P Lamalle, A Messiaen et al ICRH CT matching satellite meeting, 23rd SOFT, Venice, 21 September 2004 Mutual coupling effects on ICRH Conjugate-T matching

P Lamalle, A Messiaen et al ICRH CT matching satellite meeting , 23rd SOFT, Venice, 21 September 2004

Mutual coupling effects on ICRH Conjugate-T matching

P Lamalle, A Messiaen, P Dumortier, F Durodié, M Evrard, F Louche

Plasma Physics Laboratory - Partner in TEC

Royal Military Academy

Brussels, Belgium

ICRH Conjugate-T antennas matching satellite meeting

23rd SOFT, Venice, 21 September 2004


Outline

Investigation of mutual coupling effects on conjugate T (CT) matching:

• General properties of ICRH array impedance matrices;

• Model of several CT circuits mutually coupled via the antenna straps;

• New results: optimization of load tolerance for the single CT circuit

with mutual;

• Specific problems resulting from coupling between several CTs,

possible remedies. [NB parts of this section already presented at Ringberg

meetings in April and JET-EP design review in June 2004]


Properties of ICRH array impedance matrices

• N-port antenna array n by n input impedance matrix ZA = RA+ jXA

• Input voltages and currents: arrays VA, IA , VA = ZA IA

• Mutual impedances ZAik=RAik+jXAik between straps i and k (ik):

– Nonzero resistive part RAik, due to interference of the field components

radiated by each strap, contributes to the radiated active power.

Responsible for the strong experimental dependence of antenna port

loading on toroidal phasing of the straps observed on ICRH arrays;

– XAik induces power transfer between straps i and k.

– Complex power Pi delivered by the external sources to strap i:

2* *2 ( ) ( )i Ai Ai Aii Aii Ai Aik Aik Ai Akk i

P I V R jX I R jX I I


Active power

The active power delivered to strap i, Re(Pi) = PRi + Pxi , includes

(1) Radiation and/or ohmic dissipation:

(2) Active power exchanged between straps:

22 cosRi Aii Ai Aik Ai Ak ik

k i

P R I R I I

2 sinXi Aik Ai Ak ikk i

P X I I

arg( / )ik Ai AkI I

NB: within a typical CT at nominal loading !12sin ~ 0.5 0.7


Mutual coupling is generally responsible for

• An unbalance of power distribution between antenna ports (see next fig.)

(even when the straps are geometrically identical);

• Coupling between the power sources if the array is simultaneously fed

by different sources.


Example

Single ITER external CT circuit with line stretchers in presence of mutual coupling

Stretcher 1

Stretcher 2 Adjustable impedance transformer(stretcher+stub)


Active power sharing between the 2 CT branches, showing the large power exchange due to the mutual reactance.

Exchanged

Total active (branch 2)

Total active (branch 1)

Radiated (branch 1)

Radiated (branch 2)

Active power sharing

Load scaling factor(ELM-like perturbation)


Other features of ZA= RA+j XA

• RA is positive definite.

• The signs of the mutual coefficients depend on the relative orientations of the radiating current loops.

• Simulations including plasma gyrotropic effects (Colas, EPS 2004) may yield nonsymmetric ZA (with the main effect on RA) and enhance

dissymmetries.

• The antenna data used in the following simulations were either obtained by 3D electromagnetic simulations of the ITER array (Louche, EPS 2004), or from measurements on a scaled mockup (see A. Messiaen, P3T-B-211)


Models of ELM-induced loading perturbations

• Experimental observations (JET, see I. Monakhov, 15th RF Topical):

- large transient increase of resistive loading (factor of 4 to 5),

- smaller relative decrease of the reactance (~10 to 25%, depending on magnetic equilibrium

• Two phenomenological models are used for the analysis of load tolerance:

(1) Purely resistive with scaling of the resistance matrix:

(ELM-free loading: =1; ELM trajectory: 1)

Interest: simplicity, leads to more explicit analytical results.

(2) General first-order model: perturbation matrix A may include reactive terms

(ELM-free loading: =0; ELM trajectory: 0)

Allows a realistic account of experimental data.

A A AZ R jX

0A A AZ Z


Circuit equations for multiple CT circuits (1)

n=2p antenna ports connected in pairs through n independent matching circuits to p feed ports.

