Magnetic Induction 

and 

Magnetic resonance

Magnetic induction and resonance 

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Magnetic Induction

Progress in Magnetic induction transferIn practical use  Electric toothbrush, electric shaver, etc.

Electric devices in a wet areaCordless phone, etc.

In progress Enhancement of transfer efficiency

development of efficient coilsreduction in ohmic heating, cooling‐freehigher power: cf. cell phone

Improvement of Safetya metal detectorcertificate ID to avoid charging to wrong devices 

Sophistication and functionalitysimultaneous data & power transfer

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Qi (inductive power standard)

Output voltage is regulated by a digital control loop between the transmitter and receiver.

Communication is unidirectional from the power receiver to transmitter requesting more or less power via backscatter modulation.

In backscatter modulation, the power‐receiver coil is loaded, changing the current draw at the power transmitter. 

These current changes are monitored and demodulated into the information required for the two devices to work together.

By Wireless Power Consortium

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Contact‐less IC(Integrated circuit) card/tag 

Inductive power supply using a 13.56 MHz radio wave emitted by a card reader.

Battery‐free: card slimming, no run‐out, less toxic‐substance in disposal

Less driving‐power: FET(Field Effect Transistor)/CPU & reflecting sender. Input voltage modulation by resistance control (not by injection current control.) Nonvolatile memory.

Two‐way telecommunication by modulating the power carrier wave between a Reader/Writer and a card at 100‐400 kbps at a distance of 10 cm. 

Card and reader circuits

Felica (sony)

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High‐speed charging to EVs

IPT hybrid bus(MLIT、Hino motors)

Electric vehicle(Nissan Leaf)

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IPT: abbreviation for Inductive Power Transmitter.

A plug-less hybrid bus links Haneda Airport‘s terminals covering a distance of about 4.2 km and runs 15 km only by battery charge and 300 km with a dynamo engine.

A coil on the bottom of the bus aligns with a charging coil. A charging coil is embedded in concrete.

150 kW transmission at total charging and discharge efficiency of 80％.

Lithium ion batteries on a roof. Charge voltage 500V, Capacity 80 Ah, 40 kWh. (cf. Toyota Purius 200V, 6.5 Ah, 1.3 kWh) 30 times bigger.

IPT Hybrid Bus 

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Theory: What is Magnetic Induction? What is Magnetic Resonance?

Intel: Intel Developer Forum 2008

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Electric vehicle(Nissan Leaf)

A set of four independent equations describing the electric and magnetic fields in vacuum :

(1) (Faraday’s law of induction)

(2) (Ampere’s law)

(3) (Gauss’ law)

(4) (Gauss’ law for magnetism)

Maxwell’s equations

0

t

HE

J

t

EH

E

0B

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Consider a one-turn coil C with an area S and a magnetic flux density Bpassing though C. When B changes in time, integrate the Faraday’s law of induction on S,

(5)

By Stokes’ theorem,

(6)

The induced electromotive force, EMF ε [V] in C is

(7)

Φ[Wb] : magnetic flux in Circuit C.

Electromagnetic induction

0S S

BdS dSt

E

0C S

dr B dSt

E

ddS

B dSt t

11SB dS

If there is no flux leakage from a solenoid coil C1 to C2, EMF ε2 in C2generated by the current I1 in C1 is

(8)

From the reciprocity,（相反定理）

(9)

Mutual inductance is defined as

(10)

Mutual Inductance, M

1 1 1 2 1 12 2 2 21

d dd dd d d d

N SI N N S I IN N Mt t l l t t

1 2 2 21 12

d dd d

N N S I IMl t t

1 221 12

N N SM M Ml

Two solenoid coils C1 and C2 (N1 and N2 turns) 

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EMF ε1 is also induced by I1. In the same way as the previous, self-inductance L1 is expressed as

(11)

Then, general form of induction is written as

(12)

Generally, there is an inequality

(13)or

(14)

where, k (<1) is a “Coupling coefficient.”

Coupling coefficient, k

1 1 1

2 2 2

dd

L M ItM L It

1 2M L L

1 2M k L L

Two solenoid coils

21

1N S

Ll

13

Leakage inductance leakage flux stores energy and thus acts as an inductor in series in each of 

the primary and secondary circuits.

