LLRF feedback control system (Basics) feedback control system (Basics) Feng QIU, JAS17, ......

LLRF feedback control system

(Basics)

Feng QIU, JAS17, Hayama, Kanagawa, Japan, 2017 1

RF & Accelerators Where is RF system (and LLRF system) in an accelerator?

I. The Radio Frequency (RF) systems in particle accelerators are the hardware

complexes devote to the generation of the accelerating field.

II. RF system ：High Level RF (e.g. Klystron, waveguide, coupler…)+ control loops (LLRF).

Low Energy Beam High Energy Beam

RF System

Cav

LLRF

KlystronFeedBack

Beam

Target

RF system is very important in the accelerators!

Cavity

2Feng QIU, JAS17, Hayama, Kanagawa, Japan, 2017

I. Stabilizing the RF field

(field control)

II. Minimize the input power

(resonance control)


Introduction and Motivation

Introduction & Motivation

What is LLRF system?

Why we need LLRF feedback system?

Control theory (feedback and feedforward)

What is feedback and feedforward.

Transfer function and Bode plot.

Modeling of LLRF system

Cavity model and system model.

What will influence the performance of LLRF?

Whether the system is stable or not?

LLRF technology (will be talked by Prof. FANG in this school)

Field detection.

Field control and resonance control.

Examples

Main content



RF

Source Power

Supply

Master Oscillator

Pick-up

Waveguide

Beam

Crymodule

ADC

DAC DAC

Filed Detection

Feedback and Feedforward

Digital Signal Processing

Interlock

I/Q Modulator

Baseband

Clock

Distribution

Clock

Local Oscillator (LO)

RF

Pf

Pr

ADC ADC

Low-pass Filter

Intermediate Frequency (IF)

Pf Pr

Cavity

RF

Pre-amplifier

Cryogenic System Vacuum System

FPGAEPICSI

OC

Ethernet

Gate

Low level RF

HLRF & LLRF

High level RF

Cavity


RF

Source Power

Supply

Master Oscillator

Pick-up

Waveguide

Beam

Crymodule

ADC

DAC DAC

Filed Detection



Interlock

I/Q Modulator

Baseband

Clock

Distribution

Clock


RF

Pf

Pr

ADC ADC

Low-pass Filter


Pf Pr

Cavity

RF

Pre-amplifier


FPGAEPICSI

OC

Ethernet

Gate

ADC: LTC 2208

Xillinx Virtex5 FPGA

ADC & DAC Interface

Digital I/O

uTCA Digital Board

HLRF & LLRF

Digital Board

Cavity

Klystron

Waveguide


Why LLRF？

We have lots of disturbances…

I. High voltage power supply

ripples

II. Lorentz detuning

III. Beam-loading

IV. Microphonics

V. Phase noise of MO

VI. Drift of temperature and

humidity

RF

Source Power

Supply

Master Oscillator

Pick-up

Waveguide

Beam Loading

Crymodule

ADC

DAC DAC

I/Q Modulator

Baseband

Clock

Distribution

Clock


RF

Pf

Pr

ADC ADC

Low-pass Filter


Pf Pr

Cavity

RF

Pre-amplifier

Ripples


Microphonics

Ethernet

LFD Effects

Filed Detection



Interlock

FPGAEPICSI

OCGate


Examples (PS Ripples)

We have lots of disturbances…

I. High voltage power supply

ripples

II. Lorentz detuning

III. Beam-loading

IV. Microphonics

V. Phase noise of MO

VI. Drift of temperature and

humidity

RF

Source Power

Supply

Master Oscillator

Cavity Signal

Waveguide

Beam Loading

Crymodule

ADC

DAC DAC

I/Q Modulator

Baseband

Clock

Distribution

Clock


RF

Pf

Pr

ADC ADC

Low-pass Filter


Pf Pr

Cavity

RF

Pre-amplifier

Ripples


Microphonics

Ethernet

LFD Effects

Filed Detection



Interlock

FPGAEPICSI

OCGate

Disturbances

Vc



RF

Source Power

Supply

Pick-up

Waveguide

Beam Loading

Crymodule

Cavity

Ripples

Ripples

Actual caseIdeal case

Klystron output phase

Phas

e

Phas

e

The output voltage of the

power supply usually

contains some frequency

components (High voltage

power supply ripples).

RF

Source Power

Supply

Pick-up

Waveguide

Beam Loading

Crymodule

Cavity

Ripples

cV t



Klystron output phase

Spectrum of Cavity Voltage

Pick-up

Beam Loading

Crymodule

Cavity

cV t


Why LLRF？

Amp

Phase

W/O LLRF (not “flat”) With LLRF (“flat” voltage)

cV t

cV t

cV t

cV t

Not flat

We want them flat

We need a “flat” cavity voltage for beam accelerating.

Phase Stability

Requirement of RF system

E.g.: ΔA/A<0.1% rms,

Δθ<0.1 deg. rms

θ

Amplitude StabilityAΔA

Δθ

12

Amp

CW (Continuous Wave) mode (Continuous RF field), all the time there is RF.

Pulsed Mode: Only amount of time is used for beam acceleration.

