# Presentation: 143745

## Feedback 101

**Stuart Henderson**

**August 30, 2004**

## Outline

- Introduction to Feedback
- Block diagram
- Uses of feedback systems (dampers, instabilities, longitudinal, transverse
- System requirements
- Resources
- Simplest feedback system scheme
- Ideal conditions
- Eigenvalue problem and solution
- Loop delay, delayed kick
- Closed-orbit problem
- Filtering schemes (analog/digital)
- Two turn filtering scheme
- Type of digital filters (FIR, IIR)

- Kickers
- Concepts
- Dp and dtheta calculation
- Figures of merit
- Plots of freq response, etc.
- Complete System Response
- Estimates for damping e-p

## Resources

- Several good overviews and papers on feedback systems and kickers:
- Pickups and Kickers:
- Goldberg and Lambertson, AIP Conf. Proc. 249, (1992) p.537
- Feedback Systems:
- F. Pedersen, AIP Conf. Proc. 214 (1990) 246, or CERN PS/90-49 (AR)
- D. Boussard, Proc. 5th Adv. Acc. Phys. Course, CERN 95-06, vol. 1 (1995) p.391
- J. Rogers, in _Handbook of Accelerator Physics and Technology_, eds. Chao and Tigner, p. 494.

## Why Do We Need Feedback Systems?

- High intensity circular accelerators eventually encounter collective beam instabilities that limit their performance
- Once natural damping mechanisms (radiation damping for e+e- machines, or Landau damping for hadron machines) are insufficient to maintain beam stability, the beam intensity can no longer be increased
- There are two potential solutions:
- Reduce the offending impedance in the ring
- Provide active damping with a *Feedback System*
- A Feedback System uses a beam position monitor to generate an error signal that drives a kicker to minimize the error signal
- If the damping rate provided by the feedback system is larger than the growth rate of the instability, then the beam is stable.
- The beam intensity can be increased until the growth rate reaches the feedback damping rate

## Types of Feedback Systems

**Feedback Systems**** are used to damp instabilities**
- Typical applications are bunch-by-bunch feedback in e+e- colliders, hadron colliders to damp multi-bunch instabilities
**Dampers**** are used to damp injection transients, and are functionally identical to feedback systems**
- These are common in circular hadron machines (Tevatron, Main Injector, RHIC, AGS, ...)
**Feedback systems and Dampers are used in all three planes:**
- Transverse feedback systems use BPMs and transverse deflectors...
- Longitudinal feedback systems use summed BPM signals to detect beam phase, and correct with RF cavities, symmetrically powered striplines,...

## Elements of a Feedback System

**Basic elements:**
- Pickup
- Signal Processing
- RF Power Amplifier
- Kicker
**Pickup is BPM for transverse, phase detector for longitudinal**
**Processing scheme can be analog or digital, depending on needs**
**Transverse Kicker:**
- Low-frequency: ferrite-yoke magnet
- High-frequency: stripline kicker
**Longitudinal Kicker can be RF cavity or symmetrically powered striplines**

- Kicker

- Pickup

- RF amp

- Signal Processing

- Beam

## Specifying a Feedback System

**Feedback systems are characterized by**
- Bandwidth (range of relevant mode frequencies)
- Gain (factor relating a measured error signal to output corrective deflection)
- Damping rate
**In order to specify a feedback system for damping an instability, we must know**
- Which plane is unstable
- Mode frequencies
- Growth rates
**RF power amplifier** is chosen based on required bandwidth and damping rate.  Typical systems use amplifiers with 10-100 MHz bandwidth, and 100-1000W output power.

## Simple picture of feedback

- Take simple (but not very realistic) situation:
- -functions at pickup and kicker are equal
- 90 phase advance between kicker & pickup
- Integer tune

- X

- X

- Position measurement (coordinates x, x’)

- Kick (coordinates y, y’)

- System produces a kick proportional to the measured displacement:
- At the kicker:
- At the BPM after 1 turn:

## Simple picture of feedback, continued

- So x-amplitude after 1 turn has been reduced by
- Giving a rate of change in amplitude:
- Giving a damping rate:
- But, we made two gross simplifications:
- We don’t really operate with integer tune.  Averaging over all arrival phases gives a factor of two reduction:
- In real life, we may not be able to place the BPM and kicker 90 degrees apart in phase, and the locations will not have equal beta functions.  We need a realistic calculation.

## Realistic damping rate calculation for simple processing

- Follow Koscielniak and Tran
- Coordinates at pickup are (xn,xn) on turn n
- Coordinates at kicker are (yn,yn) on turn n
- Transport between pickup and kicker has 2x2 matrix M1 and phase 1
- Transport between kicker and pickup has 2x2 matrix M2 and phase 2
- Give a kick on turn n proportional to the position measured on the same turn:
- Where *G is the feedback gain*

- Pickup (x,x)

- Kicker (y,y)

**M****1, ********1**

**M****2, ********2**

## Simple processing, cont’d

- The coordinates one turn later follow from:

## More realistic damping rate calculation, cont’d

- After n turns the coordinates are
- This is an eigenvalue problem with solution
- The eigenvalues can be obtained from

