AN IMPROVED LOG-FORMULA FOR HOMOGENEOUSLY DISTRIBUTED IMPEDANCE

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1 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH ORGANISATION EUROPEENNE POUR LA RECHERCHE NUCLEAIRE CERN - PS DIVISION PS/RF/Note 2-1 AN IMPROVED LOG-FORMULA FOR HOMOGENEOUSLY DISTRIBUTED IMPEDANCE E. Jensen The log-formula, used to determine a distributed beam impedance with the wire method, is valid only for small impedances. We derive an improved log-formula, valid for long structures, from an "exact" expression for the transmission. For completeness, the known log-formula and lumped impedance formula are derived from the same expression. The limits of their validity are indicated. Geneva, Switzerland 27 January 2

2 An improved log-formula for homogeneously distributed impedance Erk Jensen, PS/RF CERN, Geneva, Switzerland Á Abstract The log-formula, used to determine a distributed beam impedance with the wire method, is valid only for small impedances. We derive an improved log-formula, valid for long structures, from an "exact" expression for the transmission. For completeness, the known log-formula and lumped impedance formula are derived from the same expression. The limits of their validity are indicated. Á Introduction The wire method [Caspers] is used to determine the beam impedance of an accelerator component in the laboratory. A coaxial wire replaces the beam, and the impedance is calculated from the transmission S 21 measured with a vector network analyzer (VNA). Different formulas are to be applied for localized (lumped) and for distributed impedances. For a distributed impedance, one uses the so-called "log-formula"[walling]: Z 2 log L M S 21,DUT \ ]. N S 21,REF ^ This formula is known to hold only for small values of improved log-formula in the following. (1) Z cccccc z Z. We will derive the log-formula and an Á Remark It is not generally possible to determine the longitudinal coupling impedance from the measurement of S 21 if the device under test (DUT) is considered a black box. The fact that different formulas have to be used for lumped and distributed impedances is a clear indicator for this. We derive here a formula which, in the limit of short structures yields the known lumped impedance formula (11) and, applied to long structures, the improved log-formula (9). The latter converges to the log-formula (1) for small impedances. In spite of this feature, even this formula cannot be considered general. It is however an exact description of the investigated model, i.e. a homogeneous distribution an impedance modifying the series impedance of a TEM line. This text was formatted with Mathematica. Some input was suppressed in this printed version, but the executable notebook is available under Mind that in Mathematica, "log" denotes the natural logarithm. Á Transmission through a TEM line with modified series impedance Reference line: Consider a TEM transmission line of length L, with characteristic impedance Z, and propagation constant Þ E. In the following, we will write T EL for the electrical length of this line. The transmission will be used as reference in the following and is

3 S 21,REF Æ Þ T. (2) Perturbed line: Now consider another TEM transmission line, with the same physical length, the same parallel admittance per unit length, but with a series impedance per unit length which is a factor ] 2 different. This will result in a characteristic impedance ] Z, and a propagation constant Þ E]. This modified series impedance per unit length can be interpreted as the sum of the series impedance of the unperturbed line, Þ E Z, in series with an additional impedance per unit length, s L Þ E +] 2 1/ Z. It is only this additional impedance which will contribute to the longitudinal coupling impedance which, integrated over the length L, results in. The same model is sketched in [Vaccaro] and in [Walling]. The resulting transmission through a length L of this line is, when embedded in a line of wave impedance Z and compared to the unperturbed line, given by Þ T S 21,DUT Æ. S 21,REF cos+] T/ cccc 1 2 Þ,] cccc 1 sin +] T/ ] The longitudinal coupling impedance is related to the parameter ] by (3) Þ +] 2 1/ T. Z Note that the unperturbed line is characterized by ]=1. (4) The set of equations (3), (4) gives the transmission as a function of the coupling impedance, but our task is the inverse problem, i.e. to determine from the measured transmission. We have not found a general closed form solution. Á The "exact" expression: An alternative way to write (3) or rather its logarithm is log L M S 21,DUT \ ] Þ+] 1/ Tlog L 4 ] \ M cc ]. N S 21,REF ^ N +] 1/ 2 Æ 2 Þ ]T +]1/ 2 ^ This expression is exact for the considered model. (5) Á The known log-formula: A series expansion of (5) in powers of cccccc Z starts like log L M S 21,DUT \ ] c ccc c O L N S 21,REF ^ 2 Z M - 1 N Z Neglecting higher order terms and solving for cccccc Z yields the known log-formula: 2 \ ]. ^ (6) Z 2 log L M S 21,DUT \ ]. N S 21,REF ^ (7) 2

4 Z It is clear that this is limited to small values of cccccc z Z, i.e. for impedances small as compared to the unperturbed characteristic impedance. Á The "improved" log-formula: Alternatively, a series expansion of log, S 21,DUT c S 21,REF in powers of ]-1 will start like log L M S 21,DUT \ ] Þ T +] 1/ O++] 1/ 2 / (8) N S 21,REF ^ Again, truncating after the linear term allows to solve for ] and, with (4), for ccccc Z. This results in the improved log-formula: Z S 21,DUT 2 log L M S L Þ 21,DUT \ log, c ] N S 21,REF ^ M 1 S 21,REF \ cc 2 T ]. N ^ Both expansions (6) and (8) are first order power series, but the difference is that now the parameter ]1 Z 1 Þ c z Z 1, depending only on Z T z s L, is required to be small. As (8) shows, log+s 21,DUT s S 21,REF / is (to first order) proportional to the electrical length T which is exactly what one would expect for a distributed impedance. For small values of T, (9) will fail. Equation (8) is similar but not identical to Walling's equation (11). Equation (9) is identical to Vaccaro's equation (12), but with the electrical length expressed in terms of S 21,REF. Equation (9) neglects the mismatch at the beginning and the end of the perturbed line, i.e. it is equivalent to writing S 21,DUT žæ Þ ]T. (9) Á The lumped element formula: It is interesting to expand (5) in powers of r T. log L M S L 21,DUT \ 4 Ç Z \ z Z ] log cccc ccccc O, r T. N S 21,REF ^ M 2 c Ç Zz Z N 4 Ç ] ^ Z Here it is the zero order term which allows to solve for s Z, and this results directly in the lumped impedance formula: Z (1) Z 2 S 21,REF ccc 2 S 21,DUT (11) Á Validity of the approximations: Since equations (5) cannot be solved for s Z, it is not straight forward to show how good the approximations (7), (9), and (11) are. However, it is possible to use the "exact" formula (5) to map the complex -plane to the complex log+s 21,DUT s S 21,REF /-plane. For convenience we will use the "engineering" notation in the log+s 21 /-plane, i.e. we will express the real part of S 21 in db, and its imaginary part in degrees. In the following, we show these maps for different values of T. The range for s Z in all cases is, 2 for the real and 2, 18 for the imaginary part. The grid lines are spaced.4 Z for both 3

5 Re# ' and Im# '. The loci of both Re# ' and Im# ' start at the origin Re# ' towards the left, Im# ' towards the bottom. electric length 5 O: For a relatively long structure (or high frequency), the improved log-formula is in good agreement with the "exact" expression. The log-formula is only valid for small s Z (near the origin). The lumped element map is irrelevant here and has just been added for completeness. "EXACT" LOG FORMULA IMPROVED LOG FORMULA LUMPED Z

6 electric length O: The deformation of the map for smaller electrical lengths makes the log-formula even less useful. The improved log-formula still is applicable, but differences to the exact map become visible. 2 "EXACT" 2 LOG FORMULA IMPROVED LOG FORMULA 2 LUMPED Z

7 electric length O/4: For shorter lengths, even the "improved" log-formula gives wrong results. If better estimates are needed, one might consider a graphical solution using the exact map. "EXACT" LOG FORMULA IMPROVED LOG FORMULA LUMPED Z electric length O/2: For very short structures, both log-formulas give wrong values for the impedance. For values of ««of a few Z, the lumped element formula map starts to resemble the exact map. 6

8 1 "EXACT" 1 LOG FORMULA IMPROVED LOG FORMULA 1 LUMPED Z Acknowledgements The author wishes to thank F. Capsers and H. Tsutsui for their valuable remarks. References [Caspers] F. Caspers: "Bench measurements", chapter in A. Chao, M. Tigner (eds): "Handbook of Accelerator Physics and Engineering", World Scientific, Singapore, 1999 [Vaccaro] V.G. Vaccaro: "Coupling Impedance Measurements: An Improved Method", INFN/TC-94/23, Nov [Walling] L.S. Walling. D.E. McMurry, D.V. Neuffer,H.A. Thiessen: "Transmission-Line Impedance Measurements for an Advanced Hadron Facility", Nuclear Instruments and Methods in Physics Research, Vol. A281 (1989), pp

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