PROXIMITY LOSS IN RECTANGULAR CONDUCTORS WITH OPPOSING CURRENTS

Transcription

1 PROXIMITY LOSS IN RECTANGULAR CONDUCTORS WITH OPPOSING CURRENTS When the magnetic field from one conductor cuts a second conductor currents are induced which give an increase in power loss. This loss must come from the conductor producing the field and so its resistance increases. Equations for rectangular conductors are given here and these agree well with measurements. 1. INTRODUCTION Bus bars are often rectangular and they can then be packed together with minimum space. However this will increase their AC resistance because the magnetic field from one conductor cuts the other conductor and this induces eddy currents. This is illustrated below : Figure 1.1 Two Rectangular conductors with opposing currents These eddy currents produce a power loss which must then be supplied by the conductor producing the magnetic field, resulting in an increase in its resistance. The eddy currents themselves produce a flux which opposes the incident flux and so the power loss is very much reduced, particularly at high frequencies where the eddy current flux is strongest. The author s previous work on this subject gave empirical equations but these were not very accurate particularly at low frequencies. In contrast the report here has a strong theoretical basis although there are some parameters which are very complicated analytically and these have been measured. It should be noted that the loss equations given here apply even if only one of the conductors carries current (see Figure A6.4.1), However they do not apply if the two conductors are joined at each end, because the induced currents then follow different paths to those analysed here. The most significant equations are highlighted in red bold. 1

2 2. RECTANGULAR CONDUCTORS 2.1. Isolated Solid Rectangular Conductor The AC resistance of a solid rectangular conductor with no proximity loss has been analysed by the author (Payne ref 1), and the equation is given below : Figure Single isolated conductor R O = R COND / R DC [ K C / (1- e x ) ] where K C = 1 + F (f) [1.2/ e 2.1 t/w + 1.2/ e 2.1 w/t ] F (f) = (1- e p ) p = A 0.5 / (1.26 δ) A is the cross-sectional area of the conductor δ is the skin depth = [ρ/(πfµ)] 0.5 x = [2 δ/t (1+t/w) + 8 (δ/t) 3 /(w/t)] / [(w/t) 0.33 e -3.5 t/δ +1] w is the width of the conductor in metres t is its thickness in metres f is the frequency in Hz The dc resistance is equal to R dc = ρ l / (wt) where ρ is the material resistivity in Ωm (= for copper) and l is the length of the conductor in metres. The factor 1.2/ e 2.1 w/t is extremely small for w/t above 3, and indeed at w/t above about 300 Excel is unable to calculate the value and returns an error message Configuration of Rectangular conductors In this report the proximity loss is considered for the following two configurations, width to width, and thickness to thickness : Figure Configuration1 : width to width 2

3 Figure Configuration 2 : thickness to thickness The analysis which follows is for Configuration 1. In most aspects it also applies to Configuration 2 if w and t are interchanged and this is discussed in Section 8. Notice that the gap g is defined as above and is not the centre to centre distance. The loss is much greater in Configuration 2, because a much larger proportion of the flux is intercepted by the second conductor. 3. LOSS DUE TO EDDY CURRENTS 3.1. Circular Conductor Moullin (ref 2 p95-) analysed the eddy current loss in a circular conductor when penetrated by a uniform magnetic field : Figure Flux penetrating circular conductor He says...at every instant the current flow over one half-cylinder is in the same direction and in the opposite direction over the other half-cylinder. Picture the cylinder built of imaginary planks of width dx, one of which is distance x from the median plane. He then shows that the power lost is : P = π r 4 ω 2 H 2 l / ( 8 ρ) watts where r is the radius of the conductor ω = 2 π f H is the magnetic field intensity ρ is the material resistivity l is the length of the conductor 3

4 3.2. Rectangular Conductor The same analysis can be applied to a conductor with a rectangular cross section, and this has already been done for transformer laminations, which of course have a rectangular cross-section. Figure Eddy currents in transformer laminations This analysis is based on the eddy currents which circulate within each lamination due to a flux directed as shown by the large arrow above. The power lost in one lamination P E is given by Cullwick (ref 3 p234 ) as (see Appendix 1) : P E = ω 2 B M 2 l w t 3 / (k ρ) watts where B M is the peak flux density (= µ O H M ) w is the width t is the conductor thickness k =24 or 32 (see below) In the above, the dimensions appropriate to a conductor are given rather than those of a transformer lamination. Most references give the above equation with k= 24, including Cullwick. However Winch (ref 4, p 547) is unusual in that he assumes a slightly different path for the eddy currents and the factor 24 above is replaced by 32 in his equation (Appendix 1). This gives a better agreement with the measurements here, in both Configuration 1 and 2. Notice that the loss increases as the square of the frequency and so would be extremely high at RF frequencies were it not for the neutralizing effect of magnetic field produced by induced currents. This latter field is not included above and so the equation is limited to low frequencies. The power loss above must be supplied by the input current so the input power P IN increases by the same amount. This input power will be P IN = I IN 2 R IN, and in terms of the peak value of current ( to compare with the peak flux density B M in the above ), the input power is P IN = I M 2 R IN /2, so R IN = 2 P IN / I M Putting R IN as the proximity loss R PL and assuming k = 32, then the input resistance is increased by : R PL = (B / I) 2 ω 2 l w t 3 / (16 ρ) ohms Notice that the values of B and I can be the peak, mean or RMS as long as they are the same. 4

