Patent Application: US-42132306-A

Abstract:
a method of determining material parameters associated with a conductor using four points includes injecting and extracting alternating current into the plate using current - carrying wires operatively connected to two of the four points , measuring potential drop between the remaining two of the four points , and calculating the material parameters . the conductor can be of a homogenous material , a stratified material , or other type of material . the conductor can have any number of geometries , including that of a plate , a cylinder , a tube , a stratified cylinder or other shape .

Description:
the present invention provides a method of quantifying both electrical conductivity and magnetic permeability . the methodology can be applied to objects of varying geometries , including , but not limited to plates , cylindrical rods , and other geometries . in addition , the methodology can be used for characterizing surface treatments or other forms of layered structures in metals . in other words , the object being characterized need not be homogenous . thus , the present invention provides a flexible methodology for quantifying or characterizing electrical conductivity and magnetic permeability for a variety of materials in a variety of environments or situations . measurements of voltage are taken at a number of specific frequencies . the low - frequency voltage values are entered into a theoretical formula to calculate either the metal conductivity or thickness . one of these must be known independently in order to calculate the other . higher - frequency voltage values are entered into a theoretical formula ( along with the conductivity and plate thickness ) to calculate the magnetic permeability . on samples tested , conductivity measurements are accurate to within 1 % of values obtained by an independent ( eddy - current ) method . the miz - 21a eddy - current instrument measures conductivity to at best 1 % and possibly only 20 % accuracy for low conductivity metals such as stainless steel . in order to measure magnetic permeability by an eddy - current method , low - frequency instrumentation is required because the conductivity and permeability remain coupled in their behavior until low frequencies are reached . in the proposed method , the conductivity becomes decoupled from the permeability at significantly higher frequency , making it possible to measure both parameters easily using the same technology . very small variations in magnetic permeability (± 0 . 01 ) are observable for low permeability metals . the method relies on a theoretical formula for interpretation of the measurement data and deduction of the parameters of the metal plate . this theoretical formula has not previously been derived in the case of alternating current . the alternating current potential drop ( acpd ) method measures the voltage , , between two pick - up points on the surface of a conductor . for the configuration shown in fig1 , = v + ɛ = - ∫ ( p , y , 0 ) ( q , y , 0 ) ⁢ e · ⁢ ⅆ l + ∮ c ⁢ e · ⅆ l , ( a ⁢ . 1 ) where c is a closed loop [ 1 ]. ε is the rate of change of magnetic flux within the loop . in direct current potential drop measurements there is no induction effect in the measurement circuit ( ε = 0 ) since the current does not vary with time and in this case the measured potential drop is almost exclusively due to the conductor . in acpd measurements , the contribution to from the conductor dominates when the frequency is sufficiently low , since the inductive contribution from the measurement circuit , iωl , is proportional to frequency ω . at sufficiently high frequency the inductive term dominates . in this work , both contributions to fare evaluated . the far - field approximation for e is used in calculating . this approximation gives accurate results when pick - up points at ( p , y , 0 ) and ( q , y , 0 ) are sufficiently far from the source points at (± s , 0 , 0 ), in practice a few electromagnetic skin depths ( δ ) in the conductor and the conductor is somewhat thinner than the probe spacing , as shown for direct current potential drop measurements [ 10 , 11 ]. for the configuration shown in fig1 , the electric field can be obtained by superposition of fields separately associated with the two current - carrying wires : where r ± =√{ square