ELC 2050: Fields & Wave Propagation 2021. 1. 19. · ELC 2050: Fields & Wave Propagation Department...
Transcript of ELC 2050: Fields & Wave Propagation 2021. 1. 19. · ELC 2050: Fields & Wave Propagation Department...
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ELC 2050: Fields & Wave Propagation
Department of Electronics and Electrical Communications Engineering
Introduced By:Eng. Mohamed Ossama Ashour
E-mail: [email protected] term 20-21
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Agenda
© Mohamed O. Ashour 2018 Slide 2
• Review on Electromotive Force
• Faraday's Law of Electromagnetic Induction➢ A Moving Conductor in a Static Magnetic Field
➢ A Stationary Circuit in a Time-Varying Magnetic Field
➢ A Moving Circuit in a Time-Varying Magnetic Field
• Inductances and Magnetic Energy
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Electromotive Force• When any electrical energy source (electric batteries, electric generators,
photovoltaic cells or other devices) is connected in an electric circuit, it provides a driving force for the charge carriers (electrons). This force manifests itself as an equivalent impressed electric field intensity Ei.
• In case of electric batteries, The line integral of Ei from the negative to the positive electrode inside the battery is called the electromotive force (Emf) of the battery.
• The electromotive force of the battery:
Also, remember that electric field can be represented as
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Faraday's Law of Electromagnetic Induction
𝐵 = 𝐵𝑜 𝑢𝑧
A Moving Conductor in a Static (Uniform) Magnetic Field
𝐹𝑚𝑎𝑔 = 𝑞 𝑢 × 𝐵
where 𝐹𝑚𝑎𝑔 is the applied force on the free
moving charges in the conductor.
So, the work done inside the conductor:
.𝐹𝑚𝑎𝑔ׯ 𝑑𝑙 = 𝑞 𝑢 × 𝐵. 𝑑𝑙
The electromotive force: ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = ׯ𝐹𝑚𝑎𝑔
𝑞. 𝑑𝑙 = 𝑢 × 𝐵. 𝑑𝑙
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Faraday's Law of Electromagnetic Induction
From Faraday’s law:
ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = −𝑁𝑑𝜙
𝑑𝑡(1)
A Stationary Circuit in a Time-Varying Magnetic Field
From electromotive force definition, (1) & (2):
ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝐸𝑖𝑛𝑑ׯ . 𝑑𝑙 = −𝑁𝑑
𝑑𝑡.𝐵 𝑑𝑠 = −𝑁
𝑑𝐵
𝑑𝑡. 𝑑𝑠 = −𝑁 ሶ𝐵 . 𝑑𝑠
From Stokes’ theorem: 𝛻 × 𝐸𝑖𝑛𝑑 = − ሶ𝐵 “Rotational Field”
From the definition of the time varying flux:𝜙 = .𝐵 𝑑𝑠 (2)
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Faraday's Law of Electromagnetic Induction
This is the most general case where its solution is a combination of the pervious 2 cases.
A Moving Circuit in a Time-Varying Magnetic Field
ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = −𝑁න𝑑𝐵
𝑑𝑡. 𝑑𝑠
ℰ𝑡𝑜𝑡𝑎𝑙 = ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 + ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑
ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = න𝑢 × 𝐵. 𝑑𝑙
So, The time varying magnetic field can induce an Electric field. Which means that we now have 2 sources for 𝐸 :1) Primary sources: free electric charges (produces flowing field).2) Secondary sources: time varying magnetic field (produces rotating field).
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Sheet 3
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Sheet 3 Answers
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Sheet 3 Answers
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Sheet 3 Answers
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Inductances and Inductors
• Consider two neighboring closed loops, 𝐶1 and 𝐶2 bounding surfaces 𝑆1and 𝑆2 respectively. If a current 𝐼1 flows in 𝐶1, a magnetic field 𝑩1 will be
created. Some of the magnetic flux due to 𝑩1 will link with 𝐶2.
• We can define the mutual flux between these 2 loops as:
Φ12 = න𝑆2
𝑩1. 𝑑𝑠2 = 𝐿12𝐼1
where the proportionality constant 𝐿12 is called the mutual inductance between loops 𝐶1 and 𝐶2.
• Generally, We can define the flux linkage as:Λ12 = 𝑁2Φ12ELC 2050
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Inductances and Inductors
• In case the self inductance of loop 𝐶1 is required, it’s given as:
• The procedure for determining the self-inductance of an inductor is as follows:
1. Choose an appropriate coordinate system for the given geometry.2. Find 𝑩 from 𝐼 (the current in the conducting wire) by Ampere's
circuital law or Biot-Savart law.3. Find the linking magnetic flux,Φ, from B by integration.4. Find the flux linkage Λ by multiplying Φ by the number of turns. 5. Find L by taking the ratio 𝐿 = Λ / 𝐼.
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Magnetic Energy
• The magnetic energy can be represented as
OR
• In case of linear medium, this expression can be represented as
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Sheet 3
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Sheet 3 Answers
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