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Microwave Electronics

An Introduction to the Notion of Equivalent Circuit Starting from Maxwell's Equations

in six lectures

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

Microwave Electronics I

Maxwell's Equations

and

Waveguide Modes

x Lorentz Force Law y Maxwell's Equations z Skin Depth { Orthogonal Modes | Phase & Group Velocity ¶ Quiz

xLorentz Force Law

r r r r F = q (E + v × B )

r E

r B

Newtons per Coulomb=V/m "electric field strength"

Newtons per Ampere-meter=T=Wb/m2 "magnetic flux density"

or "magnetic induction"

defines the fields & abstracts them from the sources qdescribes "test particle" motion qdescribes response of media

q

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

Example: Conductivity

r r r dv v m = qE - m dt

r q r v = E m

r J = nqv

+ + + +

+ + + +

+ + + +

+ + + +

nq r = E m r = E

2

=" DC " 5.8 × 107 mho / m for Cu

Georg Simon Ohm (b. March 16, 1789, Erlangen, Bavaria --d. July 6, 1854, Munich) P. Drude, 1900

NB This is a simplified picture of a normal conductor...occasionally this picture breaks down...

and of course this model cannot be applied to all materials...

yMaxwell's Equations

Electricity & Magnetism before Maxwell...

q q q

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

After Maxwell...

q

q q

q q

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 the transit 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

Microwave Electronics

·circuit size appreciable compared to a wavelength ·voltage between two points depends on path ·elements are "distributed", spatial phase-shifts occur between them ·if the word "impedance" is used, you may always ask 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

History

Henry 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)

(

r r ^ D2 - D1 · nA = A

V

Gauss's Law r r D · dS = dV

^ n

Medium 2

)

V

+ +++++++++++++++ Medium 1 pillbox area A

D C / m2

electric displacement

or electric flux density

surface charge density

Ampere's Law before Maxwell

r r r r H · dl = J · dS

S

l^

^ n

Medium 2

Medium 1 contour side L r surface current densityK

S

(

r r r ^ H 2 - H1 · l^ = K × n · l^

)

r r r ^ n × H 2 - H1 = K

(

)

H K A /m

magnetic field strength

or magnetic flux

In Vacuum... r r r r H = B / µ0 D = 0 E

0 8.85 × 10 -12 farad per meter

µ0 = 4 × 10 -7 henry per meter

or

1 = c = 2.9979 × 108 m / s 0 µ0 µ0 = 377 0

In Media...

r r r r D = 0E + P = E r r electric dipole moment density P = e 0E r r r r H = B / µ0 - M = B / µ r r magnetic dipole moment density M = H m

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

Faraday's Law

r r r B r E · dl = - · dS S S t

Maxwell, Ampere's Law was inconsistent with conservation of charge qAfter Maxwell, the fields didn't need charge to support them, they could propagate on their own qOf course no one believed Maxwell, but the fields didn't mind

qBefore

Charge Conservation

r + ·J = 0 t

r r dV = - J · dS t V V

Ampere's Law (before Maxwell's addition of Displacement Current) implied

or

r r = - · J = - · × H = 0 t

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

Maxwell's Equations r r Gauss's Law ·D = r r r r D Ampere's Law ×H = J + with t

Maxwell's Displacement Current

r r r B Faraday's Law ×E = - t r No Magnetic Charge · B = 0

James Clerk Maxwell

b. June 13 or Nov. 13, 1831, Edinburgh d. Nov. 5, 1879, Glenlair

Boundary Conditions

apply Maxwell's equations in integral form...

