01 fields 06 1 - rfid-systems.at · 01 fields 2nd unit in course 3 ... a resistance in series to...

01 Fields01 Fields2nd unit in course 3, 2nd unit in course 3, RF Basics and ComponentsRF Basics and Components

Dipl.-Ing. Dr. Michael Gebhart, MSc

RFID Qualification Network, University of Applied Sciences, Campus 2WS 2013/14, September 30th

ContentContent

Coupling Cases

H-field and E-field emission of a loop antenna

The magnetic momentum

Coaxial and Coplanar Antenna Orientation

Near Field and Far Field

Law of Ampere

Biot-Savart Law

( ) ∫∫ ∫∫∫ +=A AC

dAnDdt

ddAnJsdH

ro

rro

rro

r1

( ) ∫∫∫ −=AC

dAnBdt

dsdE

ro

rro

r2

( ) 03 =∫∫A

dAnBr

or

( ) ∫∫ ∫∫∫=A V

dVdAnD ρr

or

4

Michael Faraday, Law of Induction (1831)

dt

dΦusdE m

i

C

−==∫r

or

( ) ( )Aµ

dt

dH

dt

AHµd

dt

ABd

dt

dU i 0⋅−=−=−=−=

ϕ

Harmonic sine-wave flux (quasi-stationary)

( ) ( ) ( )°−=−→= 90sincos tHdt

dHtHtH ωω

AHfU i 02 µπ ⋅=

Coupling Case 1: Open LoopCoupling Case 1: Open Loop

Maxwells Equations, integral form

LA

i1H

i1

source

2nd coil

1st coil

reference signal (reference phase) i1

u2

u- 90 °

2

No current in 2nd loop, so no H-field emission

Induced voltage is related to primary H-field

Coupling Case 2: Coupling Case 2:

Closed LoopClosed Loop

LA RA

i2

i2

i1H

- 180 °i2

i1

source

2nd coil

1st coil


Current in 2nd loop generates H-field…

…that cancels out primary H-field (-180 °)

at the position of the 2nd coil

( ) Dt

JHrotrrr

∂

∂+=1

( ) Bt

Erotrr

∂

∂−=2

( ) 03 =Bdivr

( ) ρ=Ddivr

4

Maxwells Equations, differential form

A closed loop of ideal conductor shorts induced voltage to zero.

This current in the 2nd loop is shifted by – 90 °to an induced

voltage (which is – 90 °shifted to primary current), so in total the

2nd current is – 180 °shifted versus the primary current.

A resistance in series to the inductance allows some (induced)

voltage drop – the remaining voltage is shorted and

compensates partly the primary H-field at the second position.

Current in 2nd loop is active and reactive…

Phase-shift is different from 180 °…

Field does not cancel out,

Amplitude of 2nd H-field can exceed primary field

page 5

Coupling Case 3: Coupling Case 3:

Resonant AntennaResonant Antenna

LA RC AA

i2

i2C i2R

i1H

- 90 °

- 180 °i2R

i2C

i1

source

2nd coil

1st coil


-180

-135

-90

-45

0

0 5 10 15 20 25 30

Quality Factor Q

Ph

ase

i2

re

lative

to

i1

Phase Shift

0,0

0,2

0,4

0,6

0,8

1,0

0 5 10 15 20 25 30

Quality Factor Q

Am

plit

ude i2

Amplitude

Resonance at carrier frequency! Resonance at carrier frequency!

Emitted Fields from a rectangular coilEmitted Fields from a rectangular coil

- As introductive example we will have a look at the fields emitted from a

rectangular loop antenna.

o antenna size 72 x 42 mm

o track width 0.5 mm

o track thickness 36 µm

o antenna is fed from short side

o antenna current 1 A(rms)

o frequency 13.56 MHz

HH--field and field and EE--field Magnitudefield Magnitude

HH--field and field and EE--field Directionfield Direction

Near Field, transition zone, Far FieldNear Field, transition zone, Far Field

Far Field (Fraunhofer region)

distance > 4 λ

wave propagation, effective power

Transition zone (Fresnel region)

0.2 λ < distance < 4 λ

“radiating Near Field”

Near Field (Rayleigh region)

distance < 0.2 λ

inductive coupling, reactive power

Antenna center

The magnetic momentumThe magnetic momentum

Heinrich Hertz developed the method of the magnetic momentum to calculate

the H-field strength in space, in analog to the electric dipole momentum.

