the electrical resistivity of carbon fiber reinforced composites ( cfrc ) is orders of magnitude higher than that of metals . the high resistivity offers both advantages and disadvantages . on the positive side , it permits eddy current inspection to greater depths than were heretofore associated with eddy current inspection . on the other hand , the resistivities of these materials may differ widely among apparently similar materials . the matrix in many cfrc precludes the formation of the good electrical contact required by contact resistivity measurement methods . the existence of protective coatings also preclude contact , and , in addition , there are no high resistivity standards currently available for the calibration of eddy current probes . eddy current nondestructive evaluation , be it defect detection and measurement or resistivity measurement , is based on an analysis of the effect of the test material on the impedance of the eddy current probe . these effects are witnessed by the normalized impedance curve obtained by plotting the normalized imaginary ( x n ) against the normalized real ( r n ) components of the impedance . these components are given by where r m and x m are the real and imaginary components of the impedance when the probe is in contact with the test material and r o and x o are the corresponding values when the probe is in air . turning now to fig1 wherein are shown three normalized impedance curves , labeled 1 , 2 and 3 , each generated with a different separation ( lift - off ) between the ferrite cup core probe having an effective radius , r , of 7 . 11 mm , and a carbon / carbon test material having a resistivity of 840 microhm . cm . associated with each point on a normalized impedance curve is a reference number given by the dimensionless ratio of the effective radius r , and the skin depth , s . the values of this reference number for points a through e in fig1 are 1 . 5 , 2 . 2 , 3 . 1 , 4 . 4 and 6 . 2 . lift - offs associated with curves 1 , 2 , and 3 are , respectively , 1 . 32 mm , 0 . 66 mm and 0 . 0 mm . the effective radius , r , of the ferrite cup core probe is taken to be one - third of the outside diameter of the core . the relationship of the quantity r to the physical dimensions of other ferrite core geometries is not defined herein . the lift - off time 5 is the locus of impedances generated as the separation ( lift - off ) between the probe and the material is increased . note that this line , connecting points on the three curves which have the same reference number , is straight and intersects the air impedance ( 1 , 0 ). the shape of a normalized impedence curve refers to its relative extent in the real and imaginary directions . the shape is described by the relationship between the reference number and lift - off angle . all normalized impedance curves generated by ferrite cup - core probes were found to have a characteristic shape , termed herein a universal curve . the characteristic shape pertains when three conditions are met : the skin depth must be less than circa 3 / 4 the thickness of the material ; the inductance of the probe must have increased no more than 4 % above its minimum value ; and there must be no frequency dependent shielding effects . the universal normalized impedance curve is highly significant . the relationship between lift - off angle 4 and reference number is a property of the probe core design and is insignificantly affected by secondary probe design parameters such as the number and distribution of turns . the product of the probe mean radius and the square root of the frequency can be treated as a normalizing factor . regardless of the resistivity of the material , any particular point on the curve can be attained by the appropriate selection of the mean probe radius - frequency combination . this feature of the normalized impedance curve is the basis for the method of measuring material resistivity disclosed herein . for ferrite core probes , there is a linear relationship between the lift - off angle , θ l and the reference number . this relationship is given by where tan θ l is the ratio of the change , due to the test material in the normalized imaginary component to the normalized real component , or where x n and r n are the normalized imaginary and real components of the impedance , x o and r o are the corresponding components of the impedance of the probe in air , and x m and r m are the corresponding components of the impedance of the probe when it is in contact with the test material . equation 1 requires that the three conditions cited above be met . where p is the resistivity of the test material and f is the frequency in hertz . substitution of equation 2 in equation 1 and solving for resistivity , p , yields : when the probe has a ferrite cup core the values of i and m are - 0 . 158 and 0 . 7035 , respectively . given that the 3 conditions , stated above , are met , the resistivity of any material can be measured . it is only necessary to select the frequency and probe size combination such that the quantity tan θ l is between 0 . 