5 4. FLUX DENSITY 4.1. Introduction The above equation requires the flux density around a rectangular conductor. This is very difficult to determine and the resultant equations are very complicated (eg Petrescu (ref 5). The situation here is even more complicated because the flux density varies across the width of the second conductor and across its thickness and this conductor then integrates this flux. Because of these difficulties the author has produced semi-empirical equations which fit the measured values of proximity resistance Flux density at surface of Circular conductor In considering this problem it is useful to start with the magnetic intensity H for an isolated circular conductor, since this is accurately known : H = I / (2 π r) A/m where r is the radius from the centre of the wire as shown below: Figure Magnetic field intensity around a circular conductor The flux density B is equal to µ O µ R H, so therefore B = I µ O µ R /(2 π a) Tesla Approximate Flux density at surface of Rectangular Conductor The flux density above is inversely proportional to the perimeter, and so as a first approximation for the rectangular conductor we can take its perimeter : B I µ O µ R / [2 (w+t)] Tesla As a check, Petrescu gives the surface flux density as T in his figure (shown below as Figure 4.4.1) for w=60 mm, t=6 mm and for a current of 800 A. Equation gives essentially the same value at T. The above equation gives the value of the flux at the centre of the (wide) face and it reduce towards the ends as shown below : 5

6 Figure Magnetic field intensity This figure gives the magnetic intensity for a conductor with w/t= 5, and with the thickness expanded compared to the width. Of course the flux density reduces with distance from the face, and this reduction is more rapid at the ends of the face compared with the middle of the face. The second conductor therefore intercepts flux which not only reduces across its thickness (ie distance) but also across its width. The average intercepted flux will therefore to be less than that given by Equation 4.3.1, and this reduction has been determined experimentally to be 0.3. In addition the flux density will be dependent upon w/t and good agreement with experiment is given by the following equation : Bo I k F µ O µ R / [2 (w+t)] Tesla where k F = 0.3 (1+ (t/w)) for Configuration 1 (width to width) The factor k F has a value of about 0.3 for w/t greater than 3, but is twice this value for a square conductor Reduction of Flux density with distance The second conductor will be spaced from the first conductor, and so the value of the field at distance is required. Petrescu gives the following curve of the flux density versus distance at the centre of the width, for w/t=10: Figure Flux density for w/t=10 An empirical equation which matches the above curve (for x>0.003) to better than 2% is : Bx = Bo / [1+1.5 x/w (x/w) 2 ] where w is the conductor width 6

7 However this equation does not give a good correlation with experiment, even when Bo is manually adjusted to minimise the errors (NB to apply this equation to the problem here, the average flux intercepted by the second conductor is at a distance x = (g+t)). The reason for the poor correlation is that the second conductor integrates the flux over its thickness t and along its length w. The following empirical equation gives good agreement with the author s measurements : Bx = Bo / [1+1.5 g/w +12 (g/w) 2 /(w/t) 2 ] where Bo is the flux at the surface of the first conductor and given by Equation Bx is the average flux density across the surface of the adjacent conductor for a gap g 5. EDDY CURRENT FLUX The induced eddy current produces its own flux and this is in opposition to the incident flux, so that the resultant flux is reduced. The analysis of this flux reduction is given in Appendix 2 resulting in the following equation: B R / B X = 1/ [1+ {µ o µ rm ω (t - 3δ) δ/ (2ρ k R ) } 2 ] where k R 1.55 This equation gives the ratio of the resultant flux to the incident flux, and this varies from unity at low frequencies to very small values at high frequencies e.g. 1/50 at 100 MHz.. The relative permeability µ rm is unity for most conductors. This equation fails when 3δ is not small compared with t, when it will return a resultant flux which is too large. However this will happen at low frequencies where the increase in resistance is very small and this is normally not significant. For instance a proximity increase of is reduced to CAPACITIVE COUPLING Capacitive coupling reduces the measured resistance of the conductors but this is negligible for copper conductors, even for Configuration 1 and with small spacings (Appendix 8). For the nickel conductors used in some of the experiments here the effect is greater because the conductor resistance is higher but even so this reduction is less than 5% at frequencies below VHF, even when the spacing between conductors is very small. 7

8 7. OVERALL EQUATION FOR CONFIGURATION 1 (width to width) Ignoring the capacitive coupling the overall equation for the increased resistance due to proximity is the product of Equations 3.2.3, 4.3.3, 4.4.2, and 5.1: R PL = (B / I) 2 ω 2 l w t 3 / (16 ρ) ohms 7.1 B/I k F µ O µ R / [2 (w+t)] / [1+1.5 g/w +12 (g/w) 2 /(w/t) 2 ]/ [1+ {µ o µ rm ω (t - 3δ) δ/ (2ρ k R ) } 2 ] 0.5 where k F = 0.3 (1+(t/w)) (determined by experiment) k R 1.55 The three factors in B/I above are a) the flux density at the surface of conductor 1 midway across its width, b) the reduction at conductor 2 due to the gap between them (g/w), and c) the reduction due to the opposing flux produced by the eddy currents in the second conductor. The total resistance is that of the isolated conductor plus that due to proximity : R TOTAL = R COND + R PL 7.2 where R COND is given by Equation R PL is given by Equation 7.1 It is often convenient to express the proximity resistance as the ratio to that of the conductor without proximity loss. R TOTAL / R COND = 1 + R PL / R COND 7.3 where R COND = R DC K C / (1- e x ) (see Equation 2.1.1) 8. OVERALL EQUATION FOR CONFIGURATION 2 (thickness to thickness) In this Section equations are given for Configuration 2. The proximity loss of the square conductor (w/t =1) is given by Equation 7.1. However the square conductor is common to both Configuration 1 and 2. If its thickness in Configuration 1 is increased slightly (so that w/t <1) it would be surprising if Equation 7.1 did not still hold, but crucially it then becomes Configuration 2. So from this argument it can be assumed that Equation 7.1 will still apply except that generally w and t are interchanged : R PL = (B / I) 2 ω 2 l t w 3 / (16 ρ) ohms 8.1 B /I k F µ O µ R / [2 (w+t)] / [1+1.5 g/w +12 (g/w) 2 /(w/t) 2 ]/ [1+ {µ o µ rm ω (w - 3δ) δ/ (2ρ k R ) } 2 ] 0.5 where k F = 0.3 (1+(w/t)) (determined by experiment) k R 1.55 Notice that the factor in g/w is the same as for Configuration 1 8

