Power Quality Vis-à-Vis

Inductance Gary Malhoit, P.E.

February 23, 2017

Tall Tree and the Eye

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Overview Inductance? Background and History Flux Linkage Wire Size and Coil Aspect Ratio PQ Applications Comments and Questions

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History of InductancePart 1

Last of the 3 fundamental circuit characteristics

“Electro-dynamic capacity” (Lord Kelvin)

“A bugaboo” (Sir William Preece) War of Currents (1880s-1890s) Lexicographer of inductance (Oliver

Heaviside) Heaviside Condition:

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𝑮𝑪=

𝑹𝑳

History of InductancePart 2

Oersted’s discovery (1820) Ampère’s Theory (1823) Ampère’s circuital law says a

magnetic flux forms around or encircles a current (moving charges) carrying conductor

Displacement Current (1886) (James Maxwell)

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Ampère’s Circuital Law

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-

V=0

E

-

E

V>0 Φ

V=Velocity

Magnetic Flux & Current

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Current flowing through a Conductor

Lines of Flux

𝜱∝ 𝑰Φ=

DCylindrical

Inductance Defined

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= henry

If Φ changes I changes

Internal & External Inductance

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Conductor

Internal Magnetic Field

External Magnetic Field

X

Induction

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-

E

>0 ΦV=Velocity

Φ

Time

InductionInductionNo Induction

Induction:Faraday’s Law

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( )By one definition, a volt is induced via one loop (N=number of turns=1) of a coil exposed to a magnetic flux of one weber changing to zero in one second.

AC

∮𝑺

.

𝑬 .𝒅𝒍=− 𝒅𝒅𝒕∬𝒔

.

𝑩 .𝒅𝒔

Induction As a Function of Inductance

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𝑽=−𝑳 𝒅𝑰𝒅𝒕

The minus sign indicates the induced voltage opposesprior conditions and is called Lenz Law

A First Definition of Flux Linkage

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𝐋=𝐅𝐥𝐮𝐱 𝐋𝐢𝐧𝐤𝐚𝐠𝐞

𝑰 =𝑵𝜱𝑰

A First Definition of Flux Linkage

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A Second Definition of Flux Linkage

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A Second Definition of Flux Linkage

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Inductance of a Straight Wire

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Area

Magnetic Flux Line

Conductor

Flux Linkage

Current

Inductance of a Straight Wire (Edward Rosa 1907)

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Biot-Savart 𝟏𝑫𝟐

A First Point

IA Second PointL

∞R

Area of Flux Linkage

Conductor

Formula: Inductance of a Straight Wire

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𝐋=2 𝒍(𝑳𝒏 2 𝒍𝑹 −1+ 14 )

R= wire radiusl= wire lengthL= inductance

Inductance & Wire Diameter

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Wheeler’s formula

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L= 𝒂2𝒏29𝒂+10𝒃

a= coil radiusn= number of turnsb=coil lengthL= inductance

Inductance of a Coil of 200 Inches of Wire

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Inductance of a Conductor

(Rectangular Cross-Section)

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𝐋=2 𝒍¿l= wire lengthw= width of wireb= height of wireL= inductance

Inductance of a Conductor

(Rectangular vs. Round Cross-Section)

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Rectangular Cross-Section

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Mutual Inductance: Parallel Conductors

Currents Same Direction

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Mutual Inductance: Parallel ConductorsCurrents Opposite

Directions

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Mutual Inductance: Parallel

Conductors Occupy Same Space

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Inductance & Lead Length Spacing of Metal

Oxide Varistors

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Minimize Spacing

*

* Minimize Lead Length

Load

Rogowski Coil

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Coaxial Return

Rogowski Coil & Immunity

to External Magnetic Fields

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Triplen Harmonics

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Third HarmonicsOverlaid

Sum of ThirdHarmonics

Phase APhase BPhase C

Third HarmonicsSum Third Harmonics

Odd Sequence: 1 3 5 7 9 11Third Harmonic: 3 3 3 3 3 3

Triplens: 3, 9, 15, 21, 27, 33

Zigzag Transformer & Mitigation of Third

Harmonics

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Opposing FieldsSubstantially CancelZero Sequence Current!

Common Mode Signals

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Ferrite Beads

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Ferrite Beads

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Ferrimagnetic (Louis Néel, 1970)

Ferromagnetic

Ferrite Beads

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IDC IAC

Mag

nitu

de

Frequency

Low Pass

=ωL

ω

Bias

Questions & Comments

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