Interaction of Rectangular Open-Ended ... - Semantic Scholar
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 6, DECEMBER 2006
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Interaction of Rectangular Open-Ended Waveguides With Surface Tilted Long Cracks in Metals Farhad Mazlumi, Seyed H. Hesamedin Sadeghi, Senior Member, IEEE, and Rouzbeh Moini, Senior Member, IEEE
Abstract—This paper presents a modeling technique for output-signal prediction of a waveguide probe when scanning a tilted long crack. The modeling technique establishes a cavity by placing a hypothetical conductive plane within the waveguide at some distance away from the probe–crack mouth. The position of the plane is iteratively changed until the appropriate resonance frequency of the cavity equals the operating frequency. In this case, the plane lies at one of the dominant-mode standing-wave nulls inside the waveguide from which the reflection coefficient, and hence, the probe output signal are determined. The main feature of the model is its ability to solve a three-dimensional problem in a two-dimensional framework, thus reducing the degree of complexity and computation time. Several simulation results are presented to evaluate the accuracy of the model at two operating frequencies in the X and K bands. The results are compared with the experimental results and those obtained using a commercial finite-element code, demonstrating the accuracy of the modeling technique. Index Terms—Metal, microwave (MNDT), tilted crack, waveguide.
nondestructive
testing
I. I NTRODUCTION
C
YCLIC LOADING on metallic structures leads to the initiation and growth of fatigue cracks on the metal surface. Undetected active fatigue cracks in structural elements may result in unstable fracture and catastrophic final failure of the structure. Currently, there are several electromagnetic nondestructive testing (NDT) techniques for detecting surface cracks in metals, each of which possesses certain limitations and disadvantages. These are the potential-drop technique [1], the eddy-current technique [2], and the surface magnetic-field measurement technique [3]. In the last decade, a great deal of attention has been given to the microwave NDT (MNDT) of materials [4]. Yeh and Zoughi [5] demonstrated the potential use of MNDT for detection and sizing of surface-breaking cracks in metals. In their work, the metal surface is scanned by an open-ended waveguide, while its standing-wave characteristics are monitored using a slotted guide and a diode detector. The crack detection and sizing in this technique are done by analyzing the overall-reflection coefficient of the incident electric field at different crack positions beneath the open-ended waveguide aperture. Manuscript received February 14, 2006; revised August 9, 2006. This work was supported by Iran Telecommunication Research Center. F. Mazlumi is with the Civil Aviation Technology College, Tehran 13878, Iran. S. H. H. Sadeghi and R. Moini are with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran 15914, Iran (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2006.884282
Fig. 1. Open-ended waveguide interrogating a long crack. (a) Plan view. (b) Side view.
To model the field-crack interaction described above, Zoughi and coworkers [5]–[7] assume that the crack can be approximated by a rectangular notch and, hence, is modeled as a short-circuited rectangular waveguide. Mazlumi et al. [8], [9] improved the modeling by solving the problem for arbitraryshape cracks. They used a discretization algorithm to model the crack as a series of rectangular waveguides. Park et al. [10] used a combination of the Fourier transform and the mode-matching technique to analyze the long cracks with arbitrary shape in width. The modeling technique involves complex integrals and a large number of modes for complex shapes. Real cracks may run at the angles other than perpendicular to the metal surface [11], [12]. In particular, fatigue cracks created in railroads are often inclined [12]. This paper aims to present a modeling technique for output-signal prediction of a waveguide probe when scanning tilted long cracks. The structure of this paper is organized as follows. In Section II, the interaction of a waveguide probe and a tilted long crack is modeled. The model poses a two-dimensional (2-D) problem where a cavity-resonance technique combined with the finite-difference method is used to determine the probe output signal. In Section III, the proposed model is validated by comparing several simulation results with those obtained in experiments and by a commercial finite-element code. II. M ODELING An open-ended rectangular-waveguide probe with dimensions a × b, interrogating a long crack in metal, is depicted in Fig. 1. The crack has an opening width w and a uniform tilted shape along its length L; lying parallel to the broad dimension of the waveguide probe a. It is assumed that the crack length is much larger than the broad dimension of the waveguide probe (i.e., L a). Also, the waveguide probe is assumed to be in contact with the metal surface, while its flange is so large
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Fig. 2. Two-dimensional E-plane structure cavity.
Assuming that the standing-wave null positions are known, one can create a cavity by placing a hypothetical plane at a null position without changing the crack-probe boundary conditions. The cavity is surrounded with the metal surface, the crack, the probe walls, and the hypothetical plane (Fig. 2). Using the uniqueness theorem, it can be shown that one of the field solutions of this cavity belongs to the probe–crack problem posed above [13]. In other words, the correct position of the plane requires that the operating frequency of the probe equals one of the infinite numbers of resonance frequencies associated with the created cavity. Assuming that the hypothetical plane is located at the nth null position, the desired resonant frequency corresponds to the mode TE10n . A. Proposed Algorithm
Fig. 3. Dominant-mode standing-wave pattern along the waveguide probe.
that the incident wave does not leak out of the probe–crack structure. Using these assumptions, it can be shown that the crack length can be taken as equal to the broad dimension of the probe [5]. As a result, the problem is reduced to an E-plane waveguide discontinuity that is essentially 2-D (Fig. 2). The operating frequency fo is such that the dominant mode of the waveguide probe TE1,0 is the only propagating mode in the probe. This mode propagates and carries the power toward the probe aperture, where the metal under test reflects all the power back to the probe. In other words, the probe–metal interface can be approximated by a load with the dominantmode reflection coefficient Γ whose modulus is unity, i.e., |Γ| = 1. Hence, any variations in the crack shape or position will only change the phase of Γ. Since the modulus of the dominant-mode reflection coefficient at the probe aperture is unity, the corresponding incident and reflected waves produce a pure standing wave in the waveguide probe. This implies that the dominant-mode standing wave along the waveguide (Fig. 3) has periodic null positions zi , where the resultant electric-field intensity is zero, i.e., (1) |Ey (zi )|TE1,0 = 1 + Γe−j2β|zi | = 0. Here, β is the propagation constant associated with the dominant mode, and the index i (= 1, 2, . . .) denotes the number of the null position starting the enumeration from the probe aperture. However, the total electric field may have a nonzero value at the null positions located in the vicinity of the probe–crack interface due to the considerable number of excited higher order evanescent modes. It is noted that the total contributions due to these evanescent modes tends to disappear at some distance away from the probe aperture. The waveguide probe scans the metal surface along the smaller dimension of the waveguide probe (i.e., along the yaxis). In the absence of a crack at the probe aperture, the metal surface is considered as a short-circuit for which Γ = −1. However, when the crack moves into the probe aperture, the reflection coefficient at the probe aperture changes. This makes the dominant-mode standing-wave shift along the waveguide probe, changing the level of the signal sensed by the diode detector.
