9: Capacitors and Inductors
• Capacitors
• 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 (2015-7150)
9: Capacitors and Inductors
Capacitors and Inductors: 9 – 1 / 12
Capacitors
9: Capacitors and Inductors
• Capacitors
• 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
A capacitor is formed from two conducting plates separated by a thin
insulating layer.
If a current i flows, positive change, q , will
accumulate on the upper plate. To preserve
charge neutrality, a balancing negative charge
will be present on the lower plate.
E1.1 Analysis of Circuits (2015-7150)
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Capacitors
9: Capacitors and Inductors
• Capacitors
• 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
A capacitor is formed from two conducting plates separated by a thin
insulating layer.
If a current i flows, positive change, q , will
accumulate on the upper plate. To preserve
charge neutrality, a balancing negative charge
will be present on the lower plate.
There will be a potential energy difference (or voltage v ) between the plates
proportional to q .
d
v = Aǫ
q 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).
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Capacitors
9: Capacitors and Inductors
• Capacitors
• 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
A capacitor is formed from two conducting plates separated by a thin
insulating layer.
If a current i flows, positive change, q , will
accumulate on the upper plate. To preserve
charge neutrality, a balancing negative charge
will be present on the lower plate.
There will be a potential energy difference (or voltage v ) between the plates
proportional to q .
d
v = Aǫ
q 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 = Aǫ
d is the capacitance and is measured in Farads (F),
hence q = Cv .
E1.1 Analysis of Circuits (2015-7150)
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Capacitors
9: Capacitors and Inductors
• Capacitors
• 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
A capacitor is formed from two conducting plates separated by a thin
insulating layer.
If a current i flows, positive change, q , will
accumulate on the upper plate. To preserve
charge neutrality, a balancing negative charge
will be present on the lower plate.
There will be a potential energy difference (or voltage v ) between the plates
proportional to q .
d
v = Aǫ
q 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 = Aǫ
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
capacitor equation: i = dt = C dv
dt .
E1.1 Analysis of Circuits (2015-7150)
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Polystyrene: Two sheets of foil separated by a
thin plastic film and rolled up to save space.
Values: 10 pF to 1 nF.
E1.1 Analysis of Circuits (2015-7150)
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Polystyrene: Two sheets of foil separated by a
thin plastic film and rolled up to save space.
Values: 10 pF to 1 nF.
Ceramic: Alternate layers of metal and ceramic
(a few µm thick). Values: 1 nF to 1 µF.
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Polystyrene: Two sheets of foil separated by a
thin plastic film and rolled up to save space.
Values: 10 pF to 1 nF.
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.
E1.1 Analysis of Circuits (2015-7150)
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Polystyrene: Two sheets of foil separated by a
thin plastic film and rolled up to save space.
Values: 10 pF to 1 nF.
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 (2015-7150)
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Types of Capacitor
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor symbol represents the two separated
plates. Capacitor types are distinguished by
the material used as the insulator.
Polystyrene: Two sheets of foil separated by a
thin plastic film and rolled up to save space.
Values: 10 pF to 1 nF.
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 “-”.
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Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors are formed from coils of wire, often
around a steel or ferrite core.
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Inductors
9: Capacitors and Inductors
• Capacitors
• 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
Inductors are formed from coils of wire, often
around a steel or ferrite core.
µN A
The magnetic flux within the coil is Φ = l i where N is the number of
turns, 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 its
permeability. For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, for
steel, µ ≈ 4000µ0 = 5 mH/m.
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Inductors
9: Capacitors and Inductors
• Capacitors
• 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
Inductors are formed from coils of wire, often
around a steel or ferrite core.
µN A
The magnetic flux within the coil is Φ = l i where N is the number of
turns, 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 its
permeability. For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, for
steel, µ ≈ 4000µ0 = 5 mH/m.
From Faraday’s law: v = N dΦ
dt =
E1.1 Analysis of Circuits (2015-7150)
µN 2 A di
l
dt
di
= L dt
.
Capacitors and Inductors: 9 – 4 / 12
Inductors
9: Capacitors and Inductors
• Capacitors
• 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
Inductors are formed from coils of wire, often
around a steel or ferrite core.
