 • Power and Energy • Summary dv d(v1+v2) dv1 dv2 dt = dt = dt + dt = i + i = i 1 + 1 C1 C2  C1 C2  1 dv i = 1 1 dt C1 + C2

1 C1C2 Same as a single capacitor of value 1 1 = C1+C2 C1 + C2

Capacitors combine just like conductances (i.e. parallel capacitors add).

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 7 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

di Inductor: v = L dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

di Inductor: v = L dt For the current to change abruptly di dt = ∞ ⇒ v = ∞.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

di Inductor: v = L dt For the current to change abruptly di dt = ∞ ⇒ v = ∞. This never happens so ...

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

di Inductor: v = L dt For the current to change abruptly di dt = ∞ ⇒ v = ∞. This never happens so ...

The current through an inductor never changes instantaneously.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Current/Voltage Continuity

9: Capacitors and Inductors dv • Capacitors Capacitor: i = C dt • Types of Capacitor • Inductors For the voltage to change abruptly • Passive Components • Series and Parallel dv Inductors = ∞ ⇒ i = ∞. • Series and Parallel dt Capacitors • Current/Voltage Continuity This never happens so ... • Average Current/Voltage • Buck Converter The voltage across a capacitor never changes instantaneously. • Power and Energy • Summary Informal version: A capacitor “tries” to keep its voltage constant.

di Inductor: v = L dt For the current to change abruptly di dt = ∞ ⇒ 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 (2017-10110) Capacitors and Inductors: 9 – 8 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt • Series and Parallel R R Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) • Series and Parallel = [v] Capacitors t2−t1 v(t1) • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary (2) If v is bounded then the average current → 0 as (t2 − t1) → ∞.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary (2) If v is bounded then the average current → 0 as (t2 − t1) → ∞.

The average current through a capacitor is zero

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary (2) If v is bounded then the average current → 0 as (t2 − t1) → ∞.

The average current through a capacitor is zero and, likewise, the average voltage across an inductor is zero.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary (2) If v is bounded then the average current → 0 as (t2 − t1) → ∞.

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 (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Average Current/Voltage

9: Capacitors and Inductors dv • Capacitors For a capacitor i = C dt . Take the average of both sides: • Types of Capacitor • Inductors 1 t2 1 t2 dv C v(t2) • Passive Components t2−t1 t1 idt = t2−t1 t1 C dt dt = t2−t1 v(t1) dv • Series and Parallel R R R Inductors C v(t2) C • Series and Parallel = [v] = (v(t ) − v(t )) Capacitors t2−t1 v(t1) t2−t1 2 1 • Current/Voltage Continuity • Average Current/Voltage (1) If then the average • Buck Converter v(t1) = v(t2) • Power and Energy current exactly equals zero. • Summary (2) If v is bounded then the average current → 0 as (t2 − t1) → ∞.

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 (2017-10110) Capacitors and Inductors: 9 – 9 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The , S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows. When S opens, the current iL cannot change instantly and so it must flow through the (we assume the diode is ideal).

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows. When S opens, the current iL cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value of x is aV ⇒ the average value of y must also be aV .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows. When S opens, the current iL cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value of x is aV ⇒ the average value of y must also be aV . aV The average current through R is R so, since the average current through aV C must be zero, the average current iL must also be R .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows. When S opens, the current iL cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value of x is aV ⇒ the average value of y must also be aV . aV The average current through R is R so, since the average current through aV C must be zero, the average current iL must also be R . dy C dt = iL − iR ⇒ if C is large, then the variations in y will be very small.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Buck Converter

9: Capacitors and Inductors [Do not memorize this circuit] • Capacitors • Types of Capacitor A buck converter converts a high • Inductors • Passive Components voltage, V , into a lower one, Y . • Series and Parallel Inductors • Series and Parallel The switch, S, closes for a fraction a Capacitors • Current/Voltage Continuity of the time. a is the duty cycle and • Average Current/Voltage is 1 in this example. • Buck Converter 3 • Power and Energy • Summary When S is closed, x = v, and a current iL flows. When S opens, the current iL cannot change instantly and so it must flow through the diode (we assume the diode is ideal). The average value of x is aV ⇒ the average value of y must also be aV . aV The average current through R is R so, since the average current through aV C must be zero, the average current iL must also be R . dy C dt = iL − iR ⇒ if C is large, then the variations in y will be very small.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor • Inductors • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv • Average Current/Voltage W = C t=t1 v dt dt • Buck Converter R • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy 1 2 v(t2) • Summary = C v  2 v(t1)

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy v(t2) • 1 2 1 2 2 Summary = C v = C v (t2) − v (t1)  2 v(t1) 2 

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy v(t2) • 1 2 1 2 2 Summary = C v = C v (t2) − v (t1)  2 v(t1) 2 

If v(t1) = v(t2) 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 (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy v(t2) • 1 2 1 2 2 Summary = C v = C v (t2) − v (t1)  2 v(t1) 2 

If v(t1) = v(t2) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls.

1 2 The energy stored in a capacitor is 2 Cv

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy v(t2) • 1 2 1 2 2 Summary = C v = C v (t2) − v (t1)  2 v(t1) 2 

If v(t1) = v(t2) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls.

1 2 1 2 The energy stored in a capacitor is 2 Cv and likewise in an inductor 2 Li .

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Power and Energy

9: Capacitors and Inductors • Capacitors Electrical power absorbed by any component at the instant t is v(t) × i(t). • Types of Capacitor t2 • Inductors So total energy absorbed between times and is . t1 t2 W = t=t1 vi dt • Passive Components R • Series and Parallel Inductors dv • Series and Parallel For a capacitor i = C dt , so Capacitors • Current/Voltage Continuity t2 dv v(t2) • Average Current/Voltage W = C t=t1 v dt dt= C v=v(t1) vdv • Buck Converter R R • Power and Energy v(t2) • 1 2 1 2 2 Summary = C v = C v (t2) − v (t1)  2 v(t1) 2 

If v(t1) = v(t2) then there has been no nett energy absorbed: all the energy absorbed when the voltage rises is returned to the circuit when it falls.

1 2 1 2 The energy stored in a capacitor is 2 Cv and likewise in an inductor 2 Li . If v and i remain bounded, then the average power absorbed by a capacitor or inductor is always zero.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 11 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Summary

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Inductor: • Summary di ◦ v = L dt

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Inductor: • Summary di ◦ v = L dt ◦ series inductors add in value (like resistors)

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Inductor: • Summary di ◦ v = L dt ◦ series inductors add in value (like resistors) ◦ average v is zero, i never changes instantaneously.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Inductor: • Summary di ◦ v = L dt ◦ 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 (2017-10110) Capacitors and Inductors: 9 – 12 / 12 Summary

9: Capacitors and Inductors • Capacitor: • Capacitors • Types of Capacitor ◦ i = C dv • Inductors dt • Passive Components • Series and Parallel ◦ parallel capacitors add in value Inductors • Series and Parallel ◦ average i is zero, v never changes instantaneously. Capacitors • Current/Voltage Continuity ◦ average power absorbed is zero • Average Current/Voltage • Buck Converter • Power and Energy • Inductor: • Summary di ◦ v = L dt ◦ 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 Ch 7 or Irwin Ch 6.

E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 12 / 12