The resonance frequency of the cavity with the mode TE10n , fTE10n , and the hypothetical-plane location ρ are related to each other as follows: fTE10n = F (ρ)
(2)
where F is a monotonically decreasing function (see the Appendix). The problem of finding the correct value of ρ is thus reduced to solving for the crossing point of F and fTE10n = fo . The solution scheme uses an iterative algorithm, based on the Newton–Raphson method, and takes an initial estimation of the hypothetical-plane location ρ0 , the maximum allowable frequency error δf , the initial size of the segments ∆z 0 , and the corresponding null-position number n, to be fixed throughout the calculations. In each iteration, the operating frequency of the probe is estimated and compared with the actual operating frequency. Prior to this stage, the resonant frequencies of the cavity are to be computed. The resonant frequencies associated with this cavity are determined using a finite-difference-based method appropriate for a 2-D cavity. The method is described in detail in [14] and is not repeated here. The method discretizes the cavity into appropriate number of rectangular cells with dimensions ∆y × ∆z, where ∆y and ∆z are their dimensions along the y-axis and z-axis, respectively [14]. In each iteration, ∆z is changed such that the obtained ρ is an integer multiple of ∆z, i.e., ∆z i =
ρi i . ρ round ∆z 0
(3)
Among several resonant frequencies of the cavity, the one associated with the mode TE10n , fTE10n is equal to the operating frequency. To find fTE10n , we need to compute the resonance frequencies associated with another cavity created in the absence of the crack. The new cavity is a a × b × ρ cube surrounded with the metal surface, the probe walls, and the hypothetical plane (Fig. 2). The resonance frequencies of the cubic cavity are readily computed using analytical expressions [15]. Since the crack depth is smaller than the half of the guide wavelength, it can be shown that the resonant frequency no−crack associated with the mode TE10n in the cubic cavity fTE 10n is close to fTE10n [15]. In fact, if we sort the resonant
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Following the calculation of the optimum hypothetical location ρopt , the phase of the reflection coefficient Γ can be calculated as follows: Γ = π + 2βρopt .
(6)
The detection of the crack is done by monitoring the reflection coefficient of the dominant mode in the waveguide probe. This is done by measuring the squared magnitude of the electric field |Ey |2 using a diode detector placed in the waveguide probe at a distance far from the probe aperture [5]. B. Numerical Considerations The initial estimation of the hypothetical-plane location ρ0 and the null position number n are two important parameters as they affect the performance of the solution technique, including speed, accuracy, and convergence rate. Departure from the correct location of the hypothetical plane in the beginning of the solution algorithm results in more iterations, or slower convergence rate. To speed up the convergence rate, one may set the value of ρ0 to the nth null position associated with no-crack situation, i.e., ρ0 = nπ/β, assuming that the hypothetical plane lies at the nth null position. The selection of an appropriate value for n is a tradeoff between the solution accuracy and the computation speed. In fact, the farther the hypothetical plane lies from the probe aperture, fewer contributions of the evanescent modes exist at the plane location, and hence, more accurate results are obtained. On the other hand, the resultant cavity becomes more spacious, and for a given accuracy, a larger number of nodes in the discretization procedure of the finite different method are to be used in the calculation of the cavity-resonant frequencies. Fig. 4. Flowchart of the iterative algorithm to find the optimum hypotheticalplane location.
frequencies obtained for both cavities in two separate columns no−crack and fTE10n , in an ascending order, the positions of fTE 10n in their corresponding columns, will be the same. Thus, a no−crack will lead us to select the correct value knowledge of fTE 10n of fTE10n from many available resonant frequencies. Having obtained the estimated value of the operating frei , the estimated value of the quency in iteration i fTE 10n hypothetical-plane location in the next iteration ρi+1 is obtained i ) by numerically approximating the tangent of F at (ρi , fTE 10n ρi+1 = ρi + ∆ρ
(4)
where ∆ρ =
ρi − ρi−1 i fo − fTE . 10n i−1 i fTE10n − fTE10n
(5)
The iterations are continued until the difference between the operating frequency and its estimation is less than δf , i.e., |fo − fTE10n | < δf . A flowchart of the algorithm is shown in Fig. 4.
III. R ESULTS AND M ODEL V ERIFICATION Based on the model described in the previous sections, a computer code was developed that can predict the sensor output for scanning tilted long surface cracks. For each crack location beneath the probe aperture, the computer code runs an iterative algorithm to obtain the sensor output signal. It is noted that the computer code discretizes the probe–crack geometry to an appropriate number of square cells, i.e., ∆y = ∆z = ∆. For verification of the simulation results for cases, where the probe-operating frequency lies within the X band, the experimental setup shown in Fig. 5 was used. In this setup, the probe consists of a rectangular waveguide whose flanges are in close contact with the metal surface. An X-band signal generator excites the probe, and a slotted guide with a diode detector is used to produce the probe output signal. The probe is attached to a computer-controlled xy scanner. A personal computer is responsible for controlling the movement of the probe, as well as the data collection and data transfer to the memory for further processing. To examine the accuracy of the model, various tests were carried out. For brevity, we only present results associated with the X and K frequency bands. The specifications of the probe at each band, including the dimensions of the probe, the operating
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Fig. 7.
Tilted milled slot.
Fig. 5. X-band experimental setup. TABLE I PROBE SPECIFICATIONS
Fig. 8. Normalized output signal of the X-band probe when scanning a slot ) the with L = 29.8 mm and the cross section shown in Fig. 7; ( proposed method, ( ) the measurement results.