µN A
The magnetic flux within the coil is Φ = l i where N is the number of
turns, 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 its
permeability. For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, for
steel, µ ≈ 4000µ0 = 5 mH/m.
From Faraday’s law: v = N dΦ
dt =
µN 2 A di
l
dt
We measure the inductance, L =
µN 2 A
,
l
E1.1 Analysis of Circuits (2015-7150)
di
= L dt
.
in Henrys (H).
Capacitors and Inductors: 9 – 4 / 12
Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
We can describe all three types of passive component by the relationship
between V and I using, in each case, the passive sign convention.
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
We can describe all three types of passive component by the relationship
between V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
We can describe all three types of passive component by the relationship
between V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
Inductor: v = L dt
E1.1 Analysis of Circuits (2015-7150)
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
We can describe all three types of passive component by the relationship
between V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
Inductor: v = L dt
Capacitor: i = C dv
dt
E1.1 Analysis of Circuits (2015-7150)
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Passive Components
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
We can describe all three types of passive component by the relationship
between V and I using, in each case, the passive sign convention.
Resistor: v = Ri
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
Inductor: v = L dt
Capacitor: i = C dv
dt
Notes: (1) There are no minus signs anywhere whatever you were taught at
school.
(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 Parallel
v = v1 + v2
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
di
di
+ L2 dt
v = v1 + v2 = L1 dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
E1.1 Analysis of Circuits (2015-7150)
d(i1 +i2 )
dt
Capacitors and Inductors: 9 – 6 / 12
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
E1.1 Analysis of Circuits (2015-7150)
d(i1 +i2 )
di1
=
dt
dt
+
di2
dt
Capacitors and Inductors: 9 – 6 / 12
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
=
d(i1 +i2 )
di1
di2
=
+
dt
dt
dt
v
1
1
v
L1 + L2 = v L1 + L2
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
=
v=
d(i1 +i2 )
di1
di2
=
+
dt
dt
dt
v
1
1
v
L1 + L2 = v L1 + L2
1
di
1 dt
1
L +L
1
E1.1 Analysis of Circuits (2015-7150)
2
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Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
=
v=
d(i1 +i2 )
di1
di2
=
+
dt
dt
dt
v
1
1
v
L1 + L2 = v L1 + L2
1
di
1 dt
1
L +L
1
2
Same as a single inductor of value
E1.1 Analysis of Circuits (2015-7150)
1
L1
1
+ L1
2
Capacitors and Inductors: 9 – 6 / 12
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
=
v=
d(i1 +i2 )
di1
di2
=
+
dt
dt
dt
v
1
1
v
L1 + L2 = v L1 + L2
1
di
1 dt
1
L +L
1
2
Same as a single inductor of value
E1.1 Analysis of Circuits (2015-7150)
1
L1
1
+ L1
2
=
L1 L2
L1 +L2
Capacitors and Inductors: 9 – 6 / 12
Series and Parallel Inductors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
di
di
+ L2 dt
v = v1 + v2 = L1 dt
di
= (L1 + L2 ) dt
Same equation as a single inductor of value L1 + L2
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
di
dt
=
=
v=
d(i1 +i2 )
di1
di2
=
+
dt
dt
dt
v
1
1
v
L1 + L2 = v L1 + L2
1
di
1 dt
1
L +L
1
2
Same as a single inductor of value
1
L1
1
+ L1
2
=
L1 L2
L1 +L2
Inductors combine just like resistors.
