Microwave Electronics · Microwave Electronics ☞Maxwell's Equations & Modes in a Guide Equivalent...

Microwave Electronics

Maxwell's Equations & Modes in a Guide Equivalent Circuit for Waveguide Modes Modes of a Cavity Cavity with a Port & External Q Microwave Networks Slater's Perturbation Theorem

An Introduction to the Notion of Equivalent Circuit

Starting from Maxwell’s Equations

in six lectures

Microwave Electronics I

Maxwell's Equations and

Waveguide Modes

① Lorentz Force Law② Maxwell’s Equations③ Skin Depth ④ Orthogonal Modes⑤ Phase & Group Velocity Quiz

①Lorentz Force Law

Newtons per Coulomb=V/m“electric field strength”

Hendrik Antoon Lorentz b. July 18, 1853, Arnhem, Netherlandsd. Feb. 4, 1928, Haarlem

defines the fields & abstracts them from the sourcesdescribes “test particle” motiondescribes response of media

r r r rF q E v B= + ×( )

r

E

rB Newtons per Ampere-meter=T=Wb/m2

”magnetic flux density” or “magnetic induction”

Example: Conductivity

P. Drude, 1900

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

Georg Simon Ohm(b. March 16, 1789, Erlangen, Bavaria --d. July 6, 1854, Munich)

m

dvdt

qE mv

r r r= −

τ

r rv

qm

E= τ

J nqv

nqm

E

E

=

=

=

r

r

r

2τ

σ

σ σ= ≈ ×" " . /DC mho m5 8 107 for Cu

NB This is a simplified picture ofa normal conductor...occasionallythis picture breaks down...

and of course this model cannot be appliedto all materials...

②Maxwell’s Equations

Charges repel or attract Current carrying wires repel or attract Time-varying currents can induce

currents in surrounding media

Electricity & Magnetism before Maxwell...

After Maxwell...

Light is an electromagnetic phenomenon

Nature is not Galilean Thermodynamics applied to

electromagnetic fields gives divergent results

Matter appears not to be stable Questions arise concerning

gravitation...


“Ordinary” Electronics

•voltages vary slowly on the scale of thetransit time = circuit size / speed of light•circuit size small compared to wavelength•voltage between two points independent of path•may treat elements as “lumped”•unique notion of impedance of an element•bring a multimeter

•circuit size appreciable compared to a wavelength•voltage between two points depends on path•elements are “distributed”, spatial phase-shiftsoccur between them•if the word “impedance” is used, you may alwaysask how it was defined...•if any result of test & measurement is quoted, you may always ask how the equipment was calibrated•bring crystal detectors, filters, mixers, a signal generator, a spectrum analyzer, and, if you have them a network analyzer, calibration kit, vector voltmeter

HistoryHenry Cavendish(b. Oct. 10, 1731, Nice, France--d. Feb. 24, 1810, London, Eng.)

Charles-Augustin de Coulomb (b. June 14, 1736, Angoulême, Fr.--d. Aug. 23, 1806, Paris)

André-Marie Ampere(b. Jan. 22, 1775, Lyon, France--d. June 10, 1836, Marseille)

Karl Friedrich Gauss(b. April 30, 1777, Brunswick--d. 1855)

Hans Christian Ørsted(b. Aug. 14, 1777, Rudkøbing, Den.--d. March 9, 1851, Copenhagen)

Siméon-Denis Poisson(b. June 21, 1781, Pithiviers, Fr.--d. April 25, 1840, Sceaux)

Michael Faraday(b. Sept. 22, 1791, Newington, Surrey --d. August 25, 1867, Hampton Court)

James Clerk Maxwell(b. June 13 or Nov. 13, 1831, Edinburgh--d. Nov. 5, 1879, Glenlair )

Heinrich (Rudolf) Hertz(b. Feb. 22, 1857, Hamburg--d. Jan. 1, 1894, Bonn)

Guglielmo Marconi(b. April 25, 1874, Bologna, Italy--d. July 20, 1937, Rome)