Pulse mode and CW mode

International Linear Collider (ILC)

Amp RF pulseCW


Control Theory

What is a control system? A control system is a device, or set of devices, that manages, commands,

directs or regulates the behavior of other devices or systems. Industrial control

system are used in industrial production for controlling equipment or machines.

(Definition from Wikipedia).

A control system mainly include a plant, a sensor and a controller.

Requirements: maintain some characteristics or behavior of the “plant”.

Systems or

devices that need

to be controlled

A device that regulate or control the

behavior of the “Plant”


Measuring tool that measured

the response of the “Plant”

PlantController

Sensor

PlantController

Sensor

Main components (Examples)

Plant: Cavities, power source,

RF Gun, antenna,…

Sensor: Phase detector,

amplitude discriminator,

or FPGA

Controller: Electrical control

phase shifter or attenuator,

FPGA

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Cavity

Examples: Digital low-level RF system.

The field-programmable-gate-array (FPGA) plays roles of both controller and

sensor (core component, will be introduced later).

Requirements: making the RF field inside the cavity stable (most significant

goal).

FPGA

15

Target: Stabilize

the field inside

the cavity

Feng QIU, JAS17, Hayama, Kanagawa, Japan, 2017

Close Loop vs. Open loop Open loop control systems vs. closed loop control systems.

In open loop control systems output is generated based on feedforward (FF).

In close loop control systems current output is taken into consideration and

corrections are made based on feedback. A closed loop system is also called a

feedback (FB) control system (from wiki).

To analyze how does a FB control system works, let’s start from “Transfer

function”.

PlantFF

Sensor

PlantFB

Controller

Sensor

Reference

Close loop Open loop

Open loop Closed loop

Advantages Generally stable

Easier construction Good disturbances rejection

Disadvantages Poor disturbances rejection Unstable risk

Difficult construction 16

Feedback

Transfer function A Transfer Function (TF) is the ratio of the output of a system to the input of

a system.

The X(s) and Y(s) are the Laplace-transform of the input/output signal,

respectively.

Key point: The transfer function H(s) includes every information of the

system (usually can be seen as a representation of a system model). i.e. if

we know the transfer function H(s) of a specified system, we can calculate the

output Y(s) by any input X(s).

System

Y(s)

X(s)

, .

Y sH s Y s H s X s

X s

Time domain

Complex freq.

domain

Freq. domain

h t

H s

H j

Laplace-transform

Fourier-transform

Transfer function


TF & Frequency response Frequency response It is a measure of magnitude and phase of the output as a

function of frequency.

From TF to transfer function: H(s)→H(jω).

Key point: Direct magnitude (amplitude) and phase response for a specified

sinusoidal input.

H(s)

Y(s)

X(s)

j H j

s jH s H j H j e

: sin 2 50 ,

: sin 2 50

If input t

Then output H j t H j

E.g. a 50 Hz sinewave

Magnitude gain Phase delayStill 50 Hz, but,…

18

20 ms 20 ms


Examples： H(s) & H(jω)

Cavity transfer function

1

sC

Ls

R

2

2

1 1 1

1 1

s

Ls RLCs R CsC Z sZ R Ls RLs

s sRC LC

0

02 2 2 20

0 0

1,

1

Rs sC QCQ R Z sL LC

s s s sRC Q

Quality factor or Q value

Resonance radian frequency

1C

L

XsC

X Ls

19

Examples： H(s) & H(jω) cnt’d

Cavity transfer function

1

sC

Ls

R

0 0 0

22 2 2 2 2 20 0 0 0

0 0 0

1

Rs s j R j RQ Q QCZ s Z j

s s s s j jRC Q Q Q

0 , , ,if then Z j R cavity is on resonance

Resonance radian frequency

1C

L

XsC

X Ls


Bode Plot

Bode diagram ( Plots of the amplitude-frequency and phase-frequency

response of the system)

0

2 2 2 200 0

1

Rs sQCZ s

s s s sRC Q

9

0 0

6

. . 2 2 1.3 10 [ / ]

1.3 10

E g f rad s

Q

Bode Plot

• Magnitude vs. frequency

• Phase vs. frequency

• Matlab function: bode(sys)

21

Z j

Z j

Z(s)

Y(s)

X(s)

Bode plot (cont’d)

Bode Plot (Example1)

• 𝜔0 = 1.3 ∙ 2𝜋 [𝑟𝑎𝑑/𝑠]• 𝑄 = 1.3 × 106


• 𝜔0 = 1.3 ∙ 2𝜋 [𝑟𝑎𝑑/𝑠]• 𝑄 = 7000

• The half power point of an electronic

amplifier stage is that frequency at which

the output power (not voltage) has dropped

to half of its mid-band value. That is a level

of -3 dB.

• 3-dB Bandwidth (BW):

• Example1: BW1=100 [Hz], Example2:

BW2=186 [kHz]

-3 dBBandwidth

Bandwidth

𝑓1𝑓2

𝑓2 − 𝑓1

22

90 deg.

90 deg.