In the second simulation, the X-band probe scans a tilted long crack whose cross-sectional data are shown in Fig. 7. With reference to Fig. 1, other dimensions of the crack are specified as follows: L = 29.8 mm; w = 4.2 mm; and θ = 45◦ . The theoretical and experimental results are depicted in Fig. 8. A comparison of the results in Fig. 8 further substantiates the accuracy of the theoretical modeling. B. K-Band Results Fig. 6. Normalized output signal of the X-band probe when scanning a slot with w = 3.2 mm, d = 4.5 mm, L = 32.4 mm, and θ = 0◦ ; ( ) Proposed method, ( ) the measurement results.
frequency, and the detector distance from the probe aperture l are given in Table I. These parameters ensure that at each frequency band, the only propagating mode within the probe will be TE1,0 , and no evanescent modes will appear at the sensor location. A. X-Band Results We present tests results associated with the X-band probe when scanning two long slots along the y-axis. In both cases, it is assumed that ∆z 0 = 0.05 mm, δf = 0.005fo , and the hypothetical plane is placed at z2 . In the first test, the crack (Fig. 1) is assumed to be perpendicular to the metal and lies along the x-axis. With reference to Fig. 1, the crack specifications: w = 3.2 mm; d = 4.5 mm; L = 32.4 mm; and θ = 0. The variations of the normalized-sensor output signal is depicted in Fig. 6. The normalization factor is the maximum value of the sensor output signal. The simulation results are accompanied by their experimental counterparts, confirming the accuracy of the theoretical modeling.
To further examine the accuracy of the model, we present the simulation results associated with the K-band probe (Table I) when scanning several long cracks along the y-axis. In all simulations presented here, it is assumed that ∆z 0 = 2.0 × 10−5 mm, δf = 0.005fo , and the hypothetical plane is placed at z2 . In the first simulation, the crack is assumed to be perpendicular to the metal surface and lies along the x-axis. With reference to Fig. 1, the crack is specified with w = 0.84 mm, d = 1.53 mm, and θ = 0. Fig. 9 shows the normalized output signal of the sensor for various values of δ. The measurement results [5] are also plotted in Fig. 9. A comparison of the theoretical and experimental results in Fig. 9 demonstrates the accuracy of the proposed model. In the next simulation, the case of a tilted crack is examined. With reference to Fig. 1, the crack specifications are θ = 45◦ , w = 0.84 mm, and d = 1.53 mm. The normalized sensor output is plotted in Fig. 10 and compared with the results obtained using the well-known HP–high-frequency structure-simulator (HP-HFSS) finite-element code [16]. A comparison of the results shown in Fig. 10 substantiates the accuracy of the proposed model. Generally, the computation times in both methods depend on the probe specifications, crack dimension, and
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Fig. 9. Normalized output signal of the K-band probe when scanning a long crack with w = 0.84 mm, d = 1.53 mm, and θ = 0◦ ; ( ) the proposed method, ( ) the measurement results [5].
Fig. 11. Normalized output signal of the K-band probe when scanning long cracks with w = 0.84 mm, d = 1.53, and different tilt angles; ( )θ= 0◦ ,( ) θ = 26.6◦ , ( ) θ = 45◦ , and ( ) θ = 63.4◦ .
Fig. 10. Normalized output signal of the K-band probe when scanning a tilted long crack with θ = 45◦ , w = 0.84 mm, and d = 1.53 mm; ( ) the proposed method, ( ) the HFSS result [16].
crack openings (Fig. 1) when entering and exiting the crack. In the case of a right angle crack (i.e., θ = 0◦ ), the two crack openings are identical, and hence, the probe output signal will be symmetrical. In addition, there is a direct relationship between the crack–tilt angle and the sharpness of the signal as the probe passes over its edges. In fact, the output signal has a faster variation when the sharper lip of the crack enters the probe aperture. It is worth noting that for a given crack depth, an increase in the tilt angle will increase the crack effective depth as shown in Fig. 1(b) (i.e., d = d/ cos(θ)). This is reflected in crack signals shown in Fig. 11, where for small tilt angles (say θ < 26◦ ), little changes are observed in crack signals as cos(θ) is almost constant. On the contrary, for larger tilt angles (say θ > 45◦ ), crack signals show remarkable variations. To study the effect of the evanescent modes on the accuracy of the results, the probe output signal associated with a particular crack (d = 1.53 mm, w = 0.84 mm, and θ = 0) was simulated for three distinct hypothetical-plane locations, starting from the closest null position to the metal surface (i.e., z1 , z2 , and z3 in Fig. 3). Considering the fact that the evanescent modes have the least contributions at the farthest null position, the result associated with z3 is taken as the reference, and the deviations of the results associated with z1 and z2 from the reference are plotted in Fig. 12. From this figure, it is observed that the results associated with the three hypothetical locations are very close together except for the case that the hypothetical plane lies at z1 , and the crack is moving outside the probe aperture. In other words, the contribution of the evanescent modes in the vicinity of the probe aperture is rapidly damped, and hence, a low value of null position number n should provide accurate results without requiring a long computation time.
crack-probe position. Nevertheless, for a given accuracy, the proposed method is found to be always faster than the HP-HFSS code. This is due to the fact that commercial finite-element codes should solve a three-dimensional problem, whereas the proposed method utilizes the finite difference method to solve a 2-D problem (i.e., the calculation of resonance frequency of a 2-D cavity). To study the effect of crack geometry on the probe output signal, we present the results associated with three tilted cracks with θ = tan−1 (0.5) = 26.6◦ , θ = tan−1 (1) = 45◦ , and θ = tan−1 (2) = 63.4◦ . The width and depth of these cracks are w = 0.84 mm and d = 1.53 mm, respectively. Fig. 11 shows the computer simulations of normalized-sensor output signal for scanning the three aforementioned tilted cracks together with the perpendicular crack. As shown in Fig. 11, by decreasing the crack–tilt angle (θ), the crack signals tend to approach each other, and finally, reach the asymptotic signal associated with the perpendicular crack. These numerical results further substantiate the consistency of the proposed model. Further examination of the results in Fig. 11 reveals that the symmetry existing in the signal of the perpendicular crack tends to disappear as the crack inclines. This is due to the fact that the probe aperture generally observes two different
IV. C ONCLUSION A theoretical model was presented to describe the interaction of an open-ended waveguide probe and a tilted crack. The modeling technique, which is based on the cavity-resonance approach, hypothesizes a cavity by placing a conductive plane at one of the null positions inside the waveguide. To determine the correct position of the hypothetical plane, an iterative algorithm was used to ensure the equality of the cavity-resonant
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Using the Foster’s reactance theorem, one can conclude that (∂X/∂ω) > 0 [17]. Also, it can be shown that ∂β/∂ω is always positive [18]. These features lead to the fact that the right-hand side of (9) is always negative, i.e., (∂ρ/∂ω) < 0. R EFERENCES
Fig. 12. Deviation from the actual results when the first ( ) or the second ( ) null number is taken as the position of the hypothetical plane shown in Fig. 2.
