1. ECE 662 – Microwave Electronics
Cross-Field Devices:
Magnetrons
April 7, 14, 2005

2. 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

3. 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

4. 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

6. Z

7. Planar Magnetron
Let VA = potential difference
between the anode and
cathode, and E0=- VA /d. An
x z applied magnetic field is in
the x direction (into the
paper). The force on the
electrons becomes:
F = m dv/dt = e[E 0 + v × B0 ], E 0 → y, B0 → x
ˆ ˆ
d 2x dx
2
=0⇒ = v x 0 = assume to be zero (initially)
dt dt
d2y e dz e dz
2
= [E 0 − B0 ] = E 0 − ωc
dt m dt m dt
d 2z e dy dy eB
2
= [ B0 ] = ωc ; where ωc ≡
dt m dt dt m

8. Planar Magnetron
dy dz dp dq e
Let q ≡ &p≡ ∴ = ωc q & = E 0 − ωc p
dt dt dt dt m
d2p
+ ωc p = ωc (E 0 / B0 )
2 2
dt 2
∴ p = M cos ωc t + N sin ωc t + (E 0 / B0 )
dp
= ωc q = −ωc M sin ωc t + ωc N cos ωc t
dt
q = − M sin ωc t + N cos ωc t
at t = 0, p = v z0 and q = v y0 ∴ M = v z0 - (E 0 / B0 ); N = v y0
dy
∴ = v y0 cos ωc t + [(E 0 / B0 ) − v z0 ] sin ωc t
dt
dz
= v y0 sin ωc t + [(E 0 / B0 ) − v z0 ][1 − cos ωc t ] + v z0
dt

9. Planar Magnetron
∴ y( t ) = ( v y 0 / ωc ) sin ωc t + (1 / ωc )(−E 0 / B0 + v z 0 )(cos ωc t − 1)
sin ωc t E 0
z( t ) = ( v y 0 / ωc )(1 − cos ωc t ) + ( v z 0 − E 0 / B0 ) + (t)
ωc B0

10. Planar Magnetron
If v z 0 = 0 and v y 0 = v 0 , then this result can be written as
2 2 2 2
 E0   E0   v0   E0 
z −
 (t)  +  y −
   =   +
 ω  ω B 
 B0   ωc B 0   c   c 0 

This neglects space charge - tends to
make trajectory more “straight”.
Result - frequency of cycloidal motion
is ωc ∴ f ∝ B and (e/m)
KEY: average drift velocity of electrons in z direction is E0/B0 ,
independent of vz0 and vy0.

11. uox here is the
v0z of our
formulation
ref: Gerwartowski

12. Planar Magnetron
Electrons have dc motion equal to E0/B0, slow wave structure is
assumed to be a propagating wave in the direction of the
electron flow with a phase velocity equal to E0/B0

13. Planar Magnetron (ref. Hemenway)

14. Planar Magnetron (ref. Hemenway)

15. Circular Magnetron
(conventional geometry)
Electrons tend to
move parallel to
the cathode. After
a few periods in
the cylindrical
geometry the
electron cloud so
formed is known
as the Brillouin
cloud. A ring
forms around the
cathode.

16. Circular Magnetron Oscillator ref: Gewartowski

17. Brillouin Cloud
Next, compute the electron
angular velocity dθ/dt for
actual geometry. Note
region I inside the Brillouin
cloud and region II outside.
Equations of motion
2
d 2 r  dθ  e dθ  d 2z
2
− r  = −  E r + Bz r , (1); 2 = 0
dt  dt  m dt  dt
1 d  2 dθ  e  dr 
r  =  Bz , (2); by eqn (2)
r dt  dt  m  dt 
d  2 dθ  dr d  ωc 2  2 dθ ωc 2
r  = ωc r =  r  ∴ r = r + constant
dt  dt  dt dt  2  dt 2

