THE UNIVERSITY OF MICHIGAN ENGNE L f.t= LI RARY Technical Report ECOM-0138-5 May 1969 COAXIAL MICROWAVE BANDPASS FILTERS C. E. L, Technical Memorandum No'. 100 Contract No. DAAB 07-68-C-0;138 DA Project No. 1 HO 21101 A04 01 02 Prepared by <W. A! Davis COOLEY ELECTRONICS LABORATORY Department of Electrical Engfneering The University of Michigan Ann Arbor, Michigan for U. S. Army Electronics Command, Fort Monmouth, N. J. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL-S.

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ABSTRACT A review of lumped element filter synthesis techniques using the power loss ratio is presented. This leads to filter synthesis based on impedance inverters for which S. B. Cohn (Ref. 1) has given an approximate microwave realization. Here an improved method is presented which considers the distributive property of the impedance inverter. Several theoretical curves are shown comparing the two methods. Finally results from an experimental model are shown. -iii

FOREWORD This study on microwave bandpass filters was motivated by the need for a filter that could be constructed completely within a coaxial line. There is no single source in the literature which describes the principles and derives the relations used here. Also there is little information on how these principles may be applied to coaxial bandpass filters. However, after applying these principles to a coaxial filter, some empirical modification is required because the design is based on approximating a lumped-reactive element with a distributed element. Therefore, this theory was improved by accounting for the nonzero length of the reactive element. These new design formulas give a filter characteristic which resembles the desired characteristic to a much greater degree than the older method. The purpose of this study is to bring together the principles and derivations needed for the construction of a microwave bandpass filter. With this information and appropriate tables for a low-pass prototype circuit an engineer can design a coaxial microwave bandpass filter for a desired bandwidth and for Chebyshev filters a desired passband ripple. The theory developed there applies equally well to bandpass impedance matching networks. In addition to the information mentioned above, the designer needs to know only the Q of the load and the load impedance level. -iv

The formulas are simple to use and sufficiently exact so that no additional empirical modification is needed. The filter is easy to make since it consists only of a coaxial line and some disks. Both of the above considerations are important in reducing fabrication costs.

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD iv LIST OF ILLUSTRATIONS vi LIST OF TABLES x COAXIAL MICROWAVE BANDPASS FILTERS 1 I. General Discussion 1 II. Power Loss Ratio 2 III. Lumped Low-Pass Prototype Circuit 4 IV. Impedance Matching with Lumped Low-Pass Networks 6 V. Bandpass Matching Networks 8 VI. Impedance Inverters 9 VII. Microwave Realization 15 VIII. Combining K and J Inverters 22 IX. K Inverters with Disks of Nonzero Length 22 X. J Inverters with Nonzero Inductor Length 30 XI. Effects of Discontinuity Capacitance 32 XII. Design Procedure 33 XIII. 50-Ohm to 50-Ohm Bandpass Filters 34 XIII a. Six-Disk Filter Using Distributed Design 35 XIII b. Impedance Transformer 43 XIII c. Impedance Plots 48 XIV. Experimental Work 48 XV. Conclusions 55 REFERENCES 56 DISTRIBUTION LIST 57 -vi -

LIST OF ILLUSTRATIONS Figure Title Page 1 Low-pass prototype circuit starting with shunt capacitor 4 2 Low-pass prototype circuit starting with series inductor 4 3 Lossless network fed through an arbitrary generator impedance 5 4 Low-pass matching network 7 5 Bandpass prototype starting with shunt resonator 9 6 Bandpass prototype starting with series resonator 9 7 Ideal impedance inverters 10 8 Lumped bandpass impedance matching network using K inverters 10 9 Lumped bandpass impedance matching network using J inverters 11 10 Prototype section 11 11 K inverter section 11 12 Prototype with adjusted impedance level 12 13 Input section of prototype 13 14 Unnormalized input section of prototype 14 15 Input K inverter section 14 16 Generalized impedance matching network using K inverters 16 17 Generalized impedance matching network using J inverters 17 -vii

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 18 Three K inverter example 18 19 Lumped K inverter realization 20 20 Lumped J inverter realization 20 21 Alternate K and J inverters 22 22 Image impedance for ZI1 infinite network chain 23 23 Coaxial realization of the K inverter of Fig. 19(b) 23 24 Bisected disk element 25 25 The series J inverter 30 26 Theoretical comparison between two design techniques for a 1 percent bandwidth bandpass filter which operates between two 50-ohm loads 38 27 Theoretical comparison between two design techniques for a 5 percent bandwidth bandpass filter which operates between two 50-ohm loads 39 28 Theoretical comparison between two design techniques for a 10 percent bandwidth bandpass filter which operates between two 50-ohm loads 40 29 Theoretical comparison between two design techniques for a 5 percent bandwidth 5-ripple filter which operates between two 50-ohm loads 41 30 Theoretical comparison between two design techniques for a 10 percent bandwidth 5-ripple filter which operates between two 50-ohm loads 42 31 Theoretical comparison between two design techniques for a 1 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load 44

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 32 Theoretical comparison between two design techniques for a 5 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load 45 33 Theoretical comparison between two design techniques for a 10 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load 46 34 Theoretical comparison between two design techniques for a 20 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load 47 35 Theoretical impedance of a 3-ripple, 10 percent bandwidth bandpass filter operating between two 50-ohm loads 50 36 Theoretical impedance of a 3-ripple, 10 percent bandwidth 50 impedance transformer as seen from the I-ohm side 51 37 Coaxial bandpass filter 52 38 Theoretical 10 percent bandwidth bandpass filter which accounts for discontinuity capacitance 53 39 Experimental 10 percent bandwidth bandpass filter 54 -ix

LIST OF TABLES Table Title Page I. Disk parameters for bandpass filters with three ripples 36 II. Disk parameters for bandpass filters with five ripples 37 III. Disk parameters for bandpass impedance transformer using three disks 49 IV. Filter design parameters 52

