cavity bpm tutorial kek march 2005

8/12/2019 Cavity BPM Tutorial KEK March 2005

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nanoBPM Meeting, KEK, March 2005 3

Excursion Into Waveguides (1)

► electric field is shown with red lines,magnetic with blue ones

► wave is Transverse Electric - the electricfield has no longitudinal component (insome literature it is marked as H-wave)

► the direction of the propagation is givenby E x H

► nodes and antinodes of transversalcomponents of E and H coincide in caseof vacuum filling

The electromagnetic field is known topropagate through a waveguide as awave (or a mixture of a few waves)with a fixed configuration. Thisconfiguration depends on thefrequency of oscillations, waveguidetype and excitation type.




Excursion Into Waveguides (2)

► indexes show the numbers ofantinodes of the field for both axes- x, y for a rectangular, φ, r for acircular waveguide

► the number of antinodes for the φdirection is a doubled index (thefield must be continuous among φ)

► magnetic coupling uses a loop acting to the magnetic field. The coupling strengthdepends on the magnetic flux through the loop i.e. inductivity of the loop

► electric coupling uses an antenna , the coupling depends on its capacitance

► electromagnetic coupling is a sum of two – electric and magnetic, they maysometimes even cancel each other




Circular Waveguide

We solve the wave equation

in the cylindrical coordinate system

Look for the solutions in a form

integrating by parts. Solutions are:

The boundary condition gives the critical k:

Transversal componentsfollow from the Maxwell’s equations




Cylindrical Cavity Resonator

A cylindrical cavity is a piece of a circular waveguide cuttransversally with conductive planes at z=0 and z=L. At these planes the sum of the transversal componentsof the electric field has to be 0:

This boundary conditionsays us that

In that way we get the equations describing all thestanding waves possible in the cavity…

…called eigenmodes andcoinciding frequencies called

eigenfrequencies




Dipole Mode

A bunch propagating through the cavity interactswith its eigenmodes exciting electromagneticoscillations in the cavity.The excitation of the modes, which have a node atr=0, is very sensitive to the beam offset, what isused for the beam position detection.The first dipole mode TM110 is used because it is the

strongest one among the others.The phase of the excited field depends on thedirection of the offset.




Useful Definitions

L R C

n

Z

It is convenient to represent acavity as an RLC circuit, usuallyloaded to external load by meansof an ideal transformator.The impedance R is called theshunt impedance.

The voltage in the cavity iscalculated among a certain path,beam trajectory in our case.

The internal quality factor isintroduced to indicate the decayof the oscillations due to the

losses in the cavity walls.The external quality factorindicates the decay due to thepower coupled out of the cavity.




Single Bunch Excitation (1)

The excitation is proportional to thevoltage seen by the bunch

The energy given by the bunch to themode n is

The voltage excited in the cavity is twotimes higher

We use the definition of the normalizedshunt impedance

and get the excited voltage and storedenergy as

Using the definition of the external Q weget the output power

With we get the output voltage




Single Bunch Excitation (2)

The dipole mode electric field in the cavity

Extension to the beam pipe region

Fit both fields in order to get constants at r=a

Using an integral we get

And the field in the beampipe is

We need the voltage, so integratingand using * again we get

The voltage is linear vs. offset!




Excitation Summary

► The bunch excites the eigenmodes of the cavitypassing through it

► The dipole mode excitation has a significantdependence on the beam offset, the phase depends on

the offset direction► The excited signal decays exponentially, depending on

how much power is lost in the walls and coupled out

► The excitation is almost linear in the beam pipe range




Multibunch Excitation

Exponential decay of the energy stored inthe cavity is given by

Were the loaded Q value is used. It takesinto account walls losses and output power

If the mode frequency is a harmonic of thebunch repetition rate, an infinite bunchtrain produces a voltage

t

t

V

V

The sum of this series is

The error can be calculated as

A fixed error gives a high limit for theloaded Q value




Beam Incline Impact

Incline component of the dipole mode isexcited if the beam trajectory is inclinedwith respect to the z axis of the cavity.We compare the excitation calculating thevoltages for the both cases.

Approximating the Bessel function we get

The ratio of the voltages does notdepend on the bunch charge

Equivalent offset for a 5.5 GHz cavity(x’ = 0.5 mrad)




Monopole Modes Impact

Monopole modes have thehighest excitation among allother modes. The difference tothe dipole mode excitationmaybe 100 dB and more. Thefirst two monopole modessurround the dipole moderesonance.

Due to the finite Q values these modeshave components at the dipole mode

frequency. These components can notbe filtered out and need a modeselective solution. A mode selectivecoupling realizing the difference in thefield structure of dipole and monopolemodes is used in all the latest designs.




Polarization and Cross-Talk

The excited dipole mode field can be repre-sented as a combination of two polarizations.

Need to align the polarizations to x, yand separate them in frequency.




Thermal NoiseThe spectral noise power density integrated over the bandwidth of the narrowest filterin the electronics gives us the level of the noise component:

Following the path of the signal in the electronics and taking into account the lossesand the internal noise of the electronics we can estimate the resolution limit:

The final estimation has to take into account also the discretization noise.




Impacts Summary

► The energy stored in the cavity decays exponentially. Ifthe decay is not fast enough, the previous bunch signalcontributes to the next bunch signal.

► An inclined beam excites the dipole mode even if it

passes through the centre. The phase difference betweenposition and incline components is 900.

► Monopole modes are strongly excited and thereforegenerate large backgrounds.

►

Asymmetries cause a coupling between x and y signals.




Analog Signal Processing

The readings are waveforms in GHz range, so we need adownconversion electronics. Basically, two methods areavailable:

►homodyne receiver►heterodyne receiver.

An accurate direct conversion is not possible because ofthe high frequency.




Homodyne Receiver

The signal is downconverted to the “directcurrent” in one stage. Just a few componentsare needed, the losses are low.

HR is very sensitive to the isolationsbetween LO and RF ports of the mixer.I/Q mixer is usually used.




Heterodyne Receiver

Downconversion is realized in several stages. That gives a better possibility for thefiltering and amplification of the signal. The mirror frequency issue does not seem tobe really dangerous in our case.




I’m afraid that’s all I can say…

►Check also

http://www.hep.ucl.ac.uk/~liapine/

cavity bpm tutorial kek march 2005

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