Lia Merminga - CASA - Jefferson Lab

Thomas Jefferson National Accelerator Facility 

USPAS Course 

on 

Recirculated and Energy Recovered 

Linear Accelerators 

G. A. Krafft and L. Merminga 

Jefferson Lab 

and 

Ivan Bazarov 

Cornell University 

Lecture 10 

USPAS Recirculating Linacs Krafft/Merminga/Bazarov 

Operated by the Southeastern Universities Research Association for the U. S. Department of Energy 

3 May 2005

Outline 

Introduction 

Cavity Fundamental Parameters 

RF Cavity as a Parallel LCR Circuit 

Coupling of Cavity to an rf Generator 

Equivalent Circuit for a Cavity with Beam Loading 

• On Crest and on Resonance Operation 

• Off Crest and off Resonance Operation 

Optimum Tuning 

Optimum Coupling 

RF cavity with Beam and Microphonics 

Q ext Optimization under Beam Loading and Microphonics 

RF Modeling 

Conclusions 




3 May 2005

Introduction 

Goal: Ability to predict rf cavity’s steady-state response and develop a differential 

equation for the transient response 

We will construct an equivalent circuit and analyze it 

We will write the quantities that characterize an rf cavity and relate them to the 

circuit parameters, for 

a) a cavity 

b) a cavity coupled to an rf generator 

c) a cavity with beam 




3 May 2005

RF Cavity Fundamental Quantities 

Quality Factor Q 0 : 

Q 

0 

ω W 

P 

0 ≡ = 

diss 

Shunt impedance R a : 

(accelerator definition); V a = accelerating voltage 

Note: Voltages and currents will be represented as complex quantities, denoted by a 

tilde. For example: 

V = V 

where is the magnitude of and is a slowly varying phase. 

c c 


Energy stored in cavity 

Energy dissipated in cavity walls per radian 

R 

a 

≡ 

V 

P 

2 

a 

diss 

( ) Re ( ) 

in ohms per cell 

{ iωt} 

( ) 

V t V t e V t V e φ 

= = 

c c c c 

V 

c 

φ 



( ) 

i t 

3 May 2005

Equivalent Circuit for an rf Cavity 

Simple LC circuit representing 

an accelerating resonator. 

Metamorphosis of the LC circuit 

into an accelerating cavity. 

Chain of weakly coupled pillbox 

cavities representing an accelerating 

cavity. 

Chain of coupled pendula as 

its mechanical analogue. 




3 May 2005

Equivalent Circuit for an rf Cavity (cont’d) 

An rf cavity can be represented by a parallel LCR circuit: 

Impedance Z of the equivalent circuit: 

Resonant frequency of the circuit: 

Stored energy W: 


1 

W = 

CV 

2 

2 

c 

∼ ∼ 

i t 

c() = c 

V t v e ω 

ω 0 = 1/ LC 

1 1 

Z⎡ ⎤ 

= 

⎢ 

+ + iCω 

⎣R iLω 

⎥ 

⎦ 



−1 

3 May 2005

Equivalent Circuit for an rf Cavity (cont’d) 

Power dissipated in resistor R: 

From definition of shunt impedance 

Quality factor of resonator: 


P 

R 

diss 

a 

= 

1 

2 

V 

P 

V 

R 

2 

a 

2 

c 

≡ ∴ R = 2R 

diss 

ω W 

Q ≡ = ω CR 

0 

0 

Pdiss 

0 

−1 

Note: 

⎡ ⎛ ω ω ⎞⎤ 

⎡ ⎛ω −ω 

⎞⎤ 

0 

0 

Z= R⎢1+ iQ For 

0 ⎜ − ⎟⎥ 

ω ≈ω0 , 

Z≈ R⎢1+ 2iQ0⎜ 

⎟⎥ 

⎣ ⎝ω0ω ⎠ 

ω 

⎦ 

⎣ ⎝ 0 ⎠⎦ 



a 

−1 

3 May 2005

Cavity with External Coupling 

Consider a cavity connected to an rf source 

A coaxial cable carries power from an rf source 

to the cavity 

The strength of the input coupler is adjusted by 

changing the penetration of the center conductor 

There is a fixed output coupler, 

the transmitted power probe, which picks up 

power transmitted through the cavity 




3 May 2005

Cavity with External Coupling (cont’d) 

Consider the rf cavity after the rf is turned off. 