8 and 6 inclusive . this limitation is not theoretical but practical . outside of this range , small errors in measurement introduce large errors in the computed quantities . practicality limits the combination of probe size and frequency in combination with material resistivity to values which provide tan θ l values in the range given . there is a further constraint when it is desirable to measure the resistivity of the bulk material , and the resistivity of the material close to the surface is possibly different . this could be the case with some cfrc or with metals where there are oxidation or corrosion products on the surface . in this case the depth over which it is desired to integrate the resistivity measurement determines the smallest probe that can be used . when a ferrite cup core probe is used , the outside diameter should be at least 5 times this depth dimension . the effective radius r is one - third of this outside diameter and therefore , stated another way , the depth dimension should be 1 . 67 times the radius . turning to fig6 the radius 11 and the outside diameter 12 of probe 2 may be graphically visualized . the quantity tan θ l can be plotted against f 1 / 2 . if the probe has a ferrite cup core , these data fall on a straight line whose slope is given by since m and r are known , p can be calculated . if the core design is unknown , but if tan θ l versus f 1 / 2 is a straight line then the quantity mr can be determined from the slope generated with a material of known resistivity . this quantity can then be used to calculate the resistivity of an unknown material from the slope of the tan θ l versus f 1 / 2 curve generated for the material of unknown resistivity . when the resistivity is unknown and the test material is a thin plate , it is possible that the skin depth may be greater than 3 / 4 the thickness of the test material over part of the frequency range . this may also be the case when instrumentation considerations limit the maximum frequencies that can be achieved . for example , even with only a single turn , the resonant frequency of a ferrite core probe may prevent its effective use in the frequency range demanded by the resistivity and thickness of the test material . resistivity values calculated for frequencies in the lower portion of the frequency range , or possible over the entire range , will tend to be high if , at those frequencies , the skin depth is greater than 3 / 4 the thickness of the test material . when frequencies are such that the skin depth is less than 3 / 4 the thickness of the test material the calculated resistivity values scatter about a median value representing the resistivity of the test material to within plus or minus 10 %. if the calculated resistivity values continue to decrease with increasing frequency , never reaching a constant value , it can be assumed that over the entire frequency range the skin depth was too great . if the instrumentation , including the probe design , is such that the necessary high frequencies cannot be achieved , a second relationship can be used : c 3 = grt 2 f 3 / 2 /( tan θ l + c ) where r is a linear dimension , in mm , related to the physical dimensions of the probe core , t is the thickness , in mm , of the test material , f is the frequency in hertz , the constants a , b , c , d , e , and g are constants determined by the geometry of the ferrite core of the probe , and tan θ l is the tangent of lift - off angle . p e is the resistivity of the test material when conditions are such that the skin depth is greater than 3 / 4 the thickness of the test material . equation 5 can be solved for p e by a method of successive approximation . when the probe has a ferrite cup core , r is equal to 1 / 3 the outside diameter of the core and the constants are a = 0 . 0808 , b = 0 . 0526 , c = 0 . 023 , d = 0 . 1817 , e = 0 . 112 , and g = 0 . 025 . of course , it is not necessary to use a multifrequency approach . if the resistivity can be assumed with confidence to be such that skin depth at a specific frequency is less than 3 / 4 the thickness of the test material , equation 4 can be used . the data collected at that single frequency can be used to calculate the resistivity . by the same token , if the resistivity can be assumed with confidence to be such that the skin depth at a specific frequency is greater than the thickness of the test material , equation 5 can be used to calculate the resistivity . to reduce the effects of scatter the single frequency measurements can be repeated several times and averaged . when conditions are such that it cannot be assumed that only equation 4 or only equation 5 are applicable , resistivities can be calculated using both equations . when the calculated resistivities , p and p e are plotted against frequency the actual resistivity of the material can be discerned . turning now to fig2 wherein are plotted both p , curve 1 , and p e , curve 2 , against log frequency . the material was a 1 . 