Bode plot (cont’d)Bode Plot (Example1)

• 𝜔0 = 1.3 × 109 ∙ 2𝜋 [𝑟𝑎𝑑/𝑠]• 𝑄1 = 1.3 × 106

• 𝐵𝑊1 = 100 [𝐻𝑧]


• 𝜔0 = 1.3 × 109 ∙ 2𝜋 [𝑟𝑎𝑑/𝑠]• 𝑄2 = 7000• 𝐵𝑊2 = 186000 [𝐻𝑧]

0 03 0,

2dB

fBW f

Q

𝐵𝑊1 = 100 [𝐻𝑧]

𝐵𝑊2 = 186000 [𝐻𝑧]

𝑄2 = 7000𝑄1 = 1.3 × 106

Resonance frequency

23

𝐶𝑎𝑠𝑒1

𝐶𝑎𝑠𝑒2

𝑍𝑜𝑜𝑚

𝐵𝑊1

SC cavity NC cavity

Case1

Case2


Modeling (TF) of LLRF system

Cavity Model

25

Cavity is like a parallel resonance circuit.

Detuning:

1

sC

Ls

R

1C

L

XsC

X Ls

frequency

2H j f

0

RF

RF

0

2 200

RF

Rs

QP s

s sQ

Band-pass Filter

dt

dIRV

dt

dV

dt

VdLc

cc2/1

2

02/12

2

22

0

0.5

TFDiff. equation

0 RF

RF vs. Baseband

i t

c cV V e cV

2

RF

dt

dIRV

dt

dV

dt

VdLc

cc2/1

2

02/12

2

22

0.5 0.5c

c

dVj V u

dt

RFj t

e

NOT interesting, We do not care

We are interested in the

base band (envelope).

LLRF directly regulate the

base-band.


First order

Second order

33

0 RF

Cavity Model (RF to Baseband)

27

In the frequency domain.

1

sC

Ls

R

1C

L

XsC

X Ls

frequency

2H j f

0

If detuning exist

0

2 200

RF

Rs

QP s

s sQ

0.5

0.5

c

cav

V sP s

U s s j

0.5 0.5

cc

dVj V u

dt

RF ( )Baseband (0 Hz)

0

t

RF

RF

Cavity Model (detuning=0)

28

If there is no detuning, we have the simplified cavity model.

1

sC

Ls

R

1C

L

XsC

X Ls

frequency

2H j f

0 1.3RFf GHz

0

2 200

RF

Rs

QP s

s sQ

0.5

0.5

baseP ss

If detuning=0

Low-pass Filter

Band-pass Filter

0 RF

0

RFbaseband

PI controller PI control is very popular in the FB control system (& LLRF FB control system)

PI Controller

1 I

P

Y s KK s K

E s s


e tPK

IK

y t PK e t

P P Iy t K e t K K e t dt

P P I

E sY s K E s K K

s

Lapalce Transform

F sf t dt

s

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Usually performed in

the FPGA (or DSP)

1/ s

Sensor (IQ detector)


ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Usually performed in

the FPGA (or DSP)

F

F

F ss

0.5F

Cavity half bandwidth

low pass filter

detector -3dB bandwidth

Detector can be also seen as a low pass filter but with higher bandwidth

than cavity.

Analytical Study (components) Basically, we have all of the LLRF components.

F

F

F ss

31

1 IP

KK s K

s

PlantController

Sensor

0.5

0.5

0baseP ss


Loop delay

Usually, system need a time to have a response (NOT immediately).

Time delay exists in the majority control system (also in LLRF system)

T(s)P(s)In Out

Step-in Response

Step-in

Response

: dT sTF of dead time T s e

: dDelays T


dsT

d

Lapalce Transform

f t T e F s

Analytical Study (components) Now, we insert the loop delay in the system.

F

F

F ss

33

1 IP

KK s K

s

PlantController

Sensor

0.5

0.5

0baseP ss


dT sT s e

Delay

TF of overall LLRF Now, the overall model can be constructed.

But, how to obtain the transfer function of the overall system?

Reference

Closed loop

IP

KK

s 0.5

0.5s

F

Fs

dT s

e

Reference

Open loop

IP

KK

s 0.5

0.5s

F

Fs

Disconnected

dT se

r

yr


1 IP

KK

s

1 I

P

KK

s

Controller

Loop Delay

Plant (cavity)

Sensor

Block diagram transformations How to calculate the TF of the whole system if we know the TF of each

subsystem?

35

1G s 1G s 2 1G s G s

1G s

1G s

1G s

2G s

1 2G s G s

1

2 11

G s

G s G s

X s Y s

X s Y s

Y s X s

2 1

Y sG s G s

X s

1 2

Y sG s G s

X s

1

2 11

Y s G s

X s G s G s

Minus

Serial

Parallel

Feedback

TF of overall LLRF Now, the overall model can be constructed.

Over all TF (open loop).

Reference

Closed loop

IP

KK

s 0.5

0.5s

F

Fs

dT s

e

Reference

Open loop

IP

KK

s 0.5

0.5s

F

Fs

Disconnected

dT se


1 IP

KK

s

1 I

P

KK

s

0.5

0.5

dT sI P P Floop

F

K K K sH s e

s s s

1G s 1G s 2 1G s G s X s Y s

TF of disturbances to cavity output

We are interesting with the TF from disturbances to cavity voltage (most

significant)?