frequency and the probe-operating frequency. The modeling was tested, both experimentally and numerically, at two operating frequencies in the X and K bands. The numerical verifications were achieved by comparing the results of the present modeling at the K band for a tilted crack with those obtained using the well-known HP-HFSS finite-element code. It was found that although the results obtained from the two methods are consistent, the computation time required in the finite-element method is about two times more than that required in the present method. Experimental confirmation of the theoretical modeling was observed by sampling the surface of several test blocks, containing perpendicular and tilted slots. For this purpose, a waveguide probe operating at the X band was used. It was shown that the output signals from the probes have asymmetrical behaviors when crossing the tilted crack. Also, by decreasing the crack–tilt angle, the crack signals tend to approach each other, and finally, reach the perpendicular crack signal. A PPENDIX From (1), it is inferred that if the hypothetical plane lies at one of the null positions, the dominant-mode reflection coefficient at the probe aperture is expressed as follows: Γ = −ej2βρ
(7)
where ρ > 0 and β > 0. Since the probe–crack structure constructs a lossless one port, the reflection coefficient can also be expressed as follows [17]: Γ=
jX − 1 jX + 1
(8)
where X is the one-port reactance normalized to the impedance of the dominant mode. By differentiating (7), with respect to the angular frequency ω and using (8), one can reach the following expression: ∂ρ 1 1 ∂X ∂β =− + ρ . (9) ∂ω β ∂ω 1 + X 2 ∂ω
[1] M. D. Halliday and C. J. Beevers, “The dc electrical method for crack length measurement,” in The Measurement of Crack Length and Shape During Fracture and Fatigue, C. J. Beevers, Ed. Warley, U.K.: Eng. Mater. Advisory Service Ltd., 1980, pp. 85–112. [2] C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys., vol. 39, no. 6, pp. 2829–2838, May 1968. [3] D. Mirshekar-Syahkal and S. H. H. Sadeghi, “Surface magnetic field measurement technique for nondestructive testing of metals,” Electron. Lett., vol. 30, no. 3, pp. 210–211, Feb. 1994. [4] C. Huber and R. Zoughi, “Detecting stress and fatigue cracks,” IEEE Potentials, vol. 15, no. 4, pp. 20–24, Oct./Nov. 1996. [5] C. Yeh and R. Zoughi, “A novel microwave method for detection of long surface cracks in metals,” IEEE Trans. Instrum. Meas., vol. 43, no. 5, pp. 719–725, Oct. 1994. [6] C. Huber, H. Abiri, S. I. Ganchev, and R. Zoughi, “Modeling of surface hairline-crack detection in metals under coatings using an open-ended rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2049–2057, Nov. 1997. [7] ——, “Analysis of the crack characteristic signal using a generalized scattering matrix representation,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 4, pp. 477–484, Apr. 1997. [8] F. Mazlumi, S. H. H. Sadeghi, and R. Moini, “Detection of surface cracks of arbitrary shape in metals using an open-ended waveguide probe,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 21, D. O. Thompson and D. E. Chimenti, Eds. College Park, MD: Amer. Inst. Phys., 2002, pp. 491–497. [9] ——, “Analysis technique for interaction of rectangular open-ended waveguides with surface cracks of arbitrary shape in metals,” NDT E Int., vol. 36, no. 5, pp. 331–338, Jul. 2003. [10] H. H. Park, Y. H. Cho, and H. J. Eom, “Surface crack detection using flanged parallel-plate waveguide,” Electron. Lett., vol. 37, no. 25, pp. 1526–1527, Dec. 2001. [11] V. M. Babich, V. A. Borovikov, L. J. Fradkin, V. Kamotski, and B. A. Samokish, “Scatter of the Rayleigh waves by tilted surface-breaking cracks,” NDT E Int., vol. 37, no. 2, pp. 105–109, Mar. 2004. [12] Y. C. Li, “Analysis of fatigue phenomena in railway rails and wheels,” in Handbook of Fatigue Crack Propagation in Metallic Structures, vol. 2, A. Carpinteri, Ed. Amsterdam, The Netherlands: Elsevier, 1994, pp. 1497–1537. [13] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Hoboken, NJ: Wiley, 2001, ch. 3. [14] M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. Boca Raton, FL: CRC, 2001, ch. 3. [15] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 7. [16] HP High Frequency Structure Simulator 5.4, User’s Guide, HewlettPackard Co., Palo Alto, CA, 1999. HP Part No. 85180-90166. [17] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 4. [18] ——, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 3.
Farhad Mazlumi was born in Tehran, Iran, in 1974. He received the B.S. and M.S. degrees from Sharif University of Technology, Tehran, in 1997 and 1999, respectively, and the Ph.D. degree in electrical engineering from Amirkabir University of Technology, Tehran, in 2005. In the summer of 2005, he worked at Bilkent University, Ankara, Turkey, as a Research Assistant, working on integral equations for bodies with impedance surface. He is currently an Assistant Professor with the Civil Aviation Technology College, Tehran. His research interests include computational methods for electromagnetics and microwave-passive devices.
MAZLUMI et al.: INTERACTION OF RECTANGULAR OPEN-ENDED WAVEGUIDES WITH TILTED CRACKS IN METALS
Seyed H. Hesamedin Sadeghi (M’92–SM’05) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1984, and the Ph.D. degree from the University of Essex, Colchester, U.K., in 1991. Between 1980 and 1983, he was with the electrical power industry in Iran. In 1984, he was a Research Assistant at the University of Lancaster, Lancaster, U.K. He then joined the University of Essex as a Senior Research Officer. In 1992, he was appointed as a Research Assistant Professor with the Vanderbilt University, Nashville, TN. In 1996–1997 and 2005–2006, he was a Visiting Professor at the University of Wisconsin–Milwaukee. He is currently a Professor of electrical engineering with the Amirkabir University of Technology, Tehran. His current research interests include electromagnetic nondestructive evaluation of materials and electromagnetic compatibility issues in power engineering.
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Rouzbeh Moini (M’93–SM’05) was born in Tehran, Iran, in 1963. He received the B.S., M.S., and Ph.D. degrees in electronics from Limoges University, Limoges, France, in 1984, 1985, and 1988, respectively. In 1988, he joined the Electrical Engineering Department, Amirkabir University of Technology, Tehran, where he is currently a Professor of telecommunications. From 1995 to 1996, he was a Visiting Professor at the University of Florida, Gainesville. His main research interests are in numerical methods in electromagnetics, electromagnetic compatibility, and antenna theory. Dr. Moini was the recipient of the 1995 Islamic Development Bank Merit Scholarship Award.