root over (( x ± s ) 2 + y 2 + z 2 )}. in the following sections the far - field form of e is determined in the region of the pick - up circuit ( air ) and in the metal plate for a single current - carrying wire located on the axis of a cylindrical co - ordinate system . for a single wire passing current i into , or out of , a conductive plate , there are two contributions to the electric field in air . one is from the current flowing in the wire , e w , and the other is from the current density in the plate . in the far - field regime , for the closed loop c , only e w is important . assuming that the wire is perpendicular to the surface of the plate and that the current has time - dependence e − iωt , the integral form of ampere &# 39 ; s law and then faraday &# 39 ; s law yields e w ⁡ ( ρ , z ) = z ^ ⁢ ⅈω ⁢ ⁢ μ 0 ⁢ i 2 ⁢ π ⁢ ln ⁢ ⁢ ρ , ⁢ ρ → ∞ , z ≤ 0 , ( a ⁢ . 3 ) where ρ is the radial co - ordinate of a cylindrical system centered on the wire and e w has the same direction as the current density in the wire , j ={ circumflex over ( z )} j z . an expression for the electric field in the conductive plate is obtained in a manner similar to that given in reference [ 2 ] for a conductive half - space . for a current source oriented perpendicular to the surface of the plate , only the transverse magnetic ( tm ) potential , ψ ″, is required to fully describe the electric field : where ∇ z ≡∇−{ circumflex over ( z )}(∂/∂ z ) is the transverse differential operator . for a plate infinite in x and y , occupying zε [ 0 , t ], the governing equation is where k 2 = iωμσ with μ and σ being the permeability and conductivity of the plate , respectively . in the plate , only the horizontal component of the electric field , e ρ , contributes to v . it is not convenient to express e ρ in terms of ψ . rather , e ρ will be obtained from the following equation by means of relationship ( a . 5 ). e ρ ⁡ ( r ) = - ⅈ ⁢ ⁢ ωμ ⁢ ∂ 2 ⁢ ψ ″ ⁡ ( r ) ∂ ρ ⁢ ∂ z , ( a ⁢ . 7 ) where ρ and z are co - ordinates of the cylindrical system . equation ( a . 6 ) is solved for t subject to boundary conditions ψ ⁡ ( ρ , 0 ) = c ⁡ ( ρ ) ⁢ ⁢ where ⁢ ⁢ c ⁡ ( ρ ) = { i π ⁡ ( k ⁢ ⁢ a ) 2 , ρ ≤ a , 0 , ρ & gt ; a , ( a ⁢ . 8 ) these derive from the fact that , at the surface of the plate , the normal component of current density is continuous - zero everywhere apart from at the point of contact with the current carrying wire , radius α . applying the zero - order hankel transform to solve ( a . 6 ) and taking the limit α →˜ 0 yields ψ ⁡ ( ρ , z ) = i 2 ⁢ π ⁢ ⁢ k 2 ⁢ ∫ 0 ∞ ⁢ ⅇ - γ ⁢ ⁢ z ⁡ [ 1 - ⅇ 2 ⁢ γ ⁡ ( z - t ) 1 - ⅇ - 2 ⁢ γ ⁢ ⁢ t ] ⁢ j 0 ⁡ ( κρ ) ⁢ κ ⁢ ⁢ ⅆ κ , ( a ⁢ . 10 ) where y 2 = k 2 − k 2 . if t →∞, the term in square brackets tends to unity and the resulting integral is identical to that obtained for a half - space conductor [ 2 ]. it is possible to evaluate the integral ( a . 10 ) analytically by expanding the term in the denominator as a binomial series [ 4 , 3 . 6 . 10 ]: multiplying the right - hand side of ( a . 11 ) by the factor e − yz [ 1 − e 2y ( z − t ) ] and substituting the result into ( a . 10 ) yields where the order of summation and integration has been reversed . the first term in braces in ( a . 12 ), e − γz , gives rise to the result for the tm potential in a half - space conductor . the second term , − e γ ( z − 2t ) , accounts for the primary reflection of the field from the surface of the plate at z = t . other terms deal with multiple reflections between the surfaces of the plate . by analogy with the result for the half - space conductor , reference [ 2 ], or by multiple use of the analytic result given in reference [ 5 ], result 8 . 2 . 23 , the terms in ( a . 12 ) can be integrated . it is found that ψ ⁡ ( ρ , z ) = - i 2 ⁢ π ⁢ ∑ n = 0 ∞ ⁢ { ⅈ ⁢ ⁢ k ⁡ ( z + 2 ⁢ ⁢ n ⁢ ⁢ t ) ( ⅈ ⁢ ⁢ k ⁢ ⁢ r n ) 3 ⁢ ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ r n ⁡ ( 1 - ⅈ ⁢ ⁢ k ⁢ ⁢ r n ) + ⅈ ⁢ ⁢ k ⁡ [ z - 2 ⁢ ( n + 1 ) ⁢ t ] ( ⅈ ⁢ ⁢ k ⁢ ⁢ r n ′ ) 3 ⁢ ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ r n ′ ⁡ ( 1 - ⅈ ⁢ ⁢ k ⁢ ⁢ r n ′ ) } , 0 ≤ z ≤ t , ( a ⁢ . 13 ) r n =√{ square root over ( ρ 2 +( z + 2nt ) 2 )} and r ′ n =√{ square root over ( ρ 2 +[ z − 3 ( n + 1 ) t ] 2 .)} to obtain e ρ from ψ as given in ( a . 13 ) via relations ( a . 7 ) and ( a . 5 ) requires some manipulation [ 2 ]. the result is in the far field , the electric field is dominated by terms of the form e ikz / ρ and if the far - field current density is integrated over a cylindrical surface of large radius extending from z = 0 to t , the result is i └ 1 + e ik ( 2n − 1 ) t ┘ for a series truncated to n terms . this expression tends to i as n →∞, as it should . if t →∞ the far - field expression for the electric field in a half - space conductor is recovered [ 2 ] this expression was also given in reference [ 3 ] in the context of fatigue crack measurement . voltage is now calculated according to equation ( a . 1 ). for the configuration shown in fig1 the contributions are with e t given by ( a . 2 ). it is a simple matter to evaluate the last two terms on the right - hand side of equation ( a . 17 ) with e z given in equation ( a . 3 ). to neatly evaluate the first term on the right - hand side of ( a . 17 ) recognize that , at the surface defined by z = 0 , equation ( a . 15 ) can be written the first term in equation ( a . 20 ) is the contribution from the conductor and has approximately equal real and imaginary parts . the contribution from the measurement circuit is imaginary ( inductive ) and proportional to the dimension of the circuit perpendicular to the conductor surface , l . for a typical non - magnetic metal and l ˜ 1 mm , the inductive term is practically negligible for frequencies up to about 10 hz whereas at 10 4 hz the terms are of similar magnitude . the logarithmic term represents the physical arrangement of the four probe points . acpd measurements were made as a function of frequency on a brass plate whose conductivity and dimensions are given in table 1 . the brass plate was precision ground to remove surface scratches and mounted on a two - inch thick plastic support plate . electrical contact with the brass plate was made via sprung , point contacts , held perpendicular to the surface of the plate . in this experiment the four contact points were arranged in a straight line , with a common midpoint between the two current drive points and the two pick - up points . the dimensions of the probe are given in table 1 . the two current - carrying wires were held perpendicular to the plate surface for a distance of 16 inches , after which they were twisted together to reduce the effects of inter - wire capacitance . this distance was sufficient to remove any effect of motion of the current wires on the measured voltage . the two pick - up wires were arranged with the objective of minimizing l , lying as close to the plate surface as possible . they were twisted together at the midpoint between the pick - up points . in the theoretical calculation , two measured values are needed . one is the current through the plate , the other is the voltage measured by the pick - up probe . to monitor the current in the plate , a high precision resistor was connected in series with the drive current circuit and the voltage across the resistor measured . the resistance maintains one percent accuracy over the range of frequency for which it could be measured with an agilent 4294a precision impedance analyzer ; 40 hz to 40 khz . the voltage across the resistor and that of the pick - up probe were both measured using a stanford research systems sr830 dsp lock - in amplifier . in order to make both voltage measurements using the same lock - in amplifier , a switch was used activated by a control signal from the auxiliary analog output of the lock - in amplifier . it was necessary to correct the experimental data for common - mode rejection ( cmr ) error in the lock - in amplifier . this systematic error shows itself in the fact that , when the pickup terminals are reversed , the measured voltage changes by a few μv . the magnitude of the error is , therefore , similar to that of the voltage being measured , and a corrective procedure is essential . the cmr error was eliminated by taking two sets of measurements , reversing the pick - up terminals for the second . the two sets were then subtracted and the result divided by two . the drive current was produced by a kepco bipolar operational power supply / amplifier , model number bop 20 - 20m . the sine signal from the internal function generator of the lock - in amplifier was connected to the current programming input of the power supply , with the power supply working as a current drive . the conductivity of the plate was measured using a miz - 21a eddy current instrument . the error quoted in table 1 is estimated from the manufacturer &# 39 ; s literature and derives from a combination of inaccuracy in the instrument , inaccuracy in the comparative standards and probe lift - off error . in fig2 , acpd measurements are compared with theory . the average of ten data sets ( taken sequentially ) is shown . the value of i was adjusted in the calculation to obtain the best fit to the high frequency imaginary part of the data , having negligible influence on the low - frequency data . the value l = 0 . 