Download Nick’s Issue with SW Structures

Nick’s Issue with SW Structures
Note that this is done in the notes on pages 36-37…
Let’s start with the unalienable basics:
Some RF power Pin is coming from the power source
The electric field associated with the RF power in the waveguide
between source and structure can be written as P = κE2
Conservation of energy requires that, in the absence of beam, the
incoming power = the sum of outgoing power, power lost into the walls,
and time rate of change of stored energy:
dU c
Pin  Pout  Pc 
dt
I can replace the Pin term with κE2in. Similarly, I can replace Pout with
κE2out, where Eout has to be the sum of the reflected and emitted electric
fields (by superposition) -- Pout = κ(Ee+ΓEin)2
 E    Ee  Ein 
2
in
2
dU c
 Pc 
dt
If the power from the source is now turned, off, we find:
dU c
 Ee  Pc 
0
dt
2
No surprises here -- if the RF source is turned off, the time rate of change
in stored energy is given by the power into the walls plus the power that
“leaks” out of the structure. If we believe that the leakage power is
proportional to the stored energy, and we know that the power lost into
the walls is proportional to the stored energy; we therefore can see that
the wall losses and coupler losses are proportional to each other!
c 
Call the constant of proportionality βc:
 Ee2
Pc
I can now replace Pc with κEe2/βc. Since Uc = QwPc/ω, I can do
additional substitutions and find:
2

E
d Qw  Ee 
2
2
 Ein    Ee  Ein  
 

 c dt    c 
 Ee2 Qw d 2
2
   Ee  Ein  

Ee 

 c  c dt
2
e
Cancellation of the ubiquitous κ yields:
E   Ee  Ein 
2
in
2
2
e
Qw d
2


E

e 
 c  c dt
E
Let’s look a bit more closely at the equation relating the incoming
power’s electric field and the emitted electric field:
E   Ee  Ein 
2
in
2
2
e
Qw d
2


E

e 
 c  c dt
E
If Γ is real and negative (which, for now, we assert) and Ee is real and
positive (which we again assert) then the first term on the RHS starts out
large (equal to Γ2Ein2) -- in fact, for Γ = -1+δ almost all of the incoming
power reflects off. As the structure “charges up” and Ee grows, the
emitted wave interferes destructively with the reflected wave and the
power which goes back up the input waveguide goes to zero! At this
point all of the power goes into storage or into the walls.
Note that it looks like there are 2 degrees of freedom here (Γ and βc).
Actually there aren’t: Γ=-1 for βc=0 and Γ=0 for βc=. In other words,
Γ is a very insensitive function of βc.
We can put it all together into an expression for the output electric field:
Eout
 2c

 t (1  c ) / 2Qw
 Ein 
1 e
 1
1   c



If we supply a constant power (or equivalently a constant value of Ein) to
a cavity which has βc>1, the output field will fall gradually to zero and
then begin to rise again (as the emitted field exceeds the reflected field).
At the instant where the emitted field cancels the reflected field, all of the
power goes into the cavity.
Thus, the key to making a happy SW structure is to compute the
accelerating voltage at the moment Eout goes to zero; compute the current
for which V(Eout=0)I=Pin-Pc; and supply that exact current at that exact
moment.
All of the incoming power will go into the wall or the beam, the stored
energy will become constant, the voltage will be constant…
If we turn off the beam but keep the RF power turned on, then the
emitted power will grow slowly from zero again (just as if the beam had
never been turned on -- it resumes the slow growth that it was doing
prior to the beam).
If however the beam and the input power are turned off simultaneously,
then the emitted wave is no longer cancelled by the reflected wave:
Eout
 2c
 t (1 c ) / 2Qw 
 Ein 
1 e

1   c



So the observed reflected power signal = the input power at t=0; it
decays to zero, at which time the beam is turned on, and as a result the
reflected power signal stays at zero; then when the beam and the RF
power are turned off the “reflected” (actually emitted) power jumps up
and exponentially damps away to zero again. Two jumps, two gradual
decays.