COAXIAL MICROWAVE BANDPASS FILTERS I. General Discussion At low frequencies a very general and complete filter synthesis technique has been developed which utilizes lumped inductors and capacitors as the basic building blocks. At microwave frequencies distributed parameter elements are used which makes filter synthesis more complicated. Since these elements have a complex frequency dependence, no complete filter synthesis technique has been developed. However, lumped techniques have been an invaluable guide to microwave filter synthesis. The following discussion presents the necessary equations and outlines a procedure for designing bandpass impedance matching networks in coaxial transmission lines. The method of analysis starts by defining the desired frequency dependence of the reflection coefficient of the network. A low-pass lumped-prototype circuit is developed which is transformed into a bandpass circuit. The use of impedance inverters (a device which takes the reciprocal of the impedance) is found necessary to realize the lumped prototype in a distributed circuit. An impedance inverter is realized by a disk and a small length of line. Since a coaxial disk introduces discontinuity capacitance, the design is modified to take this into account. Finally, an example is given to show how these relations are used. In short, a review of filter synthesis is presented which is followed by an improved synthesis technique for the impedance inverter.

II. Power Loss Ratio The power loss ratio or transducer loss ratio is defined as the incident or available power divided by the power delivered to the load (Ref. 2, pp. 403-410). IVg2 RL 11 LR 2 = * = 2 4R IV I i-rr ' 1-p g L The symbol r is the input reflection coefficient of a lossless network terminated by a resistive load RL. This synthesis method begins by choosing the reflection coefficient of the network which for passive circuits is restricted to the range 0 < r (w) < 1. For a transmission line with characteristic impedance Z terminated with an impedance ZL = R + jX the reflection o L coefficient is Z - Z R(w) Z + jX(w) r (w) +jX) = (w) (2) L 0 Z 0 R 0 + If m and n denote the even and odd parts of the polynomials respectively, the load impedance is (Ref. 3, p. 22) m +n Z = 1 (3)' L m2 + n2 When s, the complex frequency, is jw, Eq, 4 below shows that the even parts are real and the odd parts are imaginary. -2 -

(ml + nl)(m2 - n2) L (m+n)(m -n) 2 2 2 2 L (m2+ n2)( 2 2 n2 mlm2 - n1n2 2n1 1- 2 ZL 2 2 2 2= R(o) + jX(w) (4) m2 n2 m2 -n2 Therefore R(o) is an even function and X(o) is an odd function of w. This property is used in Eq. 2 to show R(w) - Z - jX(w) (-) R( Z = r () (5) 0 2 2 Since r (w) r (-w) = p, it is apparent that p is an even function of w. The power loss ratio is the ratio of even polynomials which from Eq. 1 can be expressed as (R- Z )2 X 2 o (2 (6) For the case of a low-pass Chebyshev filter, this last ratio is chosen as PLR = 1 + k Tn2(w/w ) (7) where wc is the cutoff frequency, k is the passband tolerance, and Tn is the Chebyshev polynomial of degree n. -3 -

T(w/w) = cos [nArccos (w/cw)] (8) Tn oscillates between 1 for lw/wCI < 1 and equals one when I @/it;CC -1 III. Lumped Low-Pass Prototype Circuit A low-pass filter may be realized by either of two dual prototype lumped ladder networks as shown in Figs. 1 and 2. 92 94 gn Tgl;T g3 Tg5 GL = g1 RL = gnl Z. in Fig. 1. Low-pass prototype circuit starting with shunt capacitor Rl g3 g5 gn * * Rg= ~ 1 Lg | RL gn1 3 GLn L gn+1 Fig. 2. Low-pass prototype circuit starting with series inductor A relationship must be found between these two lumped circuits and the power loss ratio which was defined in terms of a transmission line. In particular a transmission and reflection coefficient will be found for -4 -

the more general network of Fig. 3 where the source and load are connected by an arbitrary lossless network with Z1 = V1/ I1 Rll + jX11. 11 12 I I giz Ig + Vg( V1 Lossless VI RL g 1 Network Fig. 3. Lossless network fed through an arbitrary generator impedance Since the network is lossless, the average power entering port 1 must equal the average power leaving at port 2. 'V 2Ri IV212 g 11 2 (9) I + Z I RL g 1 This expression may be rearranged to give the ratio of power delivered to the load to the available power from the generator. 2 IV21 4R 4R R 2 g _ hg (10) R IV 12 IZ + Z L g g 1 This is seen to be equivalent to the transmission coefficient as the term on the right-hand side is 1 1 g 1 (R11 Rg) + (Xg+ X11) 4R R Z1 +Zg IZg+z2 'Zg +Z12 -5 -

Therefore the reciprocal power loss ratio or transmission coefficient for the circuit of Fig. 3 is simply 2 = V2 L 1 Z g 2 Iti - 2 Z +Z - 1 - 111 (11) V2/(4R) Zl + Zg g g Thus the reflection coefficient of the low-pass prototype circuit of Fig. 1 is Z. -R in g (12) Z. +R (12) in g where Z. is the input impedance of the ladder network as seen from the in terminals of the generator resistance. To derive the particular g values for the circuit of Fig. 1, the two expressions for the power loss ratio are equated. 22 2 P 1 + k Tn (w/W) = LR n c (3 The left hand side defines the ripple factor k, the cutoff frequency cc, and the number of elements n. The right hand side is a function of the lumped prototype circuit g values which can now be found. Explicit expressions exist for the gk for both the maximally flat and the Chebyshev filters. However, tabulated values will be used subsequently and they may be found in various references (i. e., Ref. 4). A similar procedure is used for the case of the filter of Fig. 2. -6 -

