Shyue-Dar Chen, Member, IEEE, and Ching-Kuang C. Tzuang, Fellow, IEEE
Abstract—This paper experimentally and theoretically con-firms the validity of the definition proposed by Das for computing the complex characteristic impedance of the first higher order ( 1) microstrip mode. The normalized complex propagation constant and complex characteristic impedance of the microstrip obtained by the rigorous full-wave integral-equation method is also presented. To better understand the circuit behavior of the leaky mode at the respective frequencies, the results are analyzed in both frequency and transformed steepest descent plane. A differential time-domain reflectometry (TDR) experiment shows that the experimental results are in excellent agreement with the time-domain plots obtained theoretically by the inverse discrete Fourier transform of the transmission line modeled by the dispersive characteristic. The propagation characteristics of the echoed signals in the time domain, which are reflected from the open end of the leaky line, are analyzed in detail using the corresponding group velocity of the 1mode. The wide spread of the echoed signals in the time domain is the direct result of the highly dispersive group velocity. The slowest group velocity is in the leaky region. The time-to-frequency conversion of the mea-sured TDR data reveals that the reflection, leaky, and propagation zones coexist simultaneously for the 1 mode propagation. The conversion also accurately assesses the attenuation constant of the 1mode if the attenuation is not too high. The Fourier transform of the TDR responses also simultaneously yields the input reflection coefficient ( 11) and the complex characteristic impedance. The complex characteristic impedance extracted from the TDR responses also agrees closely with the theoretical data.
Index Terms—Group velocity leaky waves, impedance measure-ment, microstrip, time-domain reflectrometry.
I. INTRODUCTION
N
EW LEAKY- and surface-wave effects in various planar and nonplanar guiding structures have been reported for applications in microwave integrated circuits and devices. The undesired spurious coupling caused by leaky effects may stem from dominant-mode leakage in stripline with an air gap [1], additional leaky dominant mode [2], [3], and conductor-backed slotline [4]. The physical interpretation and discussion of these effects can facilitate new methods for suppressing the leakage effects, such as using shorting pins [5] and changing the pack-aging or geometric parameters of the guiding structures [6], [7]. However, exciting a leaky mode in a controllable fashion isManuscript received June 15, 1999; revised January 12, 2001.
The authors are with the Institute of Electrical Communication Engi-neering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: cktzuang@cc.nctu.edu.tw).
Publisher Item Identifier S 0018-9480(02)04049-8.
also very important for practical use of the leaky lines. Menzel successfully designed a leaky-wave antenna in a microstrip by properly exciting the first higher order mode via step disconti-nuities, matching circuits, and a mode suppressor [8]. Further-more, excitation of microstrips first and second higher order modes using composite microslotline and microcoplanar wave-guide (micro-CPW) structures, respectively, has exhibited very good relative power absorbed (RPA) value (typically greater than 90%) and excellent input matching (typically a reflection coefficient smaller than 15 dB) of the leaky lines [9], [10].
To excite the leaky lines properly, the layout for the three-dimensional (3-D) structure is usually optimized by adjusting the geometrical parameters through a lengthy process that is based on only limited knowledge of the complex propagation constant of the targeted leaky mode [11]. Consequently, the issue of the optimal feeding structures for exciting a specific leaky line arises. To resolve this issue, the assessment of the leaky-mode excitation embedded in the complicated radiation process, which also involves space waves and surface waves, mandates appropriate methods for extracting the leaky modes’ contribution [10], [12], [13]. In addition to the quantitative as-sessment, the characteristic impedance of a leaky line provides an insightful circuit-domain view of the leaky line, thus enabling the design of an optimal feeding network. Given the propaga-tion constant and characteristic impedance of the leaky line, the microwave circuit description and modeling of the guiding structure are fulfilled [14]. Das demonstrated that a 3-D circuit comprised of short-circuit slotline stubs could be modeled accu-rately with a one-dimensional leaky line when the leaky mode dominates in the microwave circuit [4]. He defined the charac-teristic impedance ( ) for a leaky strip-type line using the fol-lowing expression:
excluding leaky fields (1) where denotes the total modal currents and represents the bound-field portion of the Poynting power, excluding the expo-nentially growing parts of the leaky field in the modal solution of the leaky mode. Das referred to the leaky fields as being only loosely attached to the central guiding region that are respon-sible for distributed radiation loss. Thereby the leaky-fields are not associated or are only loosely associated with the circuit
Fig. 1. General microstrip line and coordinate system. TheZ-axis is along the microstrip line. The structural parameters are used through this paper. The two sidewalls are separated with a distanceb in the y-direction.
element in a circuit sense. Unless the complex characteristic impedance is experimentally verified, as with the microstrip bound mode, doubts still remain about employing the complex characteristic impedance for circuit simulation.
