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Matrix Model of the Waveguide Transformer

C. Chen

Introduction

The condition to match the input impedance of the cavity to the characteristic impedance of the waveguide changes as a function of beam current. Current designs, however, do not allow the transformer to be adjustable while it is in operation. Thus, the transformer is set to match the impedance in the case of a high beam current, where minimizing power losses is essential [1]. This paper details a mathematical model of the transformer. Based on a circuit model developed at DESY, the model allows for the calculation of phase shifts and power transfer coefficients, as well as predictions of the loaded Q of the cavity.

Description of the Transformer

The waveguide transformer consists of three cylindrical plungers in a standard WR1800 waveguide [1]. The plungers are spaced approximately λ/4 apart for the operating frequency of 500 MHz. The length of the transformer, including two end pieces that adapt the bolt holes of the transformer to the CESR system, measures 90 cm. The transformer is asymmetric, with the center of the first plunger 29.21 cm from the end, and the plungers spaced about 18.89 cm apart, center-to-center. Each plunger is adjustable to a maximum protrusion of 18.19 cm into the transformer. Each of the plungers is also held by 2 ring clamps to insure good RF contact.

The transformer can be modeled as three capacitors in parallel, separated by an electrical length equal to the physical length minus the electrical thickness of the plunger [2]. Each plunger can be modeled independently due to the λ/4 separation. Values for the capacitance and electrical thickness were taken from curves [2]. The data points were fit to curves according to methods described in Appendix 1. The curves are fit as

)1020016.1(275023.0 −⋅= xY (1)

xxT ⋅+⋅= 1182.000047898.0 2 , (2)

where Y is the normalized admittance, T the electrical thickness of the plunger, and x the protrusion into the cavity in mm. The data points may be slightly inaccurate as they were obtained from physical measurements from a chart in a printed copy of the paper. However, the fit for these curves are shown in Figure 1 and Figure 2, respectively. The curves fit quite well with the data, and we assume that these curves are reasonably reliable.

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0 20 40 60 80 100 120 140 160 180 2000

5

10

15

Plunger Height (mm)

Nor

mal

ized

Plu

nger

Adm

ittan

ceBest fit curve for Normalized Plunger Admittance

Fig. 1. Curve fitting for Admittance values as a function of plunger height.

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0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

40

45

Plunger Height (mm)

El.

Plu

nger

Thi

ckne

ss (

mm

)Best fit curve for Elec. Plunger Thickness

Fig. 2. Curve fitting for Electrical Thickness as a function of plunger height.

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Matched Load Case

Measurements of the power transfer coefficients and phase shifts as functions of plunger height were first taken with a network analyzer. The equivalent circuit for this setup is shown in Figure 3. The output to the network analyzer is modeled as a matched, terminating load. The mathematical model makes use of a matrix to represent each component of the circuit [3]. The matrix model is detailed in Appendix 2. The matrices were multiplied together using a MAPLE [4] worksheet. The worksheet also calculates the power transfer coefficients and the phase shifts as functions of plunger heights according to the equations

)log(10 211 Γ⋅=S , (3)

)1log(10 221 Γ−⋅=S , (4)

where 0

0

ZCA

ZCA

+

−=Γ

and A, B, C, and D are elements of the matrix

DC

BA.

The phase shifts are given by

ΓΓ=Φ

)Re(

)Im(tan11 a (5)

Γ+

Γ+

=Φ

A

Aa

1Re

1Im

tan21 (6)

The worksheet is shown in Appendix 3. Figures 4, 5, 6, and 7 show the fit of the experimental data to the predictions of the model. Since the values are functions of three variables, the measured data is plotted against the calculations. The y=x line thus represents an exact fit of the calculated results. The fit indicates that the model fits reasonably well with measured results. The deviation in S21 for cases of very low power transfer are probably due to inaccuracies in small measurements. The large S21 phase deviations correspond to low power transfer cases.

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Fig. 4. Fit of S11 predictions to experimental results.

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Fig. 5. Fit of Φ11 predictions to experimental results.

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Fig. 6. Fit of S21 predictions to experimental results.

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Fig. 7. Fit of Φ21 predictions to experimental results.