Each branch matching circuit is lossless & reciprocal, and has a transfer matrix

(ai, di real, bi, ci imaginary, ai di - bi ci =1)

For series capacitors: ai = di = 1, bi = 1/(jCi), ci = 0

For line stretchers:

ai = di = cosli , -jbi / Z0 = -jZ0ci = sinli

Other circuits or nonideal effects are easily

included in Ti.

ZA

1

2

2p-1

2p

1

p

A

A

V

I

in

in

V

I

V

I

30 Zc

T1

T2

T2p

T2p-1

i ii

i i

a bT

c d


• Voltages and currents at branch inputs:

A, B, C, D: n x n diagonals with e.g. A = diag(ai, i=1,n)

• Branch input admittance matrix:

• Admittance matrix at the p feeds:


ZA

1

2

2p-1

2p

1

p

A

A

V

I

in

in

V

I

V

I

30 Zc

T1

T2

T2p

T2p-1

A

A

VV A B

II C D

1

A AY Z A B D Z C

TinY PY P

• P connects the branches pairwise:

1 1 0 0 0 0 00 0 1 1 0 0 0

0 0 0 0 0 1 1

P


• Matching conditions: given - reference admittance Yc =1/Zc, - array loading (reference ),- feeding mode (e.g.Vin array, relative mag. & ),

• Complex polynomial system of order p and degree 2p for the matching transfer parameters {ai, bi, ci }

• There are 2p unknowns if each circuit has 1 adjustable parameter.

(NB Ti has max. 3 independent

parameters additional design freedom is available)


ZA

1

2

2p-1

2p

1

p

A

A

V

I

in

in

V

I

V

I

30 Zc

T1

T2

T2p

T2p-1

( )in in in c inI Y V Y V


General results for a single CT circuit

• In this case,

• Matching equation Yin=Yc : quadratic in the transfer parameters.

- For lumped capacitors: yields quadratic equation for the unknown reactances,

2 matched solutions (possibly 0 depending on ZA & Zc!);

- For line stretchers: trigonometric eqs, 4 solutions in presence of mutual coupling /Fig degenerate to 2 in absence of mutual (possibly 0 solution depending on ZA & Zc!)

• High sensitivity of load-tolerant matching to mutual impedances results from the

cancellation of large reactances within the circuit branches, which makes Y12

appreciable with respect to Y11 although at the antenna straps |Z12A|<<|Z11A|.

• The cancellation is more and more delicate as Zc (explains why this sensitivity is so

much higher with CT load resilient schemes than with standard matching)

11 22 12 21inY Y Y Y Y

1

111

2221

1211

YY

YYPYPY T

in

)(1 CZDBAZY AA


Line stretcher circuit with mutual coupling

Contours of constant reflection vs. the 2 line stretcher lengths,

showing existence of 4 distinct matched points.


Optimization of load resilience (single CT circuit)

• Dependence of input admittance on ELM load scaling :

P, Q: quadratic polynomials (P linear for series capacitor matching)

• In the complex plane the input admittance has two saddle points 1, 2

(one for passive and one for active loads) where

• At fixed circuit, there are 2 matched loads 1, 2 (in general complex) for

every Yc: the solutions of

• ‘Strategic’ importance of the ‘passive’ saddle point: it is located on a path of least reflection between the 2 matched loads.

• Optimization of load resilience can simply be stated to the RF pilot as

‘Thou shall place a Saddle Point and at least one Matched Load (near the ELM-free reference) on the

ELM trajectory.’ (using Zc and M1 M2)

• This approach leads to the 2 following general results (report being written with detailed expressions and practical algorithms).

( ) / 0indY d

( ) ( ) / ( )inY P Q

( ) ( )cP Y Q

)(1 CZDBAZY AA


Result 1: For resistive load perturbations,

0 5 10 151

1.5

2

2.5

3

1.8+0.54i2.6+0.77i3.2+1i3.8+1.3i4.1+1.6i4.4+1.8i4.6+1.9i4.6+2i

Zc (Ohm):

Lambda (model 1)

SWR

The matching circuit can be configured to set a saddle point and 2 matched

loads on the ELM trajectory. The saddle point load and the local maximum

swr can be imposed a priori. This completely determines the matched loads

and the reference input impedance Zc (which generally has a reactive

component). is then the geometric mean of the matched loads:

Optimized load-tolerant configurations at SWR<1.5. Single CT circuit with line stretchers, purely resistive ELM model.

