DR. GYURCSEK ISTVÁN Capacitorsand Inductorsvili.pmmf.hu/~gyurcsek/PUB1EA/4.10L-LCE.pdf ·...

1 [email protected] 2017. 11. 15.

DR. GYURCSEK ISTVÁN

Capacitors and InductorsSources and additional materials (recommended) q Dr. Gyurcsek – Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3q Ch. Alexander, M. Sadiku: Fundamentals of Electric Circuits, 6th Ed., McGraw Hill NY 2016, ISBN: 978-0078028229q Simonyi K.: Villamosságtan. AK Budapest 1983, ISBN:9630534134q Dr. Selmeczi K. – Schnöller A.: Villamosságtan 1. MK Budapest 2002, TK szám: 49203/Iq Dr. Selmeczi K. – Schnöller A.: Villamosságtan 2. TK Budapest 2002, ISBN:9631026043q Zombory L.: Elektromágneses terek. MK Budapest 2006, (www.electro.uni-miskolc.hu)


Progress in Content

q Capacitance (Properties, Series and Parallel Connections)q Inductance (Properties, Series and Parallel Connections)q Applications (Integrator, Differentiator)


Capacitors

𝐶 =𝑞𝑣, 𝐹 =

𝐶𝑉 𝐶 = 𝜀

𝐴𝑑

𝑖 =𝑑𝑞𝑑𝑡

= 𝐶𝑑𝑣𝑑𝑡

← 𝑙𝑖𝑛𝑒𝑎𝑟𝑒𝑙𝑒𝑚𝑒𝑛𝑡

𝑣 =1𝐶6 𝑖𝑑𝑡7

89

→ 𝑣 =1𝐶6 𝑖𝑑𝑡7

7;

+ 𝑣 𝑡=

Memory element (v depends on past history of i.)

Symbols


Properties of Capacitors

𝑤 = 6𝑝𝑑𝑡7

89

= 𝐶 6𝑣𝑑𝑣𝑑𝑡𝑑𝑡

7

89

= 𝐶 6 𝑣𝑑𝑣@(7)

@(89)

=12𝐶𝑣DE

@(89)

@(7)

𝑣 −∞ = 0 → 𝑤 =12𝐶𝑣D 𝑞 = 𝐶𝑣 → 𝑤 =

12𝐶𝑣D =

𝑞D

2𝐶

𝑖 = 𝐶𝑑𝑣𝑑𝑡

→ 𝑝 = 𝑣𝑖 = 𝑣𝐶𝑑𝑣𝑑𝑡

Propertiesq Linear elementq Energy storage (memory) elementq Open circuit on DCqq Ideal C à no dissipated energyq Real capacitor model à

𝑣 −0 = 𝑣(+0)

Energy stored in EF

Property of linearity

𝑖 = 𝐶𝑑𝑣𝑑𝑡

→


Parallel Capacitors

𝑖 = 𝑖I + 𝑖D + 𝑖J + ⋯+ 𝑖L

𝑖 = 𝐶I𝑑𝑣𝑑𝑡+ 𝐶D

𝑑𝑣𝑑𝑡+ 𝐶J

𝑑𝑣𝑑𝑡+ ⋯+ 𝐶L

𝑑𝑣𝑑𝑡

𝑖 = M𝐶N

L

NOI

𝑑𝑣𝑑𝑡

→ 𝐶PQ = M𝐶N

L

NOI


Series Capacitors

𝑣 = 𝑣I + 𝑣D + 𝑣J + ⋯+ 𝑣L

𝑣 =1𝐶I6 𝑖 𝑡 𝑑𝑡7

7=

+ 𝑣I 𝑡= +1𝐶D

6 𝑖 𝑡 𝑑𝑡7

7=

+ 𝑣D 𝑡= + ⋯+1𝐶L


7=

+ 𝑣L(𝑡=)

𝑣 = M1𝐶N

L

NOI


7=

+ 𝑣(𝑡=) →1𝐶PQ

= M1𝐶N

L

NOI

1𝐶PQ

=1𝐶I+1𝐶D→ 𝐶PQ =

𝐶I R 𝐶D𝐶I + 𝐶D


Progress in Content



Inductors

𝐿 → 𝐻 =𝑉𝑠𝐴

𝐿 = 𝜇𝑁D𝐴𝑙

𝑑𝑖 =1𝐿𝑣𝑑𝑡 → 𝑖 =

1𝐿6𝑣 𝑡 𝑑𝑡7

89

Memory element(i depends on past history of v.)