( (

r r ^ D2 - D1 · n = r r ^ B2 - B1 · n = 0

) )

^ n

Medium 2

+ +++++++++++++++ Medium 1 pillbox area A

surface charge density

r r r ^ n × H 2 - H1 = K r r ^ n × E 2 - E1 = 0

( (

) )

l^

^ n

Medium 2

Medium 1 contour side L r surface current densityK

Example to Illustrate : Unmagnetized Plasma r r r dv v m = qE - m dt

r ~ jt = 1 Ee jt + E *e - jt ~ ~ E = Ee 2

(

) (

)

q ~ 1 ~ = nqv = DC E ~ v= E J ~ ~ (1+ j ) m (1+ j )

Apply charge conservation & Gauss's Law...

r ~ + · J = 0 j + · J = 0 ~ t ~ = = - 1 · J = - 1 · DC E ~ ~ · 0E ~ j j (1+ j ) 1 DC ~ 1 DC · 0 + E = 0 = 0 + j (1+ j ) j (1 + j )

N .B .

= 1- 2 lim 0

2 p

nq 2 2 with p = m 0

Evidently...

·electric displacement depends on what you consider to be the "external" circuit, electric field does not

View #1

~ ~ · 0E = plasma + other ~ ~ ~ D = 0E

View #2

~ ~ · E = other ~ ~ D = E

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

1 DC = 0 + = ( ) j (1 + j )

Polarization in the time domain...

+ d r ~ P (t ) = e jt P ( ) - 2

d jt ~ e 0 e ( )E ( ) = - 2

+ dt r d jt - jt e 0 e ( ) e E (t ) = - 2 - 2 + r = 0 dt G (t - t )E (t ) + -

+

where the Green's function is

1 + jt G (t ) = d e e ( ) 2 -

e ( ) = dt e - jt G (t )

-

+

Example...unmagnetized plasma...

1 + G (t ) = d e jt e ( ) 2 -

2 p 1 + = d e jt j (1+ j ) 2 - 2 = p 1 - e -t / H (t )

(

)

for t>0, contour may be closed at Im+ with contributions from two poles

-plane

pole at =j/ pole at =0

contour for t<0, contour may be closed at Im- so that G=0

Susceptibility in the High Frequency Limit

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

e ( ) = dt e - jt G (t )

-

+

= dt e

-

+

- jt

G (n ) (0) n H (t ) t n! n =0

G (n ) (0) n + - jt = j dt e n! 0 n =0 G (n ) (0) n 1 = j j n! n =0

G (0) G (1) (0) G (2) (0) - +j K = -j 2 3

Example: unmagnetized plasma

e ( ) = -

2 p

2

-j

2 p

3

+K

Kramers-Kronig Relations Since G is causal e must be analytic in the Im<0 half-plane, so that

1 e ( ) e ( ) = 2j -

for points ,' and contour in the lower half-plane. Let the contour lie just below the real axis and use

1 1 + j ( - ) P = - 0 - - j lim

then

1 + e ( ) e ( ) = P j - -

1 + e ( ) e ( ) = P - - 1 + e ( ) e ( ) = - P - -

relates dispersion & absorption

Scalar & Vector Potentials r r r r v A r B = ×A E =- - t

Maxwell's Equations...

r r r r r ·B = · × A = 0 r r r v r A r r r B × E = × - - = - × A = - t t t

r r r r r r r 2 × µH = × × A = · A - A r r r r E = µ × H = µ + µJ t r r A r = µ - - + µJ t t

(

)

(

)

r r r r r r A r · D = · E = · - - = t

Gauge Invariance

r r r A A -

+ t

leaves E,B unchanged Lorentz Gauge

2

r r r A 2 2 2 A - µ 2 = - µJ - µ 2 = - / t t

evidently the characteristic speed of propagation in the medium is Coulomb Gauge

2

r · A + µ =0 t

v = ( µ )-1/ 2

r r ·A = 0

r r r r A 2 A - µ 2 = - µJ + µ t t

2 = - /

related to the Lorentz Gauge potentials via

= - µ t Lorentz Gauge

2

Hertzian Potentials

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

r r ~ ~ ·E = ·H = 0

Magnetic Hertzian potential

r ~ E = - jµ × m

r 2 2 ~ ~ × H = jE = k 0 × m with k 0 = µ 2 r 2 ~ H = k 0 m + rr ~ × E = - jµ · m - 2 m r 2 ~ = - jµH = - jµ k 0 m +