IV. Impedance Matching with Lumped Low-Pass Networks A low-pass Chebyshev impedance matching network is shown in Fig. 4. gl 93 g+ go g2 5 ~ gn Vg Load Fig. 4. Low-pass matching network This network can be designed either to provide minimum reflection or to provide a specified ripple in the passband. The first criteria is the one used in the example later. The load is considered to consist of both go and g1. An optimum impedance matching network must necessarily have a filter-like characteristic. Also, if the load has a reactive part, it is impossible to have perfect power transmission over a range of frequencies. Overall power transmission may be improved, however, if a small amount of power is reflected at all frequencies in the passband. The load is characterized by the decrement 6A 1 (14) glgoc where wc is the cutoff frequency for the low-pass circuit of Fig. 4. The published tables and curves for the gk (k = 1, 2,..., n) are normalized to wc = 1. Hence these gk values must be remultiplied by w. when used. The decrement can be written in terms of the resonated QA of Ref. 4, p. 120. -7 -

the load, which by use of a reactive element has been tuned to resonate at wo,and the fractional bandwidth w. 1 (15) wQA This definition will be convenient for bandpass matching structures described later, and it emphasizes that the decrement is the reciprocal Q of the load at the edge of the impedance matching band. Better matching and lower ripple can be achieved by using more filter elements although beyond n = 3 or 4, there is only slight improvement. The calculation of the g values for the impedance matching network of Fig. 4 is complicated so curves found in Ref. 4 (pp. 126-129) are used to get the element values. V. Bandpass Matching Networks At microwave frequencies bandpass rather than low-pass impedance matching networks are usually desired. For the bandpass case both the load and generator resistances may be specified by the designer whereas for the low-pass case optimum design occurs only for a specified load to generator impedance level ratio. A bandpass filter is easily derived by the low-pass to bandpass frequency mapping c w W w 2 w2- W1 where ~ = w w2 and w = - The low-pass frequency variable is w and the bandpass frequency variable is o. The bandwidth of the -8 -

bandpass filter is w2 - wc and the center frequency is wc. This mapping 0 transforms shunt capacitors into parallel resonant circuits and series inductors into series resonant circuits. Both of these types resonate at co. The circuits of Figs. 1 and 2 are transformed into the bandpass circuits of Figs. 5 and 6. L2 C R =g L1$ C3 L3 gn+l C L (Shunt Reactances) 0ocg0 o Fig. 6. Bandpass prototype starting with shuneries resonator The steps are carried out in greater detail in several references (Ref. 5, pp. 356-362). VI. Impedance Inverters Many kinds of microwave filters can be constructed more conveniently if they can be built from either all capacitive or all inductive o-9 -

elements. This can be done with the use of ideal impedance or admittance inverters for the low-pass case. For the bandpass case, a filter may be constructed with only series or only shunt resonant circuits when inverters are used. The ideal impedance inverter is defined in Figs. 7(a) and (b). 900 Image 2 90 Image z a aZb J (a) K inverter (b) J inverter Fig. 7. Ideal impedance inverters One obvious example of a K inverter is a quarter wavelength of transmission line with a characteristic impedance of K at all frequencies. The lumped bandpass filters shown in Figs. 5 and 6 may be built from series resonant circuits separated by K inverters or from shunt resonant circuits separated by J inverters as shown in Figs. 8 and 9. L C C C L C rl rl r2 r2 rn rn K01K K!2 RA K01 12 K K23 n,n+l R Fig. 8. Lumped bandpass impedance matching network using K inverters

GA Fig. 9. Lumped bandpass impedance matching network using J inverters The values of the K. are found on the basis that the circuit in Fig. 8 i, i+1 must have the same response as the circuit in Fig. 5. A section of the lumped prototype is given in Fig. 10 together with the equivalent section of the inverter network in Fig. 11. Lc- gw!li1. c 1-1 L A-1 Q-W~g11 CP wow 'LQS TCQZQ+1- Ctl=@o~wrgflcQ~002 Fig. 10. Prototype section (L -1 L Cr (( Lr, Ir, 1a -, > B+1 Fig, 11. K inverter section

The elements Lji and Cl1 constitute the left hand series arm needed for the circuit of Fig. 11, but the impedance level must be changed to correspond to that of Fig. 11. This is accomplished by multiplying all the inductances by Lr, e1/Ll_1 and all capacitors by L_ l/Lr, 1-i as indicated in Fig. 12. (L _1) (C-1) (L ( LLXf T (L Fig. 12. Prototype with adjusted impedance level For the circuit of Fig. 12 to be identical to that of Fig. 11, Yt must be the same for both. K2 [wLr, - 1/(wC, )J + Zl wCL ) L (L J L(Lr -I) woL (Lr, (L, 1 2 [ CC(Li_1) (L_1) ] (LXi1) () )(17) K-k j [W - r-1i WLkLr- V-1J +1 (Lr V-i { r} rk

The terms with the same frequency dependence are equated to give a solution for K. Lr, Q (L r, Q-i) 1 r, r, (L L kC = rk r - (18) The K can be written in terms of the original low.pass prototype by referring to Fig. 10. Cw L (L K0- r, +(19) gkg+ 1 g The first and last K's are obtained similarly. Figure 13 is the first section of the total bandpass circuit of Fig. 5. c 2 L o 1 ww - 1)' Fig. 13. Input section of prototype The impedance level is raised to the desired generator resistance by multiplying all impedances by RA/go as shown in Fig. 14.

R LIRA Clgo Y2go Y1 Y1 Fig. 15. Input K inverter section This must have the same response as the network with the K inverter of Fig. 15 so Y are the same in both cases. [ ogiWg I + Yg0/R = K )[ r wwL R o K(21) FK i= vo r1rA (21)i Wcgog 1 -14 -

Similarly o0w Lr nRB K (22) n, n+ 1 cgngn+ 1 The bandpass prototype circuit of Fig. 5 can therefore be replaced by the circuit of Fig. 8 with its associated K inverters. Since the K or J inverters have the ability to shift the impedance levels, the sizes of the RA and RB as well as the Lr, may be chosen arbitrarily while retaining the same response as the prototype circuit. The use of K inverters has given the designer additional flexibility especially with regard to his choice of RA and RB. VII. Microwave Realization The parameters for the circuit of Fig. 8 have been explicitly derived, but two problems still remain for realization in a microwave structure: (1) how are the series resonant circuits to be made, and (2) how are the K inverters realized. Before actually resolving the first question it is convenient to generalize the expressions for the K's and corresponding J's to make them more compatible with distributive elements. A series LC circuit or more generally a series resonator which has zero reactance at co can be described in terms of its resonant 0 frequency co and a reactance slope parameter X.