In addition to the presentation of complex characteristic impedance obtained theoretically using (1) for the first higher order mode of a microstrip line, this study applies an experimental setup based on the differential time-domain re-flectometry (TDR) to: 1) excite the first higher order odd mode of the microstrip; 2) theoretically and experimentally explore the corresponding time-domain propagation characteristics; 3) relate the group velocity of the odd-mode propagation to the time-domain step response; 4) interpret the broad-band time-domain propagation characteristics; and 5) confirm the validity of the definition for complex characteristic impedance. This approach obtains a clear and comprehensive circuit level picture of the first higher order microstrip mode.
II. CHARACTERISTICIMPEDANCE OF THEMICROSTRIP’SFIRST
HIGHERORDERMODE
The structural parameters of the microstrip line shown in Fig. 1 are applied here for analyses and experiments. The microstrip with a width of 16 mm is printed on a 30-mil-thick ( mm) ULTRALAM substrate of and the sidewall distance is set at 480 mm (30 W in Fig. 1). The microstrip is parallel to the -axis, which is normal to the transverse -plane.
Numerous previous investigations have reported the full-wave two-dimensional (2-D) modal analyses of strip-type transmission lines. In general, the procedures for obtaining the characteristic impedance using 2-D modal analyses closely resemble each other [15]. The first step is to determine the propagation constant, from which the currents on the strip are then obtained. Next, the Poynting power is computed using the cross-sectional electrical and magnetic fields derived from the known current distributions on the strip. For clarity, Ap-pendix A briefly summarizes the formulation. Cautions should be exercised when calculating the transverse fields in the case of the leaky line, and the exponentially growing components in the total field expression of the transverse plane should be
(a)
(b)
Fig. 2. (a) Normalized propagation constant and complex characteristic impedance of the first higher order microstrip mode. The characteristic impedance of dominant mode is slightly dispersive, approximate 10 . (b) Enlargement of the frequency scales on the summit region.
excluded. For example, the -component of the TE mode with respect to the -axis is expressed as follows:
(2) Given a leaky-mode solution, the bound field of a leaky mode is the summation of the harmonic terms from to , where is the upper limit for truncating the infinite series and is the first term of the bound field. The magnitude of the general-ized reflection coefficient must be less than one to ensure the term is bound field [see Appendix A for the expres-sion of ]. For
are more than one, representing the leaky field of the leaky mode and suggesting the mathematical condition for the im-proper physical solution. Recent investigations of such leaky modes that are improper (they do not satisfy the radiation condi-tion at infinity, but are physically significant) used the semiana-lytical 3-D Green’s function to obtain the excitation amplitude of a leaky mode [12], [13].
complex whenever the propagation constant is also complex [16, eqs. (13) and (14)]. Fig. 2(a) plots the normalized complex propagation constant ( ) of the first higher order ( ) mode using the vertical scales on the right axis ( ,
, ; is the free-space wavenumber). The real and imaginary parts of the complex characteristic impedance [ ) and ], obtained by the procedure described ear-lier, are superimposed in Fig. 2, with scales on the left-hand-side axis. Fig. 2(b) presents the expanded view of Fig. 2(a), in the fre-quency range from 5.2 to 5.8 GHz, for a detailed investigation of leaky properties. Notably, two conducting sidewalls, separated by a distance , are placed along the -direction in Fig. 1, thus rendering the application of the discrete Fourier series. Exten-sive numerical analysis reveals that when exceeds 30 the mi-crostrip width (30 W), the sidewalls have negligible influence on the propagating characteristics of the leaky line. The dispersion curves and complex characteristic impedance converge with an error of less then 2%.
The onset frequency of the leaky mode is at approximately 7.2 GHz, above which the normalized phase constant ( ) is greater than one and the normalized attenuation constant ( ) is negligible for the particular case study. The so-called spec-tral-gap region must reside around the onset frequency [17]. Appendix B gives the expression for estimating the bandwidth of the spectral gap using the available dispersion data at and below the onset frequencies (see and ). Since the spec-tral-gap region manifests nonphysical solutions for the propa-gation constants, the estimated bandwidth of the spectral gap is important here. Notably, the measured (dispersive) propagation constants in the spectral gap are often obscure, mainly because of the radiation in the form of a surface or space wave from the exciting source in the spectral-gap region [18]. This implies that the spectral gap will significantly impact the circuit mod-eling of the leaky mode from the characteristic impedance point of view. The normalized leakage constant at the onset fre-quency is 0.23 10 (see Table I). Appendix B reveals that the bandwidth of the spectral gap is approximately the ratio of to , where is the slope of the normalized phase constant at the onset frequency . Detailed analyses in Appendix B in-dicate that the spectral gap is only approximately 10-kHz wide, too small to be plotted and distinguishable in Fig. 2(a). Since the spectral gap is very small, it is thus negligible in our anal-ysis. Furthermore, the Tektronix CSA803A differential TDR test system has a sampling period of 1.0 ps and 5120 points in the maximum record length. These parameters are equivalent to having a best frequency resolution of approximately 100 MHz in the spectrum, which is not fine enough to discriminate the circuit effect of the spectral gap in our case study.