35 mm appears reasonable since the pick - up wire is awg 32 with diameter 0 . 2 mm . the agreement between theory and experiment is excellent . there is no obvious error in the imaginary part of v . the theory overestimates the low frequency real part of vby 3 %. applying standard error analysis to the low frequency limiting expression for v , equation ( a . 22 ), shows that errors in the plate conductivity and in the relative positions of the probe points combine to give an experimental error which is also 3 %. to take the limit k → 0 in equation ( a . 20 ), not that lim k → 0 [ ikt / sinh ( ikt )]= 1 . then → - i 2 ⁢ π ⁢ ⁢ σ ⁢ ⁢ t ⁢ ln ⁢ { [ ( p - s ) 2 + y 2 ( p + s ) 2 + y 2 ] ⁡ [ ( q + s ) 2 + y 2 ( q - s ) 2 + y 2 ] } , ⁢ k → 0 . ( a ⁢ . 22 ) it is seen that at low frequency the voltage is real , being inversely proportional to the plate thickness and conductivity . formula ( a . 22 ) is consistent with one given by yamashita and masahiro for four - point dc measurements on a finite plate [ 6 ]. in fig3 , is plotted for a number of values of plate thickness . the inverse dependence of re ( ) on the plate thickness at low frequency , predicted by equation ( a . 22 ), can be clearly seen in fig3 . at high frequency the voltage is dominated by the inductive term in equation ( a . 20 ). this term is proportional to l , the length of the pick - up wire perpendicular to the metal plate . practically it is desirable to minimize the contribution of this term by making l as small as possible . in this way the contribution to due to the plate , from which useful information may be derived , is not masked by induction in the measurement circuit . in fig4 , the effect on of varying l is shown . only im ( ) is shown since l has no influence on re ( ). therefore , a four - point method of measuring material parameters has been disclosed . the simple analytic result , equation ( a . 20 ), gives useful insight into the primary contributors in acpd measurements . it is accurate for a flat plate whose edges are several tens of skin depths from the probe , for a probe whose pick - up points are several skin depths away from the current drive points and for a conductor somewhat thinner than the probe point spacing . of course the present invention has numerous applications , including but not limited to use in non - destructive quantification of metal plates of titanium / nickel alloys such as used in aircraft engines , metal billets , and other uses where material evaluation or process monitoring is used . these applications further include applications where non - destructive quantification of metal thickness is needed and access is restricted to one side . the present invention has been used to measure a variety of different types of conductors including brass , aluminum , stainless steel , carbon steel and spring steel . the present invention is not to be limited to the specific disclosure presented herein , as one skilled in the art having the benefit of this disclosure would appreciate the far - reaching scope of the present invention . the present invention also provides for the use of four - point alternating current potential drop measurements on a metal half - space . an analytic expression is derived and used to describe the complex voltage measured between the pickup points of a four - point probe , in contact with the surface of a half - space conductor . the alternating current potential drop ( acpd ) measurements permit depth - dependent information to be obtained through the phenomenon of the electromagnetic skin effect , in which the current is confined to flow in a ‘ skin ’ at the surface of the conductor , whose depth is approximately inversely proportional to the square root