- One-turn matrix

## General solution for 2x2 real matrix 

- Giving,

- Since we have a 2x2 real matrix, we expect two eigenvalues which are complex conjugate pairs.  Writing
- Where we can identify  as the damping rate (per turn), and  as the tune, *which in general will be modified by the feedback system*
- Solution:

## Damping rate and tune shift for simple processing

- We have
- With p, p the twiss parameters at the pickup, k, k at the kicker,  the tune, 1 the phase advance between pickup and kicker, 2 the phase advance from kicker around the ring to pickup:
- Finally,

## Damping rate and tuneshift for small damping

- For weak damping,
- And
- Optimal damping rate, and no tuneshift results for 1=90 degrees

- turns-1

- sec-1

- radians

## Damping vs. Gain for 1=90 degrees

## Tuneshift vs. Gain for 1=125 degrees

## Finite Loop Delay

- Up to this point we have ignored the fact that it takes time to “decide” on the kick strength in the processing electronics
- It is not necessary to kick on the same turn
- We can kick m turns later:
- In this way we can “wait around” for the optimum turn to provide the optimum phase

## Closed-Orbit Problem: the 2-turn filter

- Our simplification ignores another problem:
- A closed orbit error in the BPM will cause the feedback system to try to correct this closed orbit error, using up the dynamic range of the system
- Solution:
- Analog: a self-balanced front-end
- Digital: Filter out the closed-orbit by using an error signal that is the difference between successive turns
- 2-turn filter constructs an error signal:

## 2-turn filter, cont’d

- With
- The transfer function of the filter is:
- This gives a “notch” filter at all the rotation harmonics, which are the harmonics that result from a closed orbit error

## 2-turn Filter Frequency Response

## 2-turn Filter Phase

## Kickers for Transverse Feedback Systems

- For low frequencies (< 10 MHz), it is possible to use ferrite-yoke magnets, but the inductance limits their bandwidth
- Broadband transverse kickers usually employ stripline electrodes
- Stripline electrode and chamber wall form transmission line with characteristic impedance ZL

## Stripline Kicker Layout

- +VL

- ZL

- ZL

- -VL

- ZL

- ZL

- Beam

*l*

- d

## Stripline Kicker Schematic Model

- VK

- Zc

- Beam In

- p

- p+ p

- Beam Out

## Stripline Kicker Analysis

- Deflection from infinite parallel plates over length *l*, separated by distance *d*, at opposite DC voltages, +/- V is:
- We need to account for the finite size of the plates (width w, separation d).  A geometry factor g  1 is introduced:
- Because we want to damp instabilities that have a range of frequencies, we will apply a time-varying potential to the plates V().
- We need to calculate the deflection as a function of frequency and beam velocity.

- +V

- -V

## Stripline g

## Deflection by Stripline Kicker

- Stripline kicker terminated in a matched load produces plane wave propagating in +z direction between the plates.
- For beam traveling in +z direction:
- For beam traveling in –z direction:
**For relativistic beams, we need the beam traveling opposite the wave propagation!**

## Deflection by Stripline Kicker

- Where
- This can be written in phase/amplitude form as:

## Powering the Stripline Kicker

- For transverse deflection, one could
- Independently power each stripline with its own source
- Power the pair of striplines from a single RF power source by splitting (e.g. with a 180 degree hybrid to drive electrodes differentially)
- Using a matched splitting arrangement, the delivered power is:
- Which equals the power dissipated on the two stripline terminations:
- So that the input voltage is:

## Figures of Merit for Stripline Kickers

- One common figure of merit seen in the literature is the *Kicker Sensitivity.*
- From which we get:
- Which can be written in the form
- Important points:
- Deflection has a phase shift relative to the voltage pulse
- sin/ shows the typical transit-time factor response

## Transverse Shunt Impedance

- In analogy with RF cavities, one can define an *effective shunt impedance* that relates the transverse “voltage” to the kicker power:
- The frequency response has notches at
- For low frequencies, (  c/l)

- So after all this, what’s the kick?
- In the low frequency limit,

## Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)

## Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)

## Multiple Kickers

- For N kickers, each driven with power P,
- Where PT=NP is the total installed power
- To achieve the same deflection (damping rate) with N kickers requires only
- Example: One kicker with P1=1000W gives same kick as two kickers each driven at 250 W

## Putting it all together

- The RF power amplifier puts out full strength for a certain maximum error signal
- The system produces the maximum deflection max for a maximum amplitude xmax
- For optimal BPM/Kicker phase, the optimal damping rate is
- For a Damper systems, xmax is large enough to accommodate the injection transient
- For a Feedback system, xmax is many times the noise floor

## Parameters for an e-p feedback system

- Bandwidth:
- Treat longitudinal slices of the beam as independent bunches
- Ensure sufficient bandwidth to cover coherent spectrum
- Choose 200 MHz
- Damping time:
- To completely damp instability, we need  200 turns
- To influence instability, and realize some increase in threshold, perhaps 400 turns is sufficient
- Input parameters:
- y = 7 meters
- Xmax = 1mm
- Stripline length = 0.5 m, separation d = 0.10, w/d = 1.0
- 800 MeV

## Damping Time vs. Power at 150 MHz

## Damping Time vs. Frequency at Fixed Power