Shunt impedance

The shunt impedance is a term used in high-frequency technology , more precisely in the area of the cavity resonators . Cavity resonators are often used in particle accelerators , where they are used to accelerate and measure the position and intensity of a particle beam.

If a particle beam passes through a resonator, the shunt impedance is the proportionality factor between the power that is withdrawn or supplied to the beam and the square of the beam current that is appropriately guided through the cavity resonator: ${\displaystyle R_{\text{S}}}$ ${\displaystyle P}$ ${\displaystyle I}$

${\displaystyle P=R_{\text{S}}\cdot I^{2}}$

The shunt impedance has the dimension of an electrical resistance .

Measurement of the shunt impedance using the disruptive body method

The so-called interfering body measurement can be used to determine the shunt impedance of a cavity resonator. The idea is that the smallest possible dielectric object is polarized by the electric or magnetic field inside the resonator , thus changing the properties of the resonator. The disruptive body usually consists of a material with high electrical permittivity, for example Teflon . There are various methods of obtaining a value for the shunt impedance from this frequency reduction, the best known are the resonant and the non-resonant interfering body method.

Resonant body measurement

Here, the resonance frequency of the resonator is considered as a function of the position of the disruptive body. Assuming a very small interfering body that does not deform the field, the amount of the electric field at the position of the interfering body results from ${\displaystyle E}$ ${\displaystyle z}$

${\displaystyle E(z)={\sqrt {{\frac {2W}{\alpha _{\text{S}}}}{\frac {\nu (z)}{\nu _{0}}}}}}$

The energy stored in the resonator field denotes the current resonance frequency of the resonator, the undisturbed resonance frequency and the disruptive body constant. This results from ${\displaystyle W}$ ${\displaystyle \nu (z)}$ ${\displaystyle \nu _{0}}$ ${\displaystyle \alpha _{\text{S}}}$

${\displaystyle \alpha _{\text{S}}={\frac {1}{2}}\,V\varepsilon _{0}(\varepsilon -1)}$

Here is the volume of the bluff body, the electric field constant and the electrical permittivity of the bluff body. ${\displaystyle V}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \varepsilon }$

If you integrate the electric field along the measured axis, you get the voltage that a charged particle passes through on this axis. ${\displaystyle U}$

${\displaystyle U=\int _{0}^{l}E(z)\,\mathrm {d} z}$

The relationship between power, voltage and resistance results in the shunt impedance

${\displaystyle R_{\text{S}}={\frac {U^{2}}{P_{\text{V}}}}}$

This is the power loss that is lost due to ohmic effects in the walls of the resonator. ${\displaystyle P_{\text{V}}}$

Non-resonant disruptive body measurement

In contrast to the resonant method, the non-resonant method does not track the resonance frequency, but rather the reflection factor at a fixed frequency. The reflection factor indicates the ratio of the amplitude of the wave entering the resonator and the wave exiting the resonator, as well as their phase relationship. ${\displaystyle \rho }$

${\displaystyle \rho ={\frac {{\underline {U}}_{\text{aus}}}{{\underline {U}}_{\text{ein}}}}}$

This is generally complex . From the deviation of the complex reflection factor when the interfering body passes through to the undisturbed case, a value for the amount of the electric field can now be determined, similar to the resonant method.

${\displaystyle E(z)={\sqrt {{\frac {(1+\kappa )^{2}}{2\kappa Q_{0}}}{\frac {W}{\alpha _{\text{S}}}}|\Delta \rho (z)|}}}$

It describes the coupling factor, which describes the adaptation of the wave resistance of the resonator to the connected cable. This results from the reflection factor at the resonance frequency. ${\displaystyle \kappa }$

${\displaystyle \kappa ={\begin{cases}{\dfrac {1+|\rho |}{1-|\rho |}}&{\text{bei }}\rho >0\\&\\{\dfrac {1-|\rho |}{1+|\rho |}}&{\text{bei }}\rho <0\end{cases}}}$

${\displaystyle Q_{0}}$denotes the unloaded circular quality . The further calculation is carried out analogously to the resonant measurement.

literature

• Manuel Schedler: Optimization of high-frequency intensity monitors at the ELSA electron accelerator . Bonn 2009.

Web links

• Optimization of high-frequency intensity monitors at the ELSA electron accelerator (PDF file; 1.66 MB)
• Damping of beam instabilities in the ELSA electron accelerator with the help of broadband resonators p. 16 ff. (Accessed on November 10, 2017)
• Theoretical and experimental investigations to determine the transverse shunt impedance and quality of interference mode-damped accelerator resonators for linear colliders and high-current accelerators in medium and high energy ranges (accessed on November 10, 2017)