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

A Tutorial on Non-intercepting Electromagnetic Monitors for

Charged Particle Beams

Bob WebberJanuary 30, 2008

Bob Webber January 30, 2008 2FermiLab

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


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


A Model Charge Distribution

xq b x t( )d

dA 1 cos 2

x c t L b

A 1 cos 2 fx

ct

Amplitude in units of A

Position in units of Lb


Beam Current of Distribution

ib cxqb

d

d c A 1 cos 2

c tLb

c A 1 cos t

where = 2f. 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.

Velocity = β·c

Amplitude in units of A

Position in units of Lb

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:


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.

)/()/1()/()(

)()()(

32321

11

1

eEieEcA

dV

dE

dE

cdA

dV

dv

dv

di

dV

diZ

obo

b


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


An Artist’s Conception of the Fields

Q Q QVelocity β 1

Isolated charge at rest Charge at rest in pipe Charge at relativistic velocity

Electric

MagneticBeam into page

x xAt DC, fields

penetrate chamber

At moderate and high frequency, no

field outside chamber


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

meterf

2

1010 3


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

1 KHz 10 KHz 100 KHz 1 MHz 10 MHz

Copper 2.1 0.66 0.21 0.066 0.021

302 Stainless Steel 13.3 4.2 1.3 0.42 0.13


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


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.


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.

qe t( )

Le

2

Le

2

xxqb x t( )d

d

d

Le

2

Le

2

xA 1 cos x

ct

d

qe t( ) A Le 1

sinLe

2 c

Le

2 c

cos t

A Le 1

sin Le

Lb

Le

Lb

cos t

The first zero in this sin(x)/x frequency response shape occurs where Le = Lb

Solving the integral gives


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.


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:

qtap t( )

Le

2

Le

2

xxqb x t

x

c

d

d

d

Le

2

Le

2

xA 1 cos x

ct

x

c

d

Le

2

Le

2

xA 1 cos 2 x

ct

d

itap t( )tqtap t( )d

dA Le

sin Le

c

Le

c

sin t

=

The first zero occurs where Le = Lb/2


Signal Power from Capacitive Pickup

In the “short electrode” limit, Le<<Lb, the tap point current is:

itap t( )tqtap t( )d

dA Le

V itap Z A Le R

1 j

V2/R = (1/R)·(A·Le/C) 2 = (A·Le) 2/(C·τ)

PwrA

2

Le lnb

a

2 r o

Signal power for >> 1/τ is

C2 r o Le

lnb

a

with capacitance

In the frequency domain, the signal voltage on load resistor R becomes

The resulting signal power for >> 1/τ is

where Z is parallel combination of R and Celectrode and τ = R * Celectrode


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


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

Conducting beam tube

Ceramic break Exploded view

IwallIbeam

Zgap

Typical Ceramic Break Installation Circuit Model

beamgapwallgapgap IZIZV


OK - wall currents cannot bypass gap to flow through monitor

NOT OK - wall currents bypass gap through grounds then proceed through

monitor

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


Impedance Measured Across Ceramic Gap on Beam Tube

Courtesy of Jim Crisp/Mike Reid

.0

100.0

200.0

.1 1.0 10.0 100.0 1000.0

MHz

ohm

s

open

shorted

1.1 uH


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

IwallIbeam

Rs

Iwall - Ishunt

ZshuntStraps or housing


The Beam Current Transformer Model

Φ

Ls

Is

Rs Ns

Ibeam

k

r o Nk ik h 2

lnb

aie t( )

tqe t( )d

dA Le

sin Le

2 c

Le

2 c

sin t

total

r o h.

2 ln

b

a

ib Ns is ie t( ) A Le

sin Le

2 c

Le

2 c

sin t A 2 c sin Le

2 c

sin t

t total

d

d

Vs

Ns

is Rs

Ns

ie t( ) A Le

sin Le

2 c

Le

2 c

sin t A 2 c sin Le

2 c

sin t

Vs j

1 j Rs

ib

Ns

Pwr

ib2

r o h lnb

a

2

Magnetic flux due to any winding ‘k’

Total flux is sum of that due to all currents, beam

and signal

Voltage on any winding

Beam acts as single-turn

primary winding

Frequency response

Signal power at mid-band

frequencies


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.

Pwrmag

Pwrcap

2r r Above low frequency corner


Magnetic Loop Pickup

Beam

Magnetic Flux

Loop


Directional Coupler

Zo

Zo

Zo

Zo

Zo

Zo

Upstream port signal also zero in steady state case if electrode electrical length is ½ bunch spacing !

Zo

Zo

Equal signal out and along line

No signals

Forward line signal and opposite prompt signal cancel at downstream

port and opposite polarity signal propagates backwards along line

Resulting signal at upstream port

2Le


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


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.


Backups


Toroidal Flux and Inductance

Φ

I

idlrB )(ie t( )

tqe t( )d

dA Le

sin Le

2 c

Le

2 c

sin t

total

r o h.

2 ln

b

a

ib Ns is ie t( ) A Le

sin Le

2 c

Le

2 c

sin t A 2 c sin Le

2 c

sin t

t total

d

d

Vs

Ns

is Rs

Ns

ie t( ) A Le

sin Le

2 c

Le

2 c

sin t A 2 c sin Le

2 c

sin t

Vs j

1 j Rs

ib

Ns

Pwr

ib2

r o h lnb

a

2

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

Voltage on any winding

Single-turn winding with current i

Frequency response

Signal power at mid-band

frequencies

idlrB )(

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


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



• 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



• 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


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

OROpen circuit at DC

R/N at high frequencyCorner frequency at ω =1/RC

R/N at DC1/jωNC at high frequency

Corner frequency at ω =1/RC


Classical Transformer Review

Steady state circuit equations in Laplace notation

Total magnetic flux in core

Load side current

Primary side loop

Φ

VgRg Lp Ls

Ip Is

Rs

s

ss

p

ppT N

iL

N

iL

js

s

sTs R

Nsi

pTgpg NsRiV



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

T1) (Eqn.)(1

1

s

s

g

p

RL

R

Lsg

ps

g

ss

g

s

sNR

NLs

V

Ri

V

V

2

2

s

p

s

p

N

N

L

L

1

s

s

g

p

R

L

R

Ls

T2) (Eqn.1

2

2

p

s

N

Ngsp

ss

g

s

RRN

NR

V

V



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.

VgRgIbeam

Rbeam

beambeamg RiV beamg RR

1pN

beams

s

RN

Rsbeam

s

Ni

i

21

11

sbeam RR s

beams N

ii


Noise Reduction Core on Coax

Vn

50ohms

50 ohmscope

Tape wound highpermeability toroidal core

Top trace - no core Second - 5 turns of coax through core

Third - 10 turns Fourth - many turns


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.

•5 turns •many turns

•No core •10 turns


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


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.

VN

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 voltageNote 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.


Directional Coupler

Zo

Zo

Zo

Zo

Zo

Zo

Upstream port signal also zero in steady state case if electrode electrical length is ½ bunch spacing !

Zo

Zo

Equal signal out and along line

No signals

Forward line signal and opposite prompt signal cancel at downstream

port and opposite polarity signal propagates backwards along line

Resulting signal at upstream port

Documents

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