1. Mike Jenkins Lancaster University and Cockcroft Institute

2. Outline

4. Equivalent Circuits Electrical properties of RF cavities can be modelled by an RLC circuit. R L C

5. Equivalent Circuits For multi cell structures: Model each cell as individual RLC circuits Cell to cell coupling represented by inductor or capacitor Effects of beam pipes on end cells described by addition of capacitor

6. Bi Periodic Equivalent Circuit Apply Kirchhoff’s voltage summation rule to each current loop giving an equation for each cell

7. Bi Periodic Equivalent Circuit

9. Effect of Beam Pipe

10. Eigenvalue Problem

11. Modelling Out of Tune Cavities

12. Testing the Equivalent Circuit Model To test the equivalent circuit model a 5 cell bi periodic structure was simulated in CST For an unperturbed structure it is field flat Added a known error into the model

13. Testing the Equivalent Circuit Model Perturbed Case First Correction Tuned

14. Extending the Model

15. Summary