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Interaction of Rectangular Open-Ended Waveguides With Surface Tilted Long Cracks in Metals Farhad Mazlumi, Seyed H. Hesamedin Sadeghi, Senior Member, IEEE, and Rouzbeh Moini, Senior Member, IEEE
Abstract—This paper presents a modeling technique for output-signal prediction of a waveguide probe when scanning a tilted long crack. The modeling technique establishes a cavity by placing a hypothetical conductive plane within the waveguide at some distance away from the probe–crack mouth. The position of the plane is iteratively changed until the appropriate resonance frequency of the cavity equals the operating frequency. In this case, the plane lies at one of the dominant-mode standing-wave nulls inside the waveguide from which the reflection coefficient, and hence, the probe output signal are determined. The main feature of the model is its ability to solve a three-dimensional problem in a two-dimensional framework, thus reducing the degree of complexity and computation time. Several simulation results are presented to evaluate the accuracy of the model at two operating frequencies in the X and K bands. The results are compared with the experimental results and those obtained using a commercial finite-element code, demonstrating the accuracy of the modeling technique. Index Terms—Metal, microwave (MNDT), tilted crack, waveguide.
nondestructive
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I. I NTRODUCTION
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YCLIC LOADING on metallic structures leads to the initiation and growth of fatigue cracks on the metal surface. Undetected active fatigue cracks in structural elements may result in unstable fracture and catastrophic final failure of the structure. Currently, there are several electromagnetic nondestructive testing (NDT) techniques for detecting surface cracks in metals, each of which possesses certain limitations and disadvantages. These are the potential-drop technique [1], the eddy-current technique [2], and the surface magnetic-field measurement technique [3]. In the last decade, a great deal of attention has been given to the microwave NDT (MNDT) of materials [4]. Yeh and Zoughi [5] demonstrated the potential use of MNDT for detection and sizing of surface-breaking cracks in metals. In their work, the metal surface is scanned by an open-ended waveguide, while its standing-wave characteristics are monitored using a slotted guide and a diode detector. The crack detection and sizing in this technique are done by analyzing the overall-reflection coefficient of the incident electric field at different crack positions beneath the open-ended waveguide aperture. Manuscript received February 14, 2006; revised August 9, 2006. This work was supported by Iran Telecommunication Research Center. F. Mazlumi is with the Civil Aviation Technology College, Tehran 13878, Iran. S. H. H. Sadeghi and R. Moini are with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran 15914, Iran (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2006.884282
Fig. 1. Open-ended waveguide interrogating a long crack. (a) Plan view. (b) Side view.
To model the field-crack interaction described above, Zoughi and coworkers [5]–[7] assume that the crack can be approximated by a rectangular notch and, hence, is modeled as a short-circuited rectangular waveguide. Mazlumi et al. [8], [9] improved the modeling by solving the problem for arbitraryshape cracks. They used a discretization algorithm to model the crack as a series of rectangular waveguides. Park et al. [10] used a combination of the Fourier transform and the mode-matching technique to analyze the long cracks with arbitrary shape in width. The modeling technique involves complex integrals and a large number of modes for complex shapes. Real cracks may run at the angles other than perpendicular to the metal surface [11], [12]. In particular, fatigue cracks created in railroads are often inclined [12]. This paper aims to present a modeling technique for output-signal prediction of a waveguide probe when scanning tilted long cracks. The structure of this paper is organized as follows. In Section II, the interaction of a waveguide probe and a tilted long crack is modeled. The model poses a two-dimensional (2-D) problem where a cavity-resonance technique combined with the finite-difference method is used to determine the probe output signal. In Section III, the proposed model is validated by comparing several simulation results with those obtained in experiments and by a commercial finite-element code. II. M ODELING An open-ended rectangular-waveguide probe with dimensions a × b, interrogating a long crack in metal, is depicted in Fig. 1. The crack has an opening width w and a uniform tilted shape along its length L; lying parallel to the broad dimension of the waveguide probe a. It is assumed that the crack length is much larger than the broad dimension of the waveguide probe (i.e., L a). Also, the waveguide probe is assumed to be in contact with the metal surface, while its flange is so large
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Fig. 2. Two-dimensional E-plane structure cavity.
Assuming that the standing-wave null positions are known, one can create a cavity by placing a hypothetical plane at a null position without changing the crack-probe boundary conditions. The cavity is surrounded with the metal surface, the crack, the probe walls, and the hypothetical plane (Fig. 2). Using the uniqueness theorem, it can be shown that one of the field solutions of this cavity belongs to the probe–crack problem posed above [13]. In other words, the correct position of the plane requires that the operating frequency of the probe equals one of the infinite numbers of resonance frequencies associated with the created cavity. Assuming that the hypothetical plane is located at the nth null position, the desired resonant frequency corresponds to the mode TE10n . A. Proposed Algorithm
Fig. 3. Dominant-mode standing-wave pattern along the waveguide probe.
that the incident wave does not leak out of the probe–crack structure. Using these assumptions, it can be shown that the crack length can be taken as equal to the broad dimension of the probe [5]. As a result, the problem is reduced to an E-plane waveguide discontinuity that is essentially 2-D (Fig. 2). The operating frequency fo is such that the dominant mode of the waveguide probe TE1,0 is the only propagating mode in the probe. This mode propagates and carries the power toward the probe aperture, where the metal under test reflects all the power back to the probe. In other words, the probe–metal interface can be approximated by a load with the dominantmode reflection coefficient Γ whose modulus is unity, i.e., |Γ| = 1. Hence, any variations in the crack shape or position will only change the phase of Γ. Since the modulus of the dominant-mode reflection coefficient at the probe aperture is unity, the corresponding incident and reflected waves produce a pure standing wave in the waveguide probe. This implies that the dominant-mode standing wave along the waveguide (Fig. 3) has periodic null positions zi , where the resultant electric-field intensity is zero, i.e., (1) |Ey (zi )|TE1,0 = 1 + Γe−j2β|zi | = 0. Here, β is the propagation constant associated with the dominant mode, and the index i (= 1, 2, . . .) denotes the number of the null position starting the enumeration from the probe aperture. However, the total electric field may have a nonzero value at the null positions located in the vicinity of the probe–crack interface due to the considerable number of excited higher order evanescent modes. It is noted that the total contributions due to these evanescent modes tends to disappear at some distance away from the probe aperture. The waveguide probe scans the metal surface along the smaller dimension of the waveguide probe (i.e., along the yaxis). In the absence of a crack at the probe aperture, the metal surface is considered as a short-circuit for which Γ = −1. However, when the crack moves into the probe aperture, the reflection coefficient at the probe aperture changes. This makes the dominant-mode standing-wave shift along the waveguide probe, changing the level of the signal sensed by the diode detector.