Stored energy W satisfies the equation: 

Total power being lost, P tot, is: 

P e is the power leaking back out the input coupler. P t is the power coming out the 

transmitted power coupler. Typically P t is very small ⇒ P tot ≈ P diss + P e 

Recall 

Similarly define a “loaded” quality factor Q L : 

Now 

Q 

∴ energy in the cavity decays exponentially with time constant: 


0 

ω0W 

≡ 

P 

diss 

dW ω W 

dt Q 

0 =− ⇒ = 

L 

dW 

P 

dt =− 

P = P + P + P 

tot diss e t 

W W e 

0 

ω0t 

− 

Q 

L 

Q 

L 

ω0W 

≡ 

P 



tot 

tot 

τ 

L 

QL 

= 

ω 

0 

3 May 2005


Equation 

suggests that we can assign a quality factor to each loss mechanism, such that 

where, by definition, 

Typical values for CEBAF 7-cell cavities: Q 0=1x10 10 , Q e ≈Q L=2x10 7. 


P P + P 

= 

ω W ω W 

tot diss e 

0 0 

1 1 1 

= + 

Q Q Q 

L 0 e 

Q 

e 

ω0W 

≡ 

P 

e 



3 May 2005


Define “coupling parameter”: 

therefore 

β is equal to: 

It tells us how strongly the couplers interact with the cavity. Large β implies 

that the power leaking out of the coupler is large compared to the power 

dissipated in the cavity walls. 


Q 

β ≡ 

Q 

0 

1 (1 + β ) 

= 

Q Q 

L 

e 

Pe 

β = 

P 

0 

diss 



3 May 2005

Equivalent Circuit of a Cavity Coupled to an rf Source 

The system we want to model: 

Between the rf generator and the cavity is an isolator – a circulator connected to a load. 

Circulator ensures that signals coming from the cavity are terminated in a matched load. 

Equivalent circuit: 

∼ ∼ 

i t 

I () t i k e ω = 


k 

RF Generator + Circulator Coupler Cavity 

Coupling is represented by an ideal transformer of turn ratio 1:k 



3 May 2005

Equivalent Circuit of a Cavity Coupled to an rf Source 

∼ ∼ 

i t 

I ( t) i k e ω = 

k 

∼ ∼ 

i t 

I () t i e ω i t 

I () t = i e ω = 

g g 

By definition, 


⇓ 

R R R 

β ≡ = ∴ Z g = 

β 

2 

Zgk Z0 


2 

Z g = k Z 0 


I 

g 

= 

I 

k 

k 

3 May 2005

Generator Power 

When the cavity is matched to the input circuit, the power dissipation in the 

cavity is maximized. 

∼ ∼ 

i t 

I () t i e ω = 

g g 

We define the available generator power Pg at a given generator current g to 

be equal to P diss max . 


2 

1 ⎛ I ⎞ 

1 

P = Z ⎜ ⎟ or P = R I ≡ P 

2 ⎝ 2 ⎠ 

16β 

max g 

max 2 

diss g diss a g g 



I 

3 May 2005

Some Useful Expressions 

We derive expressions for W, P diss , P refl , in terms of cavity parameters 

Q Q V 

W Q V 

P R I 

16β 16β 

g 

= 

0 Pdiss 

ω 0 

1 2 

RaI g 

= 

2 

0 c 

ω 0 R a 

1 

2 

RaI g 

16β 

0 = 2 

a ω 0 

2 

c 

2 

g 

V = I Z 

Z 

c g TOT 

TOT 

⎡ 1 1 ⎤ 

= ⎢ + ⎥ 

⎢⎣ Z g Z ⎥⎦ 


− 1 

Z TOT = 

R ⎡ ⎛ a 

ω ω ⎞⎤ 

0 

⎢(1 + β ) + iQ0⎜ 

− ⎟⎥ 

2 ⎣ ⎝ ω 0 ω ⎠⎦ 

∴ 

Q 

= 4 β 

ω 

1 

ω ω 

For 

W 

 

