Using ir lines s references for VNA phse mesurements Stephen Protheroe nd Nick Ridler Electromgnetics Tem, Ntionl Physicl Lbortory, UK Emil: Stephen.protheroe@npl.co.uk Abstrct Air lines re often used s impednce references to evlute the mgnitude uncertinty in vector network nlyser (VNA) clibrtions using the so-clled ripple technique. However, there is s yet no comprble technique for ssessing the ssocited phse uncertinty. By mens of prcticl mesurement, this pper demonstrtes the difficulty in ssessing phse uncertinty using ir lines nd describes why this is cused by lck of knowledge concerning the conductor loss in the lines. 1 Introduction Precision ir dielectric coxil trnsmission lines (or ir lines, for short) form key component in current recommended techniques for ssessing mgnitude uncertinty in vector network nlyser (VNA) clibrtions [1], but no similr technique currently exists for ssessing the phse uncertinty. Since ir lines re lredy used to ssess mgnitude uncertinty, it would be convenient if they could lso be used to ssess phse uncertinty. If the mechnicl length of n ir line is known ccurtely, then it is strightforwrd to clculte the expected phse shift due to its insertion in circuit. Consider length, l, of ir line connected between the port 1 nd port reference plnes of VNA: Port 1 Port Length l Figure 1: schemtic digrm of n ir connected between the reference plnes of VNA. The electricl dely is the time tken for the signl to trvel between the ports: l ε Dely( s) r (1) c 1
ε r reltive permittivity of line s dielectric, ε r 1.649 for ir t 3 ºC c speed of light.99795 1 8 m.s -1. For coxil line, the phse response, φ, is liner with frequency: φ( rd ) πf dely ω dely () ngulr frequency (rd.s -1 ) f frequency (Hz). Hence πfl ε r φ βl (3) c β phse constnt (rd.m -1 ) πf ε r. c The mesured phse (-π < θ < π) cn be found from the phse response s 1 : θ( rd ) π mod( φ + π,π). (4) Conversely, the phse response my be clculted from the mesured phse: φ( rd) mod(π θ,π) + πn (5) n is n integer (n, 1,, ). The electricl length of the line my therefore be clculted from the mesured phse using: c(πn θ) (πn θ) l. (6) πf ε β r Since there re multiple solutions to equtions (5) nd (6), the correct vlue for n cn be determined by finding length tht is within given tolernce of the nominl length. This 1 The modulo rithmetic opertor mod(φ + π, π) returns the reminder fter (φ + π) is divided by π.
tolernce is dependent on the frequency (wvelength) of the mesurement, e.g. t 18 GHz it will be pproximtely mm. Mesurements Three 5 ohm ir lines fitted with precision 7 mm connectors were selected for this investigtion: two nominl 1 mm lines from different mnufcturers (i.e. Hewlett Pckrd nd Mury Microwve) nd one nominl 3 mm line (mnufctured by Mury Microwve). The mechnicl lengths of these lines were mesured by UKAS-ccredited lbortory. The S-prmeters of the lines were mesured using NPL s primry ntionl stndrd VNA mesurement system to obtin the best possible ccurcy. The electricl length of the lines ws clculted using eqution (6) from both S 1 nd S 1 phse dt nd compred with the mechnicl length. Figure shows plot of the difference in length between the electricl nd mechnicl determintions, s function of frequency..6.5 Length difference, Electricl - Mechnicl (mm).4.3. S1 S1.1. 4 6 8 1 1 14 16 18 Frequency (MHz) Figure : showing the difference in length between the electricl nd mechnicl determintions of the length for the 3 mm line. A correction ws pplied to these length mesurements to tke ccount of the difference in temperture between the UKAS-ccredited clibrtion lbortory temperture (i.e. ºC) nd NPL s VNA lbortory temperture (3 ºC). 3
This Figure shows tht the electricl determintion of the line s length produces longer length vlue thn the mechnicl determintion t ll frequencies. The length difference is lso frequency dependent. Finlly, Figure lso shows tht the dt derived from S 1 is essentilly the sme s tht derived from S 1 (s is to be expected for reciprocl device) nd so only S 1 dt will be shown in the plots tht follow. Figure 3 shows similr plot to Figure except tht the dt for the two 1 mm lines hs been included. As before, there is systemtic difference between the electricl nd mechnicl determintions of the length the electricl determintions mesured the lines to be longer thn their mechnicl length. However, this difference is less for the 1 mm lines thn it is for the 3 mm line. Agin, s before, the length difference is frequency dependent. However the frequency dependence is less for the 1 mm lines thn it is for the 3 mm line..5.45 Length difference, Electricl - Mechnicl (mm).4.35.3.5..15 HP 1mm Mury 1mm Mury 3mm.1.5. 4 6 8 1 1 14 16 18 Frequency (MHz) Figure 3: showing the difference in length between the electricl nd mechnicl determintions of length for the two 1 mm lines nd the 3 mm line. 3. Lossless versus lossy lines The theory presented so fr in this pper hs ssumed tht the ir lines re lossless. However, in prctice, this will not be the cse. To illustrte this, Figure 4 shows mesurements of the liner mgnitude of S 1 (i.e. equivlent to the ttenution, or loss ) of the lines s function of frequency. 4
A consequence of this loss is to increse the vlue of the phse constnt from its lossless vlue, β, by n mount tht is dependent on frequency. For exmple, lthough β 11. m -1 t 1 GHz, the electricl length mesurement indictes vlue tht is somewht higher. 1..995.99.985 Mgnitude (U).98.975.97.965.96.955.95 HP 1mm Mury 1mm Mury 3mm 4 6 8 1 1 14 16 18 Frequency (MHz) Figure 4: S 1 liner mgnitude, with uncertinties shown s error brs, for ech ir line. From the mesured phse dt nd mechnicl length, eqution (6) cn be used to determine vlue for the ctul phse constnt,, tking into ccount the loss in the line: ( πn θ ) β (7) l If we now consider lossy coxil line in detil, then: ( R + jωl)( G jωc) γ α + jβ + (8) γ propgtion constnt α ttenution constnt (Np.m -1 ) R series resistnce (Ω.m -1 ) L series inductnce (H.m -1 ) 5
G shunt conductnce (S.m -1 ) C shunt cpcitnce (F.m -1 ) It hs been shown tht []: R ωl L L1 + d G ωc C C k F d 1 d k ( 1 + d k F ) F k F 1 L series inductnce of lossless line (H.m -1 ) C shunt cpcitnce of lossless line (F.m -1 ) rdius of line s centre conductor (m) b rdius of line s outer conductor (m) k ngulr wvenumber, k π/λ (rd.m -1 ) ε permittivity, ε ε ε r (F.m -1 ) ε permittivity of free spce, ε c µ (F.m -1 ) µ permebility, µ µ µ r (H.m -1 ) ( b ) µ ln π πε ln ( b ) µ permebility of free spce, µ 4π 1-7 (H.m -1 ) µ r reltive permebility of line s dielectric, µ r 1 for ir F nd d re constnts defined s: F d ( b ) δ s 4 ln 1 ( b ) ( 1 + ( b ) b ln( b ) δ s skin depth (m) nd ( b ) ln( b ) ( b ) + 1 ρ π fµ 1 b 1 + ρ resistivity (Ω.m). 6
Since the rdii nd b re known for these lines, eqution (7) nd the imginry prt of eqution (8) cn be equted nd vlue for the resistivity cn be determined. Figure 5 shows vlues of resistivity determined in this wy for the three ir lines under investigtion. 6 5 HP 1 mm Mury 1 mm Mury 3 mm Resistivity (nohm.m) 4 3 1 4 6 8 1 1 14 16 18 Frequency (MHz) Figure 5: vlues of resistivity for ech ir line s function of frequency. The uncertinties shown in Figure 5 hve been clculted by observing the chnges in the clculted resistivity due to offsetting ech of the clcultion s input prmeters, in turn, by its stndrd uncertinty. These component uncertinties re then combined using root-sum-of-squres pproch. 3 The error brs on the grph show the uncertinties t 95% level of confidence (i.e. using coverge fctor, k ). 4. Discussion Figure 5 shows the vlues of resistivity for ech ir line tht re needed to cuse the electricl nd mechnicl determintions of ech line s length to become equl (to within the stted uncertinties). It is interesting to note tht the vlues of resistivity clculted for ech ir line gree with ech other to within the stted uncertinties. The clculted resistivity lso ppers to rise slightly with frequency s perhps one might expect this to remin effectively 3 A sensitivity nlysis of the model derived from equtions (6) nd (7) hs shown tht the dominnt component to the overll uncertinty is due to the mesured phse. Other components, such s the mesured mechnicl length nd temperture effects, were found to hve negligible effect on the overll uncertinty of the resistivity determintion. 7
constnt over this rnge of frequencies. However, ny such frequency dependence is only slight nd well within the stted uncertinties. In fct, the results could be further summrised by choosing vlue of pproximtely 15 nω.m, which could be pplicble cross this entire frequency rnge. The bove experimentlly determined resistivity vlue of 15 nω.m cn be compred with textbook vlues for suitble mterils. For exmple, in [3], vlue for the resistivity of brss t room temperture is given s 63 nω.m. This is considerbly less thn the vlue determined in this pper. This could be for the following resons: 1) It is not certin tht the ir lines conductors re mde of brss, lthough brss is mteril tht cn be used to fbricte such ir lines [4]. Other suitble mterils (e.g. beryllium copper, which is lso often used to fbricte ir lines) will hve different vlues of resistivity; ) The textbook vlue of 63 nω.m hs probbly been derived from mesurements mde t DC nd therefore my not be representtive of the resistivity t high frequencies. However, the lck of significnt frequency dependence observed in this pper tends to suggest tht vlues derived t DC will not be too different from vlues t higher frequencies; 3) Textbook vlues of resistivity usully refer to bulk smples of mterils, s the metl used for ir lines is often formed by depositing lyers using electroplting nd mchining techniques. The degree of compctness nd surfce finish, long with tendency for electroplted surfces to become porous, will cuse the resistivity to vry significntly from textbook vlue of bulk smple. This will usully result in n increse in the loss (nd hence the resistivity) of the mteril used to construct the line [5]. Such n increse is consistent with the discrepncy observed bove. For the bove resons, it is desirble to estblish convenient method for determining the resistivity of prticulr line under the conditions to which it is used. The development of such method is currently work-in-progress t NPL nd is bsed on n erlier experimentl technique [6]. Erly indictions suggest tht the vlue of 15 nω.m is not unrelistic for the lines considered in this investigtion [7]. 5. Conclusion The prcticl difficulty of using ir lines s references for VNA phse mesurements hs been demonstrted. The electricl length will pper longer thn the mechnicl length of the line if the ssumption tht the line being used is lossless. Air lines re therefore only useful s ccurte phse references if their loss is considered. This requires n ccurte knowledge of the resistivity of the conductors of the line nd this is likely to be property of ech individul line in question. When method becomes vilble to relibly nd conveniently determine the loss in given line, then this pper hs shown tht these lines will become suitble reference devices for verifying VNA phse mesurements. This will provide significnt improvement to the current interntionl guidelines for evluting VNA mesurements given in [1]. 8
6. References [1] EA Guidelines on the Evlution of Vector Network Anlysers (VNA), Europen co-opertion for Accredittion, publiction reference EA-1/1, My. (Avilble from: www.europen-ccredittion.org.) [] W C Dywitt, First-order symmetric modes for slightly lossy coxil trnsmission line, IEEE Trns, MTT-38(11):1644-1651, Nov 199. [3] Kye & Lby: http://www.kyelby.npl.co.uk/generl_physics/_6/_6_1.html [4] Type 9 ir lines, The Generl Rdio Experimenter, 37(11):7-8, Nov 1963. [5] B O Weinschel, Errors in coxil ir line stndrds due to skin effect, Microwve Journl, 33(11):131-143, Nov 199. [6] G J Kilby nd N M Ridler, Comprison of theoreticl nd mesured vlues for ttenution of precision coxil lines, IEE Electronics Letters, 8(1):199-1994, Oct 199. [7] C P Eiø nd N M Ridler, Chrcterising ir lines using VNA loss mesurements, BEMC 5 conference digest, Ntionl Physicl Lbortory, November 5. 9