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
i = i1 + i2
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
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Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
E1.1 Analysis of Circuits (2015-7150)
d(v1 +v2 )
dt
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
E1.1 Analysis of Circuits (2015-7150)
d(v1 +v2 )
dv1
=
dt
dt
+
dv2
dt
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
=
d(v1 +v2 )
dv1
=
dt
dt
i
C1
E1.1 Analysis of Circuits (2015-7150)
+
i
C2
=i
+
1
C1
dv2
dt
+
1
C2
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
=
i=
d(v1 +v2 )
dv1
=
dt
dt
i
C1
1
C1
E1.1 Analysis of Circuits (2015-7150)
+
i
C2
=i
+
1
C1
dv2
dt
+
1
C2
1
dv
1 dt
+C
2
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
=
i=
d(v1 +v2 )
dv1
=
dt
dt
i
C1
1
C1
+
i
C2
=i
+
1
C1
dv2
dt
+
1
C2
1
dv
1 dt
+C
2
Same as a single capacitor of value
E1.1 Analysis of Circuits (2015-7150)
1
C1
1
+ C1
2
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
=
i=
d(v1 +v2 )
dv1
=
dt
dt
i
C1
1
C1
+
i
C2
=i
+
1
C1
dv2
dt
+
1
C2
1
dv
1 dt
+C
2
Same as a single capacitor of value
E1.1 Analysis of Circuits (2015-7150)
1
C1
1
+ C1
2
=
C1 C2
C1 +C2
Capacitors and Inductors: 9 – 7 / 12
Series and Parallel Capacitors
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
dv
i = i1 + i2 = C1 dv
dt + C2 dt
= (C1 + C2 ) dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
Same equation as a single capacitor of value C1 + C2
dv
dt
=
=
i=
d(v1 +v2 )
dv1
=
dt
dt
i
C1
1
C1
+
i
C2
=i
+
1
C1
dv2
dt
+
1
C2
1
dv
1 dt
+C
2
Same as a single capacitor of value
1
C1
1
+ C1
2
=
C1 C2
C1 +C2
Capacitors combine just like conductances (i.e. parallel capacitors add).
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 7 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Capacitor: i = C dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
Informal version: A capacitor “tries” to keep its voltage constant.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
Informal version: A capacitor “tries” to keep its voltage constant.
di
Inductor: v = L dt
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
Informal version: A capacitor “tries” to keep its voltage constant.
di
Inductor: v = L dt
For the current to change abruptly
di
dt
E1.1 Analysis of Circuits (2015-7150)
= ∞ ⇒ v = ∞.
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
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 (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
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 (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Current/Voltage Continuity
9: Capacitors and Inductors
• Capacitors
• 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
Capacitor: i = C dv
dt
For the voltage to change abruptly
dv
dt
= ∞ ⇒ i = ∞.
This never happens so ...
The voltage across a capacitor never changes instantaneously.
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 (2015-7150)
Capacitors and Inductors: 9 – 8 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt .
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
R t2
t1
idt =
1
t2 −t1
R t2
t1
C dv
dt dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
R t2
t1
idt =
1
t2 −t1
R t2
t1
C
C dv
dt
=
dt
t2 −t1
R v(t2 )
v(t1 )
dv
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
C
C dv
dt
=
dt
t2 −t1
R v(t2 )
v(t1 )
dv
v(t )
[v]v(t12 )
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current exactly equals zero.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current exactly equals zero.
(2) If v is bounded then the average current
→ 0 as (t2 − t1 ) → ∞.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current exactly equals zero.
(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 (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current 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 average
voltage across an inductor is zero.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current 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 average
voltage across an inductor is zero. The circuit symbols remind you of this.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Average Current/Voltage
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
For a capacitor i = C dv
dt . Take the average of both sides:
1
t2 −t1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
R t2
t1
idt =
=
C
t2 −t1
1
t2 −t1
R t2
t1
v(t )
C
C dv
dt
=
dt
t2 −t1
[v]v(t12 ) =
C
t2 −t1
R v(t2 )
v(t1 )
dv
(v(t2 ) − v(t1 ))
(1) If v(t1 ) = v(t2 ) then the average
current 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 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 (2015-7150)
Capacitors and Inductors: 9 – 9 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
When S is closed, x = v , and a
current iL flows.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
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).
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
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 (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
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 .
The average current through R is aV
R so, since the average current through
C must be zero, the average current iL must also be
E1.1 Analysis of Circuits (2015-7150)
aV
R
.
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
• Capacitors
• 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
A buck converter converts a high
voltage, V , into a lower one, Y .
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
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 .
The average current through R is aV
R so, since the average current through
C must be zero, the average current iL must also be
aV
R
.