18. Brillouin Cloud
dθ ω 2
At r = rc ,
= 0 ∴ constant = - c rc
dt 2
dθ ωc  rc 
2
∴ = 1 − 2 
dt 2  r 
 
Note: electrons at the outermost radius of the cloud (r = r0)
move faster than those for r < r0. The kinetic energy (of the
electrons) increase is due to drop in potential energy.
m 2 m  dr   dθ  
2 2
1 dθ dr
mv = eV or V = v =   +  r   Assume that r
2
>>
2 2e 2e  dt   dt  
  dt dt
2
( )  r − rc 
2 2 2
m ω 2 eB 2
eB
2
∴ V = r 2 c 1 − rc / r 2 = 0   , where ωc ≡
2e 4 8m  r  m
 

19. Hull Cutoff Condition
For a given B0, the maximum potential difference VA that can be
applied between the anode and cathode, for which the Brillouin
cloud will fill the space to r = ra is
2
 ra − rc 
eB0
2 2 2
VA max =  or for a given VA ∃ a minimum
 r
8m 
 a 
B required to avoid filling the anode - cathode gap :
2
8m  ra 
VA  2  , the Hull cutoff condition
2
B0min = r −r 2 
e  a c 

20. Hull Cutoff Condition
B0 < B0min direct current flows to anode
and no chance for interaction with rf.
B0 > B0min Brillouin cloud has an outer
radius r0 < ra and no direct current
flows to the anode. For a typical
magnetron, B0 > B0min therefore r0 < ra
ra
B0min = 45.5 VA 2 2
, the Hull cutoff condition where
ra − rc
VA is in volts, r in cm and B in Gauss. In designing a magnetron,
2
eB0
2
 r0 − rc
2 2

generally, V(r = r0 ) =   ≈ (0.1 to 0.2) VA
8m  r 
 0 

21. Magnetron Fields
From radial force equation (1), consider electrons following
circular trajectory in Brillouin cloud. Assume that
d 2r
2
is small, and solve for E r in region I, (r < r0 ) :
dt

rθ 2 
rθ 
ErI = 
− rθB0 =  dθ
(θ − ωc ), insert the result for θ =
e/m e/m dt
r ωc r 2 − rc  1  rc  
2 2 2
  
ErI =   1 − 2  − 1
2  r  
2
e/m 2 r   
r m (eB0 ) 2   rc   1   rc  
2 2
   
ErI = − 2 1 −    1 +   
2 e m   r  2  r  
   
e ( B0 ) 2   rc  4 
 
ErI = − r 1 −   
m 4  r 
 

22. Magnetron Fields
From Poisson’s equation the charge density:
ε0 ∂
ρ 0 = ε 0∇ ⋅ E = (rE r I )
r ∂r
∂ ∂  2  rc  
4
e ( B0 ) 2
(rE r I ) = κr (1 −   ), where κ ≡ −
∂r ∂r 
 r   m 4
  rc 
4
 rc 4    rc  4  rc  4 
= κ 2r (1 −   ) + r 2 4 5  = 2rκ 1 −   + 2  
r 

 r     r
 r  
  rc  4  e ( B0 ) 2   rc  4 
 
ρ 0 = 2ε 0κ 1 +    = − ε 0 1 +   
 r 
  m 2  r 
 
ρ0 falls slightly as r increases from rc (can increase ρ0 by increasing
ρ
B0 which follows as electrons spiral in smaller cycloidal orbits 0
about the cathode.