X 2 dX (23) A shunt resonator where the susceptance is zero at w, in turn can be described in terms of its resonant frequency and a susceptance slope parameter b. b = dBo~ (24) 0 For the series LC circuit X = w L and for the shunt LC circuit b = w C. O O Thus the Q for a circuit with resistance R in series with a series resonator or a conductance G in parallel with a shunt resonator is Q = X/R and Q = b/G respectively. The K and J values in terms of these slope parameters are given in Figs. 16 and 17. For distributed circuits, these figures should be used rather than Figs. 8 and 9. Ra3 w Ko I I i i i ixI K. w jj+1i K RBwXn Kn nK Fig. 16. Generalized impedance matching networks using K inverters KA ~~~~0~1 1nn -1

GA- J01 GB1bw12 bBn Jnn+bIG j bw 01 X Jj~j+l,2'~r >n, n+l G lW c /lbj bj+lG Fig. 17. Generalized impedance matching network using J inverters The series resonant circuits X.(o), shown in Fig. 16, can be realized as a half wavelength transmission line. Thus the choice of L e in the previous section is equivalent to the choice of characteristic impedance of the half wavelength line. Often all the X. can be made to have the same characteristic impedance of 50 ohms. The reactance slope parameter is obtained from the transmission line equation. The reactance of the transmission line is (z2 _ R2) tan j = Z + RL2 tan (25) where 0 = Ed = wd/c. The slope parameter is obtained by differentiation of X. J01 j j~~~~l -17

2 2 2 2 2 X = Co 2~ 2 R2 d 20 Z + RL tan 0 - 2R tan 0 X = Z 0 (Z - RL) 2 sec 0 2 For +a transmission line taresonant at = Zwd 2 2 2 = 00 2 2 o L (ta) Inversion networks often present a very small resistance in shunt with(26) (z2 2 0)2 o + tan For a transmission line resonant at 0 = the series resonator so that RL/Z << 1. In this case Z zr 0X 2(28) The impedance matching properties of a series of K inverters can 2 22 nvbe shown bynetworks often presenFig. 18 two unequsmal resistances are matched with three K nverters separated by half-wavelength transmission-lines. AFig. 18. Three~ ~ ~~R Bi iRi Fig. 18. Three K inverter example 18

Each K inverter inverts the impedance. 2 K2 23 2 2 23 2 2 12RB Using the values of the K's from Fig. 16, 'R2 RAWX1 RBWX2 c g1g2 1! 2 RB 0gogwc g2g3W W X1X2 B RA z = A 1 gog3 R In general with n K inverters, the input impedance is Z when RAg n+1 n is even and A n when n is odd. The parameter g is always chosen as 1 while gn+l / 1 for Chebyshev filters when n is even or for Chebyshev impedance matching networks when the load is complex. If gn+l / 1 the designer must modify the value of RA in K01 so as to obtain the correct impedance level. -19 -

The series resonant circuits can be realized in a microwave structure by a half-wavelength transmission line with a reactance slope parameter of 7rZ/ 2. The second question needing clarification is how the K inverters are realized. One inverter already mentioned is a quarter wavelength line. Much broader bandwidth may be achieved using the K inverters in Fig. 19 or the J inverters in Fig. 20. - 0x 0 x 0 J<o ~ i i (a) - (b) Fig. 19. Lumped K inverter realization 0.-0/2-.-0/2. +//2+0/_ --— A- --- B< 0 B> 0 0>0 0<0 (a) (b) Fig. 20. Lumped J inverter realization In Figs. 19(a) and 20(b) the negative line length must be absorbed by available line length between inverters. The value of K for the inverters of Fig. 19 is K = Z tan 10/21 (29) -20 -

where 0 = - Arctan (2X/ Z) (30) K/Z X/Z = (31) o 1- (K/Zo) The value of J for the J inverters in Fig. 20. is J = Y tan 1O0/21 (32) 0 where,0 = - Arctan 2B/Yo (33) J/Y B/Y = (34) o - (J/Y ) Thus when K or J is calculated by the equations in Figs. 16 or 17, 0 and the reactance or susceptance can be found. The derivation of these formulas may be found in Cohn's work (Ref. 1) and will not be rederived here. A more general set of relations based on Fig. 19(b) will be derived later which give the above results when specialized to a lumped capacitor. -21 -

VIII. Combining K and J Inverters When wavelengths are too long to space inverters of one kind every half wavelength, K and J inverters can be placed alternately every quarter wavelength as in Fig. 21. b/4 4 RFig. 21 Alternate K and J inverters Fig. 21. Alternate K and J inverters The only difference between the values here and those in Figs. 16 and 17 is a difference in slope parameter. For a quarter wavelength line X = 72ZO /4 and b = iY /4. IX. K Inverter with Disks of Nonzero Length The image impedance and the image propagation function are found useful in the derivation of the disk length and phase shift of the impedance inverter. The image impedance in a uniform transmission line is the characteristic impedance of the line, and the image propagation function is the propagation constant of the transmission line. The image impedance and propagation function are more general than that implied above for a uniform line. If ZI1 and ZI2 are the image impedances of an unsymmetrical network, then an infinite chain of these networks -22 -

connected together, as shown in Fig. 22, will present its corresponding image impedance at any junction of these networks...o... 2 i I 2 I2 I1 ZI ZI1 I2 Fig. 22. Image impedance for ZI infinite network chain Since the impedance is the same in both directions, the reflection coefficient at each junction is zero. If a wave propagates from left to right, its phase and attenuation will be affected by each network according to its image propagation function, but it will travel from one network to another without reflection (Ref. 4, pp. 49-50). 2r ~-0/2 —+ fk 0/2+ — Fig. 23. Coaxial realization of the K inverter of Fig. 19(b) -23 -

