Introduction to beam impedance

O.Berrig23/4 2015

A beam passing through an equipment, have electro-magnetic fields trailing behind it:

Beam

These electromagnetic fields are called “wake fields”, exactly like the wake behind a boat:

The more elaborate name for beam impedance is:

BEAM COUPLING IMPEDANCE

Because beam impedance is a result of the beam interacting with the walls (and other things like space charge, electron cloud, colliding beams and equipment like kickers, collimators, bellows, etc.).

Therefore a single particle passing through empty space have no beam coupling impedance:

Andrea Latina, Hao Zha

example: Measurement of transverse kick in CLIC accelerating structure in FACET

The wake fields give the beam a kick – in mathematical terms – a change in momentum: ΔpWe normally separate the kicks in longitudinal kicks and transverse kicks: Δp││ and Δp

s

Most of the wake fields are following the beam out of equipment, but some are left behind inside the equipment.The reason is that the image currents on the surface of the vacuum chamber walls tries to follow the beam, but since the beam moves with approximately the speed of light, the image currents cannot follow and breaks off. The basic theorem is that charged particles, when they are accelerated, sends out electro-magnetic radiation.

Image currentsPhotons

The wake fields will affect the bunch itself and any other bunches behind it. The way we treat this mathematically, is to imagine a bunch (driving: q1) like a Dirac delta function which moves through the equipment and with wake fields behind it*. Behind it is another bunch (test: q2), and we now want to know how this second bunch is affected by the wake fields from the first bunch.

bunch

Leading/driving (q1) and trailing/test (q2) charges in a pillbox cavity with axial symmetry

* For relativistic less than 1, then the wake fields are also in front of the driving bunch (and needs to be taken into account)

s

Finally we get to the definition of the beam impedance:

Let’s look at the longitudinal beam impedance. If we integrate the electrical field (E) along the length of the equipment, we get a voltage: We have another name for his voltage:the wake potential, except that the wakepotential is normalized with the sourcecharge i.e.

Dividing this voltage with the beam current gives us the longitudinal beam impedance: ││

NB! This definition is technically wrong, but pedagogically good. It will be cleared up on the next slide.

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sVsW

However, when we look at this definition, we see that it depends on the distance between the driving bunch and the witness bunch. But impedance is essentially a frequency domain definition, i.e. Z – by definition - must depend on frequency. Since the distance between the driving and the witness bunch depends on the speed, in reality V(s) can also be expressed as V(t). If we make a Fourier Transform of V(t), we will get V(w). Similarly, we can make a Fourier Transform of the current of the driving bunch, i.e. I(s) which again can be expressed as I(t), and we get I(w).

The 100% correct definition of the beam impedance is then:

w is the frequency [rad/second]

This transformation comes from the fact that:and that

It is also interesting to note that Fourier Transforms has nothingreally to do with time and frequency; it is just a transformationthat decompose a function into sine functions: See an example: \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\FourierApproximation.cdfSee also: http://cds.cern.ch/record/702550/files/sl-note-99-047.pdf

It is useful to mention that the frequencies are both positive and negative. This comes from the modern way of doing Fourier transforms:

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T ________ is the time period (i.e. the function s(t) repeats itself after T)

_______ is the fundamental frequency

a0 , an and bn are Fourier coefficients (real numbers)

cn _______ are Fourier coefficients (complex numbers) and cn = c-n* (complex conjungated)

Old way

Modern way

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The huge advantage of using Fourier Transforms is that it provides us with a conceptually easy way of calculating a response from an equipment.

If we as input to an equipment have a sine function, then the output will be another sine function but with a different amplitude (smaller) and a different phase:

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wwforeA

B

wwforeA

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Notice that if we had used a cos(w0t) function as input, then we would have gotten B*cos(w0t+ ) as output. That is why, when we normally analyze electrical circuits, we describe a voltage as U*eiwt instead of U*sin(wt) or U*cos(wt), because we know that the circuit will have the complex conjugated Fourier coefficient for the negative frequency, and finally we can just say that the circuit change the amplitude and the phase by certain values

So Z(w) is really just a transfer function. The reason why we call it Z(w) instead of H(w) is that it is voltage divided by current.

And now we also understand why we use a driving bunch like a Dirac delta pulse (or some people like to call it a point charge).

It is really standard electrical theory. We can analyze an electrical circuit in two ways, that are equivalent:

1) Excite it with a delta pulse. Then the Fourier Transform of the response is equal to the response function H(w) of the circuit. The reason for this is that the Fourier Transform of a Dirac pulse is 1: = 1so we really excite the circuit with sine functions with amplitude 1 and phase 0.

2) Excite it with sine functions (with amplitude 1 and phase 0) for all frequencies in the interval [0;]. Then the response function is a sine function with amplitude “a” and phase . The Fourier coefficients are equal to a*ei for the positive frequency and equal to a*e-i for the negative frequency. This is the method that a Network Analyzer use. It is also called swept frequency.

So we excite the equipment with a Dirac delta pulse(i.e. the driving bunch) and measure the responseat different times after the Dirac pulse(i.e. at different positions of the test bunch)

t

We can measure the beam impedance with wire measurements*

This is based on the assumption that a bunch interacts with an equipment in exactly the same way as a coaxial cable (i.e. a wire inside the equipment):

* It should still be investigated exactly what are the conditions for these measurements.

Ultra-relativistic beam field

TEM mode coax waveguide

See A.Mostacci: http://pcaen1.ing2.uniroma1.it/mostacci/wire_method/care_impedance.ppt

At low frequency only the fundamental TEM mode exists. So this is the only one we can measure at low frequency. At high frequencies we must continue to measure only the fundamental TEM mode in order to have self consistent results. This means that as soon as the frequency becomes so high that other modes appears e.g. TE11 or TM01 modes; they should be damped or prevented from existing at all.

The first mode to damp is the TE11 mode at: This mode does not move longitudinally. It is not really damped, but is prevented from existing by placing the wire in the exact centre of the round vacuum chamber. In this way it is not excited.

The second mode to appear is TM01 at: This mode travels longitudinally and is damped by placing microwave lossy foam at the end plates. All modes, different from the fundamental TEM node, are called higher order modes.

Microwave lossy foam

Removes reflections between the DUT and the Network Analyzer. These reflections originates because the termination of the Network Analyzer is not exactly 50

Thank you for the attention!