Gauss’s Law

Ampere’s Lawbefore Maxwell

r rD dS dV

V V•∫ = ∫

∂ρ

r r r rH dl J dS

S S∂∫ ∫• = •

electric displacementor electric flux density

r rD D nA A2 1−( ) • =ˆ Σ

Medium 2

Medium 1+ + + + + + + + + + + + + + ++

surface charge density Σpillbox area A

n

Medium 2

Medium 1

surface current densitycontour side L

n

rK

l

D C m⇔ ⇔Σ / 2

r r rH H l K n l2 1−( ) • = × •ˆ ˆ ˆ

n H H K× −( ) =r r r

2 1

H K A m⇔ ⇔ /

magnetic field strength or magnetic flux

In Vacuum...In Vacuum...

µ π074 10= × − henry per meter

ε0128 85 10≈ × −. farad per meter

12 9979 10

0 0

8

ε µ= = ×c m s. /

µε

0

0377= Ω

or

r rD E= ε0

r rH B= / µ0

In Media...In Media...

magnetic dipole moment density

r r r r

D E P E= + =ε ε0

electric dipole moment density r r

P Ee= χ ε0

r r

M Hm= χ

Finer Points:•These are really frequency domain expressions•In general ε,µ are tensors• µ may be non-linear & biased by a DCfield•H,D depend on your point of view

r r r r

H B M B= − =/ /µ µ0

Faraday’s Law

r rr

rE dl

Bt

dSS S

•∫ = − •∫∂

∂∂

Before Maxwell, Ampere’s Law was inconsistent with conservation of chargeAfter Maxwell, the fields didn’t need charge to support them, they could propagate on their own

Of course no one believed Maxwell, but the fieldsdidn’t mind

Charge Conservation

∂ρ∂t

J+ ∇ • =r

0

∂∂

ρ∂t

dV J dSV V∫ ∫= − •

r ror

∂ρ∂t

J H= −∇ • = −∇ • ∇ × =r r

0

Ampere’s Law (before Maxwell’s additionof Displacement Current) implied

...actually not a bad approximation in conductors, or a dense plasma..excellent for electrostatics, magnetostatics

Gauss’s Law

Ampere’s LawwithMaxwell’s Displacement Current

Faraday’s Law

r r∇ • =D ρ

r rr

∇ × = −EBt

∂∂

r r rr

∇ × = +H JDt

∂∂

∇ • =rB 0No Magnetic Charge

Maxwell’s Equations

James Clerk Maxwellb. June 13 or Nov. 13, 1831, Edinburghd. Nov. 5, 1879, Glenlair

Medium 2

Medium 1+ + + + + + + + + + + + + + ++

surface charge density Σpillbox area A

n

Medium 2

Medium 1

surface current densitycontour side L

n

rK

l

Boundary Conditions

r r

r rD D n

B B n

2 1

2 1 0

−( ) • =

−( ) • =

ˆ

ˆ

Σ

ˆ

ˆ

n H H K

n E E

× −( ) =

× −( ) =

r r r

r r2 1

2 1 0

apply Maxwell’s equations in integral form...

Example to Illustrate ε:Unmagnetized Plasma

rE Ee Ee E ej t j t j t= ℜ( ) = +( )−˜ ˜ ˜ *ω ω ω1

2

m

dvdt

qE mv

r r r

= −τ

˜ ˜ ˜ ˜ ˜vj

qm

E J nqvj

EDC=+( )

⇒ = =+( )

11 1ωτ

τ σωτ

Apply charge conservation & Gauss’s Law...

∂ρ∂

ωρt

J j J+ ∇ • = ⇒ + ∇ • =r

0 0˜ ˜

∇ • = = − ∇ • = − ∇ •+( )

ε ρω ω

σωτ0

1 11

˜ ˜ ˜ ˜Ej

Jj j

EDC

∇ • ++( )

= ⇒ = ++( )

εω

σωτ

ε εω

σωτ0 0

11

01

1j jE

j jDC DC˜

N B withnqm

pp. . lim

τ

εε

ωω

ωε→∞

= − =0

2

22

2

01

Evidently...

•electric permittivity is a frequency-domain concept...