23. Magnetron Fields
Outside the Brillouin cloud, r0 < r < ra, in region II, use Gauss’s
Theorem:
r0
∫ D ⋅ ds = ε E
surface
0 r II 2πrdz = Q encl = ∫ ρ0 (r )2πr dr dz
rc
r0 4 2 2 4 4
rc r0 rc 1 rc 1 rc
= κ1 ∫ (r + 3 )dr = κ1[ − − 2
+ 2
]
rc
r 2 2 2 r0 2 rc
2 4
r0 rc 2πe(B0 ) 2 ε 0
= κ1[ − 2 ], where κ1 ≡ dz
2 2r0 2m
4 4 4
2 rc e(B0 ) 2 1 r0 − rc
∴ ε 0 E r II 2πrdz = [r0 − 2 ] or E r II = − [ 2
]
2r0 4m r r0

24. Hartree Relationship
The potential difference VA between the cathode and anode to
maintain the Brillouin cloud of outer radius r0 is given by:
ra r0 ra
VA = − ∫ E r dr = − ∫ E rI dr − ∫ E rIIdr
rc rc r0
2 r0 4 2 ra 4 4
e( B 0 ) rc e( B 0 ) 1 r0 − rc
=
4m ∫ r(1 − r 4 )dr + 4m
rc
∫ r [ r0 2 ]dr
r0
 r 2 1 r 4  r0  r 4 − r 4
e( B 0 )  2
 
ra

=  + c2  + ( 0 2 c ) ln r  
4m  2 2 r  r  r0  r0 
 c 
=
e(B0 )  r0 − rc


2
2 2
( ) 2
+ 2(
4
r0 − rc
4
ra 

) ln( ), Hartree Relationship
8m  r0 2 r0
2
r0 
 

25. Hartree Relationship
VA =
2
e( B 0 )  0

(
 r 2 −r 2 2
c ) + 2(
4
r0 − rc
4
ra 
) ln( )

2 2
8m  r0 r0 r0 
 
or the Hartree Relationship maybe expressed by
ra VA − VB
ln( ) = 4 4
, where is VB is voltage at r = r0
r0 ωc B0 r0 − rc
( 2
)
4 r0
2 2
 ω r − rc ) = circular velocity at r = r
v B = r0 θ = c ( 0 0
2 r0
This vB is important since it gives the velocity of the electrons at the
outer radius of the Brillouin cloud. It is this velocity vB that is to
match the velocity of the traveling waves on the multicavity structure.

26. Anode - Cathode Spacing
Again, consider the planar version of the magnetron;
r0 − rc is small fraction of ra − rc such that
VB (r = r0 ) ≈ (0.1 to 0.2) VA
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
with

27. Anode - Cathode Spacing
These traveling waves are slow waves with the desired phase velocity,
vp ~ vB. Consider the wave equation as follows:
∂ 2E ∂ 2E ∂ 2E
∇2E + k 2E = 2 + 2 + 2 + k 2E = 0
∂x ∂y ∂z
Fields traveling in z direction e j( ωt -βz) , β = ω/v p , ∂/∂x = 0
∂ 2E ∂ 2E
∴ 2 − (β 2 − k 2 )E ≈ − β 2 E = 0, since
∂y ∂y 2
ω2 ω2
k 2 = 2 << β 2 = 2
c vp
since v p ~ v B , electron velocity << c

28. Anode - Cathode Spacing
The solution of this equation results in hyperbolic trig functions:
ω  ω 
A sinh  y z + B cosh 
ˆ y yˆ
 v  v 
∴E =  p   p  e j( ωt -βz) , d ≡ r − r
a c
ω 
sinh  d 
v 
 p 
ωd/vp → not too large, such that the E at Brillouin layer is
insufficient for interaction
ωd/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,
(ωd ) / v p = (ω / v p )(ra − rc ) ≈ 4 to 8

29. Multicavity Circuit - Slow Wave Structure
Equivalent circuit of multicavity
structure - here each cavity has been
replaced by its LC equivalent. This
circuit is like a transmission line
filter “T” equivalent.
j ωL 1
Z1 = , ω0 = Z1 = impedance of
1 − ( ω / ω0 ) 2
LC
parallel LC network representing the uncoupled cavity
1
Z2 = C c = coupling capacitance between
j ωC c
2πε0
cavity vane and cathode = ⇒ from coaxial line
ln(ra / rc )