Reference

Closed loop

IP

KK

s 0.5

0.5s

F

Fs

dT s

e

Reference

Open loop

IP

KK

s 0.5

0.5s

F

Fs

Disconnected

dT se

y

r


1 IP

KK

s

1 I

P

KK

s

Disturbances Disturbances

Cavity

outputCavity

output

d

y

dy

1G s

2G s

1

2 11

G s

G s G s Y s D s

0.5

0.5

dT sI P P Floop

F

K K K sH s e

s s s

1

1CL

loop

G sH s d y

H s


How about the TF from disturbances to cavity voltage (most significant)?

Reference

Closed loop

IP

KK

s

F

Fs

dT s

e

Reference

Open loop

IP

KK

s 0.5

0.5s

F

Fs

Disconnected

dT se

0.5

0.5s

y

r


1 IP

KK

s

1 I

P

KK

s

Disturbances

Cavity

output

d

y

dy

1G s

2G s

1

2 11

G s

G s G s Y s D s

0.5

0.5

dT sI P P Floop

F

K K K sH s e

s s s

2G s

0.5

0.5

1CL

loop

sH s d y

H s


How about the TF from disturbances to cavity voltage (most significant)?

Reference

Closed loop

IP

KK

s 0.5

0.5s

F

Fs

dT s

e

Reference

Open loop

IP

KK

s 0.5

0.5s

F

Fs

Disconnected

dT se

0.5

0.5

1CL

loop

sH s d y

H s


1 IP

KK

s

1 I

P

KK

s

0.5

0.5

OLH s d ys

Disturbances Disturbances

Cavity

outputCavity

output

d

y

dy

1G s

2G s Y s D s

1G s

Y s D s

RF

Source Power

Supply

Pick-up

Waveguide

Beam Loading

Crymodule

Cavity

Ripples

RF

Source Power

Supply

Pick-up

Waveguide

Beam Loading

Crymodule

Cavity

Ripples

cV t



F

Fs

1

IP

KK

s

cV ty

d

y

d

Closed loop Open loop

0.5

0.5s

0.5

0.5s

0.5

0.5

1CL

loop

sH s d y

H s

0.5

0.5

OLH s d ys

-

1G s

2G s Y s D s 1G s

Y s D s

Effects of the loop gain Let’s observe the frequency response.

41

1

101

1

11

1

2

1

, 0

, 0

10

1

1

CL d y loop P

P

OL d y

H j H KK

H j

0.5

0.5

0dT s Floop P I

F

H s K e Ks s

Low

KP

High

KP

0.5

0.5

1CL

loop

sH s d y

H s

0.5

0.5

OLH s d ys

OLH

TF & Frequency response Frequency response It is a measure of magnitude and phase of the output as a

function of frequency.

From TF to transfer function: H(s)→H(jω).

Key point: Direct magnitude (amplitude) and phase response for a specified

sinusoidal input.

H(s)

Y(s)

X(s)

j H j

s jH s H j H j e

: sin 2 50 ,

: sin 2 50

If input t

Then output H j t H j

E.g. a 50 Hz sinewave

Magnitude gain Phase shiftStill 50 Hz, but,…

42

20 ms 20 ms

RF

Source Power

Supply

Pick-up

Waveguide

Beam Loading

Crymodule

Cavity

Ripples



F

Fs

PK

cV ty

d

0.5

0.5s

LLRF-

0

1 10

1d y loop P

P P

H j H KK K

d yH s

d yH s

d yH s

d yH s

d yH s

Open loop

KP=1

How about KP = ∞?

KP=10

KP=100

KP=∞

KP=0

KP=1000

PS ripples (d) Vc (y)

Stability Criteria Stable is one of the most important thesis in a feedback system, if

the system is not stable, there is no meaning for any efforts.

The Stability Criteria for a feedback system includes

1. Root locus

2. Solve the characteristic equation

3. Open loop bode plot

4. Routh–Hurwitz stability criterion

All of them are important, but…

Bode plot is the most simple one in our case, it provide not only a

stability criteria but also a simple way to observe the important

specifications of a system.

Stability criteria

Open loop-based Closed loop-based

Root locus Bode plot Characteristic equation Routh–Hurwitz


Stability Criteria Definition: A stable system is a dynamic system with a bounded

response to a bounded input.


X(s)

Y(s)

H s

1 2

1 2 1 0

1 2

1 2 1 0min

m m m

m m m

n n n

n n

b s b s b s b s bnumeratorH s

deno ator s a s a s a s a

Characteristic equation

bounded inputStable

Unstable

We can not try all of the bounded input signal anyway

time time

x(t) y(t)

Stability Criteria (bode plot)


frequency

frequency

1020log 2H j f

2H j f

dB

degree

f

f

-180 deg

180f

1802 1H j f G(s): Stable

crossf

0 dB

180 deg

PlantFF

Sensor

Open loop

Find the frequency where the PHASE becomes -180 degrees (@ lower plot).

Find the GAIN, G (in dB), at this SAME FREQUENCY (@ upper plot).

If the value of G less than 0 dB, then system is STABLE, or, system is

UNSTABLE.