1 1 1

2 2 2

21 2 21 11

22 222 1 1

0d0d

dd

V L M IV M L It

k L L V Ik L MItM k Lk L L V

2leak1,leak2 1,2 1,21 0L k L L k

Leakage inductance Equivalent circuit with leakage inductance (2)

V1 L1 L2

M

V2

I1 I2

Lleak1 Lleak2

V1 V2

I1 I2

ε1 ε2

Original circuit (1)

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211 2 21 1

22 22 1 1 2

21 2 2 1 11 1 leak1 1

22 2 leak2 22 1 1 2 2

1 0 0d0d0 1

1

1

k Lk L L VV IV Itk L L V k L

k L L V k L dI tV L dI tV L dI tk L L V k L dI t

(15)

(16)

(17)

Power factor (力率) in AC power system

Power factor: P(real power)/S(apparent power) =cosφ

No delay (φ=0, cosφ=1)

Zero power factor(φ=90, cosφ=0)

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Power factor correction (力率改善)

Cancellation by compensation capacitorsIn both primary and secondary sides

1,2 20 leak1,leak2

1CL

C1 C2

Lleak1 Lleak2

C1 C2Lleak2

Series/Series type

Parallel/Parallel type

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(18) Lleak1

Equivalent circuit with power factor correction

Power factor correction condition for k≈0 is ω=ω1=ω2

AC frequency: ωCircuit resonant frequency:

Source(src)

Load (ld)

Primary Secondary

Incident wave Transmission wave

1,2 1,2 1,2 1,2 1,2( 1 / ) Z R i L C R

Z0Z0

C1 C2

R1 R2

L1 L2

Double resonating circuit

1,2 1,2 1,21 / L C

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(19)

(20)

Analysis of equivalent circuit

Kirchhoff 2nd law 

Input power from source Pin and output power at load Pwork

transfer efficiency

1 1 1src 0 2src

20 2 2 2 1 0 2

0 ( )

Z i M I IV Z ZVi M Z Z I I i MZ Z Z M

in src 1P V I

2 2work 0

2in 1 2 1 0 2 21 0 2 0 2

2 21 2 2 0

1( ) 1 1

P Z MP R R Z Z Z RZ Z Z M Z Z

M R R R Z

2work 0 2P Z I

Z0Z0

C1 C2

R1 R2

L1 L2

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(21)

(22)

(23)

Analysis of equivalent circuit, ctd.

Figure‐of‐Merit （性能指標, fom）

coupling coefficient: Q factor:

transmission efficiency η

Optimum impedance ratio (Load resistance/internal resistance)

2 21 2k M L L

1,2 1,2 1,2Q L R2 2

2 21 2

1 2

M k Q Q fomR R

2 20 2 1 2opt

1 1Z R k Q Q fom

0 22

2 0

11 1 1 1Z R

fom R Z

2

max 22

22 2

1

1 1 1 11 1 1 11

fom

fomfomfom fom

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(24)

(25)

(26)

(27)

Theoretical transmission efficiency

Transmission efficiency at optimum impedance ratio.

To achieve high efficiency← higher Q factor

← higher frequency, ω(>10kHz)

2

max 221 1

fom

fom

2 20 2 1 2opt

1 1Z R k Q Q fom at

Q L R

20

(28)

(29)

Summary of Magnetic Induction Power Transfer

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Leakage inductance due to air gap results in lowpower factor.

Power factor is compensated by adding capacitors.

Perfect compensation condition is equal to aresonant condition.

transfer efficiency is a function of the figure-of-merit kQ.

Magnetic Resonance Power Transfer

Demonstration of WPT by Magnetic Resonance

MIT demonstrationPower: 60 WDistance: 1.8 m Coil diameter: 0.6 m. Efficiency: 45%RF: 10 MHz

With a pair of high Q coils23

Quality factor of resonating circuits

Γ: band width

0 0 1

2R L

Q

01 LQ

R C R L

RLC circuit

Ohm R2 2R L R R L

Definition:

ω0ωL ωR

C

R

L

Resonance curve

2Energy Stored

QEnergy Dissipated per cycle

Low Q case; energy is rapidly dissipated through Joule heating and radiationHigh Q case; energy is stored for long period. (high quality coil or circuit) but resonance band width becomes narrow.

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Video

Task

(1)

(2)

Magnetic Induction ? or Resonance?

