A Tutorial on Non-intercepting Electromagnetic Monitors for Charged Particle Beams Bob Webber January 30, 2008

1. A Tutorial on Non-intercepting Electromagnetic Monitors for Charged Particle BeamsBob WebberJanuary 30, 2008 AD Beam Instrumentation

2. Electromagnetic Monitors for Charged Particle Beams • Electromagnetic beam monitors offer a non-disruptive means to observe and quantify properties of the beam itself or, with the beam as a probe, of the accelerator or transport line in which the beam travels • Fundamental parameters that can be measured include: • beam current • temporal distribution of particles • transverse position of the beam in the chamber • Monitors are designed to sample the beam’s electric field, magnetic field, or a combination of each • Monitors that interact primarily with the electric field are typically called electric or capacitive monitors • Monitors that interact primarily with the magnetic field are typically called magnetic monitors or current transformers Bob Webber January 30, 2008

3. Signals and Sensitivities • A capacitive pickup signal • Polarity depends on the sign of the beam particles’ charge • Amplitude is independent of the direction the beam is traveling • A magnetic pickup signal • Polarity determined by the product of particle charge and direction of travel, i.e. the sign of electric current • Amplitude is independent of the direction the beam is traveling • An electromagnetic pickup signal coupled to both the electric and magnetic fields • Polarity depends on the relative electric and magnetic coupling • Amplitude response can depend on the direction of beam travel without regard to the particle charge like a directional power coupler Bob Webber January 30, 2008

4. Amplitude in units of A Position in units of Lb A Model Charge Distribution Bob Webber January 30, 2008

5. Amplitude in units of A Position in units of Lb Beam Current of Distribution To an observer at a fixed position in x, the distribution presents a bunch frequency of f = β·c/Lb. The beam current of this charge distribution, observed at x = 0 is: where  = 2f. The zero-frequency term of magnitude β·c·A is the DC or average beam current. This simple single-frequency model presents no loss of generality, since any real charge distribution can be described by a Fourier series, a linear superposition of terms with different frequencies and phases. Bob Webber January 30, 2008

6. The Beam as a Signal Source • The beam appears as nearly an ideal current source, that is, the beam current is unaffected by interaction with the monitor.* • The source impedance, the change in terminal voltage required to effect a change in current, is described as • For 10 mA, 10 MeV beam, find 4E11 ohms for electrons, 2E9 ohms for protons! • * Significant exceptions like beam instabilities are possible when the beam interacts repeatedly with a periodic structure. V = i R = i Z Bob Webber January 30, 2008

7. The Beam Environment • The beam and its environment communicate through the electric and magnetic fields • Typically the beam travels through an evacuated chamber bounded by an electrically conducting metallic wall • The beam, an assembly of charged particles, carries an electric field which induces image charge on the chamber wall • The beam, as charged particles in motion, carries a magnetic field that, at sufficiently high frequency, induces the image charge to flow with it. The resulting “wall currents” are, to first order, equal and opposite to the beam current Bob Webber January 30, 2008

8. Q x x An Artist’s Conception of the Fields Electric Q Q Velocity β  1 Isolated charge at rest Charge at rest in pipe Charge at relativistic velocity Magnetic Beam into page At moderate and high frequency, no field outside chamber At DC, fields penetrate chamber Bob Webber January 30, 2008

9. Beam E/M Field Attenuation • To the extent that wall currents mirror the beam current, the magnetic field outside the beam tube is cancelled • This field strength reduction corresponds to the attenuation of electromagnetic waves propagating through a conductor • The characteristic length in which the fields are reduced by a factor of e (-8.69 dB) is termed the skin depth • The skin depth in a non-magnetic good conductor is • where ρ is the resistivity of the conductor in ohm-meter and f is the frequency of interest in Hz Bob Webber January 30, 2008

10. Energy Flows through the Dielectric • Chart of skin depth in millimeters • A 1/32” stainless beam tube wall is ~6.1 skin depths at 10Mhz and attenuates magnetic fields propagating to the exterior by 53dB • Not so much as to make the beam signal invisible to a sensitive radio receiver • Sufficient to clobber the sensitivity of a practical beam current monitor • Hence beam monitors generally must be placed inside the beam chamber Bob Webber January 30, 2008

