662 magnetrons

ECE 662 – Microwave Electronics

Cross-Field Devices:

Magnetrons

April 7, 14, 2005

Magnetrons• Early microwave device

– Concept invented by Hull in 1913– Initial devices in 1920’s and 30’s

• Cavity Magnetron (UK) – 10 kW– Rapid engineering and Production– Radiation Lab (MIT) established

• Relativistic Cavity Magnetron (1975) –900MW

• Advanced Relativistic Magnetrons (1986) - 8 GW

• Commercial Magnetrons (2003) - 5 MW

Magnetrons• Inherently efficient• Delivers large powers (up to GW pulsed

power and MW cw)• Limited electronic tuning, i.e., BW limited• Low cost• Industrial uses

– microwave ovens– industrial heating– drying wood– processing and bonding materials

Magnetrons

• B no longer used to confine electron beam as in a Klystron - B is an integral part of rf interation.

• Multicavity block• Coaxial cathode• Coupling - I/O- loop or Waveguide

Planar MagnetronLet VA = potential differencebetween the anode andcathode, and E0=- VA /d. Anapplied magnetic field is in the x direction (into the paper). The force on the electrons becomes:

m

eB

dt

dy

dt

dyB

m

e

dt

zd

dt

dz

m

e

dt

dzB

m

e

dt

yd

vdt

dx

dt

xd

xByBvmF

cc

c

x

where;][

E]E[

)(initially zero be toassume0

ˆ ,ˆE ,]e[Edv/dt

02

2

0002

2

02

2

0000

x z

Planar Magnetron

z0z000y0

z000y0

y000z0y0z0

00

0022

2

2

0

v]cos1][v)/E[(sinv

sin]v)/E[(cosv

vN );/E(- vvq and vp 0, t

cossin

cossin

)/E(sincos

)/E(

E& &qLet

tBtdt

dz

tBtdt

dy

BMat

tNtMq

tNtMqdt

dp

BtNtMp

Bpdt

pd

pm

e

dt

dqq

dt

dp

dt

dzp

dt

dy

cc

cc

cc

ccccc

cc

cc

cc

Planar Magnetron

)t(B

Etsin)B/Ev()tcos1)(/v()t(z

)1t)(cosvB/E)(/1(tsin)/v()t(y

0

0

c

c000zcc0y

c0z00ccc0y

Planar Magnetron

2

0c

0

2

c

0

2

0c

0

2

0

0

00y0z

B

Ev

B

Ey)t(

B

Ez

as written becan result then this,vv and 0v If

This neglects space charge - tends tomake trajectory more “straight”.Result - frequency of cycloidal motionis c f B and (e/m)

KEY: average drift velocity of electrons in z direction is E0/B0 , independent of vz0 and vy0.

uox here is the v0z of our formulation

ref: Gerwartowski

Planar Magnetron

Electrons have dc motion equal to E0/B0, slow wave structure isassumed to be a propagating wave in the direction of theelectron flow with a phase velocity equal to E0/B0

Planar Magnetron (ref. Hemenway)

Circular Magnetron (conventional geometry)

Electrons tend to move parallel tothe cathode. Aftera few periods inthe cylindricalgeometry theelectron cloud soformed is knownas the Brillouincloud. A ringforms around thecathode.

Circular Magnetron Oscillator ref: Gewartowski

Brillouin CloudNext, compute the electron angular velocity d/dt for actual geometry. Note region I inside the Brillouincloud and region II outside.

constantr2dt

drr

2dt

d

dt

drr

dt

dr

dt

d

(2)eqn y b (2); ,dt

drB

m

e

dt

dr

dt

d

r

1

0dt

zd (1); ,

dt

drBE

m

e

dt

dr

dt

rd

motion of sEquation

2c22cc

2

z2

2

2

zr

2

2

2

Brillouin Cloud

2

2cc

2c

cc

r

r1

2dt

d

r2

-constant 0dt

d ,rr At

Note: electrons at the outermost radius of the cloud (r = r0)move faster than those for r < r0. The kinetic energy (of theelectrons) increase is due to drop in potential energy.

m

eB where,

r

rr

8m

eBr/r1

4r

2e

mV

dt

dr

dt

dr that Assume

dt

dr

dt

dr

2e

mv

2e

mVor eVmv

2

1

c

22c

220

222c

2c2

2222

Hull Cutoff Condition

For a given B0, the maximum potential difference VA that can beapplied between the anode and cathode, for which the Brillouincloud will fill the space to r = ra is

condition cutoff Hull the,rr

rV

e

m8B

:gap cathode-anode thefilling avoid torequired B

minimum a Vgiven afor or r

rr

8m

eBV

2

2c

2a

aA

20

A

2

a

2c

2a

20

A

min

max

Hull Cutoff ConditionB0 < B0min direct current flows to anodeand no chance for interaction with rf.B0 > B0min Brillouin cloud has an outer radius r0 < ra and no direct currentflows to the anode. For a typicalmagnetron, B0 > B0min therefore r0 < ra

