6.
5 Inductors and Inductances
6.5 Inductors and Inductances
Module 6: Magnetostatic Force and boundary conditions
Course: ECE1003 Electromagnetic Field Theory
-Dr Richards Joe Stanislaus
Assistant Professor - SENSE
6.5 Inductors and Inductances
1.1 Flux
• Circuit(closed conducting path) carrying current 𝐼 will produce magnetic
field 𝐁 which causes the flux 𝜓 = 𝐁 ∙ 𝑑𝐒 to pass through each turn of
circuit
𝐼 𝐼
𝐁
6.5 Inductors and Inductances
1.2 Flux linkage and Inductance
• Circuit(closed conducting path) carrying current 𝐼 will produce magnetic field
𝐁 which causes the flux 𝜓 = 𝐁 ∙ 𝑑𝐒 to pass through each turn of circuit.
• Circuit has 𝑁 identical turns,
Flux linkage λ = N𝜓
𝐼 𝐼
𝐁
6.5 Inductors and Inductances
1.2 Flux linkage and Inductance
• Circuit(closed conducting path) carrying current 𝐼 will produce magnetic field
𝐁 which causes the flux 𝜓 = 𝐁 ∙ 𝑑𝐒 to pass through each turn of circuit.
• Circuit has 𝑁 identical turns,
Flux linkage λ = N𝜓
𝐼 𝐼
• For linear medium around circuit,
Flux linkage λ ∝ 𝐼
𝐁
λ = 𝐿𝐼
6.5 Inductors and Inductances
1.2 Flux linkage and Inductance
• Circuit(closed conducting path) carrying current 𝐼 will produce magnetic field
𝐁 which causes the flux 𝜓 = 𝐁 ∙ 𝑑𝐒 to pass through each turn of circuit.
• Circuit has 𝑁 identical turns,
Flux linkage λ = N𝜓
𝐼 𝐼
• For linear medium around circuit,
Flux linkage λ ∝ 𝐼
𝐁
λ = 𝐿𝐼
• Constant of proportionality: 𝐿: Inductance of circuit
(Property of physical arrangement of the circuit)
• Inductor: A circuit or part of circuit that has inductance
λ 𝑁𝜓
• Inductance is ratio of magnetic flux linkage to current, 𝐿 = =
𝐼 𝐼
6.5 Inductors and Inductances
1.3 Self inductance and Magnetic energy
λ 𝑁𝜓
•𝐿= =
𝐼 𝐼
• Unit of inductance: henry (H) or webers/ampere. Millihenrys (mH)
are used more often
• The inductance shown above is self inductance (Linkages are
produced by inductor itself).
6.5 Inductors and Inductances
1.3 Self inductance and Magnetic energy
λ 𝑁𝜓
•𝐿= =
𝐼 𝐼
• Unit of inductance: henry (H) or webers/ampere. Millihenrys (mH)
are used more often
• The inductance shown above is self inductance (Linkages are
produced by inductor itself).
• Inductance: measure of how much magnetic energy is stored in an
inductor
1
• Magnetic energy (joules) stored in an inductor: 𝑊𝑚 = 𝐿𝐼2
2
• Self inductance: 𝐿 = 2𝑊𝑚 /𝐼2
6.5 Inductors and Inductances
1.4 Mutual Inductance
𝐁
𝐼2