Fig. 3. Complex characteristic impedance of theEH microstrip mode and the locations of the leaky pole are imposed on the transformed -plane (steepest descent plane). A transformation = 0j = k sin( ) k = k 0 =
k cos( ) is used where = 0 j is the complex plane. The curve
labeled asSDP is the steepest descent path through the saddle point =
=2. The leaky poles within the region under the curve SDP and curve = 1 (cos( ) sinh( ) = = 1) contribute strongly to the far field. The
symbols and represent the leaky pole locations that > 1; < 1, respectively. The symbol2 represents the bound (real) pole locations. The list of specific frequencies corresponding to the numerical and alphabetical points is also tabulated.
If the frequency is increased further, to well above the onset frequency, the value of will asymptotically approach that of the dominant bound (even) mode. As expected, we observe the corresponding of the mode, which is real and nearly reaches that of the bound mode.
Below the onset frequency, the mode becomes leaky, with its attenuation constant increasing as the frequency de-creases. As illustrated on summit and in Fig. 2(b), the max-imum values of and are 56.8 (summit ) and 48.8 (summit ) at 5.56 and 5.46 GHz, respectively. Further-more, let the intersecting point of curves and be point at approximately 5.52 GHz. Below , increases rapidly and
quickly declines to zero. However, rises sharply from nearly zero to summit and then gradually declines to zero again. Consequently, we can observe that the leaky line, in the leaky region, depicts a strong reactive (inductive) circuit behavior below and becomes lossy above . As in Fig. 2(a), the frequency below summit , , and is approx-imately 0.08 and 1.8 , respectively, at 1.0 GHz, implying the leaky line is essentially a short circuit. The combined effect of large inductance near and below point and the short-circuit behavior below summit means the leaky mode is not easily excited below point . The above finding correlates with the finding of Lee [19] and Oliner in that, for large decay rates ( ), the contribution of the leaky mode to the total field can usually be disregarded. To clarify this phenomenon, Fig. 3 superimposes the complex characteristic impedance of the var-ious leaky poles onto the steepest descent plane. Twenty leaky poles in the steepest descent plane are numbered and listed in the inset of Fig. 3. A transformation
Fig. 4. Differential TDR experimental setup for exciting purely odd mode and investigating the propagating properties of the first higher order microstrip mode. (left-hand side) Tektronix CSA803A with a sampling head SD-24. (right-hand side) Microstrip line under test.
Fig. 5. Experimental and theoretical differential TDR step responses of the microstrip. The theoretical step response in the time domain is obtained by the inverse direct Fourier transform of the transmission line modeled by the complex propagation constant and complex characteristic impedance of Fig. 2.
CH1 is the positive-going step voltage, displaying TDR voltage 0500 mV and
corresponding to reflection coefficient = 01 = +1. Similarly, CH2 is the negative-going step voltage, displaying TDR voltage0 0500 mV and corresponding to reflection coefficient = 01 = +1. The differential TDR responses of the microstrip: measurement (- - -) and theory (—).
seven points, denoted by the symbol , located below 4.6 GHz, are the leaky poles outside the curve , and simultane-ously display almost short circuit behavior; and rapidly drop from 10 to 0. The short-circuit property makes the excitation of the first seven leaky poles extremely difficult. Points 8–16, denoted by “ ,” are leaky poles within the region bounded by the curves and . The curve labeled represents the steepest descent path through the saddle point . Notice that points 10–12 correspond to summit , point , and summit , respectively. Points 8–11 define a re-gion where complex characteristic impedance is highly disper-sive, and where a strongly inductive component such as rapidly falls to zero. Such large reactive components tend to pro-hibit the leaky mode from being excited. Points 12–16, above point , constitute the leaky-mode region of the relatively in-significant , thus defining a region that exhibits strong leakage if excited properly. Meanwhile, points – , marked with symbol , are classified as real poles that propagate in the form of the bound mode, also possess real characteristic impedance.
III. DIFFERENTIALTDR EXPERIMENTALSETUP
As shown in Fig. 4, a differential TDR experiment is con-ducted to validate the power–current definition for the char-acteristic impedance of the leaky mode. Using the Tektronix CSA803A communication analyzer, a pair of differential step
voltages with a rise time of 17.5 ps are inflicted into the left-hand side (source terminal) of a 178-mm-long microstrip line. Thus, the differential step voltages naturally excite the odd mode of the microