RF Phase and Amplitude Control

* RF PHASE AND AMPLITUDE CONTROL

Robert A. Jameson Los Alamos Scientific Laboratory, University of California

Los Alamos, New Mexico

Basic Specitications

From beam dynamics

Phase - Averag~ tank phase, measured over a period of -10 p..ser?" to remain within a:±:2 square well around the reference phase.

Since the shOrt- term phase instabilities will include random effects, the above specification must be redefined as a mean deviation such that a random phase jitter of Gaussian distribution remains in the + 20 square well some percentage of the time:

Percent of Time Within Square Well

Amplitude - .:!:. 2"70

From power amplifier

99"70 950/0 90"/.

The HPA (Coaxitron) has transit time phase shifts of the order of 2.5

0/1"70 change in plate voltage. This indicates the necessity for ampl~~

tude control to less than 0 . 5"70 , if operating open loop.

The transit time problem is further complicated by the necessity of increasing of power output to match increasing beam l.oading as the beam is turned on. The effects of various load configurations on the amplifier are being studied to determine how great this effect is .

General Layout -- Figure 1

Comments

l . After initial setup, tanks nand n' are ganged together with ~_ sensing from one tank. Work p erformed under the auspices of the U. S . Atomic Energy" Commission.

Proceedings of the 1964 Linear Accelerator Conference, Madison, Wisconsin, USA

505

2. Both fast and slow phase shift ers mp:y be req\1ired, depen ding on range available from fast shifte r. Survey of existing mechanic al, ferroelectric and varactor phase shifters in dicate fas t phase control must be done at power levels like 10 watts . CW or less.

Quot a tions have been received for development work on ferro electric shifters capable of handling 100 W CW (17 00 W a t ~% duty)' . shifting + 10

0 in 1-2 /-,-sec. This is al r eady a development task -- shifters

c apable of handling higher powe r levels would require majo r development.

Basic Control Loops - Block Diagrams--Figure 2

1. Coupling Effect s

a. Major coupling from IP A and HP A plate v oltage changes t o phase of Coaxitron output, as m entioned e a rlie r . E ffect i s n onlinear, given approximat ely by:

Transit time, cathode to plate 3 dcg + dgp sec. . 7jV"Q O 5 .93 x 10 10

where dc g and dgp are cathodtT-to·~ grid and grid-plate distances in centimeters, and ~ = Vp min = V g max .

10

b. Second-order couplings between amplitude and phase in tank . More on t h is late r.

2. Other Nonlineariiie s

a. Phase comparator outp ut p r oporiion al t o the s ine of t he phase error.

b. Nonlinear phase s hifter char a cteristics.

c. Tank behavior n onlinear for large signals.

3. The effect of beam loading on t he l oop must be included.

4. Bas i c Frequency Stability Requirem ents_

It is interesting t o note what frequency stability is required of a standard in order that t he short~term stab ility of the standard itself does not introduce ph~s e disturbance s which exceed the specified toler ances. - Assume that the characteristic time constant in the t ransfer fun ction fro m the reference phase to the t ank phase


506

BEAM PHASE MONITOR

PHASE REF. w

z ::::;

w ?:; a: 0 .

MECHANICAL <II

PHASE SHIFTER :::> rn

MAIN DRIVE LINE

FIG. I GENERAL LAYOUT

PHASE COMPARATOR COMPENSAnON

I lNETWORKS

I vl I I I LINEAR I

PHASE SHIFTER

POWER AMPS

DETECTOR_J COMPENSATION NETWORKS

COMPENSAnON HPA NETWORKS MODULATOR

I I I I

I I

FIG. 2 BASIC CONTROL LOOPS - BLOCK DIAGRAMS


507

is that of the tank--in the order of 10 fLsec. We are thus intere:;+,ed in phase perturbations that occur at rates of ,...,16 kc or less. The relation for frequency uncertair::.ty due to a short~term phase irstability is

b.f ~e -f 2 7rN

where ~9 phase -. stability, proportional to noise-to-signal ratio

f

2 7r

N ::: number of periods used for measurement

20 ~7 ------------- .;:: 7 x 10 for 10 fJ-sec meas-))r'ement 360

0 x 800 x 106 x 10- 5 at 800 Me.

-7 = 7 x 10 _;:: 5 x 10- 9 for 10 fLsee measuremer:.t at 5 Me.

100

Thus the primary freq1)ency source which will provide th,,:; reference and drive signals must be very precise-~ln fact someNhat better than any source advertised at this date.

