9: capacitors and inductors - imperial college · pdf filea capacitor is formed from two...
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![Page 1: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/1.jpg)
9: Capacitors and Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 1 / 12
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Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12
A capacitor is formed from two conducting plates separated by a thininsulating layer.
If a current i flows, positive change, q, willaccumulate on the upper plate. To preservecharge neutrality, a balancing negative chargewill be present on the lower plate.
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Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12
A capacitor is formed from two conducting plates separated by a thininsulating layer.
If a current i flows, positive change, q, willaccumulate on the upper plate. To preservecharge neutrality, a balancing negative chargewill be present on the lower plate.
There will be a potential energy difference (or voltage v) between the platesproportional to q.v = d
Aǫq where A is the area of the plates, d is their separation and ǫ is
the permittivity of the insulating layer (ǫ0 = 8.85 pF/m for a vacuum).
![Page 4: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/4.jpg)
Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12
A capacitor is formed from two conducting plates separated by a thininsulating layer.
If a current i flows, positive change, q, willaccumulate on the upper plate. To preservecharge neutrality, a balancing negative chargewill be present on the lower plate.
There will be a potential energy difference (or voltage v) between the platesproportional to q.v = d
Aǫq where A is the area of the plates, d is their separation and ǫ is
the permittivity of the insulating layer (ǫ0 = 8.85 pF/m for a vacuum).
The quantity C = Aǫd
is the capacitance and is measured in Farads (F),hence q = Cv.
![Page 5: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/5.jpg)
Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 2 / 12
A capacitor is formed from two conducting plates separated by a thininsulating layer.
If a current i flows, positive change, q, willaccumulate on the upper plate. To preservecharge neutrality, a balancing negative chargewill be present on the lower plate.
There will be a potential energy difference (or voltage v) between the platesproportional to q.v = d
Aǫq where A is the area of the plates, d is their separation and ǫ is
the permittivity of the insulating layer (ǫ0 = 8.85 pF/m for a vacuum).
The quantity C = Aǫd
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
capacitor equation: i = dqdt
= C dvdt
.
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.
Ceramic: Alternate layers of metal and ceramic(a few µm thick). Values: 1 nF to 1µF.
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.
Ceramic: Alternate layers of metal and ceramic(a few µm thick). Values: 1 nF to 1µF.
Electrolytic: Two sheets of aluminium foilseparated by paper soaked in conductingelectrolyte. The insulator is a thin oxide layeron one of the foils. Values: 1µF to 10mF.
![Page 10: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/10.jpg)
Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.
Ceramic: Alternate layers of metal and ceramic(a few µm thick). Values: 1 nF to 1µF.
Electrolytic: Two sheets of aluminium foilseparated by paper soaked in conductingelectrolyte. The insulator is a thin oxide layeron one of the foils. Values: 1µF to 10mF.
Electrolytic capacitors are polarised: the foil with the oxide layer mustalways be at a positive voltage relative to the other (else explosion).Negative terminal indicated by a curved plate in symbol
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.
Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.
Ceramic: Alternate layers of metal and ceramic(a few µm thick). Values: 1 nF to 1µF.
Electrolytic: Two sheets of aluminium foilseparated by paper soaked in conductingelectrolyte. The insulator is a thin oxide layeron one of the foils. Values: 1µF to 10mF.
Electrolytic capacitors are polarised: the foil with the oxide layer mustalways be at a positive voltage relative to the other (else explosion).Negative terminal indicated by a curved plate in symbol or “-”.
![Page 12: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/12.jpg)
Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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 are formed from coils of wire, oftenaround a steel or ferrite core.
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Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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 are formed from coils of wire, oftenaround a steel or ferrite core.
The magnetic flux within the coil is Φ = µNAl
i where N is the number ofturns, A is the cross-sectional area of the coil and l is the length of the coil(around the toroid).
µ is a property of the material that the core is made from and is called itspermeability . For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, forsteel, µ ≈ 4000µ0 = 5mH/m.
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Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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 are formed from coils of wire, oftenaround a steel or ferrite core.
The magnetic flux within the coil is Φ = µNAl
i where N is the number ofturns, A is the cross-sectional area of the coil and l is the length of the coil(around the toroid).
µ is a property of the material that the core is made from and is called itspermeability . For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, forsteel, µ ≈ 4000µ0 = 5mH/m.
From Faraday’s law: v = N dΦdt
= µN2Al
didt
= L didt
.
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Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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 are formed from coils of wire, oftenaround a steel or ferrite core.
The magnetic flux within the coil is Φ = µNAl
i where N is the number ofturns, A is the cross-sectional area of the coil and l is the length of the coil(around the toroid).
µ is a property of the material that the core is made from and is called itspermeability . For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, forsteel, µ ≈ 4000µ0 = 5mH/m.