of the excitation frequency . the acpd technique therefore has application in assessing materials whose electromagnetic parameters vary with depth , for example , in the case of electrically conductive surface treatments and coatings . one advantage of acpd over dcpd is that a lower measuring current can be applied in order to achieve a given sensitivity [ 10 ] ( section 8 ). this reduces the risk of heating of the specimen and associated changes in electrical conductivity . in previous work , mitrofanov derived an expression for the complex voltage measured between the pickup points , of a four - point probe , in contact with the surface of a half - space conductor [ 12 ]. the solution was expressed in terms of an infinite series expansion in powers of k , where δ being the electromagnetic skin depth in the conductor . in equation ( b . 1 ), ω = 2πf is the angular frequency of the injected current and μ and σ are the magnetic permeability and electrical conductivity of the half - space , respectively . the analytical expression describing the complex voltage measured between the pickup points of a four - point probe , in contact with the surface of a half - space conductor , is derived in closed form . there are two contributions to the measured voltage . one arises from the potential drop due to electric current flowing in the conductor . the other arises from induction in the loop of the pickup circuit . both terms are obtained by integrating analytic expressions for the electric field , derived previously [ 2 , 13 ], along appropriate paths . it is shown that the closed - form expression obtained here for the potential drop due to current flowing in the conductor can be expressed as a power series in k , giving the same result as that presented in [ 12 ]. the contribution to the measured complex voltage due to inductance in the pickup circuit was not analyzed in [ 12 ]. theory is compared with experimental data for co - linear and rectangular arrangements of the four probe points in contact with a thick aluminum block and very good agreement is obtained in both cases . the acpd method measures a complex voltage , v , which has two contributions : the first term , v , is the potential drop between the two points on the plate at which the measurement circuit makes contact with its surface . the source of v is the current in the plate injected by the other two points of the four - point probe . at arbitrary frequency , v is complex . the second contribution , ε , is proportional to the inductance of the measurement circuit . it arises form the changing magnetic flux within the loop of the measurement circuit due to harmonic variation of the applied current , of the form e − iωt , ε is purely inductive , therefore imaginary . in the static limit of direct current , only v remains . for the geometry given in fig5 , where c is a closed loop in the case where p ′ and q ′ coincide , as happens when the pickup wires are twisted together at their point of meeting . at low frequency , the measured potential drop is almost exclusively due to the conductor . in an acpd measurement on a conductive plate , the contribution to v from the plate is most significant at lower frequencies , with the contribution from ε becoming larger , and eventually dominant as the frequency increases . strictly , the quantities v , ν , ε and e are complex amplitudes . for brevity , the time dependence is not shown explicitly in equations ( b . 2 ) to ( b . 4 ) or in the equations that follow . for current injected into a half - space conductor by a single wire held perpendicular to the conductor surface , the components of the electric field in the conductor are [ 2 ] e ρ s ⁡ ( r ) = - i 2 ⁢ π ⁢ ⁢ σ ⁢ ⅈ ⁢ ⁢ k ρ ⁢ { ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ z - ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ r ⅈ ⁢ ⁢ k ⁢ ⁢ r ⁡ [ 1 + ( ⅈ ⁢ ⁢ k ⁢ ⁢ z ) 2 ⅈ ⁢ ⁢ k ⁢ ⁢ r ⁢ ( 1 - 1 ⅈ ⁢ ⁢ k ⁢ ⁢ r ) ] } , z & gt ; 0 , ( b ⁢ . 5 ) e z s ⁡ ( r ) = i 2 ⁢ π ⁢ ⁢ σ ⁢ z r 3 ⁢ ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ r ⁡ ( 1 - ⅈ ⁢ ⁢ k ⁢ ⁢ r ) , z & gt ; 0 , ( b ⁢ . 6 ) in which ρ and z are the variables of a cylindrical co - ordinate system centered on the current wire and r 2 = ρ 2 + z 2 . the electric field in air may be expressed [ 13 ] as e w = z ^ ⁢ i 2 ⁢ π ⁢ ⅈωμ 0 ⁢ ln ⁢ ⁢ ρ , ρ & gt ; 0 , z ≤ 0 , ( b ⁢ . 