11. A Typical Messy Environment Lines of induction around isolated circulating beam Wall current induced in beam tube attenuate external field Break in tube impedes wall current Typical complex and distributed paths available to induced currents Bob Webber January 30, 2008

12. A Capacitive (Electric Field) Pickup An isolated cylindrical electrode with length Le and radius a inside tube with radius b. Electric field lines from beam can terminate on electrode, but no loop area is present to intercept magnetic flux. Bob Webber January 30, 2008

13. Estimating the Signal • Assuming a purely transverse electric field, the charge induced on the inside of the capacitive electrode at any time is equal to the total beam charge contained within the linear extent of the electrode and opposite in sign. Solving the integral gives The first zero in this sin(x)/x frequency response shape occurs where Le = Lb Wrong High Frequency Limit Bob Webber January 30, 2008

14. When Propagation Time Matters • Missing is consideration of the time delay for a charge induced at any position along the electrode to be recognized at the tap point. This information can only travel at the speed of an electromagnetic wave, the speed of light, for a monitor with vacuum as the only dielectric. At low frequencies the effect is negligible; where transit times are a significant fraction of the period and the bunch length is comparable to the electrode length, the impact is large. • Accounting properly for signal propagation times (ignoring azimuthal effects on the assumption of cylindrical symmetry including beam position), the effective charge observed at the tap point at any time is the sum of the charges induced at all longitudinal segments of the electrode at a time earlier by an amount x/v, where x is the position of the segment relative to the tap point and v = β·c for vacuum. Bob Webber January 30, 2008

15. Mathematically • Mathematically the time at each position along the electrode is weighted by the transit time to the tap point. The expression for the induced charge as observed at the tap point becomes: • And the available signal current is: = Correct High Frequency Limit ! The first zero occurs where Le = Lb/2 Bob Webber January 30, 2008

16. Signal Power from Capacitive Pickup • In the “short electrode” limit, Le<<Lb, the tap point current is: In the frequency domain, the signal voltage on load resistor R becomes where Z is parallel combination of R and Celectrode and τ= R * Celectrode Signal power for  >> 1/τ is V2/R = (1/R)·(A·Le/C) 2 = (A·Le) 2/(C·τ) with capacitance The resulting signal power for  >> 1/τis Bob Webber January 30, 2008

17. A Window to the Beam • Since the magnetic field of the beam is severely attenuated outside a continuous conducting vacuum chamber, a practical beam current monitor must either be placed within the vacuum chamber walls or the conducting path in the chamber must be broken • To minimize the mechanical complications of inserting a device into the vacuum, a non-conducting material, often ceramic, is typically inserted in a section of the beam tube • This break in the beam tube conduction path forces wall currents to find a new path, potentially under the instrument designer’s control, outside the vacuum chamber Bob Webber January 30, 2008

18. The Ceramic Break • Beam tube capacitance, grounds, and ungrounded parallel connections may be intentional or incidental, local or distant, but something will always be present • Zgap is combination of the gap capacitance and all external parallel elements • Gap voltage will be generated Typical Ceramic Break Installation Circuit Model Bob Webber January 30, 2008

19. Location of Current Monitor Relative to Gap • OK - only beam currents, not wall currents will pass through monitor • NOT OK - wall currents bypass gap through grounds then proceed through monitor OK - wall currents cannot bypass gap to flow through monitor NOT OK - wall currents bypass gap through grounds then proceed through monitor Bob Webber January 30, 2008

20. Impedance Measured Across Ceramic Gap on Beam Tube Courtesy of Jim Crisp/Mike Reid Bob Webber January 30, 2008

21. Controlling the Environment and Taming the Gap Impedance • Zshunt is applied to control potentially high gap impedances • Strap or housing around the transformer and gap • Short circuits external currents that might flow through Zshunt and produce undesired monitor signals • Shields external world from the beam current and gap voltage Bob Webber January 30, 2008

22. The Beam Current Transformer Model Magnetic flux due to any winding ‘k’ Total flux is sum of that due to all currents, beam and signal Beam acts as single-turn primary winding Voltage on any winding Signal power at mid-band frequencies Frequency response Bob Webber January 30, 2008