0 

W P 2 g 

0 2 2 ⎛ ⎞ 0 

(1 + β ) + Q 0 ⎜ − ⎟ 

ω 0 ω 

ω ω 

0 

⇒ 

− 1 

⎝ ⎠ 

4βQ01 2 

(1 + β ) ω 0 ⎡ Q 0 ω − ω ⎤ 0 

1 + ⎢2 (1 β ) ω 

⎥ 

⎣ + 

0 ⎦ 

2 

Pg 



3 May 2005

Some Useful Expressions (cont’d) 

Define “Tuning angle” Ψ: 

Recall: 


 

4βQ1 0 

W P 

2 

2 g 

(1 + β ) ω 0 ⎡ Q 0 ω − ω ⎤ 0 

1+ ⎢2 (1 + β ) ω 

⎥ 

⎣ 0 ⎦ 

⎛ ω ω ⎞ ω−ω Ψ ≡− − ≈− ω ≈ω0 

⎝ ⎠ 

0 0 

tan QL⎜ ⎟ 2QL for 

ω0 ω ω0 

∴ 

4βQ1 W = P 

(1+ β) ω 1+tan Ψ 

P 

∴ 

diss 

= 

ω 0W 

Q 

0 

0 

2 2 

0 

4β1 P = 

P 

(1 + β ) 1 + tan Ψ 

diss 2 2 g 

g 



3 May 2005

Some Useful Expressions (cont’d) 

. Optimal coupling: W/P g maximum or P diss = P g 

which implies ∆ω = 0, β = 1 

this is the case of critical coupling 

. Reflected power is calculated from energy conservation: 

P = P −P 

refl g diss 

⎡ 4β1 ⎤ 

Prefl = Pg⎢1 

− 

2 2 

(1 β ) 1 tan 

⎥ 

⎣ + + Ψ⎦ 

. On resonance: = 

4 β Q 


0 

W P 2 g 

(1 + β ) ω 0 

4 β 

P = P 

(1 + β ) 

diss 2 g 

⎛1−β⎞ P = ⎜ ⎟ P 

⎝1+ β ⎠ 

refl g 

2 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

Dissipated and Reflected Power 

0 1 2 3 4 5 6 7 8 9 10 

β 



3 May 2005

Equivalent Circuit for a Cavity with Beam 

Beam in the rf cavity is represented by a current generator. 

Equivalent circuit: 

Differential equation that describes the dynamics of the system: 

R L is the loaded impedance defined as: 


dv v di 

iC = C , i R = , vC = L 

dt R /2 dt 

C C L 

L 

R 

L 

Ra 

= 

(1 + 

β ) 



3 May 2005

Equivalent Circuit for a Cavity with Beam (cont’d) 

Kirchoff’s law: 

Total current is a superposition of generator current and beam current and beam 

current opposes the generator current. 

Assume that vc, ig, ib 

have a fast (rf) time-varying component and a slow 

varying component: 

vc iωt = V ce 

i iωt = Ie where ω is the generator angular frequency and c g b are complex quantities. 


i + i + i = i −i 

 

L R C g b 

2 

dvc ω0 dvc 2 ω0RL 

d 

+ + ω 

2 

0vc 

= ig −ib 

dt QL dt 2QL 

dt 

 

g g 

i= Ie b b 

iωt V , I , I 

( ) 



3 May 2005


Neglecting terms of order c c we arrive at: 

where Ψ is the tuning angle. 

For short bunches: | Ib | ≈ 2I0where 

I0 is the average beam current. 