If time delay is 0

time delay ≠ 0 (actual case)

Stability Criteria (bode plot)


frequency

frequency

1020log 2H j f

2H j f

dB

degree

f

f

-180 deg

180f

1802 1H j f G(s): Stable

crossf

0 dB

180 deg

10 1800 20log 2G j f

180 2 crossG j f

Larger margin→ Better robustnessGain Margin:

Phase Margin

Gain margin

Phase margin

Effects of the loop gain If gain is too large, system may be unstable.

0.51 0

0.5

dT s FOL

F

H s K es s

frequency

frequency

1020log 2H j f

2H j f

dB

degreef

f

-180 deg

180f

Stable

crossf

0 dB

180 deg

0.52 0

0.5

10 dT s FOL

F

H s K es s

1 2OL OLH j H j

Phase margin

1OLH j

2OLH j20 dB


Unstable

Gain margin

d yH s

1,2OLH


Digital LLRF System

LLRF System (Review)

PlantController

Sensor

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Cavity

FPGA



FPGA



or FPGA


Digital LLRF Digital LLRF is the current tendency.

Both the sensor and controller are performed by FPGA (digital

devices)

PlantController

Sensor

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Cavity

FPGA

DAC

ADC



FPGA



or FPGA

51

:SF Sampling

SF


Z-transform Very similar with Laplace transform.

h t

H s

0

st

tH s h t e dt

Z-domainS-domain

kH z h k z

h k

sk n Th k h t

Impulse response

H z

Discrete time

Laplace-transform z-transform


sk n T

f k f t

f t

ST


LLRF technology

Field and Resonance control


Tuner(frequency

ctrl.)FB

(field ctrl.)

Piezo

Kly/SSACavity

Pf

Pick up

Pf

Pick up

Pick up

Internet

Communication

PC

Two main feedback loops in the LLRF.



Tuner(frequency

ctrl.)FB

(field ctrl.)

Piezo

Kly/SSACavity

Pf

Pick up

Pf

Pick up

Pick up

Internet

Communication

PC

2H j f

t

Optimize the detuning tominimize the klystron power

0.5 0.5c

c

dVj V u

dt



Tuner(frequency

ctrl.)FB

(field ctrl.)

Piezo

Kly/SSACavity

Pf

Pick up

Pf

Pick up

Pick up

Internet

Communication

PC

cA V

cP V

Stabilize the RF field (A&P) for beam acceleration

2H j f

t

Minimize the klystron power

0.5 0.5c

c

dVj V u

dt

“Flat”


PlantController

Sensor

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Cavity

FPGA


PlantController

Sensor

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

Tuner & Cavity

FPGA

Frequency

Control

Field Control

cSet V

Set


Field Detection

Field Detection


We must to measure the cavity voltage (amplitude and phase) at first.

, ( ) cosc RF RF RF RFV t A t

Beam

Cavity

cV tRF source

ControllerSensor

RF

Digital LLRF

ADC DAC

2RF

RF

T

Set

RF

,

RF RFj t

c RF RFV t A e

IQ

Mod

The phase error is

usually too large if

directly sampling

the RF signal.Fs

Field Detection

60

Usually, it is difficult to process the RF signal directly.

Beam

Cavity

cV tRF source

ControllerSensor

RF

Digital LLRF

ADC DAC

LO

IF

,

RF RFj t

c RF RFV t A e

,

IF RFj t

c IF RFV t A e

, 2 LO

j tc LOV t e

cos 2cos cos cosmixer RF RF RF LO RF IF RF RF LO RFV A t t A t t t

Filtered by LPF

LO RF IF

Set


IQ

Mod

Fs

Field Detection

61

Finally, we need to demodulate the baseband signal (A & φ) in the

FPGA.

Beam

Cavity

cV tRF source

ControllerSensor

RF

Digital LLRF

ADC DAC

LO

IF

, 2 LO

j tc LOV t e

LO RF IF

Set

,RF

jc base RFV t A e

Fs

,

RF RFj t

c RF RFV t A e

,

IF RFj t

c IF RFV t A e


IQ

Mod

How to get the baseband

I Q

-I -Q

Time [s]

IQ S

amp

ling:

2 2

cos.

sin

.arctan

RF RF

RF RF

RF

RF

I A

Q A

A I Q

Q I

Usually, it is more convenient to demodulate the in-phase (I) and quadrature

(Q) signal.

Simplest way: IQ sampling.

Controller

IQ dem.

Digital LLRF

ADC

DACIF

Controller DAC

I

Q

Fs

1

4

IF

s

f

F

64

Sensor

Set

Non-IQ

Nonlinearities in the analog front-end or the ADC generate harmonics, which will be aliased to the IF frequency.

Non-IQ sampling (higher precision).

Perfect

Real

4

15

IF

s

f M

F N 2

M

N


f k

IQ dem.

Non-IQ The demodulation algorithm is derived from the Fourier series.

1

0

1

0

cos

2cos 2 ,

2sin 2 .

SRF IF RF t k T

n

k

n

k

f k A t

MI f k k

N N

MQ f k k

N N

₁ z ₁ z +

CosSin

₁ z

...

₁ z

₁ z + ₁ z

...

IF

I

Q


Controller

Digital LLRF

ADC

DACIF

Controller DAC

I

Q

Fs

Sensor

Set

Up Conversion Recover the RF signal with IQ modulator.