ε εω

σωτ

ε ω= ++( )

= ( )01

1j jDC

•electric displacement depends on what you consider to be the “external” circuit, electric field does not

View #1

View #2

∇ • = +

=

ε ρ ρ

ε0

0

˜ ˜ ˜

˜ ˜

E

D E

plasma other

∇ • =

=

ε ρ

ε

˜ ˜

˜ ˜

E

D E

other

Polarization in the time domain...

r

r

r

P td

e P

de E

de

dte E t

dt G t t E t

j t

j te

j te

j t

( ) ˜

˜

= ∫ ( )

= ∫ ( ) ( )

= ∫ ( ) ′ ′( )∫

= ′ − ′( ) ′( )∫

−∞

+∞

−∞

+∞

−∞

+∞−

−∞

+∞

−∞

+∞

ωπ

ω

ωπ

ε χ ω ω

ωπ

ε χ ωπ

ε

ω

ω

ω ω

2

2

2 2

0

0

0

G t d e j te( ) = ∫ ( )

−∞

+∞12π

ω χ ωω

where the Green’s function is

χ ω ωe

j tdt e G t( ) = ∫ −

−∞

+∞( )

Example...unmagnetized plasma...

G t d e

d ej j

e H t

j te

j t p

pt

( )

( )/

= ∫ ( )

= ∫ +( )= −( )

−∞

+∞

−∞

+∞

−

12

12 1

1

2

2

πω χ ω

πω

ω τω ωτ

ω τ

ω

ω

τ

ω-plane

pole at ω=0

pole at ω=j/τ

for t<0, contourmay be closedat Imω→−∞so that G=0

contour

for t>0, contourmay be closedat Imω→+∞with contributionsfrom two poles

Susceptibility in the High Frequency Limit

when the Green’s function is analytic near t>0,

χ ω

∂∂ω

∂∂ω ω

ω

ω

ω

ω

ej t

j tn

n

n

n

n

nj t

n

n

n

dt e G t

dt e H tG

nt

Gn

j dt e

Gn

jj

jG

( ) = ∫

= ∑∫

= ∑

∫

= ∑

= −

−

−∞

+∞

−

=

∞

−∞

+∞

=

∞ −+∞

=

∞

( )

( )( )!

( )!

( )!

( )

( )

( )

( )

0

0

0 1

0

0

0 0

0

−− +GjG( ) ( )( ) ( )1

2

2

30 0

ω ωK

Example: unmagnetized plasma

χ ωωω

ωω τe

p pj( ) = − − +2

2

2

3 K

Kramers-Kronig Relations

Since G is causal χe must be analyticin the Imω<0 half-plane, so that

for points ω,ω’ and contour in the lower half-plane. Let the contour liejust below the real axis and use

χ ωπ

χ ωω ωe

e

j( ) = ′( )

′ −∫1

2

limε ω ω ε ω ω

π δ ω ω→ ′ − −

=′ −

+ ′ −( )

0

1 1j

P j

then

χ ωπ

χ ωω ωe

e

jP( ) = ′( )

′ −∫−∞

+∞1

ℜ ( ) = ℑ ′( )′ −∫

ℑ ( ) = − ℜ ′( )′ −∫

−∞

+∞

−∞

+∞

χ ωπ

χ ωω ω

χ ωπ

χ ωω ω

ee

ee

P

P

1

1

relatesdispersion&absorption

Scalar & Vector Potentials

Maxwell’s Equations...

r r r r r

r v rr

r r rr

∇ • = ∇ • ∇ × =

∇ × = ∇ × − − ∇

= − ∇ × = −

B A

EAt t

ABt

0

∂∂

ϕ ∂∂

∂∂

r r r r r r r

r rr

r

rr r

∇ × = ∇ × ∇ ×( ) = ∇ ∇ •( ) − ∇

= ∇ × = +

= − − ∇

+

µ

µ µ ∂ε∂

µ

µε ∂∂

∂∂

ϕ µ

H A A A

HEt

J

tAt

J

2

r r r r rr

r∇ • = ∇ • = ∇ • − − ∇

=D EAt

ε ε ∂∂

ϕ ρ

r r rB A= ∇ ×

vr

rE

At

= − − ∇∂∂

ϕ

r r rA A

t

→ − ∇

→ +

ψ

ϕ ϕ ∂ψ∂

Gauge Invariance

leaves E,B unchanged

Lorentz Gauge ∇ • + =

rA

tµε ∂ϕ

∂0

∇ − = −2

2

2

rr

rA

At

Jµε ∂∂

µ ∇ − = −22

2ϕ µε ∂ ϕ∂

ρ εt

/

evidently the characteristic speedof propagation in the medium is v = ( )−µε 1 2/