 

2 

d V dI 1 dV 

, , 

2 

dt dt QL dt 

dV ω ω R 

+ (1−itan Ψ ) V = ( I −I 

) 

dt Q Q 

c 0 0 L 

2 L 

c 

4 L 

g b 

 



3 May 2005


At steady-state: 


dV ω ω R 

+ (1−itan Ψ ) V = ( I −I 

) 

dt Q Q 

c 0 0 L 

2 L 

c 

4 L 

g b 

RL /2 RL 

/2 

V = I − 

I 

(1 −itan Ψ) (1 −itan Ψ) 

RL or V c = I g cos Ψe 2 

RL 

− I bcosΨe 

2 

or V = V iΨ cos Ψ e + V iΨ 

cosΨe 

or 

c g b 

c gr br 

iΨiΨ V = V + V 

c g b 

⎧ 

V ⎪ gr 

⎪ 

⎨ 

⎪V br ⎪⎩ R L ⎫ 

= I 

g 

2 ⎪ 

⎬ 

R L = − I 

⎪ 

b 

2 ⎪⎭ 

are the generator and beam-loading voltages on resonance 

⎧V ⎪ ⎫ g ⎪ 

and ⎨ ⎬ are the generator and beam-loading voltages. 

⎪V ⎩ b ⎪⎭ 



3 May 2005


Note that: 

| V| = R I 

br L 


2 

| Vβ gr | = PgRL ≈2 

PgRL for large β 

1+ 

β 

0 



3 May 2005


V = V cos Ψe 

g gr 

V = V cos Ψe 

b br 

iΨ 

iΨ 


Im( Vg ) 

i 

V cos e Ψ 

Ψ 

gr 

Ψ 

V 

gr 

Re( Vg ) 

As Ψ increases the magnitude of both V g and V b decreases while their phases 

rotate by Ψ. 



3 May 2005



V = V + V 

c g b 

Cavity voltage is the superposition of the generator and beam-loading voltage. 

This is the basis for the vector diagram analysis. 



3 May 2005

Example of a Phasor Diagram 

Idec 


ψ 

Vbr 

Vb 

ψ b 

I b 

 

Vg 


Iacc 

Vc 


3 May 2005

On Crest and On Resonance Operation 

Typically linacs operate on resonance and on crest in order to receive 

maximum acceleration. 

On crest and on resonance 

Vbr 

⇒ 


I b 

 

where V a is the accelerating voltage. 

Vc 

Va= Vgr −Vbr 

Vgr 



3 May 2005

More Useful Equations 

We derive expressions for W, V a , P diss , P refl in terms of β and the loading parameter K, 

defined by: K=I 0 /2 √R a /P g 

From: 

2 

| Vβ | = P R 

gr g L 

1+ 

β 

⇒ 

= 

| V| R I 

br L 


0 

V = V −V 

a gr br 

⎧⎪ 2 β ⎛ K ⎞⎫⎪ 

Va = PgRa ⎨ ⎜1− ⎟⎬ 

⎪⎩ 1+ 

β ⎝ β ⎠⎪⎭ 

4β 

Q ⎛ K ⎞ 

= ⎜1− ⎟ 

(1 ) 

0 

W P 

2 

+ β ω0⎝ β ⎠ 

4β 

⎛ K ⎞ 

P = ⎜1− ⎟ P 

(1 + β ) ⎝ β ⎠ 

diss 2 

g 

IV = I RP 

η 

0 a 0 a diss 

IV 2 β ⎛ K ⎞ 

2K⎜1 ⎟ 

Pg 

1+ 

β ⎝ β ⎠ 

0 a ≡ = − 



2 

2 

⎡( β −1) −2Kβ 

⎤ 

P = P −P −I V ⇒ P = 

⎣ ⎦ 

P 

( β + 

1) 

refl g diss 0 a refl 2 

g 

g 

3 May 2005 

2

More Useful Equations (cont’d) 

For β large, 

For P refl =0 (condition for matching) ⇒ 


1 

P ( V + I R ) 

g a 0 L 

4RL 

1 

P ( V − I R ) 

refl a 0 L 

4RL 

R 

M 

a 

L = M 

I0 

and 

P 

V 

I V ⎛ V I ⎞ 

⎜ ⎟ 

⎝ ⎠ 

M M 

g 

0 a 

4 

a 0 + M M 

VaI0 2 

2 



2 

3 May 2005

Example 

For V a =20 MV/m, L=0.7 m, Q L =2x10 7 , Q 0 =1x10 10 : 


Power I 0 = 0 I 0 = 100 µA I 0 = 1 mA 

P g 3.65 kW 4.38 kW 14.033 kW 

P diss 29 W 29 W 29 W 

I 0 V a 0 W 1.4 kW 14 kW 

P refl 3.62 kW 2.951 kW ~ 4.4 W 



3 May 2005

Off Crest and Off Resonance Operation 

Typically electron storage rings operate off crest in order to ensure stability 

against phase oscillations. 

