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Lecture 9: Inductance & CapacitanceNilsson 6.1-6.5
ENG17 : Circuits I
Spring 2015
April 28, 2015
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Inductance in the Real World
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What Does an Inductor Look Like?
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What Does an Inductor Do?
• Inductor: electrical component that opposes any change in electrical current
• Associates electrical current and magnetic fields
• Magnetic field generated by charge in motion (i.e., current)
• If current changes wrt time, mag field changes wrt time– Varying mag field results in voltage in linked
conductor
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Add’l Note
• Inductors can store energy through the mag fields– Used in spark plugs
• Inductors do not generate energy, so they are still passive devices
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Inductors in Circuits
Symbol: LUnits: [H] for henrys = [Wb/A]
𝑣=𝐿𝑑𝑖𝑑𝑡
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Determining Voltage
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Current in Inductor
𝑖 (𝑡 )= 1𝐿∫𝑡0
𝑡
𝑣 𝑑𝜏+𝑖(𝑡 0)
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Power & Energy in Inductor
𝑝=𝐿𝑖𝑑𝑖𝑑𝑡
𝑤=12𝐿𝑖2
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Capacitance in the Real World
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What Does a Capacitor Look Like?
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What Does a Capacitor Do?
• Capacitor: electrical component that consists of 2 conductors separated by dielectric
• “Stores” electric charge (like a battery)• Associated with electric fields, due to
separation of charge (voltage)• If voltage varies with time, E-field varies with
time– Varying E-field generates displacement current in
space occupied by field– Capacitance relates displacement current to voltage
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Capacitors in Circuits
Symbol: CUnits: [F] for farads = [C/V]
𝑖=𝐶𝑑𝑣𝑑𝑡
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Capacitor Voltage, Power, & Energy
𝑣 (𝑡 )= 1𝐶∫
𝑡 0
𝑡
𝑖𝑑𝜏+𝑣(𝑡0)
𝑝=𝑣𝑖=𝐶𝑣𝑑𝑣𝑑𝑡
𝑤=12𝐶𝑣2
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Example
𝑣 (𝑡 )={ 0 ,𝑡≤0 𝑠4 𝑡 ,0𝑠≤ 𝑡≤1𝑠4𝑒−(𝑡− 1) ,𝑡≥1𝑠
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ExampleObtain the energy stored in each capacitor under DC conditions.
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Example
Under DC conditions, find i, vc, and iL.Also, find the energy stored in the capacitor and inductor.
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Inductors Series-Parallel
Series Parallel
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Capacitors Series-Parallel
Series Parallel
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Example
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Magnetically Coupled Coils
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Self- and Mutual-Inductance
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Dot Convention
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Mesh Current
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Self Inductance
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Mutual Inductance i.t.o. Self Inductance
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Overview
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored
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Energy Stored in Magnetically-Coupled Coils
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Recap
• Inductors• Capacitors• Series-Parallel L/C• Mutual Inductance• Self Inductance• Energy Stored