1 2


The matching circuit can be configured to set a saddle point and 1 matched

load on the ELM trajectory. The saddle point load and the local maximum swr

can be imposed a priori. This completely determines the matched loads and the

reference input impedance Zc (which generally has a reactive component).

Result 2: For the general load scaling,

Optimized load-tolerant configurations at SWR<1.5. Single CT circuit with line stretchers, ELM model with a reactive component.

0 5 10 151

1.5

2

2.5

3

1.9+0.2i2.8+0.3i3.7+0.5i4.4+0.79i5+1.1i5.5+1.5i5.8+1.8i5.9+2.1i

Zc (Ohm):

Lambda (model 2)

SWR


• The underlying equations are valid for any lossless matching circuit and any antenna

(incl. mutual, dissymmetries, gyrotropic effects). They generalize earlier analytical

results obtained for uncoupled antenna straps and lumped capacitors.

• (Currently attempting to generalize this to several coupled CTs.)

• In both models achieving the optimal resilience settings generally requires a complex

reference impedance Zc=1/Yc, which plays the role of 2 additional tuning parameters

(i.e., the transformer settings required to match this Zc to 30).

• In such configurations the active power exchange between CT branches is reduced.

Strong interest for an adjustable transformer stage between the CT connection and

the main transmission line: allows operation of the system with a complex impedance

at the T ensuring best load tolerance at each operating frequency. To be tested on

JET-A2, TEXTOR, JET-IL where this freedom exists.

• Possibilities of implementation on ITER should be investigated with high priority.

Single CT - optimized load resilience - Discussion


• Goal of ongoing study: finding ‘all’ matched ELM-free configurations.

• Tools: (1) basic continuation methods; (2) advanced polynomial system solvers (spinoff from robotics, economics, chemistry and pure math.)

• The number of solutions depends on strap dissymmetries, strength of mutual coupling (Fig.), array phasing (all features able to lift degeneracies from the system), and on the reference Zc. This can make obvious tracking

methods (e.g. decreasing Zc from a high initial value, for which mutual

effects are less critical) unreliable to find matched settings.

• For large ratios |mutual|/ Zc there may be no solution; for the whole array of 4 coupled CTs tens of solutions may be found. Only a few, sometimes just one, of these solutions are physically attractive (i.e. within operational limits and with proper strap current phasing).

• This raises basic practical questions: selection and implementation of the best solution on the ICRH plant, with only approximate knowledge of the antenna impedance matrix; avoiding attraction of automatic control system by undesired solutions, etc.

Specific effects for several CT circuits


0 0.5 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Solution #2 - not OK

0 0.5 1-0.5

-0.25

0

0.25

0.5Solution #6 - OK

x x

Seeking matching solutions for 4 coupled CTs: basic continuation method

Four CT circuits with 8 capacitors, antenna dipole toroidal phasing: two attempts to obtain matched solutions with mutual coupling (at x=1) starting from two solutions in absence of mutual between CTs (at x=0). The deformation paths are shown for the capacitor currents.

Unsuccessful: solution disappears en route

Successful: x=1 reached


• Appearance and disappearance of solutions is also observed as antenna loading varies, e.g. from vacuum to plasma: a good vacuum match may prove of little help on plasma!

• Strong interaction between matching systems via the array makes matching procedure much more complex than in the case of a single CT. A typical pathology of constant reflection surfaces for such higher dimensional systems is shown on next Fig.

• Only a few - sometimes just one - of the solutions are physically attractive

(i.e. within operational limits, with proper strap current phasing)

[Note also different behaviours with respect to 3dB hybrid performance on JET, M. Evrard simulations.]

• For large ratios |mutual terms|/ Zc there may be no solution at all.


x2

x3

x1x2

x3

x1

Constant reflection surfaces for 2 coupled CTs

Feed line 1 (x1, x2):

SWR=1.5

Feed line 2 (x3, x4):

SWR=1.5

Two coupled CT circuits with 4 capacitors (reactances xi): Surfaces of constant input SWR=1.5 at each feed vs (x1,x2,x3). x4 is linearly constrained by x1, x2, x3 to locate 4 matched solutions in a 3D view. Note the strong influence of x3 (located in CT#2) on feed line #1 in a limited interval. Dipole antenna phasing, Zc=3.

