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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)
2 [email protected] 2017. 11. 15.
Progress in Content
q Capacitance (Properties, Series and Parallel Connections)q Inductance (Properties, Series and Parallel Connections)q Applications (Integrator, Differentiator)
3 [email protected] 2017. 11. 15.
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
4 [email protected] 2017. 11. 15.
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
→
5 [email protected] 2017. 11. 15.
Parallel Capacitors
𝑖 = 𝑖I + 𝑖D + 𝑖J + ⋯+ 𝑖L
𝑖 = 𝐶I𝑑𝑣𝑑𝑡+ 𝐶D
𝑑𝑣𝑑𝑡+ 𝐶J
𝑑𝑣𝑑𝑡+ ⋯+ 𝐶L
𝑑𝑣𝑑𝑡
𝑖 = M𝐶N
L
NOI
𝑑𝑣𝑑𝑡
→ 𝐶PQ = M𝐶N
L
NOI
6 [email protected] 2017. 11. 15.
Series Capacitors
𝑣 = 𝑣I + 𝑣D + 𝑣J + ⋯+ 𝑣L
𝑣 =1𝐶I6 𝑖 𝑡 𝑑𝑡7
7=
+ 𝑣I 𝑡= +1𝐶D
6 𝑖 𝑡 𝑑𝑡7
7=
+ 𝑣D 𝑡= + ⋯+1𝐶L
6 𝑖 𝑡 𝑑𝑡7
7=
+ 𝑣L(𝑡=)
𝑣 = M1𝐶N
L
NOI
6 𝑖 𝑡 𝑑𝑡7
7=
+ 𝑣(𝑡=) →1𝐶PQ
= M1𝐶N
L
NOI
1𝐶PQ
=1𝐶I+1𝐶D→ 𝐶PQ =
𝐶I R 𝐶D𝐶I + 𝐶D
7 [email protected] 2017. 11. 15.
Progress in Content
q Capacitance (Properties, Series and Parallel Connections)q Inductance (Properties, Series and Parallel Connections)q Applications (Integrator, Differentiator)
8 [email protected] 2017. 11. 15.
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
9 [email protected] 2017. 11. 15.
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→
10 [email protected] 2017. 11. 15.
Series Inductors
𝑣 = 𝑣I + 𝑣D + 𝑣J + ⋯+ 𝑣L
𝑣 = 𝐿I𝑑𝑖𝑑𝑡+ 𝐿D
𝑑𝑖𝑑𝑡+ 𝐿J
𝑑𝑖𝑑𝑡+ ⋯+ 𝐿L
𝑑𝑖𝑑𝑡
𝑣 = M𝐿N
L
NOI
𝑑𝑖𝑑𝑡→ 𝐿PQ = M𝐿N
L
NOI
11 [email protected] 2017. 11. 15.
Parallel Inductors
𝑖 =1𝐿I6𝑣 𝑡 𝑑𝑡7
7=
+ 𝑖I 𝑡= +1𝐿D
6𝑣 𝑡 𝑑𝑡7
7=
+ 𝑖D 𝑡= + ⋯+1𝐿L
6𝑣 𝑡 𝑑𝑡7
7=
+ 𝑖L(𝑡=)
𝑖 = 𝑖I + 𝑖D + 𝑖J + ⋯+ 𝑖L
𝑖 = M1𝐿N
L
NOI
6𝑣 𝑡 𝑑𝑡7
7=
+ 𝑖(𝑡=) →1𝐿PQ
= M1𝐿N
L
NOI
1𝐿PQ
=1𝐿I+1𝐿D→ 𝐿PQ =
𝐿I R 𝐿D𝐿I + 𝐿D
12 [email protected] 2017. 11. 15.
Progress in Content
q Capacitance (Properties, Series and Parallel Connections)q Inductance (Properties, Series and Parallel Connections)q Applications (Integrator, Differentiator)
13 [email protected] 2017. 11. 15.
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.)
14 [email protected] 2017. 11. 15.
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 →
15 [email protected] 2017. 11. 15.
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)