𝐼1
𝐁
6.5 Inductors and Inductances
1.4 Mutual Inductance
𝐁
𝜓12 𝐼2
𝜓21
𝜓22
𝜓11
𝐶𝑖𝑟𝑐𝑢𝑖𝑡 1 𝐶𝑖𝑟𝑐𝑢𝑖𝑡 2
6.5 Inductors and Inductances
1.4 Mutual Inductance
• Magnetic interaction between two circuits
• Mutual inductance 𝑀12 and 𝑀21 are not magnetization vector 𝐌
𝐁
𝜓12 𝐼2
𝜓21
𝜓11 𝜓22
𝐶𝑖𝑟𝑐𝑢𝑖𝑡 1 𝐶𝑖𝑟𝑐𝑢𝑖𝑡 2
6.5 Inductors and Inductances
1.4 Mutual Inductance
• Magnetic interaction between two circuits
• Mutual inductance 𝑀12 and 𝑀21 are not magnetization vector 𝐌
• Self inductance of circuits 1 and 2. 𝐁
λ11 𝑁1 𝜓1 λ22 𝑁2 𝜓2 𝜓12 𝐼2
• 𝐿1 = = and 𝐿2 = =
𝐼1 𝐼1 𝐼2 𝐼2
𝜓21
• 𝜓1 = 𝜓11 + 𝜓12 and 𝜓2 = 𝜓22 + 𝜓21
𝜓11 𝜓22
𝐶𝑖𝑟𝑐𝑢𝑖𝑡 1 𝐶𝑖𝑟𝑐𝑢𝑖𝑡 2
6.5 Inductors and Inductances
1.4 Mutual Inductance
• Magnetic interaction between two circuits
• Mutual inductance 𝑀12 and 𝑀21 are not magnetization vector 𝐌
• Self inductance of circuits 1 and 2. 𝐁
λ11 𝑁1 𝜓1 λ22 𝑁2 𝜓2 𝜓12
•𝐿 =
1 = and 𝐿 = = 2 𝐼2
𝐼1 𝐼1 𝐼2 𝐼2
• 𝜓1 = 𝜓11 + 𝜓12 and 𝜓2 = 𝜓22 + 𝜓21 𝜓21
• Total energy in magnetic field 𝜓11 𝜓22
𝑊𝑚 = 𝑊1 + 𝑊2 + 𝑊12
1 1
= 𝐿1 𝐼12 + 𝐿2 𝐼22 ± 𝑀12 𝐼1 𝐼2
2 2
Positive: when currents strengthen MF
Negative: with currents: opposing MF 𝐶𝑖𝑟𝑐𝑢𝑖𝑡 1
𝐶𝑖𝑟𝑐𝑢𝑖𝑡 2
6.5 Inductors and Inductances
1.5 Inductors
• Inductors are: Toroid, solenoids, coaxial transmission lines, parallel
wire transmissions.
6.5 Inductors and Inductances
1.5 Inductors
• Inductors are: Toroid, solenoids, coaxial transmission lines, parallel
wire transmissions.
• To find self inductance 𝐿
a) Choose suitable coordinate system
b) Inductor carry current 𝐼
c) Find B from Biot Savart’s law
(or Ampere’s law if symmetry exists)
6.5 Inductors and Inductances
1.5 Inductors
• Inductors are: Toroid, solenoids, coaxial transmission lines, parallel
wire transmissions.
• To find self inductance 𝐿
Mutual inductance is also
a) Choose suitable coordinate system
calculated in similar way
b) Inductor carry current 𝐼
c) Find B from Biot Savart’s law
(or Ampere’s law if symmetry exists)
d) Calculate 𝜓 = 𝐁 ∙ 𝑑𝐒\
𝜆 𝑁𝜓
e) Find 𝐿 = =
𝐼 𝐼
6.5 Inductors and Inductances
1.6 External and Internal inductance
• Co-axial or parallel wire transmission:
Internal inductance 𝐿𝑖𝑛 : inductance due to flux internal to the conductor
External inductance 𝐿𝑒𝑥𝑡 : inductance due to flux external to conductor
6.5 Inductors and Inductances
1.6 External and Internal inductance
• Co-axial or parallel wire transmission:
Internal inductance 𝐿𝑖𝑛 : inductance due to flux internal to the conductor
External inductance 𝐿𝑒𝑥𝑡 : inductance due to flux external to conductor
• Total inductance: 𝐿 = 𝐿𝑖𝑛 + 𝐿𝑒𝑥𝑡
• If 𝐶 is known, then 𝐿𝑒𝑥𝑡 may be calculated from 𝐿𝑒𝑥𝑡 𝐶 = 𝜇𝜀
6.5 Inductors and Inductances
1.7 Cases of Inductances
𝑙
𝜇0 𝑙
1. Wire: 𝐿 =
8𝜋
2𝑎
6.5 Inductors and Inductances
1.7 Cases of Inductances
𝑙
𝜇0 𝑙
1. Wire: 𝐿 =
8𝜋
2𝑎
2. Hollow cylinder
𝑙
𝜇0 𝑙 2𝑙
𝐿= ln −1
2𝜋 𝑎 2𝑎
𝑙≫𝑎
6.5 Inductors and Inductances
1.7 Cases of Inductances
𝑙
𝜇0 𝑙
1. Wire: 𝐿 =
8𝜋
2𝑎
2. Hollow cylinder
𝑙
𝜇0 𝑙 2𝑙
𝐿= ln −1
2𝜋 𝑎 2𝑎
𝑙≫𝑎
3. Parallel wires 𝑙 𝑑
𝜇0 𝑙 𝑑
𝐿= ln
𝜋 𝑎
𝑙 ≫ 𝑑, 𝑑 ≫ 𝑎
6.5 Inductors and Inductances
1.7 Cases of Inductances
4. Coaxial
𝜇0 𝑙 𝑏
𝐿= ln 𝑎
𝜋 𝑎
𝑙
𝑏
6.5 Inductors and Inductances
1.7 Cases of Inductances
4. Coaxial
𝜇0 𝑙 𝑏
𝐿= ln 𝑎
𝜋 𝑎
𝑙
𝑏
5. Circular loop 𝜌0
𝜇0 𝑙 4𝑙
𝐿= ln − 2.45 with 𝑙 = 2𝜋𝜌0 , 𝜌0 ≫ 𝑑
2𝜋 𝑑
𝑑
6.5 Inductors and Inductances
1.7 Cases of Inductances
4. Coaxial
𝜇0 𝑙 𝑏
𝐿= ln 𝑎
𝜋 𝑎
𝑙
𝑏
5. Circular loop 𝜌0
𝜇0 𝑙 4𝑙
𝐿= ln − 2.45 with 𝑙 = 2𝜋𝜌0 , 𝜌0 ≫ 𝑑
2𝜋 𝑑
𝑑
6. Solenoid 𝑙
𝜇0 𝑁 2 𝑆 𝑎
𝐿= with 𝑙 ≫ 𝑎
𝑙
6.5 Inductors and Inductances
1.7 Cases of Inductances
7. Torus (of circular cross section
𝐿 = 𝜇0 𝑁 2 𝜌0 − 𝜌02 − 𝑎2
2𝑎 𝜌0
6.5 Inductors and Inductances
1.7 Cases of Inductances
7. Torus (of circular cross section
𝐿 = 𝜇0 𝑁 2 𝜌0 − 𝜌02 − 𝑎2
2𝑎 𝜌0
8. Sheet
2𝑙
𝐿 = 𝜇0 2𝑙 ln + 0.5
𝑏+𝑡
𝑡 𝑎
𝑏
6.5 Inductors and Inductances
1.8 Energy in Magnetostatic field
• The region is filled with different volumes. Each volume has
Δ𝜓 Δ𝑧
inductance Δ𝐿 = = 𝜇𝐻 Δ𝑥 but Δ = HΔ𝑦
Δ𝐼 Δ𝐼
6.5 Inductors and Inductances
1.8 Energy in Magnetostatic field
• The region is filled with different volumes. Each volume has
Δ𝜓 Δ𝑧
inductance Δ𝐿 = = 𝜇𝐻 Δ𝑥 but Δ = HΔ𝑦
Δ𝐼 Δ𝐼
1 1
• Δ𝑊𝑚 = Δ𝐿Δ𝐼 = 𝜇𝐻 2 Δ𝑥
2
Δ𝑦 Δ𝑧 ,
2 2
1
• Δ𝑊𝑚 = 𝜇𝐻 2 Δ𝑣
2
6.5 Inductors and Inductances
1.8 Energy in Magnetostatic field
• The region is filled with different volumes. Each volume has
Δ𝜓 Δ𝑧
inductance Δ𝐿 = = 𝜇𝐻 Δ𝑥 but Δ = HΔ𝑦
Δ𝐼 Δ𝐼
1 1
• Δ𝑊𝑚 = Δ𝐿Δ𝐼 = 𝜇𝐻 2 Δ𝑥 Δ𝑦 Δ𝑧 ,
2
2 2
1
• Δ𝑊𝑚 = 𝜇𝐻 2 Δ𝑣
2
_ 1 2 1 𝐵2
• 𝑤𝑚 = limΔ𝑣→0 𝜇𝐻 = 𝐁 ∙ 𝐇 =
2 2 2𝜇
6.5 Inductors and Inductances
1.8 Energy in Magnetostatic field
• The region is filled with different volumes. Each volume has
Δ𝜓 Δ𝑧
inductance Δ𝐿 = = 𝜇𝐻 Δ𝑥 but Δ = HΔ𝑦
Δ𝐼 Δ𝐼
1 1
• Δ𝑊𝑚 = Δ𝐿Δ𝐼 = 𝜇𝐻 2 Δ𝑥 Δ𝑦 Δ𝑧 ,
2
2 2
1
• Δ𝑊𝑚 = 𝜇𝐻 2 Δ𝑣
2
_ 1 2 1 𝐵2
• 𝑤𝑚 = limΔ𝑣→0 𝜇𝐻 = 𝐁 ∙ 𝐇 =
2 2 2𝜇
1 1
• Energy 𝑊𝑚 = 𝑤𝑚 𝑑𝑣 = 𝐁 ∙ 𝐇 𝑑𝑣 = 𝜇𝐻 2 𝑑𝑣
2 2