(

(

)

)

choice of gauge

r 2 = · m 2 + k 0 = 0

2

2 m + k 0 m

=0

rr 2 ~ H = · m + k 0 m

Electric Hertzian potential

r ~ H = j × e

rr 2 ~ E = · e + k 0 e

2

2 e + k 0 e

=0

Energy Conservation

r r -J · E = rate of work done on fields r D r r r = - × H ·E t r D r r r r r r r r r r = · E - × H ·4 +4444 - × E · H E H 1444 2 × E ·3 r r r t r r D r r r r B r ·H = ·E + · E × H + t t r r r S = E × H = Poynting Flux

· E ×H

(

)

(

)

In a linear medium, with and independent of frequency:

µ

r r r r u -J · E = · S + t

where

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

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

r ~ jt = 1 Ee jt + E *e - jt ~ ~ E = Ee 2

(

) ( )

)

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

1 ~ ~ S = E × H 2

(

r r D r B r r= ·E + ·H t t

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

Questions arise...is this integrable?

r=

u =? t

and if so, what is the average stored energy density

u

?

To address this problem we first compute take

r ~ ~ ~ E (t ) = E ( ,t )e jt = E 0 ( ) + E 1( )t e jt r r ~ D (t ) = 0E (t ) + P ( ,t )e jt

{

{

}

{(

r D r ·E t

}

) }

and compute P...

~ P ( ,t )e

jt

~ ~ = 0 dt G (t - t ) E 0 ( ) + E 1( )t e jt

- +

+

(

)

~ ~ = 0E 0 ( ) dt G (t - t )e jt + 0E 1( ) j

-

+

-

dt G(t - t )e

jt

~ ~ = 0E 0 ( ) e ( )e jt + 0E 1( )

e ( )e jt j

or

~ ~ ~ P ( ,t ) = 0E 0 ( ) e ( ) + 0E 1( ) et - j e ~ ~ = 0 e ( )E ( ,t ) - 0 j e E 1( )

r r ~ D E P ~ = 0 + 0 jP ( ,t )e jt + e jt t t t r ~ ~ E E ( ,t ) ~ = 0 + 0 j e ( )E ( ,t ) - j e E 1( ) e jt + e jt e ( ) t t ~ E ( ,t ) ~ ~ = e jt + jE ( ,t ) + 0E 1( ) e t ~ E ( ,t ) ( ) ~ = e jt + jE ( ,t ) t

then

finally

r ~ 2 D r 1 ( ) E ( ,t ) ~ ~ · E = · E ( ,t ) + j E ( ,t ) t 2 t

Finally

r ~ 2 2 ( i ) E ( ,t ) ~ D r 1 ( r ) ~ ~ ·E = · E ( ,t ) E ( ,t ) - i E ( ,t ) - 2 t 4 t t

so that, in the absence of losses,

r r D r B r u r= ·E + ·H = t t t

where the field energy density is

2 (µ ) 1 ( ) ~ ~ ( ,t ) 2 u= E ( ,t ) + H 4

Energy conservation takes the form

r r u -J · E = · S + t

with time-averaged Poynting flux

1 ~ ~ S = E × H 2

(

)

zSkin Depth

Start from Maxwells Equations

r ~ ~ = D + J J = E ~ ~ ~ ×H t r ~ B ~ ~ ×E = - = - jB t r r ~ ~ ·E = ·B = 0

Fields exp(jt)

Reduce to an equation for H

r r r r ~ ~ ~ ~ × × H = · H - 2H = -2H r ~ ~ ~ = × E = - jB = - jµH

(

)

(

) ( )

~ - j sgn 2 H = 0 ~ H 2

2

=

2 =" skin - depth " µ

2 µm at 1GHz in Cu

Let's solve for fields in conductor...