- It delivers good results in the far field of an antenna

- The magnetic momentum for a conductor loop of any shape is given by the

alternating current times the area inside the loop

∫⋅= AdImd

rr

x

y

z

L o o p -

a n te n n a

D ip o le -

a n te n n a

…current density times velocity

…direction from origin to space element

page 11

The magnetic momentumThe magnetic momentum

Heinrich Hertz developed the method of the magnetic momentum to calculate

the H-field strength in space, in analog to the electric dipole momentum.

- It delivers good results in the far field of an antenna

- The magnetic momentum for a conductor loop of any shape is given by the

alternating current times the area inside the loop

∫⋅= AdImd

rr

( ) IrNAINmd ⋅=⋅⋅= π2r

- In practice we find for the absolute of the momentum for a planar loop:

- for a circular loop: - for a rectangular loop:

( ) IblNAINmd ⋅⋅=⋅⋅=r

dVJrmd

rrr×⋅= ∫2

1

- More general, for arbitrary current distribution in the space volume:

…space volume element( ) Φ= dddrrdV θθsin2

rr

vJvr

ρ=

…radian wavelength,

angular wavelength

page 12

++−+Φ

+= θθθ

π

)DD)DD

D

rr

sin1cos24 2

2

2

2

2 rj

rrj

rr

mH d

Φ

−

⋅=

)D

D

rv

θπ

µsin1

4 2

0

rj

r

mcE d

H

EZ =

π

λ

2=D

- The H-field and the E-field (of a current-carrying conductor loop) in space can

be derived from the magnetic momentum for any point in space in spherical

coordinates by

z

x

y

H

E

r

θ

Φ

I

Point in space

current-carrying

conductor loop

0

H-field expressed by the magnetic momentum

- The relation of E-field to H-field gives

the field impedance Z

Specific Antenna OrientationsSpecific Antenna Orientations

Coaxial Orientation- Center points of both antenna

conductors are on an axis

perpendicular to the antenna plane

Coplanar Orientation- both antenna conductors are in the

same plane

++−=

rj

rr

mH d DD

D

rr

2

2

21

4π

( )( ) 1sin

0cos90

=⇒

=⇒≡

θ

θθ

−

⋅=

rj

r

mcE d D

D

rr

14 2

0

π

µ

HH--field and field and EE--field for Coplanar Orientationfield for Coplanar Orientation

Coplanar Orientation

- angle to the normal axis is zero

- cosine term disappears and a simplified equation for H- and E-field remains

4224

22

22

2

2

2

1

4

14

DDD

DD

D

rr

+−=

=

+

+−=

rrrr

m

rrr

mH

d

d

π

π

( )

22

2

0

2

2

2

0

1

4

14

DD

D

D

rr

+⋅

=

=

−+

⋅=

rrr

mc

rr

mcE

d

d

π

µ

π

µ

( )4224

34

0

2

20

2

2

2

2

0

1

1

14

14

rr

jrrZ

rj

r

rj

Z

rj

rr

m

rj

r

mc

H

EZ

d

d

+−

+=

==

++−

−

⋅=

=

++−

−

⋅

=

==

DD

D

KDD

D

DD

D

D

D

π

π

µ

Field Impedance Field Impedance ZZ for Coplanar Orientationfor Coplanar Orientation

- As E-field and H-field are non zero, the field impedance Z can be calculated

from the magnitude of E and H:


- We can separate the real part and

the imaginary part of Z:

( )( )4224

4

0Rerr

rZZ

+−=

DD

( )( )4224

3

0Imrr

rZZ

+−=

DD

D

0.01 0.1 1 100.01

0.1

1

10

100

1 .103

Real & Imaginary part of Wave Impedance

Distance over radian wavelength

Imp

edan

ce i

n O

hm

c

r

m

r

mc

H

EZ

d

d

0

2

2

0

0

4

4µ

π

π

µ

=

⋅

==

D

D


( ) ( )

( )4224

66

0

22 ImRe

rr

rrZ

H

EZZZ

+−

+=

==+=

DD

D

- The magnitude of Z can be calculated

from real and imaginary part:

Ω≅Ω==⋅

=⋅= 3771201

0

0

00 πε

µ

εµ

ccZ

- At the far field the field impedance Z

approximates the wave impedance Z0

0.01 0.1 1 1010

100

1 .103

Magnitude of Field Impedance coplanar


Imp

edan

ce i

n O

hm

( )( ) 0sin

1cos0

=⇒

=⇒≡

θ

θθ

( )

+==

rj

rr

mH d DD

D

rr

2

2

24

20

πθ( ) 00 ≡=θE

r

HH--field and field and EE--field for Coaxial Orientationfield for Coaxial Orientation

Coaxial Orientation

- angle to the antenna axis is zero

- the E-field component disappears and only the H-field component remains

- consequently the field impedance Z becomes zero for coaxial orientation!

( )

+=°=

rj

rr

mH d DD

D

rr

2

2

24

20

πθ

224

32

22

2

2

2

22

1

2

4

2

ImRe

rr

m

rrr

m

H

d

d

DDD

KDD

D

r

+=

==

−+

=

=+=

π

π

( )

++−=°=

rj

rr

mH d DD

D

rr

2

2

21

490

πθ

4224

22

22

2

2

2

1

4

14

DDD

DD

D

rr

+−=

=

+

+−=

rrrr

m

rrr

mH

d

d

π

π

HH--field comparison for field comparison for

coplanar and coaxial orientationcoplanar and coaxial orientation

- magnitude - magnitude

Coplanar Orientation

- complex field vector

Coaxial Orientation

- complex field vector

Comparison of the H-Field for coaxial and coplanar Antenna-Orientation

Far FieldNear Field

Decrease of the H-field with

disance for coaxial and

coplanar antenna orientation:

Near Field

- Coaxial & coplarnar: H ~ 1/d³

Far Field

- Coaxial: H ~ 1/d³

- Coplanar: H ~ 1/d1

λ at 13.56 MHz = 22.124 m

0.1 1 10 1001 .10

6

1 .105

1 .104

1 .103

0.01

0.1

1

10

100coplanar

coaxial

distance in meters

H-f

ield

str

en

gth

in

A/m

λ at 13.56 MHz = 22.124 m

0 2 4 6 8 10

150

100

50

0

50

100

150

Near field Phase over distance

Distance in m

Angle

in d

egre

es

page 21

E-field Phase trace

H-field Phase trace

H far field Phase trace

0 2 4 6 8 10

150

100

50

0

50

100

150

Near field Phase over distance


An

gle

in

de

gre

es

Phase trace in the Near Field Hertz also investigated the phase trace over distance. John Wheeler defined

the “radiation sphere” r = λ / 2 π for the near-field.

- In the Near Field close to the conductor, the field is directly linked to current.

- Wave propagation in the Far Field follows the relation c = λ f.

x

y

z

L o o p -

a n te n n a

D ip o le -

a n te n n a

page 22

Wireless power transmission

2

0

0

2

HZZ

ES ⋅==

( ) 22422

0 /377.0/10377/1377 mmWmWmmAHZS =⋅=⋅Ω=⋅= −

HESrrr

×=

ISOTROPHS

SD =

2

0

0

2

HZAZ

EAAdSP

A

⋅=== ∫r

or

- Power density can be derived with the Poynting Vector concept

- for field magnitudes in the Far Field this means

- antenna directivity D is

For a loss-less antenna,

directivity is equal to gainGain of a planar loop in the far field

e.g. an H-field emission limit of 60 dB(µA/m) in Far Field is related to a power density of…

Contactless power transmission

i1

L1u1

i2

L2u2

k, M

SIN SOUT

An ideal transformer is a good model for contactless power transmission

- we neglect losses, resonances and inductances are linear and time-invariant

( ) ( ) ( ) nAtBntt =⋅= φψ- coil flux

( ) ( ) ( )tiLtiLt 2121111 −=ψ

( ) ( ) ( )tiLtiLt 2221212 −=ψ

- branches

ωjdt

d→- using harmonic sinewave signals…

2111 IMjILjU ωω −=

2212 ILjIMjU ωω −=

- network

equations…where U and I are root-mean-square (rms)

values of the signals u(t) and I(t).