5. )VIeasu:r:-ement Technl~~

The statE'-of-the--&.rt nat:rc<', of the stability reqmremt?Fts and

some of the hardware promise diffic'lHies in obtainirg accurai,e ar,.d repeatable measurernents.

1. Phase Comparator.

The simplest comparators use the ri' signals di!ecU y,wHnout convert:ing to intermediate freqHe:'1cies, de. Because of. J:b,,,:r simplicity and reliability, q'lis type is being considc;red u; Lhe eacly studies.

The comparator might th.1S be a q;)adra~UremLx.E'r WhOE,l~ f/v\)

outputs are Jinearly detected and differ!::.~r::ccd to foY'm ~he er :oo:r signal :

.±. k.j2 E2 sin E q"

where }; d i· ,.,.. .

e .. ei.. ';or etflclfTlCleS

.E 1 ;: uEknown ,signal E 2 :: reference signal

E1 » E2

for ~f' « 1 .


508

e(t, 0)

2. Phase Shifter

At thj s iime, it is proposed to use a varactor controlled phase shifter. Indications from industr"y are that these devices will soon be available at the desired pO'wer levels (I'V 10 W CW), with phase shift ranges of greater tha::-l :!: 90

0• The phase shift vs. bias voHage

characteristic is nonHr:ear 0 tending to saturate at the high end. A linear approximation may be used for smaJl phase s·wings.

3. Power AmI2s

The chain is treated as a sing} e resonant cavity, with the envelope responses characterized by an expone-r.tial rise to a step input.: e. g.

Output Amplitude Input Amplitude

Output Phase Input Phase

s +~ ot. "---'

8+0(.

The phase transfer function is good for sman signals only, becoming an inverse tangent. function ir' the ti nl~' domain for 1 arge sign.aJs, which irltrodu ;::es a nod.inear:"Y.

4. . 1

TanK

ar;alys~s of a lumped par':S.met:er eqcivaJimt :ircnit for the acce1·· F:rator ca";"[ i,y:

"11'he-8e results are s1;tmmarized from LASL Tech Memos p~11/HAJ·3 and P·~11/HAJ'"4.


509

The steady~state theory for this circuit has been developed by Nagle and Knapp--some results of this were given in Knapp1s talk earlier.

The matrix equations for the circuit were written using Laplace transforms, transforming first with respect to ~ime, and then, . afte r using jump functions to express th", repetitive nature of the cells , again with respect to the cell number., The solution of the homogeneous equation gives the characteristic equation and the n dependence. The inhomogeneous equation is then:;;olved using an orthogonality relation based on the n dependence ; The com-

. plete solution for the current in any cell is :

Nd 9 w 1 (q) .s E (s , d)

1(s, n) =

where d = index on rf drive, e. g. E (s, d) is a drive in the dth cell

q = .mode number n = cell number E (s, d) " drive t erm normalized by 1/2 L

1

The transfer function with respeet to a particular drive is

I (8, n)

E (s , d) = G (s, n, d) -

f 2 cos ~ng cos -;.ct..g w 1 (q) s . --~~~----~------~--~

q=o N(1+kcos -W ) [s2+ Wo s+ WQ2 _] Q (1 + k cos if) 1 + k cos ';l

For example the time response to a single sinusoidal drive in the d th cell ~tarted at t = 0:

e (t, d) = u (t) sin wt)

N

i (t, n) ~ )"' Kq [Sin ( M


510

where 2 w (i Magnitude of driv e ~ } w 1 (q)

0( q

w q

-1 tan

- Q [W 2 (1 + k cos ~9,) -

= tan -1 -/4 Q2 (1 + k

( 11' n .

2 Q 1 + k cosN~)

(1 + k cos ~) N

~1 and the approximation results :rom re~:.ainlng only the fin~.t, term

. 1 of the exp anSlOn of the factor ... ---~.---~--... ~.--= .. '=-====;;;:: 11 ~4_1Q'.'2" (1'+ k cos-'1?:..q \ V~ 4 ~~.!

which modified the trans.!ent P<';1'* of Une SO}U;,lOD.

Discussion

1. The steady-state boll.Jior say"" ~hat the steady~·::d,ate curt'e;:.t at any particular' drive ir eq'.:eL'.yi6 the SJm of componpr,.~,i:) from each mode. If the dri\,," s at a rnOdf? resonant fr'equtncy, that component has itsmax.im1J!n amp~Lt1)de and is in phase with the drive. The othe>' cornponen1:s art': present bQcauso? (,f the losses in the system. They are dr1;eE far i'rom re~.o~.a;ice and thus have small ampLitlliJes acd are? aboi:t 900 out of phase with the drive.