The design of the K inverters used in the previous designs is based on one of Cohn's procedures (Ref. 1), which is here designated as a " lumped-design procedure. " In that procedure, the filter design calls for a K inverter of the form shown in Fig. 23. The inverter is realized by selecting a disk of nonzero length, so that the disk capacitance is equal to that of the capacitor in Fig. 19(b). A more accurate procedure has been developed here, and is designated as a "distributed-design procedure. " This procedure follows the general form of Cohn's analysis, but considers the coaxial K inverter shown in Fig. 23 to be a section of low-Z line rather than a lumped capacitor. The dimensions of this structure are thus chosen with regard to the distributed character of the disk. Typically, the disk diameter is specified, and the length and characteristic impedance are determined. The image propagation constant can be expressed as i "\ _ SC tanh y = Z11Y11 -oc and the image impedance as Z K = Z Z Ii ~ Y oc sc Zll = Zoc is the input impedance of a two-port network when the output is open circuited, and Yl = 1/Z is the input admittance of a twoport network when the output is short circuited (Ref. 3, pp. 186-191). _ 9AL_

If the circuit element is bisected, as shown in Fig. 24, the open circuit and short circuit impedances can readily be found. V 0/2~ I/2 Z Z b a Fig. 24. Bisected disk element For the open circuit case Zaoc= -jZc cot 2 where 3 =(/c)r, and er is the relative dielectric constant of the material between the disk and the outer conductor while Z is the c characteristic impedance of the coaxial disk. Now Z is translated a aoc distance 0/2 down the coaxial line (which has characteristic impedance Z ). Since this circuit is assumed lossless, the impedance at Zb is 0 o imaginary. tan /2 - Z cot Z 2 Xboc o Z 1+ -c cot - tan~ -25 5 -

Thlis can be expressed as the tangent of the sum of two angles by a trigonometric identity. Oa 0 - cot Xboc tan 2 Arctan cot 2 If instead of an open circuit, as shown in Fig. 24, there is a short circuit at the plane of bisection, then Zasc = jZ tan 2 asc c 2 Translating this 0/2 radians, the short circuit reactance is found. Xbsc= Z tan + Arctan ctan Xbsc o2 z 2 Since this circuit is lossless, the propagation constant consists of only the imaginary part P'. -Xbsc -jy = 3' = 2 Arctan IXb Xboc -Z tan [ +Arctan( tan p': 2 Arcta n o V Z tan - Arctan ( cot)] -26 -

As in the quarter-wave transformer, the electrical length is j' = ~ 90~. This means the argument of the Arctan is 1. tan - ArctanZ tan -2 tan Arctan cot (2 scot) A (Z2j) - Arctan r cot -2 Arctan 2 Using the trigonometric identity, Arctan A ~ Arctan B = Arctan [ (A ~ B)/ (1 + AB)], the phase angle 0 may be expressed in terms of the yet unknown length k. 0 = Arctan (36) If 83/2 << 1, this 0 reduces to the value obtained by Cohn as indicated in Eq. 30. The image impedance or K value can be found also. K -= -Xboc Xbsc K = Z jti+ Arctan -cot + Arctan c tan -27 -

Eliminating the cotangent term by use of Eq. 35 gives the value of K in terms of the unknown length Q. K = Z tan[0 + Arctan tan (37) This again reduces to Cohn's value for K in Eq. 29 when i/ 2 0. An expression for 0/2 in terms of K is found by applying the trigonometric identity for the tangent of the sum of two angles to Eq. 37. 7z tan + tan K o 1 - tan 72 tan 2 Solving this for 0/2 gives K c tan KZ - Z tan 1 + tan 2 2 Equations 36 and 38 are two equations in the two unknowns 0 and k. The unknown 0 will be eliminated, and the result will be one equation in the unknown F - tan 1l3/2. -28 -

From Eq. 36 Ztan.2 tan - tan - Zo tan (Z c 1 Equation 38 is substituted into the right-hand side of the above expression. c/(1 F 2 4 + F (0 2 o o This is a quartic equation which after some algebra can be written in the following form: 4 K 3 c ~( o - + (Z c/Z- ) 1 - (K/Z) It should be noted thequation which after some algebra c rpositive. This expression cathe -29 - -29 -

be factored to give four roots. Z G jz o KP KP= 2 -The second two roots are the roots of interest. The reciprocal of G gives an explicit solution for f in terms of known quantities. F=- tan F KP (40) X. J Inverter with Nonzero Inductor Length The same general theory described in the previous section may be applied in an analogous fashion to the series inductor J inverter shown in Fig. 25. B<O I I I I I I __ _..... I I I >/2 0/2 (a) Lumped J inverter (b) DistributedJ inverter Fig. 25. The series J inverter -30 -

In this case the image propagation constant is tanh y - Z11Y11 Sc and the image admittance is Y =J- Y11 - YIi ~ Z V sc oc The derivation follows exactly as before with Z replaced by Y, SC oc Zoc replaced by Y s Zc replaced by Yc and ZO replaced by YO The final answer is tan = Y - (40') C: ) where 1 - (Y /Y o) P = (39') 1- (J/Y ) and [cot( 3 / 2) - tan ( 3/ 2) (Yc/Y ) = Arctan c (36 ') 1+(Y /Yo) The remaining two impedance inverters shown in Figs. 19(a) and 20(b) are -31 -

not amenable to this kind of analysis since a structure approximating a shunt inductance or a series capacitance would not be a TEM structure. XI. Effects of Discontinuity Capacitance A sudden change in diameter of the center conductor introduces fringing capacitance which has not yet been accounted for in the design. Also higher order propagating and evanescent modes around the disks will have the effect of storing more energy than the theory presented would indicate. P. I. Somlo (Ref. 6) has derived curves for discontinuity capacitance which takes both these effects into account. For computer calculations he has given the approximate formula 2__+ 1+ 4a C E=.n a 1 - 21nn 2 d 0 [ 1- aI + 0. 111 ~ (1 - c)(T - 1) 2a b ~ 10 (41) where b is the radius of the outside conductor, a is the radius of the center conductor, r is the radius of the disk, a = - r)/ (b -a) and =b/ a. This formula introduces a maximum error of ~ 0. 03 (27rb) pF when 0.01< aK< 1.0 and 1.0< T< 6.0. The length of the disks must be shortened to account for this added capacitance. The capacitance of a TEM line is C = o (42) c -32 -