23. Compare Signal Power from Capacitive and Magnetic Pickups of Same Time Constant • • In the absence of magnetic or dielectric materials, μr = εr = 1, the power from the magnetic monitor can never exceed that from the capacitive monitor. • For non-relativistic beams, the capacitive monitor provides greater signal power than the magnetic monitor by a factor of 1/ β2. • For relativistic beams the available power from the two monitors is identical. • • The addition of magnetic or dielectric material enhances the relative performance of the magnetic monitor, an advantage that can be dramatic as commonly available magnetic materials can offer permeability > 10,000. Hence, the predominance of magnetic type beam current monitors. • • Capacitive monitors, in the relativistic beam regime, can offer benefits in instances where the signal power is adequate and at high frequencies where the advantage of magnetic materials can be lost. Above low frequency corner Bob Webber January 30, 2008

24. Magnetic Loop Pickup Bob Webber January 30, 2008

25. Directional Coupler Equal signal out and along line Zo Zo Zo Zo No signals 2Le Zo Zo Zo Zo Forward line signal and opposite prompt signal cancel at downstream port and opposite polarity signal propagates backwards along line Resulting signal at upstream port Upstream port signal also zero in steady state case if electrode electrical length is ½ bunch spacing ! Bob Webber January 30, 2008

26. Conclusion • A conceptual introduction to electromagnetic beam monitors was presented from basic principles • Models for estimating signal strength from both capacitive and magnetic monitors were shown and the relative signal strength from the two types was given • A model for understanding a directional coupler monitor was outlined • Hopefully the talk stimulates each listener to develop his/her own understanding of these basic processes in addition to the formal mathematics and to ask ‘how does it work’ at a fundamental level Bob Webber January 30, 2008

27. Previous Tutorials • First Workshop, 1989, BNL, Upton, NY --- Accelerator Instrumentation, AIP Conference Proceedings 212, New York: American Institute of Physics, 1990. • “Longitudinal Emittance: An Introduction to the Concept and Survey of Measurement Techniques Including Design of a Wall Current Monitor”, pp. 85-125. • Sixth Workshop, 1994, Vancouver, BC, Canada --- Beam Instrumentation Workshop, AIP Conference Proceedings 333, New York: American Institute of Physics, 1995. • “Charged Particle Beam Current Monitoring Tutorial”, pp. 3-23. • Ninth Workshop, 2000, Cambridge, MA --- Beam Instrumentation Workshop 2000, AIP Conference Proceedings 546, New York: American Institute of Physics, 2000. • “Tutorial on Beam Current Monitoring”, pp. 83-104. Bob Webber January 30, 2008

28. Backups Bob Webber January 30, 2008

29. Toroidal Flux and Inductance Lines of magnetic induction, B, encircle a current i. In a homogeneous region near the current, B is a function of radius from the current is given by Ampere’s Law. constant at any i is …wihre B is constant and any radius and the path of intergration is a circule around the wire and has magnitude 2pir. Assume toroid of inner radius a, outer radius b, and height (thickness) h Assume toroid of inner radius a, outer radius b, and height (thickness) h Total flux is sum of that due to all currents, beam and signal Single-turn winding with current i Voltage on any winding Signal power at mid-band frequencies Frequency response Bob Webber January 30, 2008

30. Common Types of Beam Current Transformers • Integrating Current Transformer (ICT) • Passive current transformer depending on short (< 1 nsec), isolated (±50 nsec) beam bunch to drive impulse response of transformer • Output pulse shape is fixed by design and independent of shape of sufficiently short beam pulse • Output amplitude is directly proportional to charge of beam pulse • Useful in synchrotrons, storage rings, and transport lines provided short isolated bunch criteria are met • Advantage • Simple, relatively inexpensive, stable passive calibration • Output stretched in time relative to very short beam pulse • Disadvantage • Bunch shape information is not available Bob Webber January 30, 2008