The resonance frequency of the cavity with the mode TE10n , fTE10n , and the hypothetical-plane location ρ are related to each other as follows: fTE10n = F (ρ)
(2)
where F is a monotonically decreasing function (see the Appendix). The problem of finding the correct value of ρ is thus reduced to solving for the crossing point of F and fTE10n = fo . The solution scheme uses an iterative algorithm, based on the Newton–Raphson method, and takes an initial estimation of the hypothetical-plane location ρ0 , the maximum allowable frequency error δf , the initial size of the segments ∆z 0 , and the corresponding null-position number n, to be fixed throughout the calculations. In each iteration, the operating frequency of the probe is estimated and compared with the actual operating frequency. Prior to this stage, the resonant frequencies of the cavity are to be computed. The resonant frequencies associated with this cavity are determined using a finite-difference-based method appropriate for a 2-D cavity. The method is described in detail in [14] and is not repeated here. The method discretizes the cavity into appropriate number of rectangular cells with dimensions ∆y × ∆z, where ∆y and ∆z are their dimensions along the y-axis and z-axis, respectively [14]. In each iteration, ∆z is changed such that the obtained ρ is an integer multiple of ∆z, i.e., ∆z i =
ρi i . ρ round ∆z 0
(3)
Among several resonant frequencies of the cavity, the one associated with the mode TE10n , fTE10n is equal to the operating frequency. To find fTE10n , we need to compute the resonance frequencies associated with another cavity created in the absence of the crack. The new cavity is a a × b × ρ cube surrounded with the metal surface, the probe walls, and the hypothetical plane (Fig. 2). The resonance frequencies of the cubic cavity are readily computed using analytical expressions [15]. Since the crack depth is smaller than the half of the guide wavelength, it can be shown that the resonant frequency no−crack associated with the mode TE10n in the cubic cavity fTE 10n is close to fTE10n [15]. In fact, if we sort the resonant
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Following the calculation of the optimum hypothetical location ρopt , the phase of the reflection coefficient Γ can be calculated as follows: Γ = π + 2βρopt .
(6)
The detection of the crack is done by monitoring the reflection coefficient of the dominant mode in the waveguide probe. This is done by measuring the squared magnitude of the electric field |Ey |2 using a diode detector placed in the waveguide probe at a distance far from the probe aperture [5]. B. Numerical Considerations The initial estimation of the hypothetical-plane location ρ0 and the null position number n are two important parameters as they affect the performance of the solution technique, including speed, accuracy, and convergence rate. Departure from the correct location of the hypothetical plane in the beginning of the solution algorithm results in more iterations, or slower convergence rate. To speed up the convergence rate, one may set the value of ρ0 to the nth null position associated with no-crack situation, i.e., ρ0 = nπ/β, assuming that the hypothetical plane lies at the nth null position. The selection of an appropriate value for n is a tradeoff between the solution accuracy and the computation speed. In fact, the farther the hypothetical plane lies from the probe aperture, fewer contributions of the evanescent modes exist at the plane location, and hence, more accurate results are obtained. On the other hand, the resultant cavity becomes more spacious, and for a given accuracy, a larger number of nodes in the discretization procedure of the finite different method are to be used in the calculation of the cavity-resonant frequencies. Fig. 4. Flowchart of the iterative algorithm to find the optimum hypotheticalplane location.
frequencies obtained for both cavities in two separate columns no−crack and fTE10n , in an ascending order, the positions of fTE 10n in their corresponding columns, will be the same. Thus, a no−crack will lead us to select the correct value knowledge of fTE 10n of fTE10n from many available resonant frequencies. Having obtained the estimated value of the operating frei , the estimated value of the quency in iteration i fTE 10n hypothetical-plane location in the next iteration ρi+1 is obtained i ) by numerically approximating the tangent of F at (ρi , fTE 10n ρi+1 = ρi + ∆ρ
(4)
where ∆ρ =
ρi − ρi−1 i fo − fTE . 10n i−1 i fTE10n − fTE10n
(5)
The iterations are continued until the difference between the operating frequency and its estimation is less than δf , i.e., |fo − fTE10n | < δf . A flowchart of the algorithm is shown in Fig. 4.
III. R ESULTS AND M ODEL V ERIFICATION Based on the model described in the previous sections, a computer code was developed that can predict the sensor output for scanning tilted long surface cracks. For each crack location beneath the probe aperture, the computer code runs an iterative algorithm to obtain the sensor output signal. It is noted that the computer code discretizes the probe–crack geometry to an appropriate number of square cells, i.e., ∆y = ∆z = ∆. For verification of the simulation results for cases, where the probe-operating frequency lies within the X band, the experimental setup shown in Fig. 5 was used. In this setup, the probe consists of a rectangular waveguide whose flanges are in close contact with the metal surface. An X-band signal generator excites the probe, and a slotted guide with a diode detector is used to produce the probe output signal. The probe is attached to a computer-controlled xy scanner. A personal computer is responsible for controlling the movement of the probe, as well as the data collection and data transfer to the memory for further processing. To examine the accuracy of the model, various tests were carried out. For brevity, we only present results associated with the X and K frequency bands. The specifications of the probe at each band, including the dimensions of the probe, the operating
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Fig. 7.
Tilted milled slot.
Fig. 5. X-band experimental setup. TABLE I PROBE SPECIFICATIONS
Fig. 8. Normalized output signal of the X-band probe when scanning a slot ) the with L = 29.8 mm and the cross section shown in Fig. 7; ( proposed method, ( ) the measurement results.