As a consequence, the rf cavities must be detuned off resonance in order to 

minimize the reflected power and the required generator power. 

Longitudinal gymnastics may also impose off crest operation operation in 

recirculating linacs. 

We write the beam current and the cavity voltage as 

The generator power can then be expressed as: 

P 


I b 

iψ 

b = 2I0e 

i c V ψ 

= V e and set ψ = 0 

c c c 

2 2 

2 

Vc (1 + β ) ⎧⎪ ⎡ I0R ⎤ ⎡ L I0R ⎤ ⎫ 

L ⎪ 

g = ⎨⎢1 + cosψb⎥ + ⎢tan Ψ− sinψb⎥ 

⎬ 

RL 4β 

Vc Vc 

⎪⎩⎣⎦ ⎣ ⎦ ⎭⎪ 



3 May 2005

Off Crest and Off Resonance Operation (cont’d) 

Condition for optimum tuning: 

Condition for optimum coupling: 

Minimum generator power: 


IR 

V 

0 L tan Ψ= sin 

c 

IR 

ψ 

0 a β0 = 1+ cosψb 

Vc 

P 

g,min 

2 

Vc 

β0 

= 

R 

a 

b 



3 May 2005

RF Cavity with Beam and Microphonics 

δ f ± δ f δ f 

0 m 

0 

The detuning is now: tan Ψ=− 2Q tanψ 0 =−2Q 

L L 

f f 

Frequency (Hz) 

10 

8 

6 

4 

2 

0 

-2 

-4 

-6 

-8 

-10 

90 95 100 105 110 115 120 

Time (sec) 


0 

0 0 

where δ f is the static detuning (controllable) 

and δ f is the random dynamic detuning (uncontrollable) 

m 

Probability Density 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

Probability Density 

Medium β CM Prototype, Cavity #2, CW @ 6MV/m 

400000 samples 

-8 -6 -4 -2 0 2 4 6 8 


Peak Frequency Deviation (V) 


3 May 2005

Q ext Optimization under Beam Loading and Microphonics 

Beam loading and microphonics require careful optimization of the external Q 

of cavities. 

Derive expressions for the optimum setting of cavity parameters when 

operating under 

a) heavy beam loading 

b) little or no beam loading, as is the case in energy recovery linac cavities 

and in the presence of microphonics. 




3 May 2005

P 

Q ext Optimization (cont’d) 2 2 

2 

Vc(1 + β ) ⎧⎪⎡ ItotRL = ⎨⎢1+ cosψ RL 4β 

⎪ V 

⎩⎣ c 

⎤ 

⎥ 

⎦ 

⎡ ItotRL + ⎢tanΨ− sinψ 

⎣ Vc 

⎤ ⎫⎪ 

⎥ ⎬ 

⎦ ⎪⎭ 

δ f 

g tot tot 


tan Ψ=−2QL 

f 

where δf is the total amount of cavity detuning in Hz, including static detuning and 

microphonics. 

Optimization of the generator power with respect to coupling gives: 

βopt = 

2 ⎡ δ f ⎤ 

( b+ 1) + ⎢2Q0+ btanψtot 

⎥ 

⎣ f0 

⎦ 

ItotRa where b ≡ 

cosψ 

tot 

V 

c 

where I tot is the magnitude of the resultant beam current vector in the cavity and ψ tot is the 

phase of the resultant beam vector with respect to the cavity voltage. 

0 



2 

3 May 2005

P 

Q ext Optimization (cont’d) 

2 2 

2 

Vc(1 + β ) ⎪⎧⎡ItotR ⎤ ⎡ L ItotR ⎤ 

L ⎪⎫ 

g = ⎨⎢1+ cosψtot⎥ + ⎢tan Ψ− sinψtot⎥ 

⎬ 

RL 4β 

Vc Vc 

where: 

To minimize generator power with respect to tuning: 

⇒ 


⎪⎩⎣⎦ ⎣ ⎦ ⎪⎭ 

δ f + δ f 

0 m 

tan Ψ=−2QL 

f0 

f 

δ = − tan Ψ 

0 f0b 2Q0 

2 

2 

c (1 + β) ⎧⎪ 2 ⎡ δ ⎤ ⎫ 

m ⎪ 

⎨(1 β ) ⎢20⎥ ⎬ 

L 4β 

0 

V f 

Pg= + b+ + Q 

R ⎪ ⎣ f 

⎩ ⎦ ⎭⎪ 



3 May 2005

Q ext Optimization (cont’d) 

Condition for optimum coupling: 

and 

In the absence of beam (b=0): 

and 


β 

opt 

⎛ δ f ⎞ 

2 

m 

= ( b+ 1) +⎜2 Q0⎟ f0 

⎝ ⎠ 

2 

2 ⎡ 

opt V 2 ⎛ c δ f ⎞ 

⎤ 

m 

g = ⎢ + 1 + ( + 1) +⎜2 ⎥ 

0 ⎟ 

2R 

⎢ a 

f ⎥ 0 

P b b Q 

⎣ 

⎝ ⎠ 

⎦ 

β 

opt 

⎛ δ f ⎞ m 

= 1+⎜2Q0⎟ ⎝ f0 

⎠ 

2 

2 ⎡ 

opt V ⎛ c δ f ⎞ 

⎤ 

m 

g = ⎢1+ 1+⎜2 0 ⎟ 

⎥ 

2R 

⎢ a 

f ⎥ 0 

P Q 

⎣ 

⎝ ⎠ 

⎦ 

2 



2 

3 May 2005

Homework 

Assuming no microphonics, plot β opt and P g opt as function 

of b (beam loading), b=-5 to 5, and explain the results. 

How do the results change if microphonics is present? 




3 May 2005

Example 

ERL Injector and Linac: 

δf m =25 Hz, Q 0 =1x10 10 , f 0 =1300 MHz, I 0 =100 mA, V c =20 MV/m, L=1.04 m, 

R a /Q 0 =1036 ohms per cavity 

ERL linac: Resultant beam current, I tot = 0 mA (energy recovery) 

and β opt =385 ⇒ Q L =2.6x10 7 ⇒ P g = 4 kW per cavity. 

ERL Injector: I 0 =100 mA and β opt = 5x10 4 ! ⇒ Q L = 2x10 5 ⇒ P g = 2.08 MW 

per cavity! 

Note: I 0 V a = 2.08 MW ⇒ optimization is entirely dominated by beam loading. 




3 May 2005

RF System Modeling 

To include amplitude and phase feedback, nonlinear effects from the klystron 

and be able to analyze transient response of the system, response to large 

parameter variations or beam current fluctuations 

• we developed a model of the cavity and low level controls using 

SIMULINK, a MATLAB-based program for simulating dynamic systems. 

Model describes the beam-cavity interaction, includes a realistic representation 

of low level controls, klystron characteristics, microphonic noise, Lorentz 

force detuning and coupling and excitation of mechanical resonances 




3 May 2005

RF System Model 




3 May 2005

RF Modeling: Simulations vs. Experimental Data 

Measured and simulated cavity voltage and amplified gradient error signal (GASK) in 

one of CEBAF’s cavities, when a 65 µA, 100 µsec beam pulse enters the cavity. 




3 May 2005

Conclusions 

We derived a differential equation that describes to a very good 

approximation the rf cavity and its interaction with beam. 

We derived useful relations among cavity’s parameters and used phasor 

diagrams to analyze steady-state situations. 

We presented formula for the optimization of Q ext under beam loading and 

microphonics. 

We showed an example of a Simulink model of the rf control system which 

can be useful when nonlinearities can not be ignored. 




3 May 2005

Lia Merminga - CASA - Jefferson Lab

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