Coulomb Gauge r r∇ • =A 0

∇ − = − + ∇2

2

2

rr

r rA

At

Jt

µε ∂∂

µ µε ∂ϕ∂

∇ = −2ϕ ρ ε/

∇ = −

2ψ µε ∂ϕ∂t Lorentz Gauge

related to the Lorentz Gauge potentials via

Hertzian Potentials

in a homogeneous, isotropic source-free region...

Magnetic Hertzian potential

r r∇ • = ∇ • =˜ ˜E H 0

E j m= − ∇ ×ωµr

Π

choice of gauge

∇ + =202 0Π Πm mk

∇ × = = ∇ × =

⇒ = + ∇

∇ × = − ∇∇ • − ∇( )= − = − + ∇( )

˜ ˜

˜

˜

˜

H j E k with k

H k

E j

j H j k

m

m

m m

m

ωε µεω

ψ

ωµ

ωµ ωµ ψ

02

02 2

02

2

02

r

r

r r

r

Π

Π

Π Π

Π

ψ ψ ψ= ∇ • ⇒ ∇ + =r

Πm k202 0

Electric Hertzian potential H j e= ∇ ×ωεr

Π

∇ + =202 0Π Πe ek E ke e= ∇∇ • +

r rΠ Π0

2

H km m= ∇∇ • +r r

Π Π02

Energy ConservationEnergy Conservation

In a linear medium, with ε and µ independent of frequency:

− • =

= − ∇ ×

•

= • −∇ × • + ∇ × • − ∇ × •

= • + ∇ • ×( ) +

∇• ×( )

r r

rr r r

rr r r r r r r1 24444 34444

r r r

rr r r r

r

r r r

J E rate of work done on fields

Dt

H E

Dt

E H E E H E H

Dt

E E HB

E H

∂∂

∂∂

∂∂

∂∂tt

H•r

r r rS E H Poynting Flux= × =

− • = ∇ • +

r r r rJ E S

ut

∂∂

where

u E D B H field energy density= • + • =r r r r

When ε and µ are not independent of frequency we should work in the frequency domain...

rE Ee Ee E ej t j t j t= ℜ( ) = +( )−˜ ˜ ˜ *ω ω ω1

2and similarly for J, H, etc...let us compute averagesover the rapid rf oscillation...can show that

S E H= ℜ ×( )∗12

˜ ˜

somewhat more challenging is the calculation of therate of change of field energy density

Questions arise...is this integrable?

r

Dt

EBt

H= • + •∂∂

∂∂

rr

rr

rut

= =∂∂

?

and if so, what is the average stored energy density ?u

To address this problem we first compute

take

and compute P...

then

∂∂

rrD

tE•

r

r r

E t E t e E E t e

D t E t P t e

j t j t

j t

( ) ˜( , ) ˜ ( ) ˜ ( )

( ) ( ) ˜( , )

= ℜ = ℜ +( ) = + ℜ

ω ω ω

ε ω

ω ω

ω

0 1

0

or

˜( , ) ( ) ˜ ( ) ˜ ( )

˜ ( ) ( ) ˜ ( ) ( )

˜ ( )

P t e dt G t t E E t e

E dt G t t e Ej

dt G t t e

E

j t j t

j t j t

ω ε ω ω

ε ω ε ω ∂∂ω

ε ω χ

ω ω

ω ω

= ′ − ′ + ′( )

= ′ − ′ + ′ − ′

=

−∞

+∞′

−∞

+∞′

−∞

+∞′

∫

∫ ∫

0 0 1

0 0 0 1

0 0 eej t

ej te E

jeω ε ω ∂

∂ωχ ωω ω( ) + ( )0 1

˜ ( )