Coupling coefficient k >0.5. → Magne c induc on with 

‐ Compensation capacitor (power factor correction)‐High AC frequency >10KHz (high Q)

Coupling coefficient k =0.1~0.001. → Magne c Resonance coupling with

‐ very high Q of the order of 1000‐ Higher AC frequency ~10MHz ‐ low resistance and radiation‐ narrow resonance frequency band

transfer distance and efficiency

2opt 1 21r r k Q Q

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High Q coils (1)

Resonance frequency Designed：10.56±0.3MHz Measured：9.90MHz

Q factor Designed：2500(σ=5.9×107m/Ω

assumed) Measured：950±50(Due to 

surface oxidization?)

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Helical type (MIT) Spiral type (Ryukoku univ.)

Resonance frequency 21MHz (diameter 5.5cm)

Q factor Measured：1000 with pitch 5mm

High Q coils (2)

Resonance frequency Designed：13.54MHz Measured：13.43MHz

Q factor Designed：308 Measured：341

Mica Condenser

Cupper wire

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Loop type (UT)

Energy losses in a coil

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1. Ohmic loss:  1/Qohm

2. Dielectric material loss: 1/Qd

3. Radiation loss: 1/Qr

1/Q=1/Qohm+1/Qd+1/Qr = (Rohm+Rd +Rr)/ωL

1. Ohmic loss:  1/Qohm

ohmic 2l lRa a

Skin depth (resistivity: ρ)

Skin effect (表⽪効果）High frequency AC is distributed within a conductor such that the current density is largest near its surface and this “skin” becomes thinner with frequency.

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2

Resistance (cable diam.: a, cable length: l)

2. Dielectric material loss: 1/Qd

Equivalent series resistance in capacitor (寄⽣抵抗） Rp

rP

c d

1tan I CRI Q

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Dissipation factor, tanδ

tanδ= 0.02-0.002 (@1GHz) for good capacitors.

Need care for covers, coatings and surface oxidization for solenoid coil wires.

(3)

3. Radiation loss: 1/Qr

Radiation loss of a one-turn loop coil (coil diam. D, wavelength λ)

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Then,

470

rad8

3c DR

Inductance L is also a function of D as,

0 2ln 2.2022

l DLa

3 3

rr

3 2ln 2.202 1.338

L D D DQR a

(for D/a=50)

(4)

(5)

(6)

How to measure Q factorExcitation 

coil

Rogowskicoil

resonator

TG outputSA input

3dB3dB

f1 f0 f2

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12

0

fff

QL

Resonance curve

Transmission efficiency analysis (1)

2 2

0 2 0 21

,1 1

k Q rZ Z r Z Rix ix

1,2 01,2 1,2 1,2 1,2 1,2

1,2 1,2 1,2 0

1 11 1L

Z R i L R i R ixC C R

Assume, power source freq. ω is slightly different from circuit resonance freq. ω0Then,

Where

1,2 0 0

1,2 1,2 0 0

1Lx Q

C R

Impedance Ratio

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(7)

(8)

(9)

Double resonating circuit

1 01 1 1src 0 2src2

2 02 2 2 1 01 2 020 ( )( ) ( )

Z Z i M I IV Z ZVi M Z Z I I i MZ Z Z Z M

Kirchhoff's voltage law

Efficiency2 2 2

2 01 02 1 2 1 221 21 2 22 2

1 01 2 02 1 2 1 2

4 4

( )( ) ( ) (1 )(1 )

M Z Z k Q Q r rs

Z Z Z Z M ix r ix r k Q Q

R1

Vsrc

Z01

C1

L1

R2

Z02

C2

L2

Transmission efficiency analysis (2)

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(10)

(11)

Efficiency Map at high kQ（>1）

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0

0.2

0.4

0.6

0.8

1

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

(ω-ω0)/ω0

0

0.2

0.4

0.6

0.8

1

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

(ω-ω0)/ω0

Frequency Matching & Impedance Matching

r=5

p

0 2

k

kQ=10k=0.01Q=1000

kQ=10k=0.01Q=1000

r=13

Frequency Matching Impedance Matching

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Summary of Magnetic Resonance Power Transfer

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High transfer efficiency, ie. High kQ, is achievableby high Q even at low k.

High Q is available in high frequency range ofseveral MHz.

Bandwidth in which two high Q coils can resonateis very small.

Input & output impedance matching is importantfor efficient power transfer.

QuestionWe want to design a high quality coil of Q=1000 with 1 m cupper wire. Available minimum resistance R=0.1 Ω and maximum inductance L=10 μH.

How much angular frequency of AC is required?

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