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Transformer - Cavity System

The transformer-cavity system consists of the transformer connected to the cavity by a piece of waveguide. Power is fed in through the network analyzer and coupled to a waveguide. From the waveguide, input power passes to the waveguide transformer, the ceramic window, the waveguide step, and then to the cavity. There are also waveguide pieces on each end of the step. The ceramic window is matched to the waveguide system so it can be modeled as just a section of waveguide. The cavity is modeled as a resistor, capacitor, and inductor in parallel. The equivalent circuit is shown in Figure 8. The program is shown in Appendix 4. The values for these components are calculated from the loaded Q of the cavity. This value is somewhat variable between the values of 1.8×105 and 2.5×105. The value of 2.2×105 was chosen to best conform to experimental results. This gives Rs = 2.27×106 Ohm.

The impedance of the waveguide step is given by calculations from the Waveguide Handbook [5]. The length of the waveguide between the step and the cavity is also somewhat variable. Due to the bends in the waveguide, the effective waveguide length could be longer or shorter than an approximated 0.54λ. The length was adjusted 0.24 λ to conform to the experimental results. Table 1 compares the measured Qext and the calculated Qext. The fit of the data is pretty good, particularly after adjustment of the loaded Q and the waveguide length. While data lends credence to the model, the insufficiency of measured data does not allow us to confirm the model.

Table 1. Calculated Q values predicted by the model are quite close to those measured. All three plungers are at the same height.

Plunger height (mm) Q measured (105) Q calculated(105)

38.1 1.65 1.74

63.5 2.04 2.00

88.9 2.50 2.50

114.3 4.00 8.00

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Calculations were also made for various combinations of plunger heights. Appendix 5 provides 4 charts, which detail 4 different values for the height of plunger 1. The Qext is plotted versus the height of plunger 3, and multiple lines are drawn to represent different heights of plunger 2.

Conclusions

The matrix model of the waveguide transformer provides a simple, quantitative method for adjusting the match conditions of the cavity. The attached programs as well as the charts in Appendix 5 provide a straightforward method for controlling the transformer to couple the input power to the cavity for a variety of beam conditions.

References:

1. B. Dwersteg, “SC-Cavity Operation Via WG-Transformer,” Proc. of the 4th SRF Workshop, KEK Report 89-21, Vol. 2, pp. 593-604.

2 . B. Dwersteg and Q. Yufang, “High RF Power Waveguide Transformer,” Preprint DESY-M-89-08, August 1989.

3 . G.L. Matthaei, L. Young, E.M.T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Mcgraw-Hill Inc, 1964.

4 . MAPLE 6, Waterloo Maple Inc.

5 . N. Marcuvitz, Waveguide Handbook, M.I.T. Radiation Lab. Series No.10, Boston Tech Pub. (1964).

Appendix 1: Curve fitting Values of the electrical thickness and the normalized admittance were read as a function of plunger height off the graph in [2]. Preliminary curve fitting methods indicated that an exponential curve would best fit the admittance values while a 2nd order power series would best fit the electrical thickness values. However neither of these cases will in general pass through the origin point, thus giving 0 admittance and 0 electrical thickness with the plunger fully removed. This results in inaccuracies for low values of plunger height. It is also important to see that the model provides proper results for the case where all plungers are extracted, as this is a simple calculation to perform. Thus the data were fit instead to equations of the form

Y = a×(bx – 1) for the normalized admittance and

T = c×x2 + d×x for the electrical thickness.

X is the plunger height in mm, and a, b, c, and d are the curve fitting constants.

Normalized Admittance

To fit the data to the function, we take the logarithm of both sides

Y = a×(bx – 1) becomes

Y + a = a×bx

and ln (Y + a) = ln a + x×ln (b).

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Now we use an iterative process whereby the equation is transformed into

ln (Y + a0) = ln a1 + x×ln (b).

We initially set a0 =0 and use a linear fit for the logarithms of the data. This gives us a set of a1 and b values. These a1 values are then placed back in for a0 and the process is repeated until the new a and b values stop changing. While there was no guarantee of convergence, this process happened to converge and the final values obtained were

a = 0.275023 ,

b = 1.020016 .

Electrical Thickness

We want to fit the electrical thickness to

T = c×x2 + d×x.