VACUUM

t angent ial H normal H

CONDUCTOR

select coordinate along outward normal

d2 ~ 2 ~ H - j sgn 2 Ht = 0 2 t d

~ ~ Ht ( ) = Ht (0) exp - (1 + j sgn ) ~ H 1r ~ 1 ~ ^ ^ ~ E = ×H n × = - Z s n × Ht

"impedance boundary condition"

1 + j sgn Zs = = Rs (1 + j sgn ) 1 Rs = = surface resistance 8.3m at 1GHz in Cu

Power per m2 into the conductor...

^ S ·n = = = = 1 ~ ~ ^ E × H ·n 2 1 ~ ^ ~ ^ - Z s n × Ht × H · n 2 1 ~t 2 ^ Z s n × H 2 1 ~ 2 = 1 Rs K 2 ~ Rs n × H 2 2

(

)

( (

)

)

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

~ ~ ^ ~ K = Jd = -n × H (0)

0

{Orthogonal Modes

(Uniform Waveguide)

From Maxwell's Equations, with fields

e jt - jk z z

r r ~ ~ ·E = ·B = 0 r ~ ~ × E = - jB r ~ ~ × H = jD

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

r r ~ ~ × × H = -2H r ~ ~ = j × D = j - jB

(

)

(

)

~ = 2µH

and similarly for E, so that

( ) 2 ~ 2 + k c )E = 0 (

where

2 k 0 = 2µ

2 ~ 2 + k c H = 0

2 2 2 kc = k 0 - k z

The divergence conditions take the form

1 ~ ~ Ez = · E jk z

1 ~ ~ Hz = · H jk z

1 r ~ ~ E= × Z 0H jk 0

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

1 r ~ ~ Z 0H = - ×E jk 0

Z0 =

µ

r ~ ~ - jk 0Z 0H = - z × E z - jk z z × E ^ ^ ~ r ~ ~ ~ jk 0E = - z × Z 0H z - jk z z × Z 0H ^ ^

or

to express the transverse components in terms of the longitudinal

r k0 kz r ~ ~ ~ E = - 2 z × Z 0H z - E z ^ k0 jk c r k0 kz r ~ = - ~ z - Z 0H z ~ Z 0H ^ - z × E 2 k0 jk c

TE Modes Ez=0

kz r ~ ~ Z 0H = 2 Z 0H z jk c r k0 ~ = - ~ z = - k 0 z × Z 0H ~ E z × Z 0H ^ ^ 2 kz jk c

longitudinal field satisfies

transverse fields may be determined from Hz

(

2 ~ 2 + k c H z = 0

)

with boundary condition

r kz ~ ~ ^ ^ 0 = n · Z 0H = 2 n · Z 0H z jk c

~ H z =0 i .e. n

TM Modes Hz=0

transverse fields may be determined from Ez

kz r ~ ~ E = 2 E z jk c

r k0 k ~ ~ Z 0H = 2 z × E z = 0 z × E ^ ^ ~ kz jk c

2 ~ 2 + k c E z = 0

longitudinal field satisfies

(

)

with boundary condition

~ Ez = 0

Cut-Off & Characteristic Impedance Zc

Boundary conditions restrict the permissible values of cut-off wavenumber kc to a discrete set. Each mode has a corresponding minimum wavelength c beyond which it is "cut-off" in the waveguide.

c = 2 / k c

The guide wavelength is

g = 2 / k z = 0 / 1 - 2 / 2 0 c

0 = 2 / k 0

In general for a given mode we have

~ Z cH = z × E ^ ~

where

g k0 Z 0 k = Z 0 0 z Zc = Z 0 k z = Z 0 0 g k0

TE mod e TM mod e

Modal Decomposition

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

r ~t = E a (r ) a (z , ) E V

a a

r ~ t = H a (r )I a (z , )Z ca ( ) H

where

Z caH a = z × E a ^

and we adopt the normalization

r r d rE a (r ) · E a (r ) = 1

2

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

Va (z , ) = Va+e - jk za z + Va-e jk za z Z caI a (z , ) = Va+e - jk za z - Va-e jk za z

