1. 22/May/2009LHCb Upgrade1 Skin effect Skin effect: –Tendency of AC current to distribute itself within a conductor so that the current density near the surface of it is greater than at its core –f↑ → R↑ –Define penetration or “skin” depth δ: Signal attenuation due to skin effect –Resistance per unit length R S : –It can be shown that: –Transmission line characteristic impedance: R.L. Chase, C. de La Taille, S, Rescia and N. Seguin, Nucl. Instr. and Meth. A 330 (1993) 228

2. 22/May/2009LHCb Upgrade2 Signal attenuation due to skin effect (continued) –Transfer function of a length of line: –Propagation constant: –Inverse Fourier transform: –And the step response of the transmission line: Skin effect with Introduction of long time constants → strong attenuation

3. 22/May/2009LHCb Upgrade3 Impedance seen at position x towards the detector when Z d =1/jCw: Skin effect: noise contribution 50 Ohm

4. 22/May/2009LHCb Upgrade4 Impedance seen at position x towards the detector when Z d =1/jCw: Skin effect: noise contribution 20 m 12 m 1 m 5 m 50 Ohm

5. 22/May/2009LHCb Upgrade5 Phase seen at position x when Z d =1/jCw: Skin effect: noise contribution 20 m 12 m 1 m 5 m

6. 22/May/2009LHCb Upgrade6 Impedance seen at position x towards the detector when Z d =1/jCw: Skin effect: noise contribution 100 MHz 50 MHz 1 MHz 10 MHz 50 Ohm

7. 22/May/2009LHCb Upgrade7 Phase seen at position x when Z d =1/jCw: Skin effect: noise contribution 100 MHz 50 MHz 1 MHz 10 MHz

8. 22/May/2009LHCb Upgrade8 Impedance seen at position x towards the detector when Z d =R d : Skin effect: noise contribution 50 Ohm 25 Ohm

9. 22/May/2009LHCb Upgrade9 Impedance seen at position x towards the detector when Z d =R d : Skin effect: noise contribution 20 m 12 m 1 m 5 m 50 Ohm 25 Ohm

10. 22/May/2009LHCb Upgrade10 Phase seen at position x when Z d =R d : Skin effect: noise contribution 20 m 12 m 1 m 5 m

11. 22/May/2009LHCb Upgrade11 Resistance per unit length R S : Skin effect resistor Rs values: –Freq high enough to suppose current only on the cable surface –T = 300 K Skin effect: noise contribution Rs per length unitRs for a 12 m cable

12. 22/May/2009LHCb Upgrade12 Skin effect: noise contribution Noise generator per unit length: –Propagation constant (rearranged): Plot: e n (nV/√Hz): Cable length: 12 m Fast shaping times approximation: -Skin effect noise ~ single noise generator at preamp input -Aproximate R S at preamp+shaper central frequency

13. 22/May/2009LHCb Upgrade13 Impedance seen at the preamp input for very high frequencies is R 0 (50 Ohm) Impedance R s due to skin effect on the cable is about 10 to 15 Ohm For further realistic analysis on V n 2: –The noise current per unit length at position x: –From which we can obtain V n 2 multiplying by the amp+shaper transfer function and integrating over all the frequencies. Conclusions