˜( , ) ˜ ( ) ˜ ( )

˜( , ) ˜ ( )

P t E E t j

E t j E

e ee

ee

ω ε ω χ ω ε ω χ ∂χ∂ω

ε χ ω ω ε ∂χ∂ω

ω

= ( ) + −

= ( ) −

0 0 0 1

0 0 1

∂∂

ε ∂∂

ε ω ω ∂∂

ε ∂∂

ε ω χ ω ω ∂χ∂ω

ω χ ω ∂ ω∂

ω ω

ω ω

r r

r

Dt

Et

j P t e ePt

Et

j E t j E e eE t

t

j t j t

ee j t j t

e

= + ℜ +

= + ℜ ( ) −

+ ( )

0 0

0 0 1

˜( , )˜

˜( , ) ˜ ( )˜( , )

= ℜ + +

= ℜ ( ) +

eE t

tj E t E

eE t

tj E t

j t e

j t

ω

ω

ε ∂ ω∂

εω ω ε ω ω ∂χ∂ω

∂ ω∂

∂ ωε∂ω

εω ω

˜( , ) ˜( , ) ˜ ( )

˜( , ) ˜( , )

0 1

∂∂

∂ ωε∂ω

∂ ω∂

ω εω ωr

rDt

EE t

tE t j E t• = ℜ ( ) • +

∗12

2˜( , ) ˜ ( , ) ˜( , )

finally

∂∂

∂ ωε∂ω

∂∂

ω ε ω ω∂ ωε

∂ω∂ ω

∂ω

rrD

tE

tE t E t

E tt

E tri

i• = ( ) − − ( ) ℑ •

∗1

42

2 2˜( , ) ˜( , )˜( , ) ˜ ( , )

Finally

so that, in the absence of losses,

where the field energy density is

Energy conservation takes the form

S E H= ℜ ×( )∗12

˜ ˜

with time-averaged Poynting flux

r

Dt

EBt

Hut

= • + • =∂∂

∂∂

∂∂

rr

rr

u E t H t= ( ) + ( )

14

2 2∂ ωε∂ω

ω ∂ ωµ∂ω

ω˜( , ) ˜ ( , )

− • = ∇ • +

r rJ E S

ut

∂∂

③Skin DepthStart from Maxwells Equations

Fields ∝ exp(jωt)

Reduce to an equation for H

δµω σµ

= = −

≈

2

2 1

" "skin depth

m at GHz in Cu

r

r

r r

∇ × = + ≈ =

∇ × = − = −

∇ • = ∇ • =

˜˜

˜ ˜ ˜

˜˜

˜

˜ ˜

HDt

J J E

EBt

j B

E B

∂∂

σ

∂∂

ω

0

r r r r

r∇ × ∇ ×( ) = ∇ ∇ •( ) − ∇ = −∇

= ∇ × ( ) = − = −

˜ ˜ ˜ ˜

˜ ˜ ˜

H H H H

E j B j H

2 2

σ ωσ µωσ

∇ − =22

20˜ sgn ˜H j Hω

δ

ξ

CONDUCTORVACUUMt angent ial H

normal H

Let’s solve for fields in conductor...

Zj

R js s= + = +( )11

sgnsgn

ωσδ

ω

select coordinate alongoutward normal

“impedance boundary condition”

R

at GHz in Cu

s = =

≈

1

1σδ

surface resistance

8.3mΩ

dd

H j Ht t

2

2 22

0ξ

ωδ

˜ sgn ˜− =

˜ ˜ exp ( sgn )H H jt tξ ξδ

ω( ) = ( ) − +

0 1

˜ ˜ ˆ

˜ˆ ˜E H n

HZ n Hs t= ∇ × ≈ × = − ×1 1

σ σ∂∂ξ

r

Power per m2 into the conductor...