9 Resistance Ohms Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 9. SUMMARY OF MEASUREMENTS 9.1. Introduction Measurements have been made on conductors in Configuration 1 for w/t = 1, 2.7, and 10, and in Configuration 2 for w/t =1 (same as Configuration 1) and w/t =10. Details of the measurements are given in Appendices 3 and 4, and below are some examples of these measurements Configuration 1 The following shows the measurements compared with Equation 7.1 for a two nichrome conductors with w=3 mm, t= 0.3 mm and g= 0.8 mm. Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.26) Calculated Rin Frequency MHz Figure Measurements of Configuration 1 (w/t =10) 9.3. Configuration 2 The following shows the measurements compared with Equation 8.1 for a two nickel conductors with w=3 mm, t= 0.3 mm and g= 2.54 mm. Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.85) Calculated Rin Frequency MHz Figure Measurements of Configuration 2 (w/t =0.85) 9

10 The highest frequencies were subject to a large correction for SRF (see Appendix 7), and thus have a much larger error than the error bars indicate. 10. DISCUSSION It is seen that the measurements agree well with Equations 7.1 and 8.1. However these equations are inaccurate in at least the following ways : a) They are based on Equation 3.2.3, which assumes that the induced current is not affected by the reducing skin depth. This will increase the path length of the current and thus its resistance, and thereby reduce the input resistance. Given this flaw the good correlation with measurements suggests that there is a compensating effect, and one possibility is that the flux density increases with frequency. This may result from the crowding of current towards the edge of the currentcarrying conductor at high frequencies. b) Equations or give flux density independent of frequency, and this seems unlikely. 11. ERROR ANALYSIS An error analysis is given in Appendix 9, and assesses the overall measurement accuracy to be ± 6.2%. Generally Equations 7.1 and 8.1 agree with the measurements to within this error, and so this is the likely accuracy of these equations. 10

11 Appendix 1 : LOSS DUE TO EDDY CURRENTS A1.1. Analysis by Cullwick The power loss in transformer laminations has been calculated by many researchers and the following procedure by Cullwick (ref 3 ) is typical (the following uses dimensions applicable to conductors rather than laminations). Consider a conductor of length L, width w and thickness t : Figure A1.1.1 Eddy Currents in rectangular conductor The flux from the adjacent conductor is assumed to enter the narrow face and is assumed to be constant across the face. This flux induces an emf in the conductor and this leads to a circulating eddy current, the magnitude of which is determined by the resistance of the path which the current follows. Assuming the incident magnetic field is normal to the thickness and is given by : B = B M Sin ωt A1.1.1 Consider the e.m.f induced in a closed circuit consisting of two parallel elementary laminae of thickness δx, each distant x from the centre line of the conductor, joined at each end of the conductor of length L by short paths of length 2x. The flux linking this elementary circuit is : Φ = 2 Lx B M Sin ωt A1.1.2 So the induced e.m.f is : e = - dφ/dt = - 2 Lx ω B M Cos ωt A

12 The resistance of the circuit is; R ρ 2L/ ( w δx) A1.1.4 (this neglects the lengths 2x in comparison with L ). The power loss is thus given by : where ρ is the resistivity of the conductor e 2 / R = 2 L x 2 ω 2 (B M ) 2 (Cos ωt) 2 w δx / ρ A1.1.5 the mean rate of this loss, since the mean value of (Cos ωt) 2 = 1/2, is : δ P E = L x 2 ω 2 (B M ) 2 w δx / ρ A1.1.6 hence the power lost in the whole conductor due to eddy current is : t/2 P E = [L ω 2 (B M ) 2 w / ρ ] [ x 3 /3] 0 P E = L w ω 2 (B M ) 2 t 3/ (24 ρ) A1.1.7 A1.2. Analysis by Winch (ref 4) The analysis by Winch is unusual in that he assumes a different current path as shown below : Figure A1.2.1 Eddy Currents in rectangular conductor 12

13 Here the elemental path always has the same ratio of sides as the conductor itself. This leads to the same equation as Equation A1.1.7 except that the factor 24 in the denominator is replaced by 36. Appendix 2 A2.1. FLUX REDUCTION DUE TO EDDY CURRENTS Flux Reduction In this Appendix the reduction in flux density produced by the flux from the eddy currents is calculated. Consider a conductor of length L, width w and thickness t : Figure A2.1.1 Eddy Currents in rectangular conductor The flux from the adjacent conductor is assumed to enter the thickness t (see figure 1.1) and is assumed to be constant across the face. This flux induces an emf in the conductor and this leads to a circulating eddy current, the magnitude of which is determined by the resistance of the path which the current follows. In turn this eddy current generates a magnetic field which opposes the incident field, and partially cancels it. The resultant field B R is therefore lower than the incident field and is given by B R = B X -B E Since we are interested here in the ratio of the resultant field to the incident field : B R / B X = (1- B E /B X ) A2.1.1 where B X is average incident flux density across the face B E is the flux produced by the eddy current 13