31. Common Types of Beam Current Transformers • Direct Current Transformer (DCCT, PCT, etc.) • A strong well-controlled magnetizing force is applied to one or more toroids enabling sampling of magnetic bias imposed by beam • Operates in zero flux mode, a feedback current equal and opposite to the beam is driven through the toroidal cores of the device • Practical DCCTs for particle beams are a combination DC section and AC transformer to prevent aliasing and extend bandwidth • Useful in synchrotrons and storage rings, not transport lines • Advantages • Measures 0 Hz (DC) component of bunched or unbunched beams • Long term stability and <1 microampere DC resolution • Disadvantage • Relatively expensive for applications not requiring DC response Bob Webber January 30, 2008

32. Common Types of Beam Current Transformers • Classical AC Transformer • Beam current couples magnetic flux to toroidal transformer core inducing current in sense winding on same core • Output signal can provide hi fidelity representation of beam current pulse shape over wide bandwidth (10’s of Hz to few MHz) • Passive device that can be supplemented with various active circuits to modify performance (e.g. Hereward and ‘active-passive’ configurations • Advantages • Simple and available in many configurations to suit application • Disadvantages • NOT DC coupled, provides NO DC output component Bob Webber January 30, 2008

33. Zshunt • Circuit elements as depicted are often used to realize Zshunt • Multiple elements in parallel should be distributed across the gap more or less uniformly around the circumference • Zshunt must be sufficiently high impedance at beam current frequencies to be measured so as not to short circuit the gap as seen by the current monitor, typically >10 ohms is acceptable • Series RC network blocks low frequency external noise currents from flowing across gap and through monitor • Parallel RC network exhibits lower impedance at high frequencies Bob Webber January 30, 2008

34. Classical Transformer Review • Steady state circuit equations in Laplace notation • Total magnetic flux in core • Load side current • Primary side loop Bob Webber January 30, 2008

35. Classical Transformer Review • Simultaneous solution and use of , where Np and Ns are the number of primary and secondary winding turns respectively, yields • In mid-band where this simplifies to Bob Webber January 30, 2008

36. Classical Transformer Review • In the case of the beam current it is appropriate to replace the voltage generator by an equivalent current generator • where and • Substituting into Eqn. T1 with , find • Given that , the familiar result is obtained. Bob Webber January 30, 2008

37. Noise Reduction Core on Coax Top trace - no core Second - 5 turns of coax through core Third - 10 turns Fourth - many turns Bob Webber January 30, 2008

38. Noise Signals and Spectra w/Core • Signals at 50mV/div and 40msec/div and spectra at 10dB/div and 125 Hz/div, except no core is 500 Hz/div. No core 10 turns 5 turns many turns Bob Webber January 30, 2008

39. How Does the Core Help? To Z or not to Z? • Noise measurements showed very low impedance source • A perfectly conducting cable shield could short out the noise source, thereby eliminating the noise • Yet apparently adding impedance to the shield in the form of a core also reduces the noise • Dilemma - • Increase shield impedance to reduce noise currents? • Reduce shield impedance to attenuate noise source voltage? • Solution - • Low impedance noise is not completely overcome, core acts as transformer coupling equal voltage to shield and center conductor Bob Webber January 30, 2008

40. Toroid on a Coaxial Cable • Shield and center conductor circuits, each looping core N times, link same magnetic flux in core and will therefore experience same induced voltage. At frequencies above ω = R/L, where R is the shield resistance and L is the inductance of the shield winding on the core, the end-to-end center conductor voltage will be identically equal to the shield voltage, in this case Vnoise! Therefore the differential shield-to-center voltage at both ends can be independent of the noise voltage Note that low shield resistance is still a good thing. It reduces the corner frequency at which the transformer action becomes effective! Dilemma resolved! The core cannot influence desired signals propagating inside coax. With equal and opposite in shield and center conductor currents, these signals present no net current to core, effectively removing the it from the picture. Bob Webber January 30, 2008

41. Directional Coupler Equal signal out and along line Zo Zo Zo Zo No signals Zo Zo Zo Zo Forward line signal and opposite prompt signal cancel at downstream port and opposite polarity signal propagates backwards along line Resulting signal at upstream port Upstream port signal also zero in steady state case if electrode electrical length is ½ bunch spacing ! Bob Webber January 30, 2008