where Zc is the characteristic impedance of the line at the disk given by Zc 2=- in (b/r) (43) Since there are two discontinuities for the disk, the total discontinuity capacitance C = 2 Cd and the length V' by which the disk must be shortened is easily found. For the case of the series inductor J inverter, the shunt discontinuity capacitance cannot be absorbed in the inductor as is possible for the disk K inverter. XII. Design Procedure The procedure for the design of a disk K inverter for a bandpass filter in a coaxial line follows: (1) A value for K is calculated on the basis of the formulas under Fig. 16. Usually tables for the gj are normalized so that c=1. (2) A diameter is assumed for the disk. The larger the disk diameter, the smaller its length will be, and the greater its discontinuity capacitance. (3) The disk characteristic impedance is found from the relation 43 (Ref. 2, p. 81) where?"r" is the center conductor (in this case, disk) radius and "b" is the outer conductor radius. -33 -

(4) The value for P is found from Eq. 39. (5) At this point a test must be made. If the value KP/ Z < 1, the disk diameter is too small, and a larger one must be assumed. (6) Use Eq. 40 to find the disk length. (7) Use Eq. 36 or 38 to find the phase angle 0. (8) The discontinuity capacitance is found from Eq. 41 and a length k' from Eq. 42 is subtracted from the disk length. The disk length found in step 6 must of course be larger than '. The assumed disk diameter, and the resulting disk length and phase angle completely specify the disk K inverter. This design approach accounts for the nonzero disk length. XIII. 50-Ohm to 50-Ohm Bandpass Filters Several curves were derived for various bandpass filters centered at 8. 5 GHz with different fractional bandwidths for the purpose of comparing the distributed design described in the previous section with the lumped design. The following filters were based on a Chebyshev low-pass prototype circuit with 0. 1 dB ripple. For the three-ripple case this means that go = 1.0, gl = 1. 0315, g2 = 1.474, g3 = 1. 0315, and g4 1.0. The design was carried out using 50-ohm coaxial line with an outer conductor diameter of 0. 5626 in. and a center conductor diameter of 0. 24425 in. In the theoretical curves which follow, the effect of discontinuity capacitance has been neglected. However, when a coaxial bandpass filter -34 -

is built, the design should be modified to account for the discontinuity capacitance as described in Section XII. Figures 26, 27, and 28 show that there is considerable improvement in using the distributed design. The distributed design filters are all centered at the design frequency although the bandwidth is narrower than the design bandwidth. The maximum ripple in the passband is smaller for the distributed case although it is still larger than the design ripple of 0. 1 dB or I rI 2= 0. 023. The design data for these curves are shown in Table I. Since the bandpass filter is symmetrical, the first two disks are identical with the last two, so only the values for the first two are given. XIII a. Six-Disk Filter Using Distributed Design. A 5-percent and a 10-percent bandwidth filter were designed with six disks using the distributed design technique. The iesulting curves are shown in Figs. 29 and 30. The same improvements over the lumped design are evident for the five-ripple case. A comparison of the five-ripple case and the three-ripple case for the distributed design shows that the maximum ripple is still higher for the five-ripple case. The design parameters for the filters are shown in Table II. Since the filter is symmetrical, the first three disks are identical to the last three. Tables I and II show that the difference in disk lengths between the lumped and distributed case is not large. As would be expected, the difference in the phase angles 0 between the two cases is greatest -35 -

Lumped Disk D'sk Lumped Bandwidth K K DikDs or 1 2 Diameter Diameter 21c f cm rad. rad. Distributed D in. D in. L 1% 6.1701 0.72194 0.502 0.554 0.3011 0. 3496 0.24556 0.02888 D 1% ft 0. 3471 0. 4302 0. 15534 0.01330 L 5% 13.7969 3. 6097 0.502 0.532 0.1263 0. 2547 0. 53847 0.14414 D. 5% t 0. 1298 0.2777 0.50663 0.10963 L lo% 19.5117 7. 2193 0.502 0. 502 0.0820 0.2559 0.74412 0.28679 D 10% 0.0833 0.2816 0.72378 0.21529 Table I. Disk parameters for bandpass filters with three ripples

Lumped Bandor width Distributed w K K2 K3 D1 in. D2 in. D3 in. L 5% 13. 0849 3.1316 2.3863 0.502 0.532 0. 532 D 5% "t "" L 10o% 18.5049 6. 2632 4.7726 0. 502 0. 502 0.532 D o% Lumped Bandor width Distributed, w f1 cm |2 c m |c 01 rad. 02 rad ra2 L 5% 0.134 3 0.2940 0.3864 0. 51192 0.12510 0. 09538 D 5% 0.1384 0.3328 0.5462 0.47793 0.82929 0.01747 L 10o% 0.0880 0.2965 0.1919 0.70893 0.24923 0.19033 D io% 0. 0896 0. 3399 0.2009 0.68706 0.16114 0.16587 Table II. Disk parameters for bandpass filters with five ripples

I rl 2 Distributed Design Lumped Design 1.0 0.9 I I 0. 8 ~0.7~I I 0.7 '1-4 I '0o.!0 rl I W I 0.4 0.3 0.2 0.1 0 J 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 26. Theoretical comparison between two design techniques for a 1 percent bandwidth bandpass filter which operates between two 50-ohm loads -38 -

I r 2 - Distributed Design - - Lumped Design 1. 0 0.9 0.8 0.7 I 4 — 0. 6 0.6 0.4 0.3 0. 2 0.1 I I 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 27. Theoretical comparison between two design techniques for a 5 percent bandwidth bandpass filter which operates between two 50-ohm loads -39 -

r 2 Distributed Design Lumlped Design 1.0 0. 9 // \: 0. 7 F 0.6' 0.5 0.4 0. 3I 0. 2 0 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 28. Theoretical comparison between two design techniques for a 10 percent bandwidth bandpass filter which operates between two 50-ohm loads -40 -