In general, the standard deviation is defined as

E(a,b) = Σ yi – T(xi, c, d)2

To minimize this, we take the partial derivatives with respect to c and d and set these to 0

(∂E/∂c) = 0 = 2×Σ [ yi – T(xi, c, d) ] × (-∂f/∂c),

this becomes

2×Σ [ c×xi4 + d×xi

3 – xi2yi] = 0,

thus

c×avg(x4) + d×avg(x3) = avg(x2y).

Similarly, from the partial derivative with respect to d we obtain

c×avg(x3) + d×avg(x2) = avg(xy).

Solving this system of equations we obtain

d = [ avg(xy)-avg(x3) ×avg(x2y) / avg(x4) ] / [ avg(x2) – (avg(x3))2 / avg(x4) ],

c = [ avg(x2y) / avg(x4) ] – [ avg(x3) ×d/avg(x4) ].

Thus we obtain

c = 0.00047898

and d = 0.1182 .

Special thanks to Greg Werner for his assistance in deriving these fits.

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Appendix 2: Matrix model

The model for the waveguide transformer is based on the matrix methods described in the Waveguide Handbook [5]. Each component of the equivalent circuit is represented by a matrix of the form

⋅

=

2

2

1

1

I

V

DC

BA

I

V ,

where the voltages and currents are shown in Figure A1.

Matrices representing various components are also found in the Waveguide Handbook. For example, in the case of terminating resistor the matrix is represented by

=

1101

1101

RZ .

Thus, in Figure A2, I2 = 0 so we get the equations

V1 = V2

and I1 = V2 / R

as expected for a standard resistor circuit.

Thus for a combination of circuit elements, each element is broken up into a component which is represented by a matrix. Thus the overall transfer function is just given by multiplication of the matrices representing the components. The matrices that represent the components are

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( ) ( )( ) ( )

=

kLZ

kLikLiZkL

WaveguideLosslesscos

sinsincos

_

=

1

01

iYPlunger

=

1101

_Z

LoadMatched

=

1101

Rresistor

=

1

01

CiCapacitor

ω

−=

1

01

LiInductorω

=

1

01_

iYStepWaveguide

Appendix 3: MAPLE worksheet for matched load case

Appendix 3

> restart;

h2, h4, h6 in mm. Everything else in SI units

> h2:=61.9;h4:=91.9;h6:=91.9; := h2 61.9

:= h4 91.9

:= h6 91.9

> u:=1.25663706144*10^(-6); := u .1256637061 10-5

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> e:=8.85418781762*10^(-12);

:= e .8854187818 10-11

> c:=3*10^8; := c 300000000

> f:=5*10^8; := f 500000000

> a:=.4572; := a .4572

> fc:=3*10^8/(a*2); := fc .3280839896 109

> w:=2*Pi*f; := w 1000000000 π

> b:=evalf(sqrt(w^2*a^2-Pi^2*c^2)/(c*a));

:= b 7.902308312

> Z:=sqrt(u/e)/sqrt(1-(fc/f)^2);

:= Z 499.2352187

> L1o:=.2921;L3o:=.1889125;L5o:=.1889125;L7o:=.2301875; := L1o .2921

:= L3o .1889125

:= L5o .1889125

:= L7o .2301875

> lp2:=.00047898*h2^2+.1182*h2;

:= lp2 9.151844558

> lp4:=.00047898*h4^2+.1182*h4; := lp4 14.90785828

> lp6:=.00047898*h6^2+.1182*h6; := lp6 14.90785828

> L1:=L1o-.0005*lp2; := L1 .2875240777

> L3:=L3o-.0005*(lp2+lp4); := L3 .1768826486

> L5:=L5o-.0005*(lp4+lp6); := L5 .1740046418

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> L7:=L7o-.0005*lp6;

:= L7 .2227335709

> A:=matrix(2,2,[cos(b*L1), I*Z*sin(b*L1), I/Z*sin(b*L1), cos(b*L1)]);

:= A

-.6452172303 381.4152908 I

.001530339081 I -.6452172303

> Y2:=.275023/Z*(1.020016^h2-1);

:= Y2 .001327719176

> Y4:=.275023/Z*(1.020016^h4-1);