14 The first thing to be determined here is the flux produced by the eddy current B R. In the above diagram consider a closed circuit consisting of two parallel elementary current carrying laminae of thickness δx, each a distance x from the centre of the face, joined at each end of the length L by short paths of length 2x. There are therefore two parallel current sheets of width w carrying current in opposite directions and the current density in each sheet will be J x = i/w. For large values of w compared with t this will equal the magnetic intensity H (Jordan ref 6 p78), so the flux density will be B E = µ o µ rm i/w. However, w is not always large compare with t and so we have to include a correction factor k R, giving : B E = µ o µ rm i/ (k R w) A2.1.2 k R is greater than unity and its value is considered in Section A2.2 below. The relative permeability µ rm is unity for most metals, such as copper and aluminium. Notice that since we are taking the ratios of the flux densities it does not matter whether these are mean or peak or RMS values as long as they are all the same. At high frequencies the current diffuses exponentially into the thickness, with an average depth equal to δ, the skin depth. This is given by : δ = [ρ/(πfµ)] 0.5 metres A2.1.3 µ= µ r µ o µ r is the material relative permeability (= 1 for copper) µ o = 4π 10-7 H/m ρ = resistivity (ohm-metres) (= for copper The average width of the current path is therefore equal to δ. The average width of the conducting loop will then be t - 3δ ( for δ <<2t), and the area of loop will be L (t - 3δ). The emf induced in this loop is due to the resultant flux φ R : e = d φ R / dt A2.1.4 which for sinusoidal excitation is e = j ω φ R = jω L (t - 3δ) B r A2.1.5 For a loop resistance R, the current i induced in the loop will be i= e/r : i = j ω L (t - 3δ) B r / R A2.1.6 Substituting this into Equation A2.1.2 : B E = j µ o µ rm ω L (t - 3δ) B R / (k R R w) A2.1.7 So B E /B X = j µ o µ rm ω L (t - 3δ) B R /B X / (k R R w) A2.1.8 The resistance R of the loop is : Subs in A2.1.8 : R = ρ 2 L/ (w δ ) B E /B X = j µ o µ rm ω (t - 3δ) B R /B X δ/ (k R 2ρ) A2.1.9 A Equation A2.1 : B R / B X = (1- B E /B X ) 14

15 Fringing Payne : Proximity Loss in Rectangular Conductors issue 2 Substituting A2.10 : B R / B X = 1 - j µ o µ rm ω (t - 3δ) B R / B X δ/ (k R 2ρ) B R / B X =1+ j µ o µ rm ω (t - 3δ) δ/ (k R 2ρ) B R / B X = 1/ [1+ j µ o µ rm ω (t - 3δ) δ/ (k R 2ρ) ] A B X is the flux density at the second conductor. To deduce this from the flux at the surface of the first conductor B O, this must be reduced by the factor given in Section Taking the modulus gives : B R / B X = 1/ [1+ {µ o µ rm ω (t - 3δ) δ/ (k R 2ρ) } 2 ] 0.5 A where k R 1.55 (See Section A2.2 below) The reduction in the flux density B R / B X can be very large, for example ranging from unity to 50 or more, depending on the frequency. So the resultant flux in this case will range down to less than 2% of the incident flux. This equation fails when 3δ is not small compared with t, when it will return a ratio which is too large. However this will happen at low frequencies where the increase in resistance is very small. So this error will reduce a very small increase and this is normally not significant eg an increase of is reduced to A2.2. Flux fringing The flux produced by the eddy current is given by Equation A2.1.2 above. This includes a factor k R which is the reduction in flux density due to fringing of the field when w is not large compared with t. No equations have been found for this magnetic fringing but it is analogous to the capacitive fringing field. For two strip conductors the increase in capacitance due to fringing is (see Appendix 8) : C/C 0 = 1+ [g/(π w)] [Ln (2 π w/g )] A2.2.1 It is assumed here that the magnetic fringing field is the same and that this reduces the flux density by the above amount. So for a distance between current paths of t, the above becomes for the reduction in magnetic flux density : k R = 1+ [t/(π w)] [Ln (2 π w/t )] A2.2.2 This is plotted below Fringing Fringing w/t Figure A2.2.1 Fringing In practice it was found that good agreement with experiment was achieved if k R was assumed to be 1.55 and independent of w/t. 15

16 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Appendix 3 MEASUREMENTS : CONFIGURATION 1 (width to width) A3.1. W/T =10 A Conductor Copper conductors have a very low resistance often less than 30 mω, and the accuracy of test equipment is then very poor. To raise the resistance nichrome conductors were used because their resistivity is around 70 times that of copper. However nichrome is an alloy and its composition can vary across the conductor width and across the thickness, so that the resistivity varies with depth from the surface and thus with skin depth. In addition there can be impurities such as iron which raise its permeability above unity. The nichrome tape used here had a width of 3 mm and a thickness of 0.3 mm. Its permeability was unity or very close to unity since it was not attracted to a powerful permanent magnet. The DC resistivity was measured as Ωm (using a milli-ohm meter) however the resistivity changed with skin depth. For instance a straight isolated conductor gave the following measurements ( in Jig 2 : Appendix 6 ) compared with the prediction given by Equation : Resistance of Single conductor Theoretical Rac, isolated conductor Measured Resistance Frequency MHz Figure A Measurements of Single Conductor If the measured values were multiplied by the following factor then good agreement was given between measurements and Equation : x = ( f) 0.5 where f is in MHz A