Relation between Power, V & I

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

r r d rE a (r ) · E b (r ) = ab r r 2 Z ca Z cb d rH a (r ) · H b (r ) = ab

2 2 -1 ^ d r z · E a × H b = ab Z ca

(

)

This requires Green's Theorem,

r r 2 2 dl 1 2 + 1 · 2 d r = 1 n

(

2

)

and the eigenvalue equations for Hz & Ez As a result, one may express the power flow in the waveguide, in terms of V & I according to

1 ~ ~ P = d 2r Et × Ht 2 1 = VaI a a 2

(

)

( )

Meaning of V,I

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

r r ~t (r , z ) · E a (r ) Va (z , ) = d rE r r 2 ~ t (r , z ) · H a (r ) I a (z , ) = Z ca d rH

2

and this is enough to determine the solution everywhere in the uniform guide, since this fixes the right & left-going amplitudes.

Given the uniqueness of V,I, their relation to power, and the units (volts, amperes) it is natural to refer to 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

|Phase & Group Velocity

consider a narrow-band drive at z=0

V (t , 0) = f (t )e j 0t dt j - t ~ f (t )e ( 0 ) V ( , 0) = - 2 d ~ j - 0 )t f (t ) = V ( , 0)e ( - 2

compute the voltage down-range

V (t , z ) =

-

d ~ V ( , 0)e jt - jk z z 2

j ( - 0 )t - jk z ( 0 )z - j dk z ( )( - 0 )z d 0

d ~ V ( , 0)e - 2

=e =e

j 0t - jk z ( 0 )z j 0t - jk z ( 0 )z

d ~ V ( , 0)e - 2

t - dk z z f d

dk j ( - 0 ) t - z z d

can see that constant phase-fronts travel at

v =

= phase - velocity kz

while the modulation f travels at

vg =

d = group - velocity dk z

For a mode in uniform guide,

v =

2 2 k 0 - kc

=

1 c c > 2 2 µ 1- k c / k 0 µ

c kz c c 2 2 1- kc / k 0 < vg = = µ k 0 µ µ

Summary

Lorentz Force Law Maxwell's Equations Skin Depth Modes in a Waveguide Phase Velocity v & Group Velocity vg

Acknowledgements

Encyclopedia Brittanica http://www.eb.com

100 Years of Radio http://www.alpcom.it/hamradio/

Dept. Materials Science, MIT http://tantalum.mit.edu/

Varian Associates http://www.varian.com/

Nikola Tesla's Home Page http://www.neuronet.pitt.edu/~bogdan/tesla/

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

qDept.

of Physics and Astronomy, Michigan State University qUniversity of Guelph

Vocabulary

q q q q q q q q q

Electric Field E Magnetic Field H Energy U, Power P Frequency f, or Angular Frequency Conductivity, or Resistivity Phase Velocity v, Group Velocity vg Distributed vs Lumped Elements E & H Fields, Charged Particles Behavior of Fields in Media

For More Information...

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

W3 Virtual Library of Beam Physics

links to all accelerator labs on the planet ...conferences...schools...news...jobs... companies...vendors...databases... researchers...preprints...

Recommended Reading

q

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.

q

Related Texts

qClassical

Electrodynamics, Jackson A graduate level electrodynamics text.

qField

Theory of Guided Waves, Collin A modern introduction to microwave electronics. for Microwave Engineering, Collin An overview of the elements of microwave electronics.

qFoundations

Measurements, Ginzton An introduction to practical microwave work. qPrinciples of Microwave Circuits, Montgomery, Dicke, Purcell An introduction to common network elements.

qMicrowave qWaveguide

Handbook, Marcuvitz Analysis of the circuit parameters for network elements.