8 ) e c = i 2 ⁢ πσ ⁢ ∫ 0 ∞ ⁢ γⅇ κ ⁢ ⁢ 2 ⁡ [ ρ ^ ⁢ j 1 ⁡ ( κρ ) - z ^ ⁢ j 0 ⁡ ( κρ ) ] ⁢ ⁢ ⅆ κ , z ≤ 0 , ( b ⁢ . 9 ) in equation ( b . 9 ), y 2 = k 2 − k 2 and j i ( x ) is the i - th order bessel function of the first kind . e w is the electric field in air due to the current flowing in the injection wire . e c is the electric field in air due to the current flowing in the half - space conductor . for a system of two current - carrying wires in contact with the metal surface at co - ordinates (± s , 0 , 0 ) as shown in fig5 , the electric field e can be obtained by the superposition of the field due to a single wire , e s , whose components are given above : with r ± =√{ square root over (( z ± s ) 2 + y 2 + z 2 )} closed form . in general , the line of the pickup points may be off - set from the line of the current injection points . let y = c = constant and then choose the path of the integral in equation ( b . 3 ) such that ν =−∫ p q e x ( x , c , 0 ) dx ( b . 11 ) e x ⁡ ( x , c , 0 ) = ( x + s ) ρ + ⁢ e ρ s ⁡ ( p + , 0 ) - ( x - s ) ρ - ⁢ e ρ s ⁡ ( ρ - , 0 ) , ( b ⁢ . 12 ) ρ ± =√{ square root over (( x ± s ) 2 + c 2 )}. combining the above two equations and making the change of variable i ± = ∫ ρ ± s q ± s ⁢ x x 2 + c 2 ⁢ e ρ s ⁡ ( x , c , 0 ) ⁢ ⁢ ⅆ x . ( b ⁢ . 14 ) putting e ρ s ( x , c , 0 ) from equation ( b . 5 ) into the integrand of equation ( b . 14 ) gives i ± = ⅈ ⁢ ⁢ k ⁢ ⁢ i 2 ⁢ π ⁢ ⁢ σ ⁢ ∫ ρ ± s q ± s ⁢ [ x x 2 + c 2 - x ⁢ ⁢ ⅇ ⅈ ⁢ ⁢ k ⁢ x 2 + c 2 ⅈ ⁢ ⁢ k ⁡ ( x 2 + c 2 ) 3 / 2 ] ⁢ ⅆ x . ( b ⁢ . 15 ) integration of the first term in equation ( b . 15 ) is straightforward . the second term in equation ( b . 15 ) may be evaluated by making a further change of variable , α =√{ square root over ( x 2 + c 2 )}, and using the following identity ( equation 2 . 325 . 2 ) in [ 14 ]): ∫ ⅇ ax x 2 ⁢ ⅆ x = - ⅇ ax x - ae 1 ⁡ ( - ax ) , ( b ⁢ . 16 ) in which e 1 ( z ) is the exponential integral function , defined ( equation ( 5 . 1 . 1 ) in [ 15 ]) as e 1 ⁡ ( z ) = ∫ c ∞ ⁢ ⅇ - t t ⁢ ⁢ ⅆ t ,  arg ⁢ ⁢ z  & lt ; π . ( b ⁢ . 17 ) v = i 2 ⁢ ⁢ π ⁢ ⁢ σ ⁡ [ f i ⁡ ( s + q , c ) - f i ⁢ ⁡ ( s - q , c ) - f i ⁡ ( s + p , c ) + f i ⁢ ⁡ ( s - p , c ) ] ( b ⁢ . 18 ) where , as will be shown subsequently , f i ( x , y ) can take several forms . in exact , closed form , f exact ⁡ ( x , y ) = f exact ⁡ ( ρ = x 2 + y 2 ) = ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ ρ ρ + ⅈk ⁡ [ ln ⁢ ⁢ ρ + e 1 ⁡ ( - ⅈ ⁢ ⁢ k ⁢ ⁢ ρ ) ] ( b ⁢ . 19 ) the result presented in equations ( b . 18 ) and ( b . 19 ) can be expressed in terms of a power series in k . in this way it can be shown that the result is in agreement with that of an independent calculation [ 12 ]. the two relations ( equations ( 4 . 2 . 1 ) and ( 5 . 1 . 11 ) in [ 15 ] e 1 ⁡ ( z ) = - γ e - ln ⁢ ⁢ z - ∑ n = 1 ∞ ⁢ ( - 1 ) n ⁢ z n nn ! ,  arg ⁢ ⁢ z  & lt ; π , ( b ⁢ . 21 ) applied to the exponential and exponential integral functions in equation ( b . 19 ) give f exact ( ρ )=− ik [ γ e + ln (− ik )− 1 ]+ f series ( ρ ) ( b . 22 ) f series ⁢ ⁡ ( ρ ) = 1 ρ ⁡ [ 1 - ∑ n = 1 ∞ ⁢ ( ⅈ ⁢ ⁢ k ⁢ ⁢ ρ ) n + 1 n ⁡ ( n + 1 ) ! ] ( b ⁢ . 23 ) note that the terms present in the relation between f exact and f series ( equation ( b . 22 )) are independent of ρ . this means that they drop out when inserted into equation ( b . 18 ). hence f series ( equation ( b . 23 )) may be inserted directly into equation ( b . 19 ) as an alternative to f exact ( equation ( b . 19 )). the resulting series representation for ν given by combining equations ( b . 18 ) and ( b . 23 ) agrees with that presented in [ 12 ]. one commonly - used probe configuration is that in which the four probe points are arranged along a straight line , with the voltage pickup points positioned symmetrically about the midpoint between the current injection points . in the case of this co - linear , symmetric probe , p =− q and c = 0 . equations ( b . 18 ) reduces to for a rectangular probe configuration , in which the line between the current injection points forms one side of the rectangle and that between the voltage pickup points forms the opposite side , p =− s and q = s so that v r = i πσ ⁡ [ f i ⁡ ( 2 ⁢ s , c ) - f i ⁡ ( 0 , c ) ] ( b ⁢ . 25 ) f exact dc ⁡ ( ρ ) = f series dc ⁢ ⁡ ( ρ ) = 1 ρ , ( b ⁢ . 