S n E H n

Z n H H n

Z n H

R n H R K

s t

s t

s s

• = ℜ × •( )= − ℜ ×( ) × •( )= ℜ ×

= × =

∗

∗

ˆ ˜ ˜ ˆ

ˆ ˜ ˜ ˆ

ˆ ˜

˜ ˜

12

12

1212

12

2

2 2

˜ ˜ ˆ ˜K Jd n H= = − ×∫ ( )∞

ξ0

0

where in the last line we have made useof the result for surface current density,

④Orthogonal Modes(Uniform Waveguide)

From Maxwell’s Equations, with fields

r r

r

r

∇ • = ∇ • =

∇ × = −

∇ × =

˜ ˜

˜ ˜

˜ ˜

E B

E j B

H j D

0

ω

ω

r r

r∇ × ∇ ×( ) = −∇

= ∇ × = −( )=

˜ ˜

˜ ˜

˜

H H

j D j j B

H

2

2

ω ωε ω

ω εµ

we can see that E & H satisfy the wave equation,

and similarly for E, so that

where k02 2= ω εµ

∝ −e j t jk zzω

∇ +( ) =

∇ +( ) =

⊥

⊥

2 2

2 2

0

0

k H

k E

c

c

˜

˜

k k kc z2

02 2= −

The divergence conditions take the form

˜ ˜Ejk

Ezz

= ∇ •⊥ ⊥1 ˜ ˜H

jkHz

z= ∇ •⊥ ⊥

1

so that the longitudinal field components may be determined from the transverse components. In addition,we may write the curl equations

to express the transverse components in termsof the longitudinal

˜ ˜ ˜ ˜Ejk

Z H Z Hjk

E Z= ∇ × = − ∇ × =1 1

00 0

00

r r µε

− = − × ∇ − ×

= − × ∇ − ×⊥ ⊥ ⊥

⊥ ⊥ ⊥

jk Z H z E jk z E

jk E z Z H jk z Z H

z z

z z

0 0

0 0 0

˜ ˆ ˜ ˆ ˜

˜ ˆ ˜ ˆ ˜

r

r

or

˜ ˆ ˜ ˜

˜ ˆ ˜ ˜

Ekjk

z Z Hkk

E

Z Hkjk

z Ekk

Z H

cz

zz

cz

zz

⊥ ⊥ ⊥

⊥ ⊥ ⊥

= − × ∇ − ∇

= − − × ∇ − ∇

02 0

0

002

00

r r

r r

TE Modes Ez=0transverse fields may be determined from Hz

Z Hkjk

Z H

Ekjk

z Z Hkk

z Z H

z

cz

cz

z

0 2 0

02 0

00

˜ ˜

˜ ˆ ˜ ˆ ˜

⊥ ⊥

⊥ ⊥ ⊥

= ∇

= − × ∇ = − ×

r

r

0 00 2 0= • = • ∇ =⊥ ⊥ˆ ˜ ˆ ˜ . .

˜n Z H

kjk

n Z H i eHn

z

cz

zr ∂

∂

∇ +( ) =⊥2 2 0k Hc z

˜

longitudinal field satisfies

with boundary condition

∇ +( ) =⊥2 2 0k Ec z

˜

TM Modes Hz=0transverse fields may be determined from Ez

˜ ˜

˜ ˆ ˜ ˆ ˜

Ekjk

E

Z Hkjk

z Ekk

z E

z

cz

cz

z

⊥ ⊥

⊥ ⊥ ⊥

= ∇

= × ∇ = ×

2

002

0

r

r

longitudinal field satisfies

with boundary condition Ez = 0

Cut-Off & Characteristic Impedance Zc

In general for a given mode we have

Z H z Ec˜ ˆ ˜⊥ ⊥= ×

where

Boundary conditions restrict the permissible valuesof cut-off wavenumber kc to a discrete set. Each modehas a corresponding minimum wavelength λc beyond which it is “cut-off” in the waveguide.