SI Units

Handy Numbers

10 log10 (1 / 2) -3dB 10 log10 (1 / 3) -5dB 20 log10 (0.99) -0.1dB 1mW 0 dBm

401 W / ° K m C 385 J / ° K kG

1.7 × 10 -5 / ° K 1.56 × 10 -8 - m

Copper

Joint Accelerator School

RF Engineering for Particle Accelerators

To

¶Understand ¶Invent ¶Design ¶Build ¶Operate

RF Systems

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 Guide 3 M.E. 3 - Modes of a Cavity 4 Cavity Design 5 M.E. 4 - Cavity with a Port & External Q 6 M.E. 5 - Microwave Networks 7 M.E. 6 - Slater's Perturbation Theorem 8 Superconducting Cavities 9 Beam-Cavity Interaction, Beam-Loading 10 Klystron 1 - Space-Charge Limited Flow, Guns 11 Structure 1-Standing-Wave 12 SLED Pulse Compression 13 Wakefields 1 - Fundamentals 14 Klystron 2 - Bunching, Space-Charge 15 Structure 2-Travelling Wave 16 Ferrite Loaded Cavity 1 17 Wakefields 2 - in SW & TW Structures 18 Klystron 3 - Simulation 19 Structure 3-Fabrication and Conditioning 20 Structure 4 -Surface fields, Breakdown, Multipactor, Dark Current 20 Wakefields 3 - Other Sources of Impedance 21 Other RF Sources 22 High Gradients in Superconducting Cavities 23 Modulators 24 Windows & High-Power Transmission 25 Ferrite Loaded Cavity 2 26 Design for System Stability - Heavy Beam Loading

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 Guide 3 M.E. 3 - Modes of a Cavity 4 Cavity Design 5 M.E. 4 - Cavity with a Port & External Q 6 M.E. 5 - Microwave Networks 7 M.E. 6 - Slater's Perturbation Theorem 8 Superconducting Cavities 9 Beam-Cavity Interaction, Beam-Loading 10 Klystron 1 - Space-Charge Limited Flow, Guns 11 Structure 1-Standing-Wave 12 SLED Pulse Compression 13 Wakefields 1 - Fundamentals 14 Klystron 2 - Bunching, Space-Charge 15 Structure 2-Travelling Wave 16 Ferrite Loaded Cavity 1 17 Wakefields 2 - in SW & TW Structures 18 Klystron 3 - Simulation 19 Structure 3-Fabrication and Conditioning 20 Structure 4 -Surface fields, Breakdown, Multipactor, Dark Current 20 Wakefields 3 - Other Sources of Impedance 21 Other RF Sources 22 High Gradients in Superconducting Cavities 23 Modulators 24 Windows & High-Power Transmission 25 Ferrite Loaded Cavity 2 26 Design for System Stability - Heavy Beam Loading

Microwave Electronics

An Introduction to the Equivalent Circuit Starting from Maxwell's Equations

in six lectures

Maxwell's Equations & Modes in a Guide t Equivalent Circuit for Waveguide Modes Ë Modes of a Cavity Ë Cavity with a Port & External Q Ë Microwave Networks Ë Slater's Perturbation Theorem

Microwave Electronics

An Introduction to the Equivalent Circuit Starting from Maxwell's Equations

in six lectures

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

Microwave Electronics

An Introduction to the Equivalent Circuit Starting from Maxwell's Equations

in six lectures

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

Why are you here?

What do you want?

Understanding of fundamentals... ·Distributed vs. Lumped Elements ·The Meaning of Current & Voltage ·Transit Time & Retardation Familiarity with the language... ·V, I, Z, , Rs, Qw, Qe, R/Q... · 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?

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