Lesson 13

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Lesson 13 (W, 9/20/2022): Capacitance and Capacitors ( con’t ); Inductance and Inductors Fall 2023 Prof: J. D. McKinney, mckinnjd@purdue.edu OH: TRF, 3 – 4 pm 1
1. Capacitance and capacitors. 2. Inductors and inductance. 3. C and L combinations. Outline
Capacitors: i-v characteristic 3 ? 𝑡 = ? ( 𝑡 0 ) + 1 ? න ? 0 ? 𝑖 𝑡 ′ 𝑑 𝑡 ′ 𝑖 ( 𝑡 ) = ? 𝑑? ( 𝑡 ) 𝑑𝑡 ? = 2𝐹
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Capacitors: DC & Steady State Behavior 4 A capacitor in steady-state or DC conditions behaves as an: 𝑖 ( 𝑡 ) = ? 𝑑? ( 𝑡 ) 𝑑𝑡 R 1 + V s R 2 - + v C - open circuit (with a voltage across it!)
Capacitors: Energy Storage 5 𝑝 𝑡 = ? 𝑡 𝑖 𝑡 = ? 𝑡 ? 𝑑? ( 𝑡 ) 𝑑𝑡 ? 𝑡 = ? 𝑡 0 + න ? 0 ? 𝑝 𝑡 ′ 𝑑 𝑡 ′ = ? 𝑡 0 + ? න ? 0 ? ? ( 𝑡 ′ ) 𝑑? ( 𝑡 ′ ) 𝑑 𝑡 ′ 𝑑 𝑡 ′ ? 𝑡 = 1 2 ? ቚ ? 2 ( 𝑡 ′ ) ? 0 ? = 1 2 ? ? 2 𝑡 − ? 2 ( 𝑡 0 ) ? 𝑡 = 1 2 ? ? 2 𝑡
Capacitors: Energy Storage – Example 1 6 For the following circuit find: 1. The energy stored in the capacitor at 𝑡 = ∞ . 2. The stored energy change in the capacitor from 𝑡 = 0 to 𝑡 = ∞ . 1. Circuit at 𝑡 = ∞ . ? 𝑐 = 24 ? 10 Ω 10 Ω + 2 Ω = 20 ? ? 𝑐 = 1 2 ? ? 𝑐 2 = 1 2 100 ?𝐹 20 ? 2 = 20 ?
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Capacitors: Energy Storage – Example 1 7 1. Circuit at 𝑡 = 0 − : ? 𝑐 = 15 ? 10 Ω 10 Ω + 5 Ω = 1 0 ? ? 𝑐 = 1 2 ? ? 𝑐 2 = 1 2 100 ?𝐹 10 ? 2 = 5 ?
Inductors – Basics 8 • Inductors = passive elements that store magnetic field energy • Typically from nH to mH • Phones, radios, electric cars, aircraft … https://upload.wikimedia.org/wikipedia/commons/a/a1/Electronic_component_ inductors.jpg
Inductance: Basic Physics 9 Φ = න 𝑆 ? ∙ 𝑑𝑆 Units Magnetic field: Tesla (T) Flux: : Weber (Wb) Area : m 2 𝐿 = Φ ? 𝐿 = 𝑁 Φ ? = ? ? Units Inductance : Henry (H) T = Wb/m 2 H = W s
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Inductors: Basic Physics – Solenoid Results 10 𝐿 = 𝑁 Φ ? = ? 𝑁 2 ? 𝑆
Inductors: i-v Characteristic 11 ? 𝑡 = 𝐿 𝑑𝑖 ( 𝑡 ) 𝑑𝑡 The current of an inductor is always continuous: 𝑖 𝑡 − = 𝑖(𝑡 + ) 𝑖 𝑡 = 1 𝐿 න ? 0 ? ? 𝑡 ′ 𝑑 𝑡 ′ + 𝑖 𝑡 0
Inductors: i-v Characteristic 12 ? 𝑡 = 𝐿 𝑑𝑖 ( 𝑡 ) 𝑑𝑡 𝑖 𝑡 = 1 𝐿 න ? 0 ? ? 𝑡 ′ 𝑑 𝑡 ′ + 𝑖 𝑡 0 L = 5?
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Inductors: DC Circuits and Steady-State Behavior 13 ? 𝑡 = 𝐿 𝑑𝑖 ( 𝑡 ) 𝑑𝑡 An inductor in steady-state or DC conditions behaves as a : short circuit (with a current through it!)