Ir l- ---;Distributed Design -- -- Lumrped Design 1.0 0.9 1 0. 8 0.7 _ I 4-) 0.4 0. 3 0. o 7...I 2 X 1 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 29. Theoretical comparison between two design techniques for a 5 percent bandwidth 5-ripple filter which operates between two 50-ohm loads -41

rl2 _ Distributed Design Lumped Design 1.0 0.9 0. 8 0.7 0. 6 0 0.5 o 0.4 0. 2 Ir, I/I I 0 v 3 7.6 7.8 8.0 -8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 30. Theoretical comparison between two design techniques for a 10 percent bandwidth 5-ripple filter which operates between two 50-ohm loads -42 -

when the disk length is large. The different disk diameters used in Tables I and II were the result of increasing the diameter to make the expression for tan fl /2 real. XIII b. Impedance Transformer. The distributed design approach was used to design an impedance transformer operating between 50 ohms and a complex load whose real part was 1 ohm. The load was resonated with an appropriate reactance so that the resonated Q was QA. The load was assumed to have a QA such that the decrement 6 was constant. 6 = w - 0. 0571 wQA 17.5 In this case the load acts as the first series resonator so the first K inverter is K12 which is given by 1 wRAX 2 12 w g g26 The low-pass prototype g values can be obtained from a graph (Ref. 4, p. 128), and the values for K can be found from the above expression and the formulas found in Fig. 16. For an n pole low-pass prototype circuit, only n K inverters are needed for the matching network whereas n + 1 K inverters were needed for the 50-ohm to 50-ohm filter. The results of the 50:1 impedance transformers are shown in Figs. 31 through 34 for different bandwidths. Again the passband was centered around f0 for the distributed design case. The distributed design -43 -

2 rl 2 Distributed design - Lumped design 1.0 0.9 0.8 I 0.7 1 ~0.6 ~ 0.5 0.4 0.3 0. 2 0. 1 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9. 2 9.4 Frequency in GHz Fig. 31. Theoretical comparison between two design techniques for a 1 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load -44 -

Distributed Design Lumped Designl 1.0 0.9 0.8 0.7 0.003 4 0.6 0.001 0 rD 0 8.4 8.5 8.6 0..5 0.4 0.3 0.1 1 7. 6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 32. Theoretical comparison between two design techniques for a 5 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load -45 -

Peol tuqo- e pue auqo-o0 e uaoalaq 2uil-Taado waIuojsuel oDourpadL p qplpueq luavaad 01 r loj sanbTuowal ug sap ofl uama4aq uosiaodloz Uoo OT~laloaoqJ '*g ' ZHO uT,ouanbaaJi ~6 Z 6 0'6 8'8 9'8 '*8 Z 8 0 '8 8'L 9'L 0 /ile I 1 I 1 I I 1 1 I I,1 Aooo \ — 19 ' | ~~~~/. 1~~8\\ L'0 /o ~ ~ ~ ~ ~ - 6'0 000~0 '0 pdn u TJsaa padrnq S --- I — II I

2 Distributed Design. -— Lumped Design 1.0 0.9 0.8 0.7 0.003 0.002 0.6 0. 001 0 I 8.1 8.3 8.5 8. 7 0.5 / 0.4 0. 3 0.2 0 I 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 34. Theoretical comparison between two design techniques for a 20 percent bandwidth impedance transformer operating between a 50-ohm and a 1-ohm load -47 -

transformers generally had a flatter passband and a slightly narrower bandwidth than the design goal. Table III summarizes the design data for these curves. XIII c. Impedance Plots. The impedance of the 10 percent bandpass filter shown in Fig. 28 is plotted in Fig. 35. The impedance seen from the 1-ohm side of the transformer of Fig. 33 is plotted in Fig. 36. An ideal filter would have an impedance of 50 + jO ohms in the passband in Fig. 35, and an ideal impedance matching network would have an impedance of 1 + jO ohms in the passband of Fig. 36. XIV. Experimental Work A 50-ohm to 50-ohm bandpass filter with a 10 percent bandwidth was constructed using the distributed design technique. The same ripple and g values were used as before, and the center frequency was chosen as 8. 5 GHz. The design procedure of Section XII was used to build a filter of the form shown in Fig. 37. For ease of handling the outer two disks were made thicker by using an air dielectric between the outer conductor and the disk. The center two disks had a Teflon dielectric (r = 2. 03) between the two conductors to increase the capacitance and to provide mechanical support for the disks. Here the disk lengths, Ci, have been shortened in accord with Eq. 42 to account for the discontinuity capacitance. The final design parameters are shown in Table IV. -48 -

Band- Disk Diameters Lumped or width Distributed w K1 K K3 D1 in. D2 in. D3 in. I1 cm | Cm0c rad rac. rad~~~~~~~~1c 2 c 3 cm ~1rd2 ra3 23 r a 2 L | o 0.7338 0. 6973 6.5221 0.554 0.554 0.502 0.3439 0.3620 0.2843 0.02935 0.02789 0.25942 D 1 % " " " " |" " 0.4183 0.4591|0.3215 0.01431 0.011009.17667 rp | L 5% 1.6409 3.4864 14'5839 0.542 0.532 0.502 0.3747 0.2638 0.1184 0.06561 0.13923 0.56761 CP D | 5% " " " | 0.4963 0.2899 0.1214 0.02006 0. 103070.53787 L 10% 2.3206 6.9727 20.6247 0.542 0.502 0.502 0.2647 0.2653 0.0759 0.0276 0. 27712 0.78246 D |,10%|" T,, O.|'. |"0.2906 0.2944 0.0771 0.06861 0.20206 0.76366 L 20% 3.2818 13.9454 26.1678 0.532 0o502 0.502 0.2804 0.1248 0.0427 0'13108 0.54399 1.05618 D 20o% |,,,,, " 0. 3129 0. 1282 0. 0432 0.09173 0. 5125 i.0456 8 Table III. Disk parameters for bandpass impedance transformer using three disks