C dy
dt = iL − iR ⇒ if C is large, then the variations in y will be very small.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Buck Converter
9: Capacitors and Inductors
[Do not memorize this circuit]
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
A buck converter converts a high
voltage, V , into a lower one, Y .
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
The switch, S , closes for a fraction a
of the time. a is the duty cycle and
is 13 in this example.
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 .
The average current through R is aV
R so, since the average current through
C must be zero, the average current iL must also be
aV
R
.
C dy
dt = iL − iR ⇒ if C is large, then the variations in y will be very small.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 10 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Electrical power absorbed by any component at the instant t is v(t) × i(t).
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
E1.1 Analysis of Circuits (2015-7150)
R t2
t=t1
v dv
dt dt
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
E1.1 Analysis of Circuits (2015-7150)
R t2
v dv dt=
t=t1 dt
C
R v(t2 )
v=v(t1 )
vdv
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
E1.1 Analysis of Circuits (2015-7150)
R t2
v dv dt=
t=t1 dt
C
R v(t2 )
v=v(t1 )
vdv
2 v(t2 )
2 v v(t1 )
1
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
E1.1 Analysis of Circuits (2015-7150)
R t2
v dv dt=
t=t1 dt
C
1
2 v(t2 )
=
v
2
2C
v(t1 )
1
R v(t2 )
v=v(t1 )
2
vdv
2
v (t2 ) − v (t1 )
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
R t2
v dv dt=
t=t1 dt
C
1
2 v(t2 )
=
v
2
2C
v(t1 )
1
R v(t2 )
v=v(t1 )
2
vdv
2
v (t2 ) − v (t1 )
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 (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
R t2
v dv dt=
t=t1 dt
C
1
2 v(t2 )
=
v
2
2C
v(t1 )
1
R v(t2 )
v=v(t1 )
2
vdv
2
v (t2 ) − v (t1 )
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.
The energy stored in a capacitor is 21 Cv 2
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
R t2
v dv dt=
t=t1 dt
C
1
2 v(t2 )
=
v
2
2C
v(t1 )
1
R v(t2 )
v=v(t1 )
2
vdv
2
v (t2 ) − v (t1 )
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.
The energy stored in a capacitor is 21 Cv 2 and likewise in an inductor 21 Li2 .
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Power and Energy
9: Capacitors and Inductors
• Capacitors
• 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
Electrical power absorbed by any component at the instant t is v(t) × i(t).
R t2
So total energy absorbed between times t1 and t2 is W = t=t vi dt.
1
For a capacitor i = C dv
dt , so
W =C
=C
R t2
v dv dt=
t=t1 dt
C
1
2 v(t2 )
=
v
2
2C
v(t1 )
1
R v(t2 )
v=v(t1 )
2
vdv
2
v (t2 ) − v (t1 )
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.
The energy stored in a capacitor is 21 Cv 2 and likewise in an inductor 21 Li2 .
If v and i remain bounded, then the average power absorbed by a
capacitor or inductor is always zero.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 11 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
• Capacitor:
◦ i = C dv
dt
Inductors
• Series and Parallel
Capacitors
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
• Capacitor:
◦ i = C dv
dt
◦ 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 (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• Types of Capacitor
• Inductors
• Passive Components
• Series and Parallel
Inductors
• Series and Parallel
Capacitors
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
• Current/Voltage Continuity
• Average Current/Voltage
• Buck Converter
• Power and Energy
• Summary
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
di
◦ v = L dt
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
di
◦ v = L dt
◦ series inductors add in value (like resistors)
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
di
◦ v = L dt
◦ series inductors add in value (like resistors)
◦ average v is zero, i never changes instantaneously.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
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 (2015-7150)
Capacitors and Inductors: 9 – 12 / 12
Summary
9: Capacitors and Inductors
• Capacitors
• 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
• Capacitor:
◦ i = C dv
dt
◦ parallel capacitors add in value
◦ average i is zero, v never changes instantaneously.
◦ average power absorbed is zero
• Inductor:
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 et al. Chapter 7.
E1.1 Analysis of Circuits (2015-7150)
Capacitors and Inductors: 9 – 12 / 12