From Faraday’s law: v = N dΦdt
= µN2Al
didt
= L didt
.
We measure the inductance, L = µN2Al
, in Henrys (H).
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.
Resistor: v = Ri
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductor: v = L didt
![Page 19: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/19.jpg)
Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductor: v = L didt
Capacitor: i = C dvdt
![Page 20: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/20.jpg)
Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductor: v = L didt
Capacitor: i = C dvdt
Notes: (1) There are no minus signs anywhere whatever you were taught atschool.
(2) We use lower case, v, for time-varying voltages.
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
![Page 24: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/24.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
![Page 26: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/26.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
![Page 27: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/27.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
= vL1
+ vL2
= v(
1L1
+ 1L2
)
![Page 28: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/28.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
= vL1
+ vL2
= v(
1L1
+ 1L2
)
v = 11
L1+ 1
L2
didt
![Page 29: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/29.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
= vL1
+ vL2
= v(
1L1
+ 1L2
)
v = 11
L1+ 1
L2
didt
Same as a single inductor of value 11
L1+ 1
L2
![Page 30: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/30.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
= vL1
+ vL2
= v(
1L1
+ 1L2
)
v = 11
L1+ 1
L2
didt
Same as a single inductor of value 11
L1+ 1
L2
= L1L2
L1+L2
![Page 31: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/31.jpg)
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
v = v1 + v2= L1didt
+ L2didt
= (L1 + L2)didt
Same equation as a single inductor of value L1 + L2
didt
= d(i1+i2)dt
= di1dt
+ di2dt
= vL1
+ vL2
= v(
1L1
+ 1L2
)
v = 11
L1+ 1
L2
didt
Same as a single inductor of value 11
L1+ 1
L2
= L1L2
L1+L2
Inductors combine just like resistors.
![Page 32: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/32.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2
![Page 33: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/33.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
![Page 34: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/34.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
![Page 35: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/35.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
![Page 36: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/36.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
![Page 37: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/37.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
![Page 38: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/38.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
= iC1
+ iC2
= i(
1C1
+ 1C2
)
![Page 39: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/39.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
= iC1
+ iC2
= i(
1C1
+ 1C2
)
i = 11
C1+ 1
C2
dvdt
![Page 40: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/40.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
= iC1
+ iC2
= i(
1C1
+ 1C2
)
i = 11
C1+ 1
C2
dvdt
Same as a single capacitor of value 11
C1+ 1
C2
![Page 41: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/41.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
= iC1
+ iC2
= i(
1C1
+ 1C2
)
i = 11
C1+ 1
C2
dvdt
Same as a single capacitor of value 11
C1+ 1
C2
= C1C2
C1+C2
![Page 42: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/42.jpg)
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
i = i1 + i2= C1dvdt
+ C2dvdt
= (C1 + C2)dvdt
Same equation as a single capacitor of value C1 + C2
dvdt
= d(v1+v2)dt
= dv1dt
+ dv2dt
= iC1
+ iC2
= i(
1C1
+ 1C2
)
i = 11
C1+ 1
C2
dvdt
Same as a single capacitor of value 11
C1+ 1
C2
= C1C2
C1+C2
Capacitors combine just like conductances (i.e. parallel capacitors add).
![Page 43: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/43.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
![Page 44: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/44.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ i = ∞.
![Page 45: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/45.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ i = ∞.
This never happens so ...
![Page 46: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/46.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
![Page 47: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/47.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.Informal version: A capacitor “tries” to keep its voltage constant.
![Page 48: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/48.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ 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 didt
![Page 49: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/49.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ 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 didt
For the current to change abruptly
didt
= ∞ ⇒ v = ∞.
![Page 50: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/50.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ 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 didt
For the current to change abruptly
didt
= ∞ ⇒ v = ∞.
This never happens so ...
![Page 51: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/51.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ 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 didt
For the current to change abruptly
didt
= ∞ ⇒ v = ∞.
This never happens so ...
The current through an inductor never changes instantaneously.
![Page 52: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/52.jpg)
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Capacitor: i = C dvdt
For the voltage to change abruptly
dvdt
= ∞ ⇒ 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 didt
For the current to change abruptly
didt
= ∞ ⇒ v = ∞.
This never happens so ...
The current through an inductor never changes instantaneously.Informal version: An inductor “tries” to keep its current constant.
![Page 53: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/53.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
.
![Page 54: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/54.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt
![Page 55: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/55.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
![Page 56: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/56.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
![Page 57: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/57.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
![Page 58: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/58.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
![Page 59: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/59.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.
![Page 60: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/60.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.
The average current through a capacitor is zero
![Page 61: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/61.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.
The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero.
![Page 62: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/62.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.
The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero. The circuit symbols remind you of this.
