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Derivation of the: Improved Log Formula O.Berrig Thanks to: H.Day, V.Vaccaro, C.Vollinger The longitudinal impedance is defined as a serial impedance dZ/dl=The beam impedance per length L The inductance per length C The capacity per length The L and the C are characteristics of the transfer line A moving pulse through a lossless transmission line The movement of the pulse is described by the equation: Where: l = the length of the cable π½ = the propagation1 factor. For a lossless cable: π½ = π β πΏ β πΆ 1The propagation factor is also called the wave number and then is named βkβ A moving pulse through a transmission line with losses The movement of a pulse through a line with losses, is described by the same equation as moving through a line without losses (NB! |V1+|>|V2+| and |V2-|>|V1-|) : Where: π = The length of the cable π½ = Propagation factor. The propagation factor is a complex number for a lossy line: π½ =πβ πΆβπΏβ 1β πβπ πβπΏ β 1β πβπΊ π€βπΆ See ref. [1] [1] http://pcwww.liv.ac.uk/~awolski/Teaching/Liverpool/PHYS370/AdvancedElectromagnetism-Part6.pdf A moving pulse through a transmission line with losses The general formula: π½ = π β πΆπ·ππ β πΏπ·ππ β π β π π·ππ π β πΊπ·ππ 1β β 1β π β πΏπ·ππ π€ β πΆπ·ππ Can be rewritten as: π½ = π β πΆπ πΈπΉ β πΏπ πΈπΉ β ππ 1+ ππ π β π β πΏπ πΈπΉ See ref. [2] The last formula corresponds to an additional βbeam impedance per lengthβ : dZ/dl=The beam impedance per length [2] \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\Verification_Erk_ImprovedLogFormula.nb Getting to the expression S 21 ο½ e ο iοο’ οl The scattering parameters are defined in the following way: The movement of the pulse is described by the equation: Where the waves and their voltages are defined as: Combining these three equations give: S 21 ο½ e ο i οο’ οl NB! The waves must be matched to the two-port network Derivation of the Improved Log Formula ππ π21π·ππ = π βπβπβ πΆπ πΈπΉ βπΏπ πΈπΉ β π21π πΈπΉ = π βπβπ β The formula is proved, but the devil is in the details ππΏπππππ‘π’πππππ = ππ 1+ πβπβπΏπ πΈπΉ βπ πΆπ πΈπΉ βπΏπ πΈπΉ β π π21 ππ ππ β π = β2 β π0 β πΏπ π21π·ππ π21π πΈπΉ β 1+ πβπΏπ π21π·ππ π πΈπΉ 2βπ where: π0 = The characteristic impedance of the REF line π0 = πΏπ πΈπΉ πΆπ πΈπΉ ΞΈ = The electrical length of the REF line π = π½π πΈπΉ β π = π β πΆπ πΈπΉ β πΏπ πΈπΉ β π In conclusion: when doing measurements, the VNA cables must be matched to the device under test Matching resistors β prevent reflections inside the DUT ZDUT Z0=50 Ξ© Yes, but this is not what is really done !!! The real setup Matching resistors β Matched to vacuum pipe ZDUT Problem: Reflection when the wave goes from vacuum pipe to DUT. The DUT is embedded in the vacuum pipe, and we need to modify the βimproved log formulaβ to take account of this. The real setup Derivation from Vaccaro From Vaccaro: The real setup In order to solve the measurement problem, use Vaccaroβs formula [4] : Where X is R (for reference) or D (for device under test) and SC is the measurement of the spacers: Transformations of the improved log-formula 1. The improved log-formula gives the impedance for a specific length of wire. However, in reality the impedance is specific for a given position on the wire. The improved log-formula therefore gives an average for the specific length. 2. The improved log-formula is exact: ππΏπππππ‘π’πππππ = ππ ππ β π = β2 β π0 β πΏπ π21π·ππ π21π πΈπΉ β 1+ πβπΏπ It transforms to the log-formula for ππΏπππππ‘π’πππππ << π0 : ππΏπππππ‘π’πππππ = β2 β π0 β πΏπ 3. The improved log-formula is transformed into Vaccaroβs formula: ππΏπππππ‘π’πππππ = π0 β πΏπ π21π πΈπΉ π21π·ππ β 1 + πΏπ π21π·ππ π21π πΈπΉ by the equality: πΏπ[π12π πΈπΉ ] = βπ β ΞΈ 4) For wavelengths smaller than the length of the βdevice under testβ (DUT); the lumped impedance formula is exact. It can be shown that the improved log formula can be transformed into the lumped formula, see [3]: ππΏπππππ‘π’πππππ = 2 β π0 β π21π πΈπΉ π21π·ππ β1 π21π·ππ π21π πΈπΉ 2βπ π21π·ππ π21π πΈπΉ The Lumped formula, can be derived in the following way (freely adapted from F.Caspers). The first derivation of the lumped formula was made in βOn coaxial wire measurement of the longitudinal coupling. H.Hahn and F.Pedersenβ: http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?197810003 : ZL a1 b1 I V1 V2 b2 V1 ο Z L ο I ο½ V2 We can then calculate the transmission coefficient: S 21, DUT ο½ S 21, REF ο½ ο¨ ο¨ ο© ο© ο¨V ο« Z 0 ο I ο© ο½ ο¨V2 ο« Z 0 ο I ο© ο½ ο¨Z 0 ο I ο« Z 0 ο I ο© ο½ 2 ο Z 0 b2 ο¨V2 ο« Z 0 ο I ο© / 2 ο Z 0 ο½ ο½ 2 ο¨V1 ο« Z 0 ο I ο© ο¨V2 ο« Z L ο I ο« Z 0 ο I ο© ο¨Z 0 ο I ο« Z L ο I ο« Z 0 ο I ο© 2 ο Z 0 ο« Z L a1 ο¨V1 ο« Z 0 ο I ο© / 2 ο Z 0 ο¨V2 ο« Z 0 ο I ο© / ο¨2 ο ο¨V1 ο« Z 0 ο I ο© / ο¨2 ο Z0 Z0 ο© ο½1 ο© Because ZL is zero And from S21, we get the lumped impedance: S 21, DUT S 21, REF ο½ 2 ο Z0 2 ο Z0 ο« Z L ο S ZL ο½ 2 ο 21, REF ο 2 Z0 S 21, DUT [3] http://cdsweb.cern.ch/record/960162/files/cer-002626446.pdf E.Jensen [4]\\cern.ch\dfs\Websites\o\OEBerrig\Impedance\COUPLING_IMPEDANCE_MEASUREMENTS_AN_IMPROVED _WIRE_SCANNER_METHOD_Vaccaro.pdf [5] \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\Validity_84.pdf H.Hahn Thank you ! Extra Material Lossy transmission line: No losses