A

2

0

220

20

0

220

V 0.2) to(0.1 8m

eB)rV(r generally,

magnetron, a designing Gauss.in B and cmin r in volts, is

wherecondition cutoff Hull the, 5.45Bmin

r

rr

InV

rr

rV

c

A

ca

aA

Magnetron FieldsFrom radial force equation (1), consider electrons following circular trajectory in Brillouin cloud. Assume that

4

c2

0Ir

2

c

2

c2

20

Ir

2

2c

2

2c

22c

Ir

c0

2

Ir

0r2

2

r

r1r

4

)B(

m

eE

r

r1

2

1

r

r1

m

)eB(

e

m

2

rE

1r

r1

2

1

r

rr

2m/e

rE

dt

dfor result insert the ),(

m/e

rBr

m/e

rE

:)r(r I,region inEfor solve and small, sidt

rd

Magnetron FieldsFrom Poisson’s equation the charge density:

4

0

20

4

00

44

5

42

4

20

42

r

r0

00

12

)(12

2124)1(2

4

)(

m where,)1()E(

)E(

r

rB

m

e

r

r

r

r

r

rr

r

rr

r

rr

Be

r

rr

rr

r

rrr

E

cc

cccc

cI

I

0 falls slightly as r increases from rc (can increase 0 by increasingB0 which follows as electrons spiral in smaller cycloidal orbitsabout the cathode.

0

Magnetron Fields

]r

rr[

r

1

4m

)B(eEor ]

r2

rr[rdz2E

dz2m

)B(e2 where],

r2

r

2

r[

]r

r

2

1

r

r

2

1

2

r

2

r[dr)

r

rr(

dz dr r2)r(Qrdz2E dsD

20

4c

40

20

IIr20

4c2

0IIr0

02

012

0

4c

20

1

2c

4c

20

4c

2c

20

13

4c

r

r

1

r

r

0

surface

enclIIr0

0

c

0

c

Outside the Brillouin cloud, r0 < r < ra, in region II, use Gauss’sTheorem:

Hartree Relationship

ipRelationsh Hartree ,)

r

rln()

r

rr(2

r

rr

8m

)B(e

rln)r

rr(

r

r

2

1

2

r

4m

)B(e

dr]r

rr[

r

1

4m

)B(edr)

r

r1(r

4m

)B(e

drEdrEdrEV

0

a2

0

4c

40

20

22c

20

20

r

r

20

4c

40

r

r

2

4c

220

r

r2

0

4c

40

20

r

r4

4c

20

r

r

rII

r

r

rI

r

r

rA

a

0

0

c

a

0

0

c

a

0

0

c

a

c

The potential difference VA between the cathode and anode tomaintain the Brillouin cloud of outer radius r0 is given by:

Hartree Relationship

00

2c

20c

0B

B

20

4c

400c

BA

0

a

0

a2

0

4c

40

20

22c

20

20

A

rrat velocity circular)r

rr(

2rv

rrat voltageis V is where,

)r

rr(

4

B

VV)

r

rln(

by expressed maybe ipRelationsh Hartree theor

)r

rln()

r

rr(2

r

rr

8m

)B(eV

0

This vB is important since it gives the velocity of the electrons at theouter radius of the Brillouin cloud. It is this velocity vB that is to match the velocity of the traveling waves on the multicavity structure.

Anode - Cathode Spacing

Desire microwave field repetition with spatial periodicity of the structure. This field will have traveling wave components the most important of which is a component traveling in the same direction withabout the same velocity, vB , as the outer ones in the Brillouin layer.

A0B

cac0

V 0.2) to1.0()rr(V

such that rr offraction small is rr

magnetron; theofsion planar ver heconsider t Again,

Anode - Cathode Spacing

These traveling waves are slow waves with the desired phase velocity, vp ~ vB. Consider the wave equation as follows:

celocityelectron v ,v~ vsince

vc

k

since ,0Ey

E E)k(

y

E

0x/ ,/v ,edirection zin travelingFields

0Ekz

E

y

E

x

EEkE

Bp

2p

22

2

22

22

222

2

2

pz)-tj(

22

2

2

2

2

222

Anode - Cathode SpacingThe solution of this equation results in hyperbolic trig functions:

caz)-tj(

p

pprrd ,e

dv

sinh

yyv

cosh Bzyv

sinh A

E

d/vp not too large, such that the E at Brillouin layer is insufficient for interactiond/vp not too small such that the E is so large that fields exert large force on electrons and cause rapid loss to the anode thereby reducing efficiency. Typically,

8 to4)rr)(v/(v/)d( capp

Multicavity Circuit - Slow Wave Structure

Equivalent circuit of multicavity structure - here each cavity has been replaced by its LC equivalent. Thiscircuit is like a transmission linefilter “T” equivalent.

line coaxial from)r/rln(

2cathode and ecavity van

between ecapacitanc couplingC Cj

1Z

cavity uncoupled thengrepresentinetwork LC parallel

of impedance ZLC

1 ,

)/(1

LjZ

ca

0

cc

2

1020

1


The circuit acts like low-loss filter interactive impedance = input impedance of an infinite series ofidentical networks.

