4.1 Introduction.
To generate an accelerating voltage at the gap power has to be coupled into the cavity. This can be done in different ways. One solution is by directly connecting the inner conductor of the input transmission line to the inner conductor of the cavity. This is shown in Fig. 4.la and is described in detail in appendix B. There it is shown that in order to have matched coupling, the coupling point must be very close to the shorting plate. A disadvantage of this is that the VSWR (Voltage Standing Wave Ratio, described in paragraph 4.4) is sensitive to changes of the coupling position. This leads to the necessity of a movable connection with the inner conductor. The current on the inner conductor is high near the shorting plate, so losses can occur which lead to undesired heating.
A second solution is based on inductive coupling with a loop. This is described in paragraph 4.2, where the matching conditions are derived. In paragraph 4.3 these conditions are used to find the required properties of the loop like area, number of windings, position and rotation angle.
4.2 Inductive coupling.
We have seen that direct coupling is not so convenient for a cavity employing longitudinal transmission line folding. The best solution is found in inductive coupling with a small loop.
This is shown in Fig. 4.1 b.
a b
Figure 4.1: a: Direct coupling to the inner conductor. b: Inductive coupling with a little loop.
The complete rf system can be simulated by the equivalent circuit, given in Fig. 4.2. The rf generator is represented by an ideal rf voltage V 0 in series with the characteristic impedance
Ro
of the input transmission line. The resonator is represented by a series resonant RLC lumped-element circuit. The quantitylo
is chosen to represent the current at the shorting plate.The coupling between the input line and the resonator is represented by the mutual inductance M. First we calculate the input impedance seen by the generator.
17
11
I
VtRo I - Vo I Li
I I
L--7z,n.
Figure 4.2: Equivalent circuit for the complete rf system.
For the primary circuit we get the following equation V1 =jwL/1 +jroM/0 ,
and for the secundary circuit
Also
V =-f Y - _ lo
2 cJ''s jroC '
Io
where ~ is the series resistance in the circuit. Elimination of V 2 gives jroM/1
+!+ml,+ j.'.c +R}O ,
Next, elimination of
lo
givesand with substitution of the angular resonance frequency
c
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
Eq. ( 4.5) becomes
Now the impedance as seen from the generator can, with the use of
be written as (assuming O<<l)
z
='wl + (.I) 2M2pr '} I -R-s (-1-+ 2-j-Q...,oo""'")
In Fig. 4.3 ~ is drawn in the complex plane.
Im (Z)
1 - - - ; - - -6 - 0
z I )">0
Re (Z)
Figure 4.3: The complex input impedance seen by the generator.
(4.6)
(4.7)
(4.8)
(4.9)
The diameter of the circle is co2M2
fRs.
The quantity co2M2fRs(l
+ 2jQ,o) is the impedance of the cavity transformed to the primary circuit. The primary impedance is a complex quantity, having resistive and reactive components. If we compare this with the input impedance for an isolated resonator given in Eq. 2.16 and drawn in Fig. 2.5 we see that the circle has a vertical displacement due to the reactance roL1 of the coupling loop.To obtain impedance matching of the cavity seen from the transmission line, the voltage reflection coefficient
r
as measured on the transmission liner=(Z.,.-Ro)/(Z.,.+Ro)
must be zero.Thus ~=Ro=50 Q.
From Eq. (4.9) two conditions for zero reflection can be derived. The imaginary part of ~r
must be equal to zero, and the real part of ~ must be equal to
Ro.
This leads to a requirement for the detuning factor(4.10)
and for the mutual induction
(4.11)
The first condition can be satisfied if the resonator is detunable. The second condition can be satisfied for example if the loop is rotatable, such that the mutual induction M can be varied from zero to a maximum. This maximum must be larger than M given in Eq. (4.11)
4.3 Requirements for the coupling loop.
The power dissipated in the lumped element circuit of Fig. 4.2 is Pdi. =.!Jo2R •
SS 2 S
which must be equal to the power dissipated in the cavity
v
2 1v2P -
diss_T_2_·
g - gsh Rsh
From these two equations and with the use of Eq. (4.11) we get
The mutual induction M is determined by the equation
<l>=M/o '
(4.12)
(4.13)
(4.14)
(4.15)
where <I> is the magnetic flux enclosed by the coupling loop. If we assume that the loop is not too large, then the magnetic field can be taken constant across the loop. This gives us another equation for the flux
Q:>=B A=µ c ct•c 1-1 A ' (4.16) with A the area of the loop and Be the magnetic induction at the position of the loop at a distance z from the shorting plate. The magnetic field strength is detennined by
(4.17) where ~ is the current on the inner conductor at a distance z from the shorting plate, where the loop is located, and r the radial distance from the cavity axis to the middle of the loop.
Together with Eqs. (4.15) and (4.16) we get for M
(4.18)
For the matching condition we get by using Eq. (4.14)
(4.19)
Suppose we rotate the loop over an angle 0, and the loop consists of N windings, Eq. (4.16) becomes
Q:>=B AN c cos0 , (4.20)
and the matching condition becomes
(4.21)
In order to be able to produce always matched coupling the area of the loop must be larger than A given by Eq. (4.21) with 0=0.
The ratio ~N 8 can again be calculated with the transmission line matrix theory treated in chapter 3. When coupling close to the shorting plate lc=Io. If we approximate the cavity by an ideal 1AA.-system with a characteristic impedance
Zo.
the ratio ljV 8 is lfZo.4.4 Voltage standing wave ratio.
The quantity V maxN min=lma/Imm=VSWR is called the voltage standing wave ratio and is a measure of the mismatch between a line and its terminating impedance. It is related to the
voltage reflection coefficient by
VSWR-l+Jfl
1-Jrl '
(4.22)The VSWR is a real quantity and since OS:lrlS:l, lS:VSWIL<oo, where VSWR=l implies a matched load. The magnitude of the reflection coefficient on the line is found from the standing wave ratio as
If'!-
VSWR-l VSWR+l(4.23)