Inductors: Energy Storage 14 𝑝 𝑡 = ? 𝑡 𝑖 𝑡 = 𝐿 𝑑 𝑖 ( 𝑡 ) 𝑑𝑡 𝑖 𝑡 ? 𝑡 = 1 2 𝐿 𝑖 2 𝑡 ? 𝑡 = 𝐿 𝑑𝑖 ( 𝑡 ) 𝑑𝑡 ? 𝑡 = ? 𝑡 0 + න ? 0 ? 𝑝 𝑡 ′ 𝑑 𝑡 ′ = ? 𝑡 0 + 𝐿 න ? 0 ? 𝑖 ( 𝑡 ′ ) 𝑑𝑖 ( 𝑡 ′ ) 𝑑 𝑡 ′ 𝑑 𝑡 ′ ? 𝑡 = 𝐿 2 ቚ 𝑖 2 ( 𝑡 ′ ) ? 0 ? = 1 2 𝐿 𝑖 2 𝑡 − 𝑖 2 ( 𝑡 0 )
Inductors: Energy Storage – Example 1 15 Find the total energy stored in both inductors in the following circuit.
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Inductors: Energy Storage – Example 1 16 ? 𝐿1 = 20 ? 8 Ω + 2 Ω = 2 ? ? 𝐿1 = 1 2 𝐿 1 ? 𝐿1 2 = 1 2 1 ?? 2 ? 2 = 2 ?? ? 𝐿2 = 2? 3 Ω 3 Ω + 6 Ω = 2 3 ? ? 𝐿2 = 1 2 𝐿 2 ? 𝐿 2 2 = 1 2 18 ?? 2 3 ? 2 = 4 ??
Capacitors: Connected in Series 17 1 ? 𝑒𝑞 = 1 ? 1 + 1 ? 2 + ⋯ + 1 ? 𝑁 Initial voltage across ? 𝑒𝑞 : ? 1 𝑡 0 + ? 2 𝑡 0 + ⋯ + ? 𝑁 𝑡 0 ? ? (𝑡) = ? 1 𝑡 + ? 2 𝑡 + ⋯ + ? 𝑁 𝑡 𝑑 ? ? ( 𝑡 ) 𝑑 𝑡 = 𝑑 ? 1 ( 𝑡 ) 𝑑𝑡 + 𝑑 ? 2 ( 𝑡 ) 𝑑𝑡 + ⋯ + 𝑑 ? 𝑁 ( 𝑡 ) 𝑑𝑡 𝑖 ? 𝑒𝑞 = 𝑖 ? 1 + 𝑖 ? 2 + ⋯ + 𝑖 ? 𝑁
Capacitors: Connected in Parallel 18 𝑖 𝑡 = ? 𝑒𝑞 𝑑? ( 𝑡 ) 𝑑𝑡 ? 𝑒𝑞 = ? 1 + ? 2 + ⋯ + ? 𝑁 Initial voltage across ? 𝑒𝑞 : ? 𝑡 0 𝑖 𝑡 = 𝑖 1 𝑡 + 𝑖 2 𝑡 + … + 𝑖 𝑁 𝑡 ? 𝑒𝑞 𝑑? ( 𝑡 ) 𝑑𝑡 = ? 1 𝑑? ( 𝑡 ) 𝑑𝑡 + ? 2 𝑑? ( 𝑡 ) 𝑑𝑡 … + ? 𝑁 𝑑? ( 𝑡 ) 𝑑𝑡
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Inductors: Connected in Series 19 𝐿 𝑒𝑞 = 𝐿 1 + 𝐿 2 + ⋯ + 𝐿 𝑁 Initial current through 𝐿 𝑒𝑞 : 𝑖 𝑡 0 ? ? (𝑡) = ? 1 𝑡 + ? 2 𝑡 + ⋯ + ? 𝑁 𝑡 𝐿 𝑒𝑞 𝑑 𝑖 ( 𝑡 ) 𝑑𝑡 = 𝐿 1 𝑑 𝑖 ( 𝑡 ) 𝑑𝑡 + 𝐿 2 𝑑 𝑖 ( 𝑡 ) 𝑑𝑡 … + 𝐿 𝑁 𝑑 𝑖 ( 𝑡 ) 𝑑𝑡
Inductors: Connected in Parallel 20 ? 𝐿 𝑒𝑞 = ? 𝐿 1 + ? 𝐿 2 + ⋯ + ? 𝐿 𝑁 Initial c urrent through 𝐿 𝑒𝑞 : 𝑖 1 𝑡 0 + 𝑖 2 𝑡 0 + ⋯ + 𝑖 𝑁 𝑡 0 𝑖 𝑡 = 𝑖 1 𝑡 + 𝑖 2 𝑡 + … + 𝑖 𝑁 𝑡 𝑑 𝑖 ( 𝑡 ) 𝑑 𝑡 = 𝑑 𝑖 1 ( 𝑡 ) 𝑑𝑡 + 𝑑 𝑖 2 ( 𝑡 ) 𝑑𝑡 + ⋯ + 𝑑 𝑖 𝑁 ( 𝑡 ) 𝑑𝑡 1 𝐿 𝑒𝑞 = 1 𝐿 1 + 1 𝐿 2 + ⋯ + 1 𝐿 𝑁
Passive Components 21
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Passive Components 22 Resistor Capacitor Inductor
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Passive Components 23
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