Capacitors and Inductors in Dynamic Circuits

Lecture 10 Capacitor and Inductor Hung-yi Lee

Outline Capacitor (Chapter 5.1) Inductor (Chapter 5.2) Comparison of Capacitor and Inductor Superposition for Dynamic Circuits

Outline Capacitor (Chapter 5.1) Inductor (Chapter 5.2) Comparison of Capacitor and Inductor Superposition for Dynamic Circuits

Capacitor i-v characteristics ( ) dv t reference current should flow from + to - = ( ) i t C Dynamic dt (t ) i i=0 If v is constant (t ) v Open circuit =1 1 ( ) d ( ) t 0 ( ) d t t t = + ( ) v t i v i C C 0

Capacitor i-v characteristics ( ) t v ( ) dv t = ( ) i t C dt

( ) dv t Continuity - Capacitor = ( ) i t C dt The voltage of capacitor should be continuous Current changes Infinite current

Capacitor Power and Energy (t ) vc Instantaneous consumed power of a capacitor: ( ) dv t = = ( ) ( ) ( ) ( ) c p t v t i t Cv t (t ) ic c c c dt Total energy consumed: ( ) t dv ( ) t ( ) t ( ) t t t = = E p dt (t ) p c Cv dt c dt 1 Depend on voltage at t = 2t ( ) Cvc 2 E t ( ) The energy stored

Capacitor - Application Automated external defibrillators (AED)

Capacitor - Application Automated external defibrillators (AED) ( = t vc ( = t ) 0 vc ) 5000 5000V 1 1 1 = = = = 2 2 2 E ( ) 0 Cvc t E ( ) 5000 Cvc t C 2 2 2

Capacitor - Series = + + + v v v v 1 2 N dv dv dv = = = = 1 2 N i C C C 1 2 N dt dt dt ( ) + + + d v v v dv = 1 2 N dt dt dv dv dv = + + + 1 2 N dt dt dt 1 1 1 = + + + i C 1 C C 1 1 2 N = C dv ser 1 1 1 = = i i C + + + ser Cser dt C C C 1 2 N

Capacitor - Parallel Cpar=C1+C2+ +CN = + + + i i i i 1 2 N dv dv dv = + + + C C C 1 2 N dt dt dt dv ( ) = + + + C C C 1 2 N dt dv = C par dt

Outline Capacitor (Chapter 5.1) Inductor (Chapter 5.2) Comparison of Capacitor and Inductor Superposition for Dynamic Circuits

Inductor i-v characteristics ( ) di t reference current should flow from + to - = ( ) v t L Dynamic dt (t ) i If i=constant v=0 (t ) v short circuit 1 1 ( ) ( ) t i = ( ) d t t = + ( ) i t v d v 0 L L t 0

Inductor i-v characteristics ( ) t i ( ) di t = ( ) L v t dt

( ) di t Continuity - Inductor = ( ) v t L dt The current of inductor should be continuous Voltage changes Infinite Voltage

Shock by Inductor http://www.allaboutcircuits.com/worksheets/ind.html

Inductor Power and Stored Energy Instantaneous consumed power of an inductor di ( ) t = = L p v i L i ( ) ( ) ( ) ( ) t t t t L L L dt Total Energy consumed ( ) t E ( ) t di ( ) t ( ) t t t = = p dt L L dt i dt L 1 Depend on current at t = 2 Lt i L ( ) 2 The energy stored

Inductor - Series = + + + v v v v 1 2 N di di di = + + + L L L 1 2 N dt dt dt di ( ) = + + + L L L 1 2 N dt di = L ser dt Lser=L1+L2+ +LN

Inductor - Parallel = + + + i i i Ni 1 2 di di di = = = = 1 2 N v L L L 1 2 N dt ( i dt dt ) + + + d i i di = 1 2 N dt dt di di di = + + + 1 2 N dt dt 1 dt 1 1 = 1 L = + + + v par 1 1 1 + + + L 1 L L 1 2 N L L L 1 2 N di = v = v L L par dt par

Outline Capacitor (Chapter 5.1) Inductor (Chapter 5.2) Comparison of Capacitor and Inductor Superposition for Dynamic Circuits

Summary - i-v characteristics ( ) ( ) dv t di t = = ( ) ( ) i t C v t L dt dt L C V V

Summary - Series and Parallel Capacitor Resistor Inductor 1 1 = L L = = R R Series s i s i C C s i 1 1 1 1 = = = C C Parallel L L R R p i p i p i ( ) ( ) dv t di t ( ) t = = = ( ) ( ) ( ) i t C v t L v t Ri i-v dt dt

Outline Capacitor (Chapter 5.1) Inductor (Chapter 5.2) Comparison of Capacitor and Inductor Superposition for Dynamic Circuits

Review i = y a ix This equation only for circuits with sources and resistors. i y: any current or voltage for an element xi: current of current sources or voltage of voltage sources Proportionality Principle, Superposition Principle Can be used in any circuit in this course

Linearity A circuit is a multiple-input multiple-output (MIMO) system Input: current of current sources or voltage of voltage sources Output: the current or voltage for the elements + Circuit (System) v i output input -

Linearity All linear circuits are linear system Linear Circuit: Sources Linear Elements: Resistor, Capacitor, Inductor All circuits in this course are linear circuits. v i = R

Linearity Linear System: Property 1: Input: g1(t), g2(t), g3(t), output: h1(t), h2(t), h3(t), Input: Kg1(t), Kg2(t), Kg3(t), output: Kh1(t), Kh2(t), Kh3(t), Proportionality Principle

Linearity Linear System: Property 2: Input: b1(t), b2(t), b3(t), output: y1(t), y2(t), y3(t), Input: a1(t), a2(t), a3(t), output: x1(t), x2(t), x3(t), Input: a1(t)+ b1(t), a2(t)+ b2(t), a3(t)+ b3(t), output: x1(t)+y1(t), x2(t)+y2(t), x3(t)+y3(t), Superposition Principle

Linearity Linear System: Property 2: Input: b1(t), b2(t), b3(t), output: y1(t), y2(t), y3(t), Input: a1(t), a2(t), a3(t), output: x1(t), x2(t), x3(t), Input: a1(t)+ b1(t), a2(t)+ b2(t), a3(t)+ b3(t), output: x1(t)+y1(t), x2(t)+y2(t), x3(t)+y3(t), Superposition Principle

( ) t Linearity v ( ) t 1 g 0 ( ) t v ( ) t g ( ) t i 0 ( ) t ( ) t ( ) t ( ) t = + g g g 2 g 1 2 ( ) t i Superposition Principle can be applied on all circuits in this course (Textbook: Chapter 6.5).

Announcement 10/22 ( ) Ch1. Circuit Variables and Laws (1.4, 1.5) Ch2. Properties of Resistive Circuits (2.3, 2.4, 2.5) Ch3. Applications of Resistive Circuits (3.2) Ch4. Systematic Analysis Methods (4.1, 4.2, 4.3, 4.4) PM6:30~8:30 : d03921009@ntu.edu.tw : PM6:30~8:30 : 146

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Appendix

Capacitor Series (2) i=0 If v is constant Open circuit What are v1, v2 ?

Capacitor Application How Capacitive Liquid Level Sensors Work https://www.youtube.com/watch?v=0du-QU1Q0T4

Acknowledgement (b02) Equation