27 mm thick aluminum plate having a resistivity of 4 . 1 microhm . cm . the calculated values of p e are constant , with some scatter , with frequency between 1 and 8 khz . in this frequency range the skin depth varies from 2 . 5 times the thickness of the test material to 0 . 9 the thickness . over this same range , p values are decreasing with frequency . the p e and p curves cross at the frequency where the skin depth is 3 / 4 the thickness . at frequencies above this crossover point , for example at 16 khz and at 32 khz , p values are constant with frequency and p e values increase . an average of the data points which scatter about the straight line provides an accurate measure of the resistivity . the resistivity would also be calculated by assuming that , at the crossover frequency , the resistivity is such that the skin depth is equal to 3 / 4 the thickness of the test material . the structure of graphite epoxy requires the use of a correction factor , h . in equations 4 and 5 , the geometric quantities ( r and t ) must be multiplied by this factor . an effective value of this factor for ferrite cup core probes is 1 . 414 , although slight variations in this factor are acceptable without departing from the scope of applicants &# 39 ; invention . the resistivity of some graphite epoxy materials can be frequency dependent . it is generally constant with frequency from dc to some frequency where it begins to increase before again leveling off . if the resistivity of the test material increases over the frequency range where the skin depth / thickness ratio is greater than 0 . 75 then the calculated values of p e are not level , but also increase with frequency . since the calculated values of p are decreasing over this range , it can be assumed that the crossover point has not been reached and , consequently , the calculated value of p e , at each frequency , is the resistivity of the test material at that frequency . these effects are illustrated in fig3 wherein are plotted the calculated values of p ( curve 1 ) and p e ( curve 2 ) for a graphite epoxy whose resistivity was frequency dependent . the curves cross at 6 mhz ( point 3 ) where the estimated resistivity was 15 , 000 microhm . cm and the skin depth was 3 / 4 the thickness of the material . when the resistivity of such a material is measured over a higher frequency range , where the skin depth / thickness ratio is less than 0 . 75 , then both p and p e increase with frequency . it is obvious the frequency is above the crossover point and the values of p , calculated at each frequency , is the resistivity of the test material at the frequency . in summary , both p and p e can be plotted against frequency and the frequency at which the two curves cross ( crossover point ) identified . at frequencies below this point the value of p e calculated at each frequency is the resistivity of the test material at that frequency . at frequencies above the crossover point , the value of p , calculated at each frequency , is the resistivity of the test material at that frequency . the entire method , including selection of appropriate values , can be implemented in software so the method is transparent to the technician . a resistivity measuring device to perform the above techniques can be constructed by commercially available electronic components once the herein disclosed methods are known . fig4 is a block diagram of one such instrument . therein , a means for measuring impedance 1 , capable of accurate measurement of the impedance of an eddy current probe 2 over a wide range of frequencies is shown . the impedance measuring device 1 may be an impedance analyzer , a precision bridge circuit , or a hetrodyne circuit , as long as the circuitry is capable of measuring the absolute impedance over a wide frequency range covering at least 10 khz to 50 mhz . probe 2 must be one of a set capable of operating over a frequency range adequate to fully apply the methods above to the material of interest . the means for measuring impedance 1 of fig4 should also be capable of measuring over the same frequency range . this probe will be preferably of the ferrite cup core type and physically much larger than those generally associated with the eddy current inspection art . the impedance measuring device 1 is in electrical connection with a means for computation 3 . the means for computation must accept the probe impedance measurements in air and on the test material and be capable of performing the algorithms and calculations required for performing the herein disclosed methods . computation means 3 may be an electrical interface with probe 2 or the operator can manually insert the probe parameters . this could be automatically accomplished by electrical identifiers on each probe . this computation means can be a microprocessor or discrete logic circuits . a display 4 is connected to the computation means 3 whereby the resistivity values may be discerned by an operator . it is worthy to note that probe 2 will vary in size depending upon the depth over which it is desired to integrate the resistivity measurement . it is anticipated that a field device of comprehensive scope will have several probes available . finally , an operator input interface 5 such as a keypad or control panel allows the operator to signal the computation means 3 whether the probe is in air or on the test material , the probe selected if not determined automatically by probe connectors , and information about the test material . in addition to performing the required calculations and algorithms , computation means 3 will also command the means for measuring impedance 1 to operate at the appropriate frequencies . in the preferred embodiment the device of fig4 will have a power supply 6 that will be battery powered . the device should also be lightweight to allow field testing such as would be needed on a flight line . the means for measuring impedance 1 of fig4 must provide capability of measurement over a wider range of frequencies and be able to measure lower impedance values than eddy current systems currently in the art . this capability may be obtained by various electronic methods . one such method is illustrated in fig5 wherein a block diagram of an electrical circuit capable of the frequencies and sensitivities required , is disclosed . it is considered to be the preferred embodiment to use a hetrodyne technique such as illustrated in fig5 in order to provide the required accuracy in phase detection over a wide range of frequencies . this hetrodyne technique is generally not used in currently available eddy current systems although laboratory instruments are known to hetrodyne . continuing with fig5 a frequency source 20 provides a variable frequency as required . this frequency is higher than the desired eddy current test frequency by a fixed amount related to a chosen intermediate frequency ( if ) which is 100 mhz in this embodiment . 21 provides this 100 mhz if frequency , which is used to down convert the output of frequency source 20 by means of mixer 22 and filter 23 which is a low pass device . the output of low pass filter 23 is fed to an appropriate probe 2 via an interconnecting line 24 . a current sensing circuit 25 monitors current flow in line 24 and outputs a proportional voltage to a second mixer 26 . mixer 26 hetrodynes the signal from current sensing device 25 with the frequency source on line 27 which arrives from 20 . the output of mixer 26 is passed through a 100 mhz if filter 28 where it is fed via line 29 to amplifier 30 . filter 28 also outputs the signal on line 29 to a clipping amplifier 31 . it is important to note that the signal fed from filter 28 to amplifiers 30 and 31 is a signal at the chosen if frequency with an amplitude proportional to the current into probe 2 and a phase ( at 100 mhz ) in fixed relation with the phase of the current into probe 2 . linear amplifier 30 outputs a signal to a magnitude detector 32 which in turn provides a dc signal on line 33 proportional to the magnitude of current fed to probe 2 . the output of amplifier 31 provides a reference signal 34 corresponding to the phase of the signal fed to probe 2 . signal 34 is a reference input to mixer 35 discussed hereinbelow . signal 34 also is fed to a 90 ° phase shifting circuit 36 where it is phase shifted and outputted to mixer 37 as a reference . amplifier 38 senses the voltage across probe 2 and outputs a signal proportional to that voltage to mixer 39 wherein it is mixed with the signal from line 27 . output of mixer 39 is passed through band pass filter 40 where the output is a 100 mhz signal on line 41 whose amplitude is proportional to the amplitude of the voltage across probe 2 with a phase proportional to the phase of that voltage . this signal on line 41 is received and amplified by linear amplifier 42 where it is outputted to mixers 35 and 37 . mixer 35 outputs a signal through low pass filter 43 which is a signal proportional to the in - phase component of the voltage across the probe and outputted on line 44 . it should be noted that the signal on line 44 is proportional to the component of the voltage across probe 2 that is in phase with probe current . this signal of line 44 is fed to an analog divider device 45 wherein the voltage from line 44 is divided by the voltage from line 33 giving the resultant output which is the real component of the impedance of probe 2 . similarly , the output of mixer 37 is filtered in low pass filter 46 which becomes an input to analog divider 48 on line 47 . this produces an output from divider 48 which is the imaginary component of the impedance of probe 2 . therefore , this basic device is capable of measuring all impedance values necessary to perform the methods resulting in a value equal to the actual resistivity of a material including those constructed from cfrc . modifications and variations of the present methods and device are possible in light of the above teachings and within the purview of the appended claims without departing from the spirit and scope of the invention .