λ πc ck= 2 /

The guide wavelength is

λ π λ λ λg z ck= = −2 10 02 2/ / /

λ π0 02= / k

Z

Zkk

Z TE e

Zkk

Z TM ec

z

g

z

g

==

=

00

00

00

00

λλλλ

mod

mod

Modal Decomposition

A general solution for a given geometry may berepresented as a sum over modes

where

and we adopt the normalization

˜ ( , )

˜ ( , )

E E r V z

H H r I z Z

t aa

a

t aa

a ca

= ( )∑

= ( )∑ ( )

⊥ ⊥

⊥ ⊥

r

r

ω

ω ω

Z H z Eca a a⊥ ⊥= ×ˆ

d r E r E ra a2 1⊥ ⊥ ⊥ ⊥ ⊥∫ ( ) • ( ) =

r r

where the integral is over the waveguide cross-section.We choose the sign of Z for positive kz.The coefficients V,I take the forms

V z V e V e

Z I z V e V e

a ajk z

ajk z

ca a ajk z

ajk z

za za

za za

( , )

( , )

ω

ω

= +

= −

+ − −

+ − −

This requires Green’s Theorem,

d r E r E r

Z Z d r H r H r

d r z E H Z

a b ab

ca cb a b ab

a b ab ca

2

2

2 1

⊥ ⊥ ⊥ ⊥ ⊥

⊥ ⊥ ⊥ ⊥ ⊥

⊥ ⊥ ⊥−

∫ ( ) • ( ) =

∫ ( ) • ( ) =

∫ • ×( ) =

r r

r rδ

δ

δˆ

One can also show, for non-degenerate modes, that

and the eigenvalue equations for Hz & Ez

ψ ψ ψ ψ ψ ∂ψ

∂12

2 1 22

12∇ + ∇ • ∇( )∫ = ∫

r rd r

ndl

Relation between Power, V & I

As a result, one may express the power flow in thewaveguide, in terms of V & I according to

P d r E H

V I

t t

aa a

= ∫ ℜ ×( )= ∑ ℜ( )

∗

∗

2 12

12

˜ ˜

Meaning of V,I

Given the orthogonality relations, one can determine V,I from the transverse fields at a point z

V z d r E r z E r

I z Z d r H r z H r

a t a

a ca t a

( , ) ˜ ,

( , ) ˜ ,

ω

ω

= ∫ ( ) • ( )= ∫ ( ) • ( )

⊥ ⊥ ⊥ ⊥

⊥ ⊥ ⊥ ⊥

2

2

r r

r r

and this is enough to determine the solutioneverywhere in the uniform guide, since thisfixes the right & left-going amplitudes.

Given the uniqueness of V,I, theirrelation to power, and the units(volts, amperes) it is natural to referto them as voltage & current.

It is important to keep in mind however that they appear as complex mode amplitudes,not work done on a charge or time rate of change of charge.

at the same time, for particular geometries and applications, V & I can often be related to these more conventional concepts

⑤Phase & Group Velocityconsider a narrow-band drive at z=0

V t f t e

Vdt

f t e

f td

V e

j t

j t

j t

( , ) ( )

˜( , ) ( )

( ) ˜( , )

0

02

20

0

0

0

=

= ∫

= ∫

−∞

∞ −( )

−∞

∞ −( )

ω

ω ω

ω ω

ωπ

ωπ

ω

compute the voltage down-range

V t zd

V e

dV e

ed

V e

j t jk z

j t jk z jdkd

z

j t jk z j tdk

z

zz

z

( , ) ˜( , )

˜( , )

˜( , )

= ∫

≈ ∫

= ∫

−∞

∞ −

−∞

∞ −( ) − ( ) − ( ) −( )

− ( )−∞

∞ −( ) −

ωπ

ω

ωπ

ω

ωπ

ω

ω

ω ω ωω

ω ω ω

ω ω ω ω

20

20

20

0 0 0 0

0 00

zz

z

dz

j t jk z ze f tdkd

z

ω

ω ω

ω

− ( )= −

0 0

can see that constant phase-frontstravel at

vk

phase velocityz

ϕω= = −

while the modulation f travels at

vddk

group velocitygz

= = −ω

For a mode in uniform guide,

vk k

c

k k

c

vc k

kc

k kc

c c

gz

c

ϕω

µε µε

µε µε µε

=−

=−

>

= = − <

02 2 2

02

0

202

1

1

1

/

/

Summary

Lorentz Force Law Maxwell’s Equations Skin Depth δ Modes in a Waveguide Phase Velocity vφ & Group