Controller

IQ dem.

Digital LLRF

ADC

DACIF

Controller DAC

I

QFs


2 2 1

cos sin cos

, tan

RF RF RFOut I t Q t A t

QA I Q

I

90

RF

I

QIQ

Modulator

RF source

I

Q

RF

OUT-

To cavity


Field Control

IQ control vs. A&P control


A&P Control: control the amplitude and

phase of the cavity field separately

I/Q Control: control the I and Q components

of the cavity field separately

0

I

QA

Re V

Im V

IQ control vs. A&P control


• I/Q control

– Set point can cover the four quadrants including zero

– Good for the control of large errors and large beam loading

– Coupling between I/Q channels if the loop phase is wrong

• A&P control

– No loop phase problem

– If there is large phase error, the cavity output may be driven to wrong

quadrants

Closed loop

dT s

e

K(s) P(s)

F(s)

d(t)

n(t)

w(t)

y(t)

r(t)

TF of LLRF system

dT s

olH s K s P s e F s

1d y

ol

Y s P sH

D s H s

1n y

ol

N s F s K s P sH

D s H s

Transfer function of disturbances

d(t) to cavity output y(t)

Transfer function of disturbances

n(t) to cavity output y(t)


1r y

ol

K s P sH s

H

Disturbance and Noise Transfer

Parameters for the bode plot:

• Cavity detuning = 0

• Half bandwidth = 217 Hz

• Loop gain = 100

• Detector bandwidth = 500 kHz

Conclusion:

• Low frequency disturbances are

suppressed by feedback gain

(~40 dB)

• FB gain only suppress d(t), not

n(t).

• Reducing the detector noise n(t)

will be essential to get highly

stable cavity field!

1

100


d yH

n yH

Increase

Gain

Cavity

71


d yH s

d yH s

d yH s

d yH s

Open loop

KP=1

KP=10

KP=100

KP=0

PS ripples Vc

n yH s

n yH s

n yH s

n yH s

Open loop

Noise Vc

cV t

KPSensorADC DAC

LO

Noise (n)

LO RF IF

Set

IQ

Mod

Ripples (d)

Suppression of Microphonics

The measurement of Microphnics at KEK cERL.

No Feedback (Gain=0)

Gain ≈ 10

Gain ≈ 150

Closed loop

dT s

e

K(s) P(s)

F(s)

d(t)

n(t)

w(t)

y(t)

r(t)

FFT of Microphonics

72

Gain

Gain-scanning In the real case, the loop gain is decided by gain-scanning experiment.

Experiment can be performed online.

Closed loop

dT s

e

K(s) P(s)

F(s)

d(t)

n(t)

w(t)

y(t)

r(t)

Step 1: Increasing the controller gain

(KP) gradually (e.g. from 10 to 400)

Step 2: Measuring the field amplitude

and phase by field detector

Step 3: Plot ΔA/A vs. KP,

and Δθ vs.KP, then you will

got a curve.


Effects of the loop gain If gain is too large, system may be unstable.

0.51 0

0.5

dT s FOL

F

H s K es s

frequency

frequency

1020log 2H j f

2H j f

dB

degreef

f

-180 deg

180f

Stable

crossf

0 dB

180 deg

0.52 0

0.5

10 dT s FOL

F

H s K es s

1 2OL OLH j H j

Phase margin

1OLH j

2OLH j20 dB


Unstable

Gain margin

d yH s

1,2OLH

I gain is set to be zero, optimum P gain is about 125.

Gain Scanning @ KEK cERL

KI＝0


Gain-scan with integral gain

P Gain

I G

ain

Measured

Performance

O Pmax

Imax

I=0

Two-D Mapping

P Gain

I G

ain

Optimized

Gains

Bad

Good

Imax

O

e tPK

IK

y t PK e t

Integral gain

Sometimes we are also interested in the integral gain, KI, which is

significant in the normal conducting cavity LLRF (think about why?).

We will get an 2-D map in this case if we scan both KP and KI.

I=10



Resonance Control

Resonance control

78

Tuner(frequency

ctrl.)FB

(field ctrl.)

Piezo

Kly/SSACavity

Pf

Pt

Pf

Pt

Pt

Internet

Communication

PC

cA V

cP V

Stabilize the RF field (A&P) for beam acceleration

2H j f

t

Minimize the cavity input power

related to t

0.5 0.5c

c

dVj V u

dt


Main Detuning

79

2H j f

0

t

RF

Cavity

Deformation

of the cavity

Repulsive

magnetic force

Attractive

Electric force

MicrophonicsLorentz detuning

Lorentz detuning: usually effects

the PS mode machine.

Microphonics: effects both of

pulse mode and CW mode.

It is difficult to apply the FB

control for Lorentz detuning (in

PS mode), feedforward is

recommended (think about

why?).


Tuner control

Tuner system is used to compensate the cavity detuning.