:= Y4 .002853551072

> Y6:=.275023/Z*(1.020016^h6-1);

:= Y6 .002853551072

> B:=matrix(2,2,[1,0,I*Y2,1]);

:= B

1 0

.001327719176 I 1

> C:=matrix(2,2,[cos(b*L3), I*Z*sin(b*L3), I/Z*sin(b*L3), cos(b*L3)]);

:= C

.1721532149 491.7817295 I

.001973158440 I .1721532149

> E:=matrix(2,2,[1,0,I*Y4,1]);

:= E

1 0

.002853551072 I 1

> F:=matrix(2,2,[cos(b*L5), I*Z*sin(b*L5), I/Z*sin(b*L5), cos(b*L5)]);

:= F

.1945101132 489.7000820 I

.001964806319 I .1945101132

> G:=matrix(2,2,[1,0,I*Y6,1]);

:= G

1 0

.002853551072 I 1

> H:=matrix(2,2,[cos(b*L7), I*Z*sin(b*L7), I/Z*sin(b*L7), cos(b*L7)]);

:= H

-.1881842354 490.3157551 I

.001967276563 I -.1881842354

matrix J is the terminating load

> J:=matrix(2,2,[1, 0, 1/Z, 1]);

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:= J

1 0

.002003063811 1

with(linalg);

multiply(A,B,C,E,F,G,H,J);

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> K:=multiply(A,B,C,E,F,G,H,J);

:= K

+ -.3980455222 .6230435814 I + -0. 311.0452987 I

+ -.002177708914 .001823687076 I + -1.087188986 0. I

> > x:=K[1,1];

:= x + -.3980455222 .6230435814 I

> y:=K[2,1]; := y + -.002177708914 .001823687076 I

> ref:=(x/y-Z)/(x/y+Z);

:= ref − -.3212872883 .1382181590 I

> S11:=10*log10(ref*conjugate(ref));

:= S11 + -9.124678014 0. I

> S21:=10*log10(1-Re(ref*conjugate(ref)));

:= S21 -.5666863787

> stuff:=(1+ref)/x; := stuff − -.6517730488 .6729503050 I

> p21:=evalf(180/Pi*arctan(Im(stuff)/Re(stuff)));

:= p21 45.91586220

> p11:=evalf(180/Pi*arctan(Im(ref)/Re(ref)));

:= p11 23.27743041

>

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Appendix 4: MAPLE program for transformer - cavity case

> restart;

h2, h4, h6 in mm. Everything else in SI units

> h2:=0;h4:=0;h6:=0; := h2 0

:= h4 0

:= h6 0

> QL:=2.2*10^5; := QL 220000.0

> Qo:=10^9; := Qo 1000000000

> RdivQ:=500/QL; := RdivQ .002272727272

> u:=1.25663706144*10^(-6): > e:=8.85418781762*10^(-12): > c:=299792458: > fo:=5*10^8: > a:=.4572: > wo:=2*Pi*fo: > fc:=c/(a*2): > L1o:=.2921:L3o:=.1889125:L5o:=.1889125:L7o:=.2301875: > RS:=RdivQ*Qo;

:= RS .2272727272 107

> S21:=matrix(81,1,0): > step:=125;

:= step 125

> for i from -40 by 1 to 40 do > f:=5*10^8+step*i: > w:=2*Pi*f: > b:=evalf(sqrt(w^2*a^2-Pi^2*c^2)/(c*a)): > wavelength:=c/sqrt(fo^2-fc^2); > La:=0.125195154105*wavelength; > Lb:=0.54*wavelength; > Z:=sqrt(u/e)/sqrt(1-(fc/f)^2):