Distributed Design 100 --- Lumped Design 90 80 70 60 50 x / 40 30 20 7.8 8.0 8.2 8.4 9. 0 9. 2 9 ~~-204~ ~Frequency in GHz -20 -30 -40 -50 -60 -70 -80 -90 -100 Fig. 35. Theoretical impedance of a 3-ripple, 10 percent bandwidth bandpass filter operating between two 50-ohm loads -50 -

Distributed Design 1.3 -- Lumped Design 1.2 1.1 0.9 0.6 0.7 0~5 i 0.4 O X u 0.3 %0.2 0.1 7.8 8. 8. 2 8.4 8.8 9. 0.2 Frequency in GHz -0. 2 -0. 3 -0.4 -0.5 -0. 6 -0.7 -0.8 Fig. 36. Theoretical impedance of a 3-ripple, 10 percent bandwidth 50 impedance transformer as seen from the 1-ohm side -51 -

c1Kdd1 2 Kd2 43 1d3 IL4K Fig. 37. Coaxial bandpass filter K1 = 19.51168 K2 = 7. 219345 01 = ~0 7019 radians = 0. 2153 radians All disk diameters = 0. 502 in. =1 k= 4 0.01456 in. 1 4 = I = ~0. 08465 in. 2 3 d =d 0. 79564 in. d2 = 0.7 4186 in, Table IV. Filter design parameters Tie theoretical batndpass characteristics for this filter are shown in Fig. 38 and the experimental curve is shown in Fig. 39. In both cases the high frequency ripple is the larger one, the ripple in the experiment filter being slightly larger than the theoretical one. The dip in the middle of the large ripple in Fig. 39 appears to be a small resonance due to a mismatch in the line, quite possibly in a connector. The attenuation in the out of band frequencies was lower in Fig. 39 because the high VSWR was limited by losses in the system. -52 -

Reflection Coefficient - O C0 0 JI 0 0 0 O L Ci~ co 0O 0 0 oo C, l) e -Jo c (. - - - or ~ w n, ~,~......r

rlz 0.8 0. 006 0. 004 / 7 0. 0. 002, I 8.1 8. 3 8.5 - 0.6 0 5 0.4 0.2 0.1 o 0 I 1. 8 a re-284J I I I I I 7.6 7.8 8.0 8.2 8.4 8. 6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 39. Experimental 10 percent bandwidth bandpass filter -54 -

XV. Conclusions The distributed design procedure developed here for bandpass filters and impedance transformers gives an improved overall characteristic over the lumped design procedure. Although the distributed design gives a lower ripple than the lumped design, it is still larger than the low-pass prototype design ripple. The design method itself is easy to use and is applicable over a wide range of microwave frequencies. -55 -

REFERENCES 1. S. B. Cohn, "Direct Coupled Resonator Filters, " Proc. IRE 45, February 1957, pp. 187-196. 2. R. E. Collin, Foundations for Microwave Engineering, McGrawHill, 1966. 3. E. A. Guillemin, Synthesis of Passive Networks, John Wiley and Sons, New York, 1957. 4. G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1965. 5. F. F. Kuo, Network Analysis and Synthesis, John Wiley and Sons, New York, 1962. 6. P. I. Somlo, "The Computation of Coaxial Line Step Capacitances," IEEE Trans. on Microwave Theory and Techniques, MTT-15, No. 1, January 1967. -56 -

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DISTRIBUTION LIST (Cont.) No. of Copies USAECOM Liaison Officer Aeronautical Systems Division Attn: ASDL-9 Wright-Patterson AF Base, Ohio 45433 USAECOM Liaison Office U. S. Army Electronic Proving Ground Fort Huachuca, Arizona 85613 20 Commanding General U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: 1 AMSEL-IM 1 AMSEL-EW 1 AMSEL-PP i AMSEL-IO-T 1 AMSEL-RD-MAT 1 AMSEL-RD-LNA 1 AMSEL-RD-LNJ 1 AMSEL-XL-D 1 AMSEL- NL-D 1 AMSEL-VL-D 1 AMSEL-KL-D 3 AMSEL-HL-CT-D 1 AMSEL-BL-D 3 AMSEL-WL 1 AMSEL-WL (ofc. of records) 1 AMSEL-SC Dr. T. W. Butler, Jr., Director Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 22 Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 -63 -

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R&D (Security clJaeification of title, body of absbtract and indexing annotation must be entered when the overall reportl I classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION Cooley Electronics Laboratory UNCLASSIFIED The University of Michigan 2b. GROUP Ann Arbor, Michigan 3. REPORT TITLE COAXIAL MICROWAVE BANDPASS FILTERS 4. DESCRIPTIVE NOTES (Typo of report and Inclusive date.) C. E. L. Technical Memorandum No. 100 Mayv 1969 S. AUTHOR(S) (Loat name. first name, initial) Davis, W. A. 6 REPORT DATE 7. TOTAL NO. OF PAGES 7b. NO. Of REFS Mav 1969 75 6 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DAAB 07-68-C-0138 b. PROJECT NO..1482-5 TM 100 1 HO 21101 A04 01 02... c. * h. O'tHEs tR#PORT NO(S) (Any other numbere that may be assigned d. ECOM-0138-5 '. A V A IL ABILITY/LIMITA'iON NOTICES This document is subject to special controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U.S. Army Electronics Command, Ft. Monmouth, N.J. Attn: AMSEL!-WL-S........................ 11. SUPPL EMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL-WL-S 13. ABSTRACT A review of lumped.element filter synthesis techniques using the power loss ratio is presented. This leads to filter synthesis based on impedance inverters for which S. B. Cohn (Ref. 1) has given an approximate microwave realization. Here an improved method is presented which considers the distributive property of the impedance inverter. Several theoretical curves are shown comparing the two methods. Finally results from an experimental model are shown. DD D|JA 41473 UNCLASSIFIED

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