Pathology due to mutual

Single CT-like behaviour:cylindrical along x3


Several coupled CTs: discussion

• General conclusion: the success of intuitive continuation methods is not guaranteed; they often only provide a subset of the possible solutions.

• Potentially high number of undesired solutions and high variability basic practical questions: selection and implementation of the best solution on an ICRH plant, with rough knowledge of the antenna impedance matrix; avoidance of undesired solutions by an automatic control system.

• Practical tuning algorithms are difficult to derive in view of such features and of the coupling of the different power sources.

• RF measurements of current or voltage amplitude and phase close to the antenna straps seem mandatory to properly diagnose and control the system.

• Using polychromatic heating, i.e. a different frequency for each CT or each pair of nonadjacent CTs, would replace the coupling between active circuits by a coupling to passive circuits. This avoids the problem of mutual coupling between generators; for one CT the other CT’s appear as passive detuned circuits the influence of which should be reduced. [… but there are drawbacks to this approach.]

• Another way to solve the problem is to symmetrically distribute the total power by passive junctions between all the straps grouped in two parts in order to remain with only one CT. This solution would require handling very large power in some components.


Conclusions

• By all means, an interesting topic!

• Significant progress in the detailed understanding and optimisation of the general single CT circuit with mutual coupling [achieved by a ‘shift of emphasis’ to the admittance saddle point instead of the matched load]

• There is much to win in relaxing the reality of Zc should develop an adjustable impedance transformer for integration on ITER. [Review after tests on present systems: possibility to design a fixed ‘nonstandard’ one?]

• This approach will soon be applied to multiple CT systems.

• Same basic existential questions as 3 months ago for the latter, but we are making progress!


Question to the Workshop

• Circuit modelling during an ELM: what is really constant?

• Do we need to model the system up to the tetrode (assuming constant

current there)?

• What happens to phase control on these timescales?



Reserve slides


Reminder: load tolerance

Conventional matching circuit

Ideal conjugate T

Input SWR at fixed matching elements

Strap input resistance

(ELM-induced variation) 55MHz, toroidal dipole, Z0=3

• Ingredients:

• conjugate branch impedances

• low reference input impedance

• Ideal conjugate T circuit (2000): symmetric branches, no mutual;

• c, d: nonsymmetric modules P & Q, no mutual (2001);• a, b: nonsymmetric modules P & Q, with mutual (2002);

• NB: the figure assumes ideal

settings of the ELM-free reference

match.

• Remainder of the presentation

addresses the finding of fair

approximations to such settings (and sending the system there)

Realistic conjugate T


-5 0 5 10

-6

-4

-2

0

2

4

6

( -1)

(

-1)

arg(I2/I1) and arg(Vf2/Vf1), degree

-100

-80 -6

0

-40

20

40

6060

8080

100

100

120

140

-100

-80

-60

20

4040

6060

8080 80

100

100

120

140

One pair of straps: voltage probe and branch currents relative phaseSimilar behaviour V probes provide key information in absence of magnetic loops

Blue: arg(Vf2/Vf1)=const

Red: arg(I2/I1)=const

NB: constant current relative phase loci: a family of hyperbolasBranch 1 shifted reactance

Branch 2 shifted reactance

Matched point


0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8Matching solution #1

Scaling factor

SW

R

Z0=3 Z0=3+0.3i Z0=3+0.6i Z0=3+0.9i Z0=3+1.2i

One pair: Matching to a complex reference input Z0

• Aim: reduce current and voltage imbalance in presence of mutual coupling

• OK on JET, where the transformer is adjustable!

• Reassessing load tolerance in this mode of operation:Purely resistive variations: improved tolerance:


Matching to Z0 = 3 + j 1.27 (*)

1.51 + j 32.72 -0.1754 + j 1.266Z=

-0.1754 + j 1.266 1.036 + j 30.45

Matching to Z0 = 3 (*)

55MHz:

One pair: Matching to a complex reference input Z0

-5 0 5 10

-6

-4

-2

0

2

4

6

( -1)

(

-1)

Some desirable properties

-140

-120

120120

140

140

P1=P2 |I1|=|I2| arg(I2/I1) Vc2=Vc1 Vf2=Vf1

-4 -2 0 2 4 6 8

-6

-4

-2

0

2

4

( -1)

(

-1)