17 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Resistance of Single conductor Theoretical Rac, isolated conductor Measured Resistance Frequency MHz Figure A Measurements multiplied by Resistivity Factor x It is significant that this equation is proportional to f, since skin depth is also a function of f. Thus the equation is effectively the change of resistivity with skin depth and indicates that as the skin depth reduces the resistivity reduces implying a lower resistivity on the surface. The resistivity increases with Chromium content so this implies a lower concentration at the surface in these samples. Soldering to nichrome requires a special flux such as Alu Flux available on e-bay. A Isolated Conductor (w=3 mm, t=0.3 mm) Prior to the measurements of proximity it was important to confirm Equation for the resistance of a single isolated conductor, and this is shown in Figure above. The correlation here is very good, and is partly the result of choosing a suitable equation for the change of the nickel resistivity with frequency. However the simplicity of this equation gives confidence in its application and therefore support for Equation In the above experiment the return current was via the screening box (Jig 2, Appendix 6) and this current will give some proximity increase. However the distance from the conductor to the box is large compared with the conductor width giving an average g/w of 10, so the increase in resistance due to proximity loss will be very small at around 0.5%. A Very small gap (w=3 mm, t=0.3 mm, g= 0.15 mm, g/w = 0.05) In this experiment the gap between the conductors was very small compared to the width, and so the proximity loss was large. The measurements are shown in brown below, along with Equation 7.1 in purple. Also the resistance of the isolated conductor is shown in red (Equation 2.1.1). 17

18 Resistance Ohms Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.05) Calculated Rin Frequency MHz Figure A Measurements for g/w=0.05 These measurements were conducted in Jig 1 (Appendix 5) at frequencies from 5.77 MHz to 128 MHz. This frequency range is too large for one length of conductor, because if it is long enough to resonate with the capacitor at low frequencies it will have too low an SRF for the high frequencies (Appendix 7). Two conductor lengths were used : meters (and folded) from 5.77 to 40 MHz and metres (folded) for 50 to 128 MHz. Resonance at the lowest frequencies was achieved by adding silver mica capacitors in parallel across the variable capacitor. The error at the highest frequency is probably due to the error in correcting for the SRF of 240 MHz. It should be noted that the above curve is of resistance in ohms, and the intercept at the x axis at close to unity is not due to normalization but due to the DC resistance of the conductor of 1.09 Ω (the measurements of the short conductor were scaled to the longer conductor). The plot below shows an expansion of the low frequency portion of the above curves Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.05) Calculated Rin Frequency MHz Figure A Measurements for g/w=

19 Resistance ratio Resistance ratio Payne : Proximity Loss in Rectangular Conductors issue 2 This shows that there is very little proximity loss for frequencies up to about 10 MHz so that the total resistance is very close to that of the isolated conductor. The increase in resistance due to proximity is shown below and this has been derived by dividing the measured resistance by that of the isolated conductor (Equation 2.1.1) Proximity Increase Measured g/w = 0.05 cf theory Calculated Rin/Ro Frequency MHz Figure A Increase in conductor resistance due to proximity The highest frequency measurement is subject to a large error because of the large SRF correction factor of 2. It is instructive to see the effect of eliminating the flux due to the eddy currents and this is shown below by the purple curve : Proximity Increase Measured g/w = 0.05 cf theory Calculated Rin/Ro Frequency MHz Figure Prediction with no eddy flux 19

20 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 For the above, the last factor in B /I in Equation 7.1 was made equal to unity. It can be seen that this factor is only significant for frequencies above about 35 MHz, and below this frequency the proximity resistance rises as f 2. A Medium gap (w=3 mm, t=0.3 mm, g= 0.79 mm, g/w = 0.26) The insulation between the conductors had a thickness of 0.74 mm, giving an effective gap of 0.79 mm ± 0.05 mm (see Errors Appendix 9),with the following results : Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.26) Calculated Rin Frequency MHz Figure A Measurements for g/w = 0.26 The correlation is very good, giving confirmation of Equation 7.1. A3.2. W/T =2.7 A Conductor description For these measurements a bronze conductor was used having a width of 1.27 mm and a thickness of 0.47 mm. This was not attracted by a strong permanent magnet and so its permeability was unity. The DC resistivity was measured as Ωm but varied with depth as the following shows. A Isolated Conductor Prior to the measurements of proximity it was important to confirm Equation for the resistance of a single conductor. As with the nichrome conductor the resistivity was found to change with frequency. If all the measurements were multiplied by the following equation a better match with the theory for a single conductor was achieved : x = ( f) 0.5 A where f is in MHz With this correction a conductor of length m was measured in Jig 2 (Appendix 6), with the following results (the error bars are ± 6.2 % (see Appendix 9). 20

21 Resistance Ohms Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Single Isolated Conductor Theoretical Rac, isolated conductor Measured Resistance Frequency MHz Figure A Measurements of Single conductor A w/g = 0.12 Two conductors each of length m were soldered together at one end to form a folded conductor (Jig1). The insulation between them had a thickness of 0.11 mm, giving an effective gap of 0.16 mm ± 0.05 mm (see Errors Appendix 9), giving w/g = Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.12) Calculated Rin Frequency MHz Figure A Measurements for w/g = 0.12 The SRF for the above configuration was measured as 147 MHz. This is shown below, and is unusual in that there was a dip in the magnitude of S21 at the SRF (green trace). Also the phase shows a rapid change in phase at the SRF. 21

