# Presentation: 575108

## Introduction to RF for Particle AcceleratorsPart 2: RF Cavities

- Dave McGinnis

## RF Cavity Topics

- Modes
- Symmetry
- Boundaries
- Degeneracy
- RLC model
- Coupling
- Inductive
- Capacitive
- Measuring
- Q
- Unloaded Q
- Loaded Q
- Q Measurements

- Impedance Measurements
- Power Amplifiers
- Class of operation
- Tetrodes
- Klystrons
- Beam Loading
- De-tuning
- Fundamental
- Transient

## RF and Circular Accelerators

- For circular accelerators, the beam can only be accelerated by a time-varying (RF) electromagnetic field.
- Faraday’s Law

- The integral

- is the energy gained by a particle with charge q during one trip around the accelerator.
- For a machine with a fixed closed path such as a synchrotron, if

- then

## RF Cavities

- New FNAL Booster Cavity

- Multi-cell superconducting RF cavity

- Transmission Line Cavity

## Cavity Field Pattern

- For the fundamental mode at one instant in time:

- Electric Field

- Magnetic Field

- Out
- In
- Wall Current

## Cavity Modes

- We need to solve only 1⁄2 of the problem

- For starters, ignore the gap capacitance.
- The cavity looks like a shorted section of transmission line

- x=L

- x=0

- where Zo is the characteristic impedance of the transmission line structure of the cavity

## Cavity Boundary Conditions

- Boundary Condition 1:
- At x=0:       V=0

- Boundary Condition 2:
- At x=L:      I=0

- Different values of n are called modes. The lowest value of n is usually called the fundamental mode

## Cavity Modes

- n=0

- n=1

## Even and Odd Mode De-Composition

## Even and Odd Mode De-Composition

- Even Mode
- No Gap Field

- Odd Mode
- Gap Field!

## Degenerate Modes

- The even and odd decompositions have the _same_ mode frequencies.
- Modes that occur at the same frequency are called degenerate.
- The even and odd modes can be split if we include the gap capacitance.
- In the even mode, since the voltage is the same on both sides of the gap, no capacitive current can flow across the gap.
- In the odd mode, there is a voltage difference across the gap, so capacitive current will flow across the gap.

## Gap Capacitance

- Boundary Condition 2:
- At x=L:

- where Cg is the gap capacitance

- Boundary Condition 1:
- At x=0:       V=0

## RF Cavity Modes

- Consider the first mode only (n=0) and a very small gap capacitance.

- The gap capacitance shifts the odd mode down in frequency and leaves the even mode frequency unchanged

## Multi-Celled Cavities

- Each cell has its own resonant frequency
- For _n_ cells there will be _n_ degenerate modes
- The cavity to cavity coupling splits these _n_ degenerate modes.
- The correct accelerating mode must be picked

## Cavity Q

- If the cavity walls are lossless, then the boundary conditions for a given mode can only be satisfied at a single frequency.
- If the cavity walls have some loss, then the boundary conditions can be satisfied over a range of frequencies.
- The cavity Q factor is a convenient way the power lost in a cavity.
- The Q factor is defined as:

## Transmission Line Cavity Q

- We will use the _fundamental mode_ of the transmission line cavity as an example of how to calculate the cavity Q.

- Electric Energy:

- Magnetic Energy:

- Both Halves:

## Transmission Line Cavity Q

- Assume a small resistive loss per unit length rL/m along the walls of the cavity.
- Also assume that this loss does not perturb the field distribution of the cavity mode.

- The cavity Q for the _fundamental mode_ of the transmission line cavity is:

- Time average

- Less current flowing along walls

- Less loss in walls

## RLC Model for a Cavity Mode

- Around each mode frequency, we can describe the cavity as a simple RLC circuit.

- Igen

- Vgap

- Leq

- Req

- Ceq

- Req is inversely proportional to the energy lost
- Leq is proportional to the magnetic stored energy
- Ceq is proportional to the electric stored energy

## RLC Parameters for a Transmission Line Cavity

- For the _fundamental mode_ of the transmission line cavity:

- The transfer impedance of the cavity is:

## Cavity Transfer Impedance

- Since:

- Function of geometry only

- Function of geometry and cavity material

## Cavity Frequency Response

- Referenced to o

- Peak of the response is at o

## Mode Spectrum Example – Pill Box Cavity

- The RLC model is only valid around a given mode
- Each mode will a different value of R,L, and C

