88 unifying physics of accelerators, lasers and plasma
Here, W s is the energy stored in the cavity, W p is the energy
dissipated per cycle divided by 2π, P d is the power dissipated
in the cavity walls and ω is the frequency of the cavity.
The stored energy over the cavity volume is
In vacuum H = B/μ 0 in SI
units.
ε 0
μ 0
W s =
|E|
2 dv +
|H|
2 dv
(5.18)
2
2
where the first integral is the energy stored in the E-field and
the second integral corresponds to the energy in the H-field.
In an EM wave in space or a cavity, the energy oscillates back
and forth between these two contributions.
The losses in the cavity are calculated by taking into account the finite conductivity σ of the cavity walls. Since the
linear density of the current j along the walls of a perfect conductor can be written as
j = n × H
(5.19)
where n is the vector normal to the surface, we can therefore
equate the power dissipated in the cavity walls to
R surf
P d =
|H|
2 ds
(5.20)
2
s
where the integral is taken over the inner surface of the conductor, and the surface resistance is given by Eq.5.8.
5.3.4 Shunt impedance — R s
The so-called shunt impedance R s relates the accelerating voltage V to the power that needs to be fed into the cavity to
compensate for the dissipation in the walls P d .
The accelerating voltage along the path followed by the
beam in an electric field E z is
V =
E z (x, y, z) de
(5.21)
pass
and is taken as peak-to-peak value. The shunt impedance is
then defined as
V 2
R s =
(5.22)
2P d
and is another important characteristic of an accelerating
cavity.
5.3.5 Energy gain and transit-time factor
The energy gain of a particle as it travels a distance through
the accelerating structure depends only on potential difference crossed by particle:
.
U = K P RF lR s
(5.23)
Here, W s is the energy stored in the cavity, W p is the energy
dissipated per cycle divided by 2π, P d is the power dissipated
in the cavity walls and ω is the frequency of the cavity.
The stored energy over the cavity volume is
In vacuum H = B/μ 0 in SI
units.
ε 0
μ 0
W s =
|E|
2 dv +
|H|
2 dv
(5.18)
2
2
where the first integral is the energy stored in the E-field and
the second integral corresponds to the energy in the H-field.
In an EM wave in space or a cavity, the energy oscillates back
and forth between these two contributions.
The losses in the cavity are calculated by taking into account the finite conductivity σ of the cavity walls. Since the
linear density of the current j along the walls of a perfect conductor can be written as
j = n × H
(5.19)
where n is the vector normal to the surface, we can therefore
equate the power dissipated in the cavity walls to
R surf
P d =
|H|
2 ds
(5.20)
2
s
where the integral is taken over the inner surface of the conductor, and the surface resistance is given by Eq.5.8.
5.3.4 Shunt impedance — R s
The so-called shunt impedance R s relates the accelerating voltage V to the power that needs to be fed into the cavity to
compensate for the dissipation in the walls P d .
The accelerating voltage along the path followed by the
beam in an electric field E z is
V =
E z (x, y, z) de
(5.21)
pass
and is taken as peak-to-peak value. The shunt impedance is
then defined as
V 2
R s =
(5.22)
2P d
and is another important characteristic of an accelerating
cavity.
5.3.5 Energy gain and transit-time factor
The energy gain of a particle as it travels a distance through
the accelerating structure depends only on potential difference crossed by particle:
.
U = K P RF lR s
(5.23)