30. Multicavity Circuit - Slow Wave Structure
The circuit acts like low-loss filter
interactive impedance = input
impedance of an infinite series of
identical networks.
Z1 [( Z1 / 2) + Z k ]Z 2
Zin = + = Z k , solve for Z k to find
2 Z 2 + ( Z1 / 2) + Z k
2
Z k = [( Z1 / 4) + Z1Z 2 ]1/2 If Zin = Z k is pure resistive, the generator " sees"
resistance load and delivers power. Otherwise no power is delivered.
2 2
If Z1 = ± jX a ; Z 2 =  jX b then Z k = [(−X a / 4) + X a X b ]1/2 ∴ X a X b > X a / 4
to be real or X b > X a / 4 or Z 2 > Z1 / 4, Z k = Z1Z 2 1 + Z1 /( 4 Z 2 ) ,
- 1 < Z1 /(4 Z 2 ) < 0 is the pass band. Phase shift per section θ = βp of filter is
θ = β p = 2 sin -1 − Z1 /( 4 Z 2 ) = 2 sin -1 ω2 LCc /{4[1 − (ω / ω0 ) 2 ]}

31. Multicavity Circuit - Slow Wave Structure
Rf field repeats with periodicity p
(spacing of adjacent cavities). Field at
distance z+np is same as z. β = phase
shift per unit length of phase constant of
wave propagating down the structure.
For a circular reentrant structure anode with N cavities, fields
are
indistinguishable for Z as for Z + np. 2πm πm
βNp = 2πm m = 0, 1, 2, ... N/2 for N = 6, βp = = , m = 0,1,2,3
6 3
for m = N/2, βp = π or π mode (per cavity)
∴θ = βp = π = 2 sin -1 ω 2 LCc /{4[1 − (ω / ω0 ) 2 ]} or
π ω 2 LCc ω0
sin = 1 = solve for ω. ω = ωπ = is the
2 4[1 − (ω / ω0 ) ]
2
1 + Cc /(4C )
operating frequency for the π mode.

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

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

35. Multicavity Circuit - Slow Wave Structure
For the m = N/2 - 1 mode
ω(N/2)-1 π 2 1 Cc
≈ 1− ≈ 0.97 to 0.99
ωπ 4 N C
∴ Competing modes - desire to increase this separation
2 methods - strapping and rising sun. Strapping adds
capacitance ∴ lowers the frequency of the π mode :
ω0 ωπ ωπ
ωπ = vp = = r0 , β 2πr0 = Nπ
1 + Cs / C + Cc /(4C ) β N /2

36. Strapped Cavities

37. Typical Magnetron Cross-Sections (after Collins)
(a) Hole and slot resonators
(b) Rectangular resonators
(c) Sectoral resonators

38. 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

41. 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.
two
two

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

44. 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 / ηVA.
Cathode radius from available current densities for type of cathodes
typically 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

45. General Design Procedures for Multicavity
Magnetrons
Emitting length of cathode (lc) < anode length, la ; Typically,
lc ~ 0.7 to 0.9 la , and la < λ/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) = ωπ r0 / (N/2) = [ωc r0 /2] [1- (rc2 / r02)]; therefore
r0 = rc / [1-(ωπ /ωc)(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.

46. General Design Procedures for Multicavity
Magnetrons
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, η < (1 - VB / VA )*100; hence
Smaller VB / VA contributes to improved efficiency
Anode radius: ln (ra/ r0) = [VA - VB ] / {[ωc 2 / 4(e/m)][(r0 4 -rc 4 ) / r0 2 ]}
Also Bmin = (45.5 VA) 1/2 [ra /(ra 2 -rc 2 )] << B0
(ω /vp)( ra - rc) ~ 4 to 8
N must be even such that Nπ phase shift around the circumference is a
whole 2 π.

48. Cutaway view of a
Coaxial Magnetron

50. NRL
Hybrid
Inverted
Coaxial
Magnetron

51. NRL Hybrid Inverted Coaxial Magnetron