In the second simulation, the X-band probe scans a tilted long crack whose cross-sectional data are shown in Fig. 7. With reference to Fig. 1, other dimensions of the crack are specified as follows: L = 29.8 mm; w = 4.2 mm; and θ = 45◦ . The theoretical and experimental results are depicted in Fig. 8. A comparison of the results in Fig. 8 further substantiates the accuracy of the theoretical modeling. B. K-Band Results Fig. 6. Normalized output signal of the X-band probe when scanning a slot with w = 3.2 mm, d = 4.5 mm, L = 32.4 mm, and θ = 0◦ ; ( ) Proposed method, ( ) the measurement results.
frequency, and the detector distance from the probe aperture l are given in Table I. These parameters ensure that at each frequency band, the only propagating mode within the probe will be TE1,0 , and no evanescent modes will appear at the sensor location. A. X-Band Results We present tests results associated with the X-band probe when scanning two long slots along the y-axis. In both cases, it is assumed that ∆z 0 = 0.05 mm, δf = 0.005fo , and the hypothetical plane is placed at z2 . In the first test, the crack (Fig. 1) is assumed to be perpendicular to the metal and lies along the x-axis. With reference to Fig. 1, the crack specifications: w = 3.2 mm; d = 4.5 mm; L = 32.4 mm; and θ = 0. The variations of the normalized-sensor output signal is depicted in Fig. 6. The normalization factor is the maximum value of the sensor output signal. The simulation results are accompanied by their experimental counterparts, confirming the accuracy of the theoretical modeling.
To further examine the accuracy of the model, we present the simulation results associated with the K-band probe (Table I) when scanning several long cracks along the y-axis. In all simulations presented here, it is assumed that ∆z 0 = 2.0 × 10−5 mm, δf = 0.005fo , and the hypothetical plane is placed at z2 . In the first simulation, the crack is assumed to be perpendicular to the metal surface and lies along the x-axis. With reference to Fig. 1, the crack is specified with w = 0.84 mm, d = 1.53 mm, and θ = 0. Fig. 9 shows the normalized output signal of the sensor for various values of δ. The measurement results [5] are also plotted in Fig. 9. A comparison of the theoretical and experimental results in Fig. 9 demonstrates the accuracy of the proposed model. In the next simulation, the case of a tilted crack is examined. With reference to Fig. 1, the crack specifications are θ = 45◦ , w = 0.84 mm, and d = 1.53 mm. The normalized sensor output is plotted in Fig. 10 and compared with the results obtained using the well-known HP–high-frequency structure-simulator (HP-HFSS) finite-element code [16]. A comparison of the results shown in Fig. 10 substantiates the accuracy of the proposed model. Generally, the computation times in both methods depend on the probe specifications, crack dimension, and
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Fig. 9. Normalized output signal of the K-band probe when scanning a long crack with w = 0.84 mm, d = 1.53 mm, and θ = 0◦ ; ( ) the proposed method, ( ) the measurement results [5].
Fig. 11. Normalized output signal of the K-band probe when scanning long cracks with w = 0.84 mm, d = 1.53, and different tilt angles; ( )θ= 0◦ ,( ) θ = 26.6◦ , ( ) θ = 45◦ , and ( ) θ = 63.4◦ .
Fig. 10. Normalized output signal of the K-band probe when scanning a tilted long crack with θ = 45◦ , w = 0.84 mm, and d = 1.53 mm; ( ) the proposed method, ( ) the HFSS result [16].
crack openings (Fig. 1) when entering and exiting the crack. In the case of a right angle crack (i.e., θ = 0◦ ), the two crack openings are identical, and hence, the probe output signal will be symmetrical. In addition, there is a direct relationship between the crack–tilt angle and the sharpness of the signal as the probe passes over its edges. In fact, the output signal has a faster variation when the sharper lip of the crack enters the probe aperture. It is worth noting that for a given crack depth, an increase in the tilt angle will increase the crack effective depth as shown in Fig. 1(b) (i.e., d = d/ cos(θ)). This is reflected in crack signals shown in Fig. 11, where for small tilt angles (say θ < 26◦ ), little changes are observed in crack signals as cos(θ) is almost constant. On the contrary, for larger tilt angles (say θ > 45◦ ), crack signals show remarkable variations. To study the effect of the evanescent modes on the accuracy of the results, the probe output signal associated with a particular crack (d = 1.53 mm, w = 0.84 mm, and θ = 0) was simulated for three distinct hypothetical-plane locations, starting from the closest null position to the metal surface (i.e., z1 , z2 , and z3 in Fig. 3). Considering the fact that the evanescent modes have the least contributions at the farthest null position, the result associated with z3 is taken as the reference, and the deviations of the results associated with z1 and z2 from the reference are plotted in Fig. 12. From this figure, it is observed that the results associated with the three hypothetical locations are very close together except for the case that the hypothetical plane lies at z1 , and the crack is moving outside the probe aperture. In other words, the contribution of the evanescent modes in the vicinity of the probe aperture is rapidly damped, and hence, a low value of null position number n should provide accurate results without requiring a long computation time.
crack-probe position. Nevertheless, for a given accuracy, the proposed method is found to be always faster than the HP-HFSS code. This is due to the fact that commercial finite-element codes should solve a three-dimensional problem, whereas the proposed method utilizes the finite difference method to solve a 2-D problem (i.e., the calculation of resonance frequency of a 2-D cavity). To study the effect of crack geometry on the probe output signal, we present the results associated with three tilted cracks with θ = tan−1 (0.5) = 26.6◦ , θ = tan−1 (1) = 45◦ , and θ = tan−1 (2) = 63.4◦ . The width and depth of these cracks are w = 0.84 mm and d = 1.53 mm, respectively. Fig. 11 shows the computer simulations of normalized-sensor output signal for scanning the three aforementioned tilted cracks together with the perpendicular crack. As shown in Fig. 11, by decreasing the crack–tilt angle (θ), the crack signals tend to approach each other, and finally, reach the asymptotic signal associated with the perpendicular crack. These numerical results further substantiate the consistency of the proposed model. Further examination of the results in Fig. 11 reveals that the symmetry existing in the signal of the perpendicular crack tends to disappear as the crack inclines. This is due to the fact that the probe aperture generally observes two different
IV. C ONCLUSION A theoretical model was presented to describe the interaction of an open-ended waveguide probe and a tilted crack. The modeling technique, which is based on the cavity-resonance approach, hypothesizes a cavity by placing a conductive plane at one of the null positions inside the waveguide. To determine the correct position of the hypothetical plane, an iterative algorithm was used to ensure the equality of the cavity-resonant
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Using the Foster’s reactance theorem, one can conclude that (∂X/∂ω) > 0 [17]. Also, it can be shown that ∂β/∂ω is always positive [18]. These features lead to the fact that the right-hand side of (9) is always negative, i.e., (∂ρ/∂ω) < 0. R EFERENCES
Fig. 12. Deviation from the actual results when the first ( ) or the second ( ) null number is taken as the position of the hypothetical plane shown in Fig. 2.