]})/(1[4/{LCsin 2)Z4/(Zsin 2 p

isfilter of psection per shift Phase band. pass theis 0)Z4/(Z1-

,)Z4/(Z1ZZZ ,4/ZZor 4/XXor real be to

4/XXX]XX)4/X[( ZthenjXZ ;jXZ If

delivered. ispower no Otherwise power. delivers and load resistance

sees""generator theresistive, pure isZZ If ]ZZ)4/Z[(Z

find tofor Z solve ,ZZ)2/Z(Z

Z]Z)2/Z[(

2

ZZ

20c

21-21

1-

21

2121k12ab

2aba

1/2ba

2akb2a1

kin1/2

212

1k

kkk12

2k11in

Multicavity Circuit - Slow Wave StructureRf field repeats with periodicity p (spacing of adjacent cavities). Field at distance z+np is same as z. = phase shift per unit length of phase constant of wave propagating down the structure.

mode. for thefrequency operating

theis )4/(1

.for solve ])/(1[4

12

sin

]})/(1[4/{sin 2p

cavity)(per mode or p N/2,mfor

0,1,2,3m ,36

m2p 6,Nfor N/2 ... 2, 1, 0,m 2

02

0

2

20

21-

CC

LC

orLC

mmNp

c

c

c

For a circular reentrant structure anode with N cavities, fields areindistinguishable for Z as for Z + np.

Fields and Charge Distributions for two Principal Modes of an Eight-Oscillator

Magnetron


NrrNCCCC

C

C

N

cs

c

00p0

21-(N/2)

2 ,2/

v)4/(/1

:mode theoffrequency thelowers ecapacitanc

adds Strapping sun. rising and strapping- methods 2

separation thisincrease todesire - modes Competing

0.99 to97.01

41

mode 1-N/2m For the

Strapped Cavities

Typical Magnetron Cross-Sections (after Collins)

(a) Hole and slot resonators(b) Rectangular resonators(c) Sectoral resonators

Typical Magnetron Cross-Sections (after Collins)

(d) Single ring strap connecting alternate vanes(e) Rising sun anode with alternate resonators

of different shapes(f) Inverted magnetron with the cathode exterior

to the anode

The unfavorable electrons hit the cathode and give up as heat excess energy picked up from the field. As a result, the cathode heater can be lowered or even turned off as appropriate.

twotwo

Rotating wheel formed by the favorable electrons in a magnetron oscillating in the mode ref: Ghandi

General Design Procedures for Multicavity Magnetrons

V,I requirements: From Power required may select VA. High VA keeps current down and strain on cathode, but pulsed high voltage supplies are needed.Note Pin = P0 / efficiency and I A = Pin / VA= P0 / VA.

Cathode radius from available current densities for type of cathodestypically used in magnetrons.

Typically J0 (A/cm2) 0.1 to 1.0 for continuous, 1 to 10 for pulsed

Smaller J0 lower cathode temperature so longer life of tube

Too low J0 requires a larger rc


Emitting length of cathode (lc) < anode length, la ; Typically, lc ~ 0.7 to 0.9 la , and la < /2 (prevents higher order modes) Smaller la is consistent with power needs less B0 needed (less weight)

Radius r0 (top of Brillouin cloud) from velocity synchronism condition:

vp (r = r0) = r0 / (N/2) = [c r0 /2] [1- (rc2 / r0

2)]; thereforer0 = rc / [1-( /c)(4/N)]1/2

For an assumed B0, r0 can be calculated for a number of values of N (typically 6 to 16) or 20 to 30 for a small magnetron.


Voltage eVB (r = r0) = (1/2) mvB2 where vB = vp (r = r0) or

VB = (vB /5.93x107) 2 ; vB in cm/sec ; Hence VB ~ 0.1 to 0.2 VA

Note efficiency, < (1 - VB / VA )*100; henceSmaller VB / VA contributes to improved efficiency

Anode radius: ln (ra/ r0) = [VA - VB ] / {[c 2 / 4(e/m)][(r0

4 -rc 4 ) / r0

2 ]}

Also Bmin = (45.5 VA) 1/2 [ra /(ra 2 -rc

2 )] << B0

( /vp)( ra - rc) ~ 4 to 8 N must be even such that N phase shift around the circumference is a whole 2 .

Cutaway view of aCoaxial Magnetron

NRL Hybrid Inverted Coaxial

Magnetron

NRL Hybrid Inverted Coaxial Magnetron

662 magnetrons

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