80∆𝑓: detuning

H(s): Desired case

freq

Detuning effect

Lorentz detuning

H’(s):

With Tuner

TunerForce

H(s): Desired case

freq

Detuning effect

H’(s):

HT(s): with tuner caseTuner Force

Fixed

Piezo & Tuner

81

Mechanic Tuner: usually for slow control or course control (slow and

low precision)

Piezo: Fast and high precision (piezoelectric effects)

Cavity side


Lorentz Detuning Compensation

82

Mainly Lorentz detuning, can be compensated by piezo feedforward

system.



Back up


Examples: LLRF system @ cERL

Examples: cERL RF systems Compact ERL (cERL) is a test facility for the future 3-GeV ERL project. It is a 1.3-

GHz superconducting system and is operated in CW mode.

HLRF

Three 2-cell in

the Injector

Injector consists of

four cavities: Buncher

(NC), Injector 1 (SC),

Injector 2 (SC),

Injector 3 (SC).

Main linac includes

two nine-cell cavities

(SC).

Layout of cERL

Two 9-cell

in ML (SC)


Main linac

8 kW SSA

Nine-cell SC

16 kW SSA

Main linac Two-cell SC

SC SC

300 kW Kly.

25 kW Kly.

8 kW SSA

Vector-sum

Controlling

~8.5 MV/m for main linac Cavities

~3 MV/m for Injector Cavities

~ 20 MeV

Dump

Examples: cERL RF systemsAt present, total four kinds of Power Sources are applied in cERL : 8-kW SSA, 16-

kW SSA, 25-kW Klystron and 300 kW Klystron.

RF requirement

0.1 % rms, 0.1 deg. rms for cERL

0.01% rms, 0.01deg.rms for 3GeV-ERL

Cavity QL RF power

Bun. 1.1e5 3 kW

Inj. 1 1.2e6 0.53 kW

Inj. 2 5.8e52.4 kW

Inj. 3 4.8e5

ML1 1.3e7 1.6 kW

ML2 1.0e7 2 kW

86

ADC: LTC 2208


ADC & DAC Interface

Digital I/O

uTCA Digital Board

LLRF (Digital System)

Down-convertor

IQ Mod.

FPGA boardsFPGA boards

Thermostatic Chamber

(0.01 deg.)

LLRF Cabinet

Digital Board type Feature

ADC LTC2208 16 bits, 130 MHz (Max.)

DAC AD9783 16 bits, 500 MHz (Max.)

FPGA

(Core)

Virtex 5 FX 550 MHz (Max.), includes a Power PC with

Linux, EPICS is installed on the Linux.

Four channel 16 bits ADCs + four channel

16 bits DACs.

Micro TCA

87

Calib.ADC

Vector sum

I

Q

I_Set

ADC

DAC

DAC

cos sin

sin cos

FPGA

Power PC

EPICS IOC

IQ M

od

.

Waveform Command

Linux

IF ≈ 10 MHz

I

AMC Card

AD9783

Dow

n

Conver

tor

IIR

RF = 1.3 GHz

LO ≈ 1310 MHz

Analog filter

PI

IIR

Fil.

Q_Set

I_FF

Calib.

Q

1300 MHz

Kly. / SSA

Pre. Amp.

Clk ≈ 80 MHz

LTC2208

ADC

ADC

ADC

PI

Q_FF

cos sin

sin cos

DIO (RF SW)SW1

LLRF @ cERL

FB (μTCA)

Sensor Controller

Plant


MicroTCA

DAC

Digital

I/OADC2

ADC1I/Q

dem.

I/QIIR

I/Q

dem.IIR

I/Q

Δθ

detector

sign

Filhold

CW

CCW

Pf

Pick up

Piezo

Mechanical

Tuner

KI

KP

Resonance control for CW mode

∆𝜃 = 𝜃𝑃𝑓 − 𝜃𝑝𝑖𝑐𝑘 𝑢𝑝

In CW mode machine, cavity is operated in CW mode, thus the cavity

field is almost constant (steady state).

The main detuning is Microphonics.

Tuner (μTCA)Fine ctrl.

Coarse ctrl.


Sensor: same with

LLRF control

Phase difference

0.5

0.5

tan tan ( )steady state

Performance (Screen Monitor) The beam momentum is measured by screen monitor and determined by the peak

point of the projection of the screen.

Momentum was determined by the peak point

of the projection of the screen.

Dispersion @ screen

monitor = 0.82m

Resolution = 53.4 m/pixel

(P/P=6.5e-5)

Screen Monitor

Attention: Vector-sum error would influence

the beam momentum jitter greatly! Thus the

phase error btw inj. 2 and inj. 3 should be

optimized at first!

90

Momentum Jitter= 0.006% rms

Result of the screen monitor

Beam Energy0 50 100 150 200 250 300 350 400 450 500

-0.3

-0.2

-0.1

0

0.1

0.2

time(s)

dP/P

(%)

FB2 LG:dP/P= 0.063607% rms HG:dP/P=0.0056858% rms

FB1HG&FB2LGFB1HG&FB2HG

-0.4 -0.2 0 0.2 0.40

20

40

60

80

Num

ber

dP/P (%)

FB2LG: dP/P= 0.063607% rms

-0.02 -0.01 0 0.01 0.020

100

200

300

400

500

Num

ber

dP/P (%)

FB2HG: dP/P= 0.0056858% rms

Performance (Beam energy)

91


Thank you for your attention

Reference


[1] S. Simrock, Z. Geng. The 8th International Linear Accelerator school.