- the apparent power S of the primary circuit is given by…

(without secondary load current, this is just reactive)

MjILjISIN ωω ⋅−⋅=2

21

2

1

MjILjISOUT ωω ⋅−⋅=2

12

2

2

- for effective power transmission, we need to consider… [ ]SP Re=

electromagnetic wave in the far field

page 24

=

REF

ABS

dBH

HH log20 2010

dBH

REFABS HH ⋅=

( ) ( )Ω≅Ω≅ dBZ dB 5.51377log20,0

mVmAZHE LIMITLIMIT µµ 377000377/10000 =Ω⋅=⋅=

( ) )/(5.111)(5.51)/(605.51,0 mVdBdBmAdBdBHZHE dBdBdBdB µµ =Ω+=+=+= Ω

Absolute and decibel values

Decibel values are logarithmic, relative to an absolute reference value.

- Power scales 10 times the logarithm, H-field and E-field (like current and voltage)

scale 20 times the logarithm, as they are square-proportional to power.

e.g. H-field emission limit 60 dB(µA/m) is absolute…

mmAmAmAHHdBH

REFLIMIT

dB

/1/100010/110 2060

20 ==⋅=⋅= µµ

e.g. for Far Field the related magnitude of the E-field is …

Same calculation in decibel values…

( )120 coscos

4αα

π

µ−⋅=Φ

r

IB

I....…antenna current

r...…distance from conductor center

µ0….permebility (magnetic field constant),

in free space 4π 10-7 Vs / Am

r

IHor

r

IB

ππ

µ

22

0 == ΦΦ

Law of Ampére

As a first step, Jean-Marie Ampére found a rule in 1820. Current in a conductor

generates an H-field around the conductor. Using todays notation of units, his

rule is…

- for the special case of an infinitely long, straigth conductor (=> α1 = -180 °, α2 = 0 °)

results the well-known simple relation…

I

z

0

α1

r

point in

space

R

α2

con

du

cto

r α

I1

r→

ds→

dH

Point in space

→

∫×

⋅=

Sr

rsdIH

34

1

rrr

π

For practical applications, the integral is problematic, as closed analytical

solutions only exist for special cases (like for a circular conductor).

The original Biot-Savart law does not take wave propagation into account, so

there is an error with increasing distance to the conductor.

The Biot-Savart law

In Paris of 1840, Jean-Baptiste Biot and his assistant Felix Savart derived a law,

which allows to calculate the H-field strength and direction from any conductor

geometry and current, for any point in space. In integral form, it is…

( ) ( ) ( )[ ] Φ

Φ⋅−+Φ⋅−+⋅

+⋅⋅⋅

⋅

⋅= ∫

⋅⋅−

dyyxxar

ir

eaIzyxH RSRS

SRSR

ri

ARRRz

SRπ β

βπ

2

0

2)sin()cos(

1

4,,

( ) ( ) ( ) Φ

+⋅⋅⋅Φ⋅

⋅

−⋅⋅= ∫

⋅⋅−

dr

ir

ezzaIzyxH

SRSR

ri

RSARRRx

SRπ β

βπ

2

0

2

1cos

4,,

( ) ( ) ( ) Φ

+⋅⋅⋅Φ⋅

⋅

−⋅⋅= ∫

⋅⋅−

dr

ir

ezzaIzyxH

SRSR

ri

RSARRRy

SRπ β

βπ

2

0

2

1sin

4,,

Biot-Savart law for circular loops extended by retardation (near-field & far field)

- Magnitudes of the cartesian H-field components are…

- HZ is especially useful for a coaxial Reader – Card couplig scenario

( )( ) ( ) ( )222)sin(cos),,,( RSRSRSRRRSR zzyayxaxzyxr −+−Φ⋅++−Φ⋅+=Φ

- The radius vector from source (S, loop center) to any point in space (R) is…

HH in Near Field and Far Field, coaxial and coplanarin Near Field and Far Field, coaxial and coplanar

The H-field decreases with distance to

the emitting loop antenna in...

near-field:

coaxial & coplanar orientation: H~1/d3

far-field:

coaxial: H ~ 1/d3

coplanar: H ~ 1/d1

( ) °≠∠ 0)(),( tHtE ( ) °=∠ 0)(),( tHtE

reactive power region effective power region

Thank you for your Thank you for your

Audience!Audience!

Please feel free to ask questions...

01 fields 06 1 - rfid-systems.at · 01 fields 2nd unit in course 3 ... a resistance in series to...

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