![Page 63: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/63.jpg)
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
For a capacitor i = C dvdt
. Take the average of both sides:
1t2−t1
∫ t2
t1idt = 1
t2−t1
∫ t2
t1C dv
dtdt= C
t2−t1
∫ v(t2)
v(t1)dv
= Ct2−t1
[v]v(t2)v(t1)
= Ct2−t1
(v(t2)− v(t1))
(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.
(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.
The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero. The circuit symbols remind you of this.
“Average” can either be over an exact number of periods of a repetitivewaveform or else the long-term average (provided v and i remainbounded).
“v is bounded” means |v| always stays less than a predefined maximumvalue.
![Page 64: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/64.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
![Page 65: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/65.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
![Page 66: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/66.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).
![Page 67: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/67.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).
The average value of x is aV ⇒ the average value of y must also be aV .
![Page 68: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/68.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).
The average value of x is aV ⇒ the average value of y must also be aV .
The average current through R is aVR
so, since the average current through
C must be zero, the average current iL must also be aVR
.
![Page 69: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/69.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).
The average value of x is aV ⇒ the average value of y must also be aV .
The average current through R is aVR
so, since the average current through
C must be zero, the average current iL must also be aVR
.
C dydt
= iL − iR ⇒ if C is large, then the variations in y will be very small.
![Page 70: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/70.jpg)
Buck Converter
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2017-10110) Capacitors and Inductors: 9 – 10 / 12
[Do not memorize this circuit]
A buck converter converts a highvoltage, V , into a lower one, Y .
The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1
3 in this example.
When S is closed, x = v, and acurrent iL flows.
When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).
The average value of x is aV ⇒ the average value of y must also be aV .
The average current through R is aVR
so, since the average current through
C must be zero, the average current iL must also be aVR
.
C dydt
= iL − iR ⇒ if C is large, then the variations in y will be very small.
![Page 71: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/71.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
![Page 72: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/72.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
![Page 73: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/73.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt
![Page 74: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/74.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
![Page 75: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/75.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)
![Page 76: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/76.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)= 1
2C(
v2(t2)− v2(t1))
![Page 77: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/77.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)= 1
2C(
v2(t2)− v2(t1))
If v(t1) = v(t2) then there has been nonett energy absorbed: all the energyabsorbed when the voltage rises isreturned to the circuit when it falls.
![Page 78: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/78.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)= 1
2C(
v2(t2)− v2(t1))
If v(t1) = v(t2) then there has been nonett energy absorbed: all the energyabsorbed when the voltage rises isreturned to the circuit when it falls.
The energy stored in a capacitor is 12Cv2
![Page 79: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/79.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)= 1
2C(
v2(t2)− v2(t1))
If v(t1) = v(t2) then there has been nonett energy absorbed: all the energyabsorbed when the voltage rises isreturned to the circuit when it falls.
The energy stored in a capacitor is 12Cv2 and likewise in an inductor 1
2Li2.
![Page 80: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/80.jpg)
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
Electrical power absorbed by any component at the instant t is v(t)× i(t).
So total energy absorbed between times t1 and t2 is W =∫ t2
t=t1vi dt.
For a capacitor i = C dvdt
, so
W = C∫ t2
t=t1v dvdtdt= C
∫ v(t2)
v=v(t1)vdv
= C[
12v
2]v(t2)
v(t1)= 1
2C(
v2(t2)− v2(t1))
If v(t1) = v(t2) then there has been nonett energy absorbed: all the energyabsorbed when the voltage rises isreturned to the circuit when it falls.
The energy stored in a capacitor is 12Cv2 and likewise in an inductor 1
2Li2.
If v and i remain bounded, then the average power absorbed by acapacitor or inductor is always zero.
![Page 81: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/81.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
![Page 82: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/82.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
![Page 83: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/83.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
![Page 84: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/84.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
![Page 85: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/85.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
◦ v = L didt
![Page 86: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/86.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
◦ v = L didt
◦ series inductors add in value (like resistors)
![Page 87: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/87.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
◦ v = L didt
◦ series inductors add in value (like resistors)
◦ average v is zero, i never changes instantaneously.
![Page 88: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/88.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
◦ v = L didt
◦ series inductors add in value (like resistors)
◦ average v is zero, i never changes instantaneously.
◦ average power absorbed is zero
![Page 89: 9: Capacitors and Inductors - Imperial College · PDF fileA capacitor is formed from two conducting plates separated ... Types of Capacitor 9: Capacitors and Inductors ... material](https://reader036.vdocuments.mx/reader036/viewer/2022062413/5aa95ce87f8b9a9a188cc6a9/html5/thumbnails/89.jpg)
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and ParallelInductors• Series and ParallelCapacitors
• 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
• Capacitor:
◦ i = C dvdt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
◦ v = L didt
◦ 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.