Some desirable properties

-140

-120

120

120

140

140

P1=P2 |I1|=|I2| arg(I2/I1) Vc2=Vc1 Vf2=Vf1

Matched point


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20Solution #1 - OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0

20Solution #2 - not OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20Solution #4 - OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0


0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10


0 0.2 0.4 0.6 0.8 1

-40

-20

0


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10


0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0Solution #9 - OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0


0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0Solution #11 - OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0

20Solution #12 - OK

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20Solution #13 - not OK

0 0.2 0.4 0.6 0.8 1

-40

-20

0


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20Solution #16 - OK

Switching the mutual on / Unknowns: capacitor reactances

Pathologies:- Attraction to (or simply outside feasible capacitance range)- Merging of solutions


0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #1 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #2 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #3 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #4 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #5 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #7 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #8 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0

0.5Solution #10 - OK

0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0


0 0.2 0.4 0.6 0.8 1-0.5

0


Switching the mutual on / Unknowns: capacitor currents

Sole remaining pathology:- Merging and vanishing of solutions in pairs (i.e. become complex)- Similar improvements to be sought for practical control algorithm


0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

0 0.05 0.1 0.15 0.2-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

0 0.02 0.04 0.06 0.08 0.1-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

0 0.05 0.1 0.15 0.2-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

0 0.05 0.1 0.15 0.2-0.4

-0.2

0

0.2

0.4

0 0.02 0.04 0.06 0.08 0.1-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

Connecting plasma to vacuum solutions?Decreasing loading towards vacuum / Unknowns: capacitor currents

Evolution of branch currents from 17 plasma solutions (55MHz, 6Ω, dipole):

Starting point (0): plasma loadingTarget end point (1): vacuum-like loadingThe numerical procedure fails in most cases!


0 0.2 0.4 0.6 0.8 1

-40

-20

0

20Solution #1 - OK

0 0.05 0.1 0.15 0.2

-40

-20

0


0 0.1 0.2 0.3 0.4 0.5-50

-40

-30

-20

-10


0 0.02 0.04 0.06 0.08 0.1-50

-40

-30


0 0.1 0.2 0.3 0.4

-40

-20

0


0 0.1 0.2 0.3 0.4

-40

-20

0


0 0.1 0.2 0.3 0.4

-40

-20

0


0 0.2 0.4 0.6 0.8 1

-40

-20

0

20Solution #8 - OK

0 0.1 0.2 0.3 0.4

-40

-20

0


0 0.1 0.2 0.3 0.4 0.5

-40

-20

0


0 0.2 0.4 0.6 0.8 1-35

-30


0 0.05 0.1 0.15 0.2-50

-40

-30

-20

-10


0 0.2 0.4 0.6 0.8 1-40

-35

-30

-25


0 0.05 0.1 0.15 0.2

-40

-20

0


0 0.02 0.04 0.06 0.08 0.1-50

-40

-30


0 0.1 0.2 0.3 0.4-40

-35

-30

-25


0 0.1 0.2 0.3 0.4 0.5-50

-40

-30

-20

-10


Decreasing loading towards vacuum / Unknowns: capacitor currents

Evolution of capacitor reactances from 17 plasma solutions (55MHz, 6Ω, dipole):

Linked to a vacuum match, reactances remain finite

Linked to a vacuum match, unbounded reactances


Comments - open issues

• Practical path must always remain within capacitor range!

• Define parameter paths avoiding ‘collisions’ of solutions?

• Abandon real Z0 (see 2-strap study): may hopefully yield easier handling of the system (at least numerically).


-2 0 2 4 6 8 10 12 14 16 18

-8

-6

-4

-2

0

2

4

6

8

Solution #2 - port #1

Illustration: constant reflection curves in cx. plane(purely resistive ELM perturbation)

0 2 4 6 8 10 12 14 16 18

-8

-6

-4

-2

0

2

4

6

8

Solution #2 - port #1

SWR=1.5

The ELM

Optimized settings

Re

Im

The ELM

Detuned settings

Re

Im


Illustration: constant reflection curves in cx. plane(ELM with reactance perturbation)

The ELM

Optimized settings

Re

Im

The ELM

Detuned settings

Re

Im

Documents

P Lamalle, A Messiaen et al ICRH CT matching satellite meeting, 23rd SOFT, Venice, 21 September 2004 Mutual coupling effects on ICRH Conjugate-T matching