[2] E. Vogel. High Gain Proportional RF Control Stability at TESLA Cavities. Physical Review Special Topics – Accelerators

and Beams, 10, 052001 (2007)

[3] M. Hoffmann. Development of A Multichannel RF Field Detector for the Low-Level RF Control of the Free-Electron Laser

at Hamburg. Ph.D. Thesis of DESY, 2008

[4 ]F. Qiu et al., Evaluation of the superconducting LLRF system at cERL in KEK, in Proceedings of the 4th International

Particle Accelerator Conference, IPAC- 2013, Shanghai, China, 2013 (JACoW, Shanghai, China, 2013).

[5] F. Qiu et al., Digital filters used for digital feedback system at cERL, in Proceedings of the 27th Linear Accelerator

Conference, LINAC14, Geneva, Switzerland, 2014 (JACoW, Geneva, Switzerland, 2014), MOPP074, p. 227

[6] T. Schilcher. Vector Sum Control of Pulsed Accelerating Fields in Lorentz Force Detuned Superconducting Cavities. Ph. D.

Thesis of DESY, 1998

[7] M. Hoffmann. Development of A Multichannel RF Field Detector for the Low-Level RF Control of the Free-Electron Laser

at Hamburg. Ph.D. Thesis of DESY, 2008

[8] L. Doolittle. Digital Low-Level RF Control Using Non-IQ Sampling. LINAC2006, Knoxville, Tennessee USA

[10] C. Schmidt, Ph.D. thesis, Technische Universität Hamburg-Harburg, 2010.

[12] B. Alexander, Ph.D. thesis, Development of a Finite State Machine for Automatic Operation of the LLRF Control at

FLASH

Laplace Transform


Cavity equation

95

0.5 0.5c

c

dVj V u

dt 0.5 0.5

cc

dVV u

dt

0.5 0.5c csV s V s U s

1 ( 0)t

CV t e t

Step

response

Laplace

Based-band

equation

0.50.5

0.5

1 ( 0)j t

V t e tj

Δω=0 Δω≠0

In the presence of the detuning, the cavity equation is some thing like:

If Δω=const.

detuning

0.5

0.5

c

cav

V sP s

U s s j

0.5 0.5c csV s j V s U s

0.5

0.5

c

cav

V sP s

U s s

TF

Similar with RC circuit


Sources of Microphonics

Scheme showing a technical drawing of a TESLA cavity welded in its cryounit.

Possible detuning sources due to external vibrations or liquid helium level changes

(microphonics) as well as the system response to external excitation are shown.

Microphonics


Microphonics

Microphonics will leads to a error in the RF field.

IRVjdt

VdLc

c

2/12/1

FFT

Phase

Detuning (Microphonic)

RF

Cavity

Deformation

of the cavity

Repulsive

magnetic force

Attractive

Electric force

2

accf K E

Lorentz detuning

98

A standing electromagnetic wave in a cavity exerts pressure on the

surrounding resonator walls, This radiation pressure is

2 2

0 0

1

4sP H E

The quantities H and E denote the magnetic and electric field on the

walls. The deformation of the cavity will result of detuning (think

about why?).

Detuning Acc. field

Const.


Results @ ACC6 of FLASH

99

Problems


Detuning ≈0 @ flat-time

Example (Loop gain effect)

0.5

0.5

dT sI P P FOL

F

K K K sH s e

s s s

4 5

0.50, 2, 2 9.3 10 / , 2 5 10 / , 0.7 [ ]I P F dK K rad s rad s T s

I. Which system is stable?

II. How does the loop gain

influence the bode plot?

4 5


100

Analytical Study(Loop Delay effect) How does time delay influence the bode plot and the system stability?

0.51

0.5

dT sI P I FOL

F

K K K sH s e

s s s

frequency

1020log 2H j f

2H j f

dB

degree

-180 deg

180f

0 1802 1OLH j f

Gain margin

crossf

0 dB

180 deg

0.50

0.5

I P I FOL

F

K K K sH s

s s s

0 11,dj T

OL OLe then H j H j

dj T

de T

Phase margin

Gain margin

1 0OL OL dH H T

0OLH

1OLH101

frequency

1 1802 1OLH j f

Time delay

Analytical Study(Loop Delay effect)

4 5


I. Which system is stable?

II. How does the loop delay

influence the bode plot?

4 5


102

0 11,dj T

OL OLe then H j H j

0 dB

Non-IQ Sampling

Fourier series decomposition of the RF signal

Demodulation algorithm:

1

0

1

0

sin2

cos2 n

i

i

n

i

i ixn

Qixn

I ，

,...2,1,

2sin2

2cos2

2sin2cos2

2sin2cos2sin

0

0

1

0

k

dttfktsT

b

dttfktsT

a

tfkbtfkaa

ts

tfQtfItfAts

T

IFk

T

IFk

k

IFkIFk

IFIFIF

2n

m


LLRF feedback control system (Basics) feedback control system (Basics) Feng QIU, JAS17, ......

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Transcript of LLRF feedback control system (Basics) feedback control system (Basics) Feng QIU, JAS17, ......