frequency and the probe-operating frequency. The modeling was tested, both experimentally and numerically, at two operating frequencies in the X and K bands. The numerical verifications were achieved by comparing the results of the present modeling at the K band for a tilted crack with those obtained using the well-known HP-HFSS finite-element code. It was found that although the results obtained from the two methods are consistent, the computation time required in the finite-element method is about two times more than that required in the present method. Experimental confirmation of the theoretical modeling was observed by sampling the surface of several test blocks, containing perpendicular and tilted slots. For this purpose, a waveguide probe operating at the X band was used. It was shown that the output signals from the probes have asymmetrical behaviors when crossing the tilted crack. Also, by decreasing the crack–tilt angle, the crack signals tend to approach each other, and finally, reach the perpendicular crack signal. A PPENDIX From (1), it is inferred that if the hypothetical plane lies at one of the null positions, the dominant-mode reflection coefficient at the probe aperture is expressed as follows: Γ = −ej2βρ
(7)
where ρ > 0 and β > 0. Since the probe–crack structure constructs a lossless one port, the reflection coefficient can also be expressed as follows [17]: Γ=
jX − 1 jX + 1
(8)
where X is the one-port reactance normalized to the impedance of the dominant mode. By differentiating (7), with respect to the angular frequency ω and using (8), one can reach the following expression: ∂ρ 1 1 ∂X ∂β =− + ρ . (9) ∂ω β ∂ω 1 + X 2 ∂ω
[1] M. D. Halliday and C. J. Beevers, “The dc electrical method for crack length measurement,” in The Measurement of Crack Length and Shape During Fracture and Fatigue, C. J. Beevers, Ed. Warley, U.K.: Eng. Mater. Advisory Service Ltd., 1980, pp. 85–112. [2] C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys., vol. 39, no. 6, pp. 2829–2838, May 1968. [3] D. Mirshekar-Syahkal and S. H. H. Sadeghi, “Surface magnetic field measurement technique for nondestructive testing of metals,” Electron. Lett., vol. 30, no. 3, pp. 210–211, Feb. 1994. [4] C. Huber and R. Zoughi, “Detecting stress and fatigue cracks,” IEEE Potentials, vol. 15, no. 4, pp. 20–24, Oct./Nov. 1996. [5] C. Yeh and R. Zoughi, “A novel microwave method for detection of long surface cracks in metals,” IEEE Trans. Instrum. Meas., vol. 43, no. 5, pp. 719–725, Oct. 1994. [6] C. Huber, H. Abiri, S. I. Ganchev, and R. Zoughi, “Modeling of surface hairline-crack detection in metals under coatings using an open-ended rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2049–2057, Nov. 1997. [7] ——, “Analysis of the crack characteristic signal using a generalized scattering matrix representation,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 4, pp. 477–484, Apr. 1997. [8] F. Mazlumi, S. H. H. Sadeghi, and R. Moini, “Detection of surface cracks of arbitrary shape in metals using an open-ended waveguide probe,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 21, D. O. Thompson and D. E. Chimenti, Eds. College Park, MD: Amer. Inst. Phys., 2002, pp. 491–497. [9] ——, “Analysis technique for interaction of rectangular open-ended waveguides with surface cracks of arbitrary shape in metals,” NDT E Int., vol. 36, no. 5, pp. 331–338, Jul. 2003. [10] H. H. Park, Y. H. Cho, and H. J. Eom, “Surface crack detection using flanged parallel-plate waveguide,” Electron. Lett., vol. 37, no. 25, pp. 1526–1527, Dec. 2001. [11] V. M. Babich, V. A. Borovikov, L. J. Fradkin, V. Kamotski, and B. A. Samokish, “Scatter of the Rayleigh waves by tilted surface-breaking cracks,” NDT E Int., vol. 37, no. 2, pp. 105–109, Mar. 2004. [12] Y. C. Li, “Analysis of fatigue phenomena in railway rails and wheels,” in Handbook of Fatigue Crack Propagation in Metallic Structures, vol. 2, A. Carpinteri, Ed. Amsterdam, The Netherlands: Elsevier, 1994, pp. 1497–1537. [13] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Hoboken, NJ: Wiley, 2001, ch. 3. [14] M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. Boca Raton, FL: CRC, 2001, ch. 3. [15] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 7. [16] HP High Frequency Structure Simulator 5.4, User’s Guide, HewlettPackard Co., Palo Alto, CA, 1999. HP Part No. 85180-90166. [17] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 4. [18] ——, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 3.
Farhad Mazlumi was born in Tehran, Iran, in 1974. He received the B.S. and M.S. degrees from Sharif University of Technology, Tehran, in 1997 and 1999, respectively, and the Ph.D. degree in electrical engineering from Amirkabir University of Technology, Tehran, in 2005. In the summer of 2005, he worked at Bilkent University, Ankara, Turkey, as a Research Assistant, working on integral equations for bodies with impedance surface. He is currently an Assistant Professor with the Civil Aviation Technology College, Tehran. His research interests include computational methods for electromagnetics and microwave-passive devices.
MAZLUMI et al.: INTERACTION OF RECTANGULAR OPEN-ENDED WAVEGUIDES WITH TILTED CRACKS IN METALS
Seyed H. Hesamedin Sadeghi (M’92–SM’05) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1984, and the Ph.D. degree from the University of Essex, Colchester, U.K., in 1991. Between 1980 and 1983, he was with the electrical power industry in Iran. In 1984, he was a Research Assistant at the University of Lancaster, Lancaster, U.K. He then joined the University of Essex as a Senior Research Officer. In 1992, he was appointed as a Research Assistant Professor with the Vanderbilt University, Nashville, TN. In 1996–1997 and 2005–2006, he was a Visiting Professor at the University of Wisconsin–Milwaukee. He is currently a Professor of electrical engineering with the Amirkabir University of Technology, Tehran. His current research interests include electromagnetic nondestructive evaluation of materials and electromagnetic compatibility issues in power engineering.
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Rouzbeh Moini (M’93–SM’05) was born in Tehran, Iran, in 1963. He received the B.S., M.S., and Ph.D. degrees in electronics from Limoges University, Limoges, France, in 1984, 1985, and 1988, respectively. In 1988, he joined the Electrical Engineering Department, Amirkabir University of Technology, Tehran, where he is currently a Professor of telecommunications. From 1995 to 1996, he was a Visiting Professor at the University of Florida, Gainesville. His main research interests are in numerical methods in electromagnetics, electromagnetic compatibility, and antenna theory. Dr. Moini was the recipient of the 1995 Islamic Development Bank Merit Scholarship Award.