## Cavity Coupling – Offset Coupling

- As the drive point is move closer to the end of the cavity (away from the gap), the amount of current needed to develop a given voltage must increase
- Therefore the input impedance of the cavity as seen by the power amplifier decreases as the drive point is moved away from the gap

## Cavity Coupling

- We can model moving the drive point as a transformer
- Moving the drive point away from the gap increases the transformer turn ratio (n)

- Igen

- Vgap

- Leq

- Req

- Ceq

## Inductive Coupling

- For inductive coupling, the PA does not have to be directly attached to the beam tube.
- The magnetic flux thru the coupling loop couples to the magnetic flux of the cavity mode
- The transformer ratio n = Total Flux / Coupler Flux

- Out
- In
- Wall Current

- Magnetic Field

## Capacitive Coupling

- If the drive point does not physically touch the cavity gap, then the coupling can be described by breaking the equivalent cavity capacitance into two parts.

- Vgap

- As the probe is pulled away from the gap, C2 increases and the impedance of the cavity as seen by the power amp decreases

## Power Amplifier Internal Resistance

- So far we have been ignoring the internal resistance of the power amplifier.
- This is a good approximation for tetrode power amplifiers that are used at Fermilab in the Booster and Main Injector
- This is a bad approximation for klystrons protected with isolators
- Every power amplifier has some internal resistance

## Total Cavity Circuit

- Vgap

- Leq

- Req

- Ceq

- Rgen

- Igen/n

- Leq

- Req

- Ceq

- n2Rgen

- Vgap

- Total circuit as seen by the cavity

- Igen

## Loaded Q

- The generator resistance is in parallel with the cavity resistance.
- The total resistance is now lowered.
- The power amplifier internal resistance makes the total Q of the circuit smaller (d’Q)

**Loaded Q**
**Unloaded Q**
**External Q**

## Cavity Coupling

- The cavity is attached to the power amplifier by a transmission line.
- In the case of power amplifiers mounted directly on the cavity such as the Fermilab Booster or Main Injector, the transmission line is infinitesimally short.
- The internal impedance of the power amplifier is usually matched to the transmission line impedance connecting the power amplifier to the cavity.
- As in the case of a Klystron protected by an isolator
- As in the case of an infinitesimally short transmission line

## Cavity Coupling

- Look at the cavity impedance from the power amplifier point of view:

- Vgap/n

- Leq/n2

- n2Ceq

- Rgen

- Igen

- Zo

- Req/n2

- Assume that the power amplifier is matched (Rgen=Zo) and define a coupling parameter as the ratio of the real part of the cavity impedance as seen by the power amplifier to the characteristic impedance.

**under-coupled**
**Critically-coupled**
**over-coupled**

## Which Coupling is Best?

- Critically coupled would provide maximum power transfer to the cavity.
- However, some power amplifiers (such as tetrodes) have very high internal resistance compared to the cavity resistance and the systems are often under-coupled.
- The limit on tetrode power amplifiers is dominated by how much current they can source to the cavity
- Some cavities a have extremely low losses, such as superconducting cavities, and the systems are sometimes over-coupled.
- An intense beam flowing though the cavity can load the cavity which can effect the coupling.

## Measuring Cavity Coupling

- The frequency response of the cavity at a given mode is:

- which can be re-written as:

## Cavity Coupling

- The reflection coefficient as seen by the power amplifier is:

- This equation traces out a circle on the reflection (u,v) plane

## Cavity Coupling

- Right edge (=0)        =

## Cavity Coupling

- The cavity coupling can be determined by:
- measuring the reflection coefficient trajectory of the input coupler
- Reading the normalized impedance of the extreme right point of the trajectory directly from the Smith Chart

- 1

- 0.5

- 0

- 0.5

- 1

- 1

- 0.5

- 0.5

- 1

## Cavity Coupling

- Under coupled

- Critically coupled

- Over coupled

## Measuring the Loaded Q of the Cavity

- The simplest way to measure a cavity response is to drive the coupler with RF and measure the output RF from a small detector mounted in the cavity.
- Because the coupler “loads” the cavity, this measures the loaded Q of the cavity
- which depending on the coupling, can be much different than the unloaded Q
- Also note that changing the coupling in the cavity, can change the cavity response significantly

- S21

## Measuring the Unloaded Q of a Cavity

- If the coupling is not too extreme, the loaded and unloaded Q of the cavity can be measured from reflection (S11) measurements of the coupler.

- At:

- Unloaded Q!

- For:

- where:

- Circles on the Smith Chart

## Measuring the Unloaded Q of a Cavity

- Measure frequency (-) when:
- Measure resonant frequency (o)
- Measure frequency (+) when:
- Compute

## Measuring the Loaded Q of a Cavity

- Measure the coupling parameter (rcpl)
- Measure the unloaded Q (Qo)

## NEVER – EVER!!

## Bead Pulls

- The Bead Pull is a technique for measuring the fields in the cavity and the equivalent impedance of the cavity as seen by the beam
- In contrast to measuring the impedance of the cavity as seen by the power amplifier through the coupler

## Bead Pull Setup

## Bead Pulls

- In the capacitor of the RLC model for the cavity mode consider placing a small dielectric cube
- Assume that the small cube will not distort the field patterns appreciably
- The stored energy in the capacitor will change

- Total original Electric energy

- Original Electric energy in the cube

- new Electric energy in the cube

- Where Ec is the electric field in the cube
- dv is the volume of the cube
- o is the permittivity of free space
- r is the relative permittivity of the cube

## Bead Pulls

- The equivalent capacitance of the capacitor with the dielectric cube is:

- The resonant frequency of the cavity will shift

## Bead Pulls

- For  << o and C << Ceq

## Bead Pulls

- Had we used a metallic  bead (r>1) or a metal bead:
- Also, the shape of the bead will distort the field in the vicinity of the bead so a geometrical form factor must be used.
- For a small _dielectric_ bead of radius a
- For a small _metal_ bead with radius a

- A metal bead can be used to measure the E field only if the bead is placed in a region where the magnetic field is zero!

## Bead Pulls

- In general, the shift in frequency is proportional to a form factor *F*

- Dielectric bead

- Metal bead

## Bead Pulls

- From the definition of cavity Q:

## Bead Pulls

- Since:

## Bead Pulls

- For small perturbations, shifts in the peak of the cavity response is hard to measure.
- Shifts in the phase at the unperturbed resonant frequency are much easier to measure.

## Bead Pulls

- Since:

## Bead Pulls

## Power Amplifiers - Triode

- The triode is in itself a miniature electron accelerator
- The filament boils electrons off the cathode
- The electrons are accelerated by the DC power supply to the anode
- The voltage on the grid controls how many electrons make it to the anode
- The number of electrons flowing into the anode determines the current into the load.
- The triode can be thought of a voltage controlled current source
- The maximum frequency is inversely proportional to the transit time of electrons from the cathode to the anode.
- Tetrodes are typically used at frequencies below 300 MHz

## Tetrodes

- Courtesy of Tim Berenc

## Klystrons

- The filament boils electrons off the cathode
- The velocity (or energy) of the electrons is modulated by the input RF in the first cavity
- The electrons drift to the cathode
- Because of the velocity modulation, some electrons are slowed down, some are sped up.
- If the output cavity is placed at the right place, the electrons will bunch up at the output cavity which will create a high intensity RF field in the output cavity
- Klystrons need a minimum of two cavities but can have more for larger gain.
- A Klystron size is determined by the size of the bunching cavities.
- Klystrons are used at high frequencies (>500 MHz))

## Klystrons

## Traveling Wave Tube

- Cutaway view of a TWT. (1) Electron gun; (2) RF input; (3) Magnets; (4) Attenuator; (5) Helix coil; (6) RF output; (7) Vacuum tube; (8) Collector.

## Traveling Wave Tubes

- Traveling wave tubes (TWTs) can have bandwidths as large as an octave (fmax = 2 x fmin)
- TWTs have a helix which wraps around an electron beam
- The helix is a slow wave electromagnetic structure.
- The phase velocity of the slow wave matches the velocity of the electron beam
- At the input, the RF modulates the electron beam.
- The beam in turn strengthens the RF
- Since the velocities are matched, this process happens all along the TWT resulting in a large amplification at the output (40dB = 10000 x)

## Power Amplifier Bias

- The power amplifier converts DC energy into RF energy.
- With no RF input into the amplifier, the Power amplifier sits at its DC bias.
- The DC bias point is calculated from the intersection of the tube characteristics with the outside load line

## Class A Bias

- In Class A, the tube current is on all the time – even when there is no input.
- The tube must dissipate
- The most efficient the power amplifier can be is 50%

## Class B Bias

- In Class B, the tube current is on 1⁄2 of the time
- The tube dissipates no power when the there is no drive
- The output signal harmonics which must be filtered
- The efficiency is much higher (>70%)