2. The natural frequencies and chimpin.g faci.ors are app,,,,,Y'en i; as the transient terms. The iy,;tJa1. ampLU1lde,s are very ne&rly the same as for the corresponding steady-s~;at.e term, arod the initial angles are almost 180

0 0 1;1, of phase wEh t.he steady"

state angles. The' ampli.tude d.r'd phase varia+,lons during thE: transient; period both die out w.ith i:hr:~,ame time ('onstani.:..::.


511

3. The cos (dq/N) <md cos (nq/N) terms add 1800 to the phase when they are negative . Note that a drive in an end cell excites all the modes, while a drive in the qenter (N evell) cell excites onl y the evell numbered modeS. Placing drives at node points to suppress various modes can be us e d effectively to increase the apparent "mode separation. "

Envelope Responses

The transfer functions of real interest are those of the amplitude and phase envelopes:

l"rom Input Amplitude to Output Amplitude l"rom Input Amplitude to Output Phase l"rom Input Phase to Output Phase l"rom Input Phase to Output Amplitude

We write the response to the desired drive (in this case amplitude or phase step changes in a sinus.oidal drive) as follows :

i (t, nl ~ ~0,J", ([ aq

(t) ei

¢q(t)] eiwt

}

where a q (t) ~ amplitude envel ope function q:; . (t) ~ phase envelope function.

q

Examples, the time response to a sinusoidal drive started at t ~ 0, us e d b efore :

i (t, n) [ i<Pw - Gt, t i [(Wq-W)t + ~w 1}

e + e '1 e · q~

Two types of approximation to the response are conside red :

.a. The components of the r e spons e are t~l{en as the driven mode component and an out- of-phase component . The latter is found by firs t assuming that each off-resonance component is exactly + 90

0 out of phase with the driven

mode component at steady state . The amplitude is then taken to be the vector sum of t.hese steady- stat.e amplitudes , and ca,lle d ~ K q. The damping term is taken to be that of the driven mode CtN for 'W - mode drive. The b e at fre quency f3 = Wq - W is taken to be that produced by the


512

nearest mode frequency to thE' drive frequency that is present in the tank.

b. Response of the driven mode component only. The steadystate amplitude of t.his component for 7r modE' drive is KN .

The results for the first appr'oximation are listed below, for if

mode drive. Simplification to t~e secon,d follows by letting 2:Kq :: O. The exact responses have been. found uSlng a computer program. The approximate responses have been superimposed on the exact in Figs. 3 .- 10.

L Response to pulse turn-on (Large am-p}i1ude step).

,_ ~ t - O'oN t ] 2 a(t) {[KN (1 - e ).~ L Kq e J. sin f3 t

cp (t) -1

tan.

[ -G!N t . 2 J}l / 2 + L Kq (1 - e . cos f3 t) -~t

- L Kq (1 - e - co:£~_. ____ _

_al N t -'~'\Tt KN (1 - e .. ) - 2: K E' L~ ~)in {3t

q

[ . --©'JNt ]

1 + k e . (1 - cos J3 t) .~~~-q KN

Small signal transfer funcUons:

6A (s)

fj, E (s)

« 1


513

3. Response to step phase change oL4? radi.us from steady stat.e:

+ 2 KN L Kq [ (1 - cos 'P j e _Ot.Nt (2 e - ~,t - 1) sin fJ t

+ sin e - ol.Nt

(1 - cos j3t)] + (LKq)2 [1 + 2 e,·ot'Nt

(sincf> sin{3t ~

-, (1 - cos ~) cos f3 -?f:J... t]} 1/2

+ 2 (1 .' cos cp ) e '"" N

Small signal transfer functions:

_AA (S)

69 (S) =:

/j,¢ (~ -~9 (S)

KN ~Kq

.jK~ + ( 2:Kq)2

2

[ f8 S

---')

'.S+ C-tl\J) fs + o<'N) ~ +

_____ bs __ . ____ _ 2 2

(S + ~i) +!3

KEANE: This is a very nice piece of work. One comment you did r ,-'r.

take into consideration the response for each particular mode. As a matter of fact for what I was talking about, the next mode wou1.d be ll:"le

TMOll . Because of my weak coupling to that modE:, J hdd v(;rv sligl-,i scalloping .. To the mode where I had stron.g coupJing, namely thf> TM012' the scallopIng wa,s much mor.E' prono1lDced.


514

.06

w 0.04 ::> I--oJ a. ~ <I:

RESPONSE OF CELL 00 TO PULSE TURN- ON ON-RESONANCE DRIVE AT CELL 00

ACTUAL RESPONSE

o L-___________________________________________________________ _

o 200 400 600 800

TIME, SECONDS x 10'

FIG. 3

RESPONSE OF CELL 00 TO PULSE TURN-ON ON-RESONANCE DRIVE AT CELL 00

90

50

w 10 en 0 ~ -10 I

a.

-50

ACTUAL RESPONSE

1000

-90 ~---------------------------------------------------------o 200 400 600 800 1000

TIME, SECONDS II 10'

FIG. 4


515

w o ::J t-...J a.

.060

RESPONSE OF CELL 00 TO + 2 PERCENT

AMPLITUDE STEP APPLIED TO STEADY-STATE

ON-RESONANCE DRIVE AT CELL 00

~ .059

w en « ::c n.

ACTUAL RESPONSE

.058 L..-. __________________ _

o 200 400 600 800

TIME, SECONDS x 10 8

FIG. 5

RESPONSE OF CELL CO TO + 2 PERCENT AMPLI TU DE STEP APPLI ED TO STEADY- STATE


APPROX.

ACTUAL RESPONSE

1000

,,0L..-. ____________________________ _

o 200 400 600 800 1000

- TIME, SECONDS x 10'

FIG. 6


516

110

16/

".

10

-lD D

RESPONSE OF CELL 00 TO 1800 PHASE

STEP APPLIED TO STEADY-STATE ON

RESONANCE DRIVE AT CELL 00

flt "S'" --

~ ~ "'" Acf(lQI ~sf''''''~ :-

r-~ t\ I"- f'f'rox.

'\f

2bD 4<-> 600

nM£. S6CDIJ&lS )( 10.

FIG. 7

RESPONSE OF CELL 00 TO 180 0 PHASE

STEP APPLIED TO STEADY-STATE ON

RESONANCE DRIVE AT CELL 00

..---t--

----~ ... /

, -- r

~ " J f-.--- Actual A €"sponSIi'.

rr i'-- 0/ 'fro><: •

ID(>tJ

V 4co 0

0 600 800 ----IDD

7''''£, .s6CoN&I.5 K 10 II

FIG. 8


517

RESPONSE OF CELL 00 TO + 2 DEGREE PHASE STEP APPLIED TO STEADY-STATE


-8 .

...-

~ V-

b? .

r '-......acn aJ '",sf ""e..

f ~ fI(,~ ( -'>< -t)

.~ K. 1+):"

t' offr K.

-10

26" +otJ 600 'liDO

,"I1fE", ;SECe>""D~ J( 10'

FIG. 9

RESPONSE OF CELL 00 TO +2 DEGREE

PHASE S TE P APPLI ED TO STEAOY-STATE ON-RESONANCE DRIVE AT CELL 00

FIG. 10


518

JAMESON: That is not true. The answer for the current in any cell shows the dependence on the location of the drive, or drives, as well as the modal dependence. It shows that if you pick the right place to drive, the mode separation problem becomes easier. For instance, at the center you don't excite any of the odd number modes; you excite only the even number modes. So -e1at helps you out by a factor two.

VOELKER: I want to mention our experience on the HILAC. We had two tanks coupled together with a coupling line and the scallop was directly related to the difference in frequency between the two cavities so that we used this as a rough tuning. When we took off the coupling line and began to drive from a separate amplifier, we found that we no longer had phase lock; we had as much as 100 between the two tanks during the pulse. On some experiments the beam would go away during the early part of the pulse. In the last few months we have put in a phase servo with a veractor in a low level stage and have reduced the phase shift from about 100 to 20 •

JAMESON: You do have to take into account the fact that these other modes are there even though the effect is very small. Below is a picture of output from an ll-cell iris-loaded tank, which we drove at signal level frequencies. That is the input pulse on the upper left-hand corner. Then I used the sampling scope to look at the output of the tank in the various cells of the tank. This allows us to look at the rf directly and you can see the scallops that arise. This tank has a Q of 8300, a coupling of 2%, N was equal to 10 in the theory, and it was driven at the center. It shows quite well what the effect is. We also drove at the end of the same tank, and in the cells furthest away from the drive we observed, at the turn-off of the pulse, the pip where the energy actually goes up for an instant before it comes back down. Jim Leiss mentioned this yesterday.

T ( .. -. <J,,! '" C ", ."'i'.M,...; ::t • .....,;J<,,.

"",u~" cr (~I.~ ~ {. .. ......;.\~. ~ .... "'/"' ...


519

Documents

RF Phase and Amplitude Control