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> p:=h2;q:=h4;r:=h6; > lp2:=.00047898*p^2+.1182*p; > lp4:=.00047898*q^2+.1182*q; > lp6:=.00047898*r^2+.1182*r; > L1:=L1o-.0005*lp2; > L3:=L3o-.0005*(lp2+lp4); > L5:=L5o-.0005*(lp4+lp6); > L7:=L7o-.0005*lp6; > Y2:=.275023/Z*(1.020016^p-1); > Y4:=.275023/Z*(1.020016^q-1); > Y6:=.275023/Z*(1.020016^r-1); > > A:=matrix(2,2,[cos(b*L1), I*Z*sin(b*L1), I/Z*sin(b*L1), cos(b*L1)]); > B:=matrix(2,2,[1,0,I*Y2,1]); > C:=matrix(2,2,[cos(b*L3), I*Z*sin(b*L3), I/Z*sin(b*L3), cos(b*L3)]); > E:=matrix(2,2,[1,0,I*Y4,1]); > F:=matrix(2,2,[cos(b*L5), I*Z*sin(b*L5), I/Z*sin(b*L5), cos(b*L5)]); > G:=matrix(2,2,[1,0,I*Y6,1]); > H:=matrix(2,2,[cos(b*L7), I*Z*sin(b*L7), I/Z*sin(b*L7), cos(b*L7)]); > J:=matrix(2,2,[cos(b*La), I*Z*sin(b*La), I/Z*sin(b*La), cos(b*La)]); > K:=matrix(2,2,[1,0,I*1.08*10^(-4),1]); > M:=matrix(2,2,[cos(b*Lb), I*320*sin(b*Lb), I/320*sin(b*Lb), cos(b*Lb)]); > > > P:=matrix(2,2,[1,0,1/RS,1]); > R:=matrix(2,2,[1,0,1/RdivQ*I*w/wo,1]); > S:=matrix(2,2,[1,0,-1/RdivQ*I*wo/w,1]); > with(linalg); > T:=multiply(A,B,C,E,F,G,H,J,K,M,P,R,S); > > > x:=T[1,1]; > y:=T[2,1]; > ref:=(x/y-Z)/(x/y+Z); > bbc:=Re(ref*conjugate(ref)); > gain:=evalf(10*log10(1-bbc)); > S21[41+i,1]:=%;

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> od: Warning, the protected names norm and trace have been redefined and unprotected

> eval(S21): > writedata("data.txt",S21,float); > middle:=0; peak:=-100;

:= middle 0

:= peak -100

> for kk to 81 do > if (S21[kk,1]>peak) then peak:=S21[kk,1]; > middle:=kk; > end if: > od: > middle;peak;

38

-30.81071844

> halfvalue:=peak-3; := halfvalue -33.81071844

> lownotch:=0; hinotch:=0; := lownotch 0

:= hinotch 0

> ldiff:=10; := ldiff 10

> for mm to (middle-1) do > lstep:=middle-mm; > if (abs(S21[lstep,1]-halfvalue)<ldiff) then lownotch:=lstep; > ldiff:=S21[lstep,1]-halfvalue; > end if: > od: > lownotch;

29

> hdiff:=10; := hdiff 10

> for mm from (middle+1) to 81 do > hstep:=mm; > if (abs(S21[hstep,1]-halfvalue)<hdiff) then hinotch:=hstep; > hdiff:=S21[hstep,1]-halfvalue; > end if:

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> od: > hinotch;

48

> deltaf:=(hinotch-lownotch)*step;

:= deltaf 2375

> peakf:=5*10^8-step*(41-middle);

:= peakf 499999625

> Qfactor:=evalf(peakf/deltaf);

:= Qfactor 210526.1579

>

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Appendix 5: Calculation results

0 50 100 1501

2

3

4

5

6

7

8

9

10x 10

5

Height of Plunger 3 (mm)

Qex

t

Qext for Plunger#1 = 0 mm

PL2=0PL2=50PL2=100PL2=150

Figure A3.

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0 50 100 1501

2

3

4

5

6

7

8

9

10x 10

5

Height of Plunger 3 (mm)

Qex

t

Qext for Plunger#1 = 50 mm

PL2=0PL2=50PL2=100PL2=150

Figure A4.

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0 50 100 1501

2

3

4

5

6

7

8

9

10x 10

5

Height of Plunger 3 (mm)

Qex

t

Qext for Plunger#1 = 100 mm

PL2=0PL2=50PL2=100PL2=150

Figure A5.

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0 50 100 1501

2

3

4

5

6

7

8

9

10x 10

5

Height of Plunger 3 (mm)

Qex

t

Qext for Plunger#1 = 150 mm

PL2=0PL2=50PL2=100PL2=150

Figure A6.