26 ) in fig6 , the real and imaginary parts of the dimensionless voltage , πσν ls / i , are plotted versus dimensionless frequency , ωμσs 2 , for various values of the ratio of pickup length to current injection length q / s , for a co - linear symmetric probe . it can be seen that the voltage increases as the pickup points approach the current injection points more closely , i . e . as q / s increases . voltage values calculated using the series representation for ν , equation ( b . 23 ), are also shown for the probe with equally - spaced probe points , q / s = ⅓ . to achieve agreement to within 2 percent of values calculated using the exact solution at the highest frequency considered , 30 terms in the series are required . as q / s increases , yet more terms are needed . in fig7 , the real and imaginary parts of the dimensionless voltage , λσν r s / i , are plotted versus dimensionless frequency , ωνσs 2 , for various values of the aspect ratio of a rectangular probe , c /( 2s ). again , the pickup voltage increases as the pickup points approach the current injection points more closely , i . e . as c / s decreases . to achieve agreement within 2 percent between values calculated using the exact solution ( equation ( b . 19 )) and the series solution ( equation ( b . 23 )) for a square - head probe ( c / s = 2 ) at the highest frequency considered , 70 terms in the series are required . in both fig6 and fig7 , it is evident that below a certain frequency , the pickup voltage is approximately real and constant . in this low - frequency regime , the measured voltage matches that obtained in the dc limit and equation ( b . 26 ) applies . hence , in the low - frequency regime , ν is independent of μ , and σ may be determined independent of μ by adjusting the value of σ until theory matches low - frequency experimental data . once σ is known , μ may be determined by fitting theory with experimental data taken at higher frequencies . this procedure is demonstrated in [ 16 ] in characterizing metal plates which are somewhat thinner than the probe length [ 17 ]. comparing results shown in fig6 and fig7 it can be seen that the co - linear and rectangular probes perform more similarly as the pickup points approach the current injection points more closely , as is to be expected . it can easily be shown that e c ( equation ( b . 9 )) is conservative (∇× e c = 0 ) and therefore does not contribute to the integral around the closed loop from which ε is derived ( equation ( b . 4 )). hence , with equations ( b . 7 ) and ( b . 10 ), ɛ = ∮ c ⁢ [ e w ⁡ ( r + ) - e w ⁡ ( r - ) ] · ⁢ ⅆ 1 ( b ⁢ . 27 ) considering the form of e w ( equation ( b . 8 )) evaluation of the integral in equation ( b . 27 ) is straightforward , yielding ɛ = i 2 ⁢ ⁢ π ⁢ ⅈ ⁢ ⁢ ω ⁢ ⁢ μ 0 ⁢ l ⁢ ⁢ ln ⁡ [ ( s + q ) 2 + c 2 ⁢ ( s - p ) 2 + c 2 ( s - q ) 2 + c 2 ⁢ ( s + p ) 2 + c 2 ] ( b ⁢ . 28 ) the self - inductance of the pickup circuit , l , may therefore be expressed as l = i 2 ⁢ ⁢ π ⁢ μ 0 ⁢ l ⁢ ⁢ ln ⁡ [ ( s + q ) 2 + c 2 ⁢ ⁢ ( s - p ) 2 + c 2 ( s - q ) 2 + c 2 ⁢ ( s + p ) 2 + c 2 ] ( b ⁢ . 29 ) combining results ( b . 18 ) and ( b . 28 ) in accordance with equation ( b . 2 ) gives , finally , v = i 2 ⁢ ⁢ π ⁢ ⁢ σ ⁡ [ f ⁡ ( s + q , c ) - f ⁡ ( s - q , c ) - f ⁡ ( s + p , c ) + f ⁡ ( s - p , c ) ] ( b ⁢ . 30 ) f ⁡ ( x , y ) = ⁢ f ⁡ ( ρ = x 2 + y 2 ) = ⅇ ⅈ ⁢ ⁢ k ⁢ ⁢ ρ ρ + ⅈ ⁢ ⁢ k ⁡ [ ( 1 - ⅈ ⁢ ⁢ kl μ r ) ⁢ ln ⁢ ⁢ ρ + e 1 ⁡ ( - ⅈ ⁢ ⁢ kρ ) ] ( b ⁢ . 31 ) and μ r = μ / μ 0 is the relative permeability of the half - space . in fig8 the effect of varying l on v is shown in the case of a co - linear , symmetric probe with equally - spaced probe points ( q / s = ⅓ ). only the imaginary part of v is shown since ε is purely imaginary and has no influence on the real part of v . it can be seen that , as l increases , im ( v ) becomes linear in frequency due to the dominance of | ε | over | im ( ν )|. from a practical point of view , it is important to minimize l so that the component of v which carries information about the specimen , ν , is not swamped by the inductive term , ε . the theoretical expression for the complex voltage ( equations ( b . 30 ) and ( b . 