Symbols

𝑖 =1𝐿6𝑣 𝑡 𝑑𝑡 + 𝑖(𝑡=)7

7;

(𝑐𝑜𝑖𝑙𝑓𝑙𝑢𝑥) → Ψ 𝑡 = 𝐿 R 𝑖 𝑡

(𝐹𝑎𝑟𝑎𝑑𝑎𝑦) → 𝑣 𝑡 =𝑑Ψ 𝑡𝑑𝑡

_ → 𝑣 𝑡 = 𝐿𝑑𝑖 𝑡𝑑𝑡


Properties of Inductors

𝑤 = 6𝑝𝑑𝑡7

89

= 𝐿 6 𝑖𝑑𝑖𝑑𝑡𝑑𝑡

7

89

= 𝐿 6 𝑖𝑑𝑖`(7)

`(89)

=12𝐿𝑖DE

`(89)

`(7)

𝑖 −∞ = 0 → 𝑤 =12𝐿𝑖D

𝑣 = 𝐿𝑑𝑖𝑑𝑡→ 𝑝 = 𝑣𝑖 = 𝐿

𝑑𝑖𝑑𝑡

𝑖

Propertiesq Linear elementq Energy storage (memory) elementq Short circuit on DCqq Ideal L à no dissipated energyq Real inductor model à

𝑖 −0 = 𝑖(+0)

Energy stored in MF

Property of linearity

𝑣 = 𝐿𝑑𝑖𝑑𝑡→


Series Inductors

𝑣 = 𝑣I + 𝑣D + 𝑣J + ⋯+ 𝑣L

𝑣 = 𝐿I𝑑𝑖𝑑𝑡+ 𝐿D

𝑑𝑖𝑑𝑡+ 𝐿J

𝑑𝑖𝑑𝑡+ ⋯+ 𝐿L

𝑑𝑖𝑑𝑡

𝑣 = M𝐿N

L

NOI

𝑑𝑖𝑑𝑡→ 𝐿PQ = M𝐿N

L

NOI


Parallel Inductors

𝑖 =1𝐿I6𝑣 𝑡 𝑑𝑡7

7=

+ 𝑖I 𝑡= +1𝐿D

6𝑣 𝑡 𝑑𝑡7

7=

+ 𝑖D 𝑡= + ⋯+1𝐿L


7=

+ 𝑖L(𝑡=)

𝑖 = 𝑖I + 𝑖D + 𝑖J + ⋯+ 𝑖L

𝑖 = M1𝐿N

L

NOI


7=

+ 𝑖(𝑡=) →1𝐿PQ

= M1𝐿N

L

NOI

1𝐿PQ

=1𝐿I+1𝐿D→ 𝐿PQ =

𝐿I R 𝐿D𝐿I + 𝐿D


Progress in Content



RLC Applications Summary

Resistors & Capacitorsq discrete formq IC form

LC special propertiesq Temporary energy storage (dc app)q C à no abrupt change in voltage (dc app)q L à no abrupt change in current (dc app)q Frequency sensitive behavior (ac app)

Inductors (coils)q discrete form onlyq limited in applications

(sensors, motors, etc.)


Applications – Integrator

𝑖a = 𝑖b →𝑣`𝑅= −𝐶

𝑑𝑣d𝑑𝑡

𝑑𝑣d = −1𝑅𝐶

𝑣`𝑑𝑡

𝑣d 𝑡 − 𝑣d 0 = −1𝑅𝐶

6𝑣`𝑑𝑡7

=

𝑣d 𝑡 = −1𝑅𝐶

6𝑣`(𝑡)𝑑𝑡7

=

𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑜𝑟 → 𝑣d 0 = 0 →


Application – Differentiator

𝑖a = 𝑖b → −𝑣d𝑅= 𝐶

𝑑𝑣`𝑑𝑡

𝑣d = −𝑅𝐶𝑑𝑣`𝑑𝑡

Comments (opamp application)q Not as popular as integratorq Electronically unstable circuit

(noise sensitive opamp circuit)


Questions

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DR. GYURCSEK ISTVÁN Capacitorsand Inductorsvili.pmmf.hu/~gyurcsek/PUB1EA/4.10L-LCE.pdf ·...

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