Velocity vg

Acknowledgements

Encyclopedia Brittanica http://www.eb.com

100 Years of Radiohttp://www.alpcom.it/hamradio/

Dept. Materials Science, MIThttp://tantalum.mit.edu/

Dept. of Physics and Astronomy, Michigan State UniversityUniversity of Guelph

Varian Associateshttp://www.varian.com/

Special Thanks to Prof. Shigenori Hiramatsu, KEK and Prof. Perry Wilson, SLAC for introducing me to this subject

Nikola Tesla’s Home Pagehttp://www.neuronet.pitt.edu/~bogdan/tesla/

Vocabulary

Electric Field E Magnetic Field H Energy U, Power P Frequency f, or Angular Frequency ω Conductivity, σ or Resistivity ρ Phase Velocity vφ, Group Velocity vg

Distributed vs Lumped Elements E & H Fields, Charged Particles Behavior of Fields in Media

For More Information...

W3 Virtual Library of Beam Physics

http://beam.slac.stanford.edu/

links to all accelerator labs on the planet...conferences...schools...news...jobs...

companies...vendors...databases...researchers...preprints...

Recommended Reading

RF Engineering for Accelerators, Turner, (CERN 92-03)

An introduction to RF as applied to accelerators.

Microwave Electronics, Slater The classic introduction to microwave electronics.

Classical Electrodynamics, Jackson A graduate level electrodynamics text.

Field Theory of Guided Waves, Collin A modern introduction to microwave electronics.

Foundations for Microwave Engineering, Collin An overview of the elements of microwave electronics.

Microwave Measurements, Ginzton An introduction to practical microwave work.

Principles of Microwave Circuits, Montgomery, Dicke, PurcellAn introduction to common network elements.

Waveguide Handbook, MarcuvitzAnalysis of the circuit parameters for network elements.

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10 1 2 3

10 1 3 5

20 0 99 0 1

1 0

10

10

10

log /

log /

log . .

( ) ≈ −( ) ≈ −( ) ≈ −

≡

dB

dB

dB

mW dBm

κ

α

ρ

≈ °≈ °

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Joint Accelerator School

Outline for Morning Lectures

1 Microwave Electronics 1 -Maxwell's Equations & Modes in a Guide 2 M.E. 2 - Equivalent Circuit Representation for Modes in a Guide3 M.E. 3 - Modes of a Cavity4 Cavity Design5 M.E. 4 - Cavity with a Port & External Q 6 M.E. 5 - Microwave Networks7 M.E. 6 - Slater's Perturbation Theorem8 Superconducting Cavities9 Beam-Cavity Interaction, Beam-Loading10 Klystron 1 - Space-Charge Limited Flow, Guns11 Structure 1-Standing-Wave12 SLED Pulse Compression13 Wakefields 1 - Fundamentals14 Klystron 2 - Bunching, Space-Charge15 Structure 2-Travelling Wave16 Ferrite Loaded Cavity 117 Wakefields 2 - in SW & TW Structures18 Klystron 3 - Simulation19 Structure 3-Fabrication and Conditioning20 Structure 4 -Surface fields, Breakdown, Multipactor, Dark Current20 Wakefields 3 - Other Sources of Impedance21 Other RF Sources22 High Gradients in Superconducting Cavities23 Modulators24 Windows & High-Power Transmission25 Ferrite Loaded Cavity 226 Design for System Stability - Heavy Beam Loading

Microwave ElectronicsMicrowave Electronics


An Introduction to the Equivalent Circuit


in six lectures



An Introduction to the Equivalent Circuit


in six lectures

Why are you here?

•Distributed vs. Lumped Elements•The Meaning of Current & Voltage•Transit Time & Retardation

What do you want?

Understanding of fundamentals...

Familiarity with the language...

•V, I, Z, δ, Rs, Qw, Qe, R/Q...• π mode, Travelling Wave,...•Tee, Load, Circulator, 3dB Coupler...

Ability to Solve Problems...

•How to design, build & tune my cavity?•What is the right power source to use?•My system isn’t working, what to do?

Microwave Electronics · Microwave Electronics ☞Maxwell's Equations & Modes in a Guide Equivalent...

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