31 )) has been validated by comparison with experimental data . two different four - point probes , one with co - linear arrangement of the probe points and one rectangular , were used . the probes were constructed by mounting four sprung , point contacts in a plastic support block . the separation of the contacts was measured using digital calipers . with reference to fig5 , the dimensions of the probe are listed in table 3 . the uncertainty in the dimensions derived primarily from some lateral play in the pin position which can occur as the springs are compressed . measurements of complex voltage were made with the probes in contact with a thick , alloy 2024 aluminum bloc , whose parameters are listed in table 4 . the conductivity of the block was measured independently using an eddy - current coil . details of the conductivity measurement and further details of the experimental procedure for the acpd measurements can be found in [ 16 , 18 ]. the dimensions of the aluminum block , with respect to the dimensions of the probes , are such that some discrepancy between theory and experiment due to edge effects is expected . for the co - linear probe placed centrally on the largest face of the aluminum block , the error due to edge effects is minimized by orienting the line of the probe so that it is parallel with the shorter side of the block face ( w = 149 mm ) [ 10 ]. the error is also reduced by employing a probe in which the four points are not equally - spaced , but in which the pickup points are closer to the current injection points . in fact , for the co - linear probe used in this experiment , ( q − p )/( 2s )≈ 0 . 88 and w /( 2s )≈ 3 . 7 w . for these ratios , and assuming that the aluminum block is ‘ infinite ’ in the direction perpendicular to the line of the probe ( dimension d ), edge effects are expected to lead to a discrepancy of approximately 2 percent between theory and experiment in the dc limit . since in practice this block is finite in the direction perpendicular to the line of the probe ( d = 202 mm ), a discrepancy a little larger than 2 percent is expected between theory and experiment in the dc limit , becoming smaller as frequency increases , due to greater confinement of the electric field in the region of the probe . according to calculations of dcpd as a function of the ratio of plate thickness to probe dimension ( here t / s = 5 [ 17 ], the thickness of the block is expected to approximate a half - space very well , with no significant error arising due to its finite thickness . experimental measurements made with co - linear and rectangular probes are compared with theory in fig9 and fig1 , respectively . in both cases , there is very good agreement between theory and experimental data . the calculated curves shown for the imaginary part of the impedance have been obtained by adjusting the value of the vertical dimension of the pickup circuit , l ( see fig5 ), to give the best fit to the experimental data . for the co - linear probe , l = 2 . 98 mm . for the rectangular probe , l = 2 . 15 mm . both these values are similar to the physical values of l for these probes . no free parameters are involved in obtaining the theoretical curves for the real part of the impedance shown in fig1 and fig1 . the discrepancy between theory and experiment in the low - frequency regime is approximately 4 percent for both sets of measurements . the fact that the measured real part of v is larger than that predicted by theory , rather than smaller , indicates that edge effects are likely responsible for the discrepancy . other significant sources of error are the uncertainty in the probe dimensions and in the conductivity of the same . thus , an exact solution for the complex , frequency - dependent voltage measured between the pickup points of a four - point probe in contact with a metal half - space has been derived . very good agreement between theory and experiment on an aluminum block has been obtained , for co - linear and rectangular arrangements of four probe points . as well as providing a method for measuring electrical conductivity and effective magnetic permeability of thick metal specimens , the present invention allows for theoretical analysis of four - point acpd on stratified planar conductors , for the practical purpose of nondestructive evaluation of conductive surface treatments and coatings . the following references are cited in the disclosure , all of which are herein incorporated by reference in their entireties . w . j . duffin , electricity and magnetism , mcgraw - hill , london , 1980 . 3rd edition . 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