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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