22 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Figure A Measurements for w/g = 0.12 A w/g = 0.46 The insulation thickness was increased to 0.54 mm, giving an effective gap of 0.59 mm ±0.05 mm (see Errors Appendix 9), giving w/g = Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.46) Calculated Rin Frequency MHz Figure A Measurements for Configuration 1 (w/g = 0.46) The SRF for the above configuration was measured as 147 MHz, similar to A A3.3. W/T =1 A Conductor The conductor was bronze and was not attracted to a powerful permanent magnet so its permeability was essentially equal to unity. Its DC resistivity was measured as Ωm. This changed with skin depth as described below. 22

23 Resistance Ohms Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 A Isolated Conductor Prior to the measurements of proximity it was important to confirm Equation for the resistance of a single isolated conductor. Measurements are shown below : Proximity Increase Theoretical Rac, isolated conductor Measured Resistance Frequency MHz Figure A Measurements of isolated conductor For the above, the measurements were multiplied by the following equation (see also Section A3.1.1) : x = ( f) 0.5 where f is in MHz A A Very small gap (w=0.65 mm, t=0.67 mm, g= mm, g/w = 0.05) The measurements are given below for a square conductor with g/w = The gap was very small and determined by one layer of cellotape. Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.05) Calculated Rin Frequency MHz Figure A Measurements for w/t =1 23

24 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 Appendix 4 MEASUREMENTS CONFIGURATION 2 (thickness to thickness) Given the good agreement between Equation 7.1 and the measurements and the logic for departing from the square conductor (Section 8) it was assumed that a single experiment would be sufficient to validate the Equation 8.1: Overall Resistance Theoretical Rac, isolated conductor Measured (g/w = 0.05) Calculated Rin Frequency MHz Figure A4.1 Measurements for Configuration 2 (g/w= 0.05) Nickel conductors were used in the measurements, having w= 3 mm, t=0.3 mm and g = 2.44 mm (see also Section 3.1.1). The highest frequencies were subject to a large correction for SRF (see Section 2) and this probably explains the error. Appendix 5 : JIG 1 A5.1. Measurement Apparatus A standard 50Ω Vector Network Analayser (VNA) is unable to measure resistance values of less than a few ohms with accuracy. The error increases substantially if there is also a large series reactance, as here, because the measurement problem then is to determine the small resistance value in series with a much larger reactance. An alternative technique is described in reference 7. This is a two port measurement and is illustrated below : Figure A5.1.1 Two Port measurement of very low impedances 24

25 Here the two ports of the VNA ( an Array Solutions UHF analyser) are connected together and the DUT is shunted across the connecting cable, thereby reducing the signal into the second port. For a 50 Ω system the value of the shunt impedance is given by : Z DUT = 25 S 21 /(1-S 21 ) ohms A5.1.1 This technique gives the total impedance, and to find the resistive component the inductive reactance has to be tuned-out with a capacitor. A practical circuit is shown below : Figure Measurement apparatus with copper tape calibration conductor (lid and ends not shown) This shows the coaxial cable connecting the two ports, with its centre conductor exposed in the middle to allow connection to the conductor under test (copper here). The other end of the conductor is connected to a terminal of the variable capacitor, which has its body grounded. A5.2. Jig Calibration A Ω Resistor A 1Ω, 5%, ¼ w metal film resistor was connected across the measurement terminal (the centre conductor of the semi-rigid coax). This measured Ω at 0.1 MHz. At higher frequencies the lead inductance became significant and for instance at 5 MHz the total impedance was Ω with a phase angle of 32 degrees. The resistive part is thus Cos θ = At higher frequencies the total impedance was even higher with an increased phase angle, but the errors in the phase angle were too large for calibration. At higher frequencies it is therefore important to minimise the inductance, and chip resistors should be used. However even then the inductance will be dominant at high frequencies : for instance a chip resistor with a length of 5 mm would have an inductive reactance of about 1.3 Ω at 50 MHz So accurate calibration of this jig at low resistance and high frequencies has not been possible. 25

26 Resistance Ohms Payne : Proximity Loss in Rectangular Conductors issue 2 A Variable Air Capacitor The resistance of the variable air capacitor had already been measured and this is described in Reference 10. This gave the following equation for the resistance of a variable capacitor : R CAP R 1 (C - C 2 ) / C + R 2 C 2 / C A where C is the total capacitance C 2 is the minimum capacitance R 1 = R C + α f 0.5 R 2 = β+ φ f 0.5 R C is the resistance of the contacts For the capacitor here the capacitance of each section was 9.29 pf pf. Parameters for above are C 2 =9.29 pf, R C = 1.6 mω, α = 0.012, β = 0, and φ = 0 The capacitor resistance varied from 14 to 85 mω over the frequency range 1-75 MHz. A Silver Mica Capacitors For measurement at 5.3 MHz the variable capacitor was shunted with silver mica capacitor having a value of 2000pf. The loss in this capacitor was measured at 1-4 MHz as 0.01 Ω. A Jig Evaluation To evaluate the accuracy of the test jig the resistance of a copper strip was measured, and compared with the theoretical value given by Equation These are shown below : Theoretical Rac, isolated conductor Measured Resistance Isolated Copper Conductor Frequency MHz Figure A Jig Calibration Measurement The resistance measurements were corrected by subtracting the jig losses, outlined above, and then factored for the SRF measured as 182 MHz. The uncertainty in the capacitor loss is ± 8 mω, and this is shown by the error bars above, and over most of the range the correlation is within these error bars. The exception is the highest frequency measurement, where there is a large correction for SRF, and errors in this correction probably account for the discrepancy. 26

27 A Unwanted propagation Figure A5.1.2 shows a folded strip conductor being measured, and its two halves form a transmission line whose reactance is tuned-out with the capacitor. However the conductor also forms a transmission line with the screening box, and the input impedance of this appears in parallel with the wanted impedance. This second transmission line becomes more obvious if the connection to the capacitor is removed, retaining the connection to the centre of the coaxial cable. It is then found that the measured input impedance is that of a transmission line formed between the conductor and the box. This impedance reduces with frequency until at resonance the impedance is purely resistive and is a minimum. To minimise this parallel impedance the Z O of the conductor transmission line must be much lower than the box transmission line. Thus the spacing between the conductors must be much less than the spacing of the conductors to the box. However even then the impedance of the box line at frequencies close to resonance is so low that it can materially reduce the measured resistance. Because of this problem isolated conductors were measured in Jig 2 (see Appendix 6).. A Loss in Spacers The conductors were spaced using strips of masking tape (3M 101E). This had a thickness of 0.11mm and was layered to give the required spacing. The conductors were stuck to the spacer using a further strip on each side. There will be loss in the spacer, and to assess this two copper conductors were placed back to back and the loss measured firstly with an air gap between them and then with the 0.97 mm thick spacer. The resistance at 75 MHz increased from to Ω, so the losses were within the repeatability error. A Effect of capacitance The capacitance between the conductors will reduce the measured resistance but this will be less than 5% for nichrome conductors at frequencies up to 100MHz (see Appendix 8). A Resistance at Resonance The resistance at the resonant frequency can be determined from the Q at resonance (Jordan ref 6 p 238) : Q RES = ωl/ R A In a test two conductors of w= 3mm, t=0.3 mm, length = m spaced by g=0.011 mm showed a resonant frequency of MHz and a Q at resonance of Determining the inductance is complicated : the value at low frequencies was measured as 53 nh, but given the small spacing between conductors, much of this would be internal inductance and this reduces with frequency. The internal inductance of a rectangular conductor is given by Payne (ref 8) as : L I µ o 2l T /(2π) (µ R / k P ) (1- e x )/ K c Henrys A where l T is the length of the twin conductor line k P (w/t) K C = 1 + F (f) [1.2/ e 2.1 t/w + 1.2/ e 2.1 w/t ] F (f) = (1- e p ) p = A 0.5 / (1.26 δ) A is the cross-sectional area of the conductor δ is the skin depth x= 2(1+t/w) δ/t 27

28 For the twin conductor line given above it will be 28.7 nh at low frequencies reducing to 2.8 nh at 128 MHz so the total inductance will be 27 nh at 128 MHz (ie = 27). So from Equation A the resistance is 5.03 Ω. To test this the resistance of a shorter conductor was used (to raise the SRF) of length 0.2 m and this gave a measured resistance of 2.13 Ω at 100 MHz. Correcting for the length gives 2.13*0.425/0.2 = 4.53 Ω. Assuming the resistance increases as f (from 100 to 128 MHz) gives 5.13 Ω, which is within 2% of that given by the Q at resonance. Appendix 6 JIG 2 A6.1. Jig One difficulty with Jig 1 (Appendix 5) is that if the resistance of an isolated conductor is needed, (ie with no proximity) then the loop spacing must be large to minimise proximity loss. This then leads to measurement errors caused by unwanted propagation (paragraph A5.2.5). This can be overcome by using a straight conductor as shown below : Figure A6.1.1 Measurement of straight conductor The physical implementation of this is shown below, and is a modification of the jig shown in Figure A5.1.2 : Figure A6.1.2 Measurement Jig 2 The capacitor is now placed at the far end (the other capacitor is not used). 28

29 A6.2. Box Resistance In this configuration the return current is via the box, and so its resistance must be considered. The current path through the box is via at one end the capacitor bolted to the metalwork and at the other a coaxial connector again bolted to the metalwork (actually two connectors close together, assumed to be one). These connections can be modelled as two conducting pins set in the box between which is an emf. The DC resistance between these pins can be determined from the known equation for capacitance between these pins. Assuming these are parallel wires the capacitance between them is given by : C = πε l / (Ln (d /2a) where l is the length of the pins (the wires) d is the distance between the pins a is the radius of the pins A6.2.1 By analogy the DC resistance is then given by (see Harnwell ref 12 p103), using l = t the thickness of the box material : R = ρ/(πt) Ln (d/a) where t is the thickness of the box material d is the distance between the pins a is the radius of the pins A6.2.2 For an aluminium box ρ = , of thickness 2.3 mm and length 0.5 m and pins of 3mm radius this gives a resistance of mω. Measurements across the box using a milli-ohmeter with crock clips gave 0.02mΩ (to an accuracy of ±0.03mΩ). DC measurements of the resistance between the two pins gave 6.2 mω, and this was mainly composed of the contact resistance across the fixings to the box. At high frequencies it is assumed that the path of current flow is the same as at DC, but that the conduction will be limited to the skin depth and so the resistance will be greater than the DC resistance by the ratio of the box thickness to the skin depth: R AC = R DC t/δ where t is the thickness of the metal enclosure δ is the skin depth = [ρ/(πfµ o ) ] 0.5 A6.2.3 NB in the above it is assumed that at high frequencies the current flows only on one surface of the box - the inner surface. As an example, at 50MHz the AC resistance of the box was calculated to be 8 mω, and so is small compared with the other losses such as the variable capacitor which is around 80 mω (depending upon capacity setting). A6.3. Capacitance The capacitance between the conductors will reduce the measured resistance (Appendix 8), but for nickel conductors this is generally less than 5% for frequencies up to 100MHz. 29

30 A6.4. Measurement of proximity loss The proximity loss can be measured in this jig configuration by bringing a second conductor close to the first as shown below. Notice that the conductors do not have to be joined, but if they are it must be one end only. Figure A6.4.1 Measurement jig This arrangement is generally more convenient physically than the folded conductor. Also the inductance is higher and so measurements can be made to around half the frequency for a given maximum value of the capacitor. However the conductor is half the length of that in Jig 1 and so its resistance is halved, as is the increase due to proximity. For two conductors of w=3mm and g =0.11mm and each of length length = m and ε r = 1.25 the capacitance is 155 pf and this has a reactance of 10.2 Ω at 100 MHz, and the resistance is around 2.4 Ω at this frequency, so from the above equation the measured resistance will be 5 % less than the real value. At higher frequencies the capacitive reactance decreases and the resistance increases, to give an error of 23% at 190 MHz for nickel conductors. It would be much lower for copper conductors. Appendix 7 SELF-RESONANCE The conductor when folded forms a two-wire transmission-line and this resonates when its length is approximately equal to nλ/4, where λ is the wavelength. The first resonance is the self-resonant frequency (SRF) and this is given by SRF 300/ [2 l (ε R ) 0.5 ] MHz A 7.1 where l is the total length of the wire (ie twice the length of the transmission line). ε R is the dielectric constant of the spacer The frequency is approximate because stray capacitance of only a few pf will reduce the resonant frequency. However it is a useful check that the correct frequency has been measured. As an example a line of length 2 l = had a measured SRF of 162 MHz. The above equation gives 175 MHz, 7% higher. (NB this was for conductors in Configuration 2, thickness to thickness, with no dielectric between the conductors, although the supporting plate had an of ε R 3). As the SRF is approached the measured resistance and inductance increase above their low frequency value and Welsby ( ref 9, p 37) has shown that the low frequency value is given by : L = L M [ 1- (f / f R ) 2 ] A7.2 30

31 Resistance ratio Payne : Proximity Loss in Rectangular Conductors issue 2 R = R M [ 1- (f / f R ) 2 ] 2 A7.3 L M and R M are the measured values, and f R is the self-resonant frequency. The low frequency values L and R are those which must be compared with the theory, since the theory does not include resonance effects since these are specific to a particular conductor length and dielectric constant. Welsby developed these equations for a lumped element tuned circuit and these equations become increasingly inaccurate as the resonant frequency is approached, and are generally not sufficiently accurate for frequencies greater than 0.4 SRF. The correction provided by the above equation is then 1.4 ie the measured value must be reduced by 1.4 to give the value in the absence of resonance. To test the accuracy of Equation A7.2 the resistance of two parallel conductors was measured with a welldefined known resistance. This was achieved by terminating the transmission line formed by the two conductors with a thin film resistor of measured DC resistance of 8.68 Ω. The conductors consisted of two parallel copper strips of width 6mm and thickness 0.32 mm and length of 150 mm with a spacing of about 8 mm. The measured SRF was 239 MHz and the change in resistance with frequency calculated from the above equation, and plotted below in blue. Self-resonance SRF with 10Ω resistor Equation Frequency MHz Figure A 7.1 SRF comparison The resistance was then measured and this is plotted in red. The discrepancy compared with Equation A7.3 was less than 3%. This was reduced to less than 1% if the SRF was assumed to be slightly higher at 250 MHz (+5%) and this improvement seems to be generally true. It is important that the SRF is measured (rather than calculated), and in the same jig as the measurements are being made. For details on the measurement procedure see Payne ref 10. Also the SRF will change with the setting of the capacitor because the path length through the capacitor changes with setting. If the Q at resonance is very low then a correction must be applied as follows : f R = f MEAS / (1-1/ Q 2 ) 0.5 A7.4 where f MEAS is the measured resonant frequency Q is the Q at resonance 31

32 Appendix 8 A8.1. : EFFECT OF CAPACITANCE Capacitance For the capacitance between strip conductors Terman (ref 11 p 113) gives for w/g <1 and for dielectric extending beyond the edges of the conductors (see below) : C = / [Log 10 (4 g/w)] pf/inch A8.1 1 = * ε r l / [2.303 Ln (4 g/w)] pf = ε r l / [Ln (4 g/w)] pf A8.1.2 For w/g >1 and for dielectric extending beyond the edges of the conductors (see below) Terman gives : C= w/g [1+ {g/(π w)} Ln (2 π w/g )] pf/inch A8.1.3 NB there is an error in Terman with brackets in the wrong place and the above is assumed to be correct. = *39.37 ε r l [1+ {g/(π w)} Ln (2 π w/g )] pf C = ε O ε r l w/g [1+ {g/(π w)} Ln (2 π w/g )] pf A8.1.4 In relation to a dielectric between the plates Terman also says : With solid dielectric between the plates but not projecting beyond the edges these equations still hold but with less accuracy. When solid dielectric between the plates extends considerably beyond the edges of the plates, fair results can be obtained by multiplying the above equations by the dielectric constant (as included in the above). A8.2. Jig 1 The capacitance is distributed (as is the resistance), and these can be approximated by the following lumped circuit : Figure A Equivalent Circuit This is likely to be a good representation for frequencies up to half first resonance ie when the conductors each have a length of around λ/8. In the circuit above the total resistance R is equal to R 1 +R 2, so each has a resistance of R/2. In the test set-up the two conductors are connected together at one end and this is shown by the shorting strip. With this in place the resistor R 2 is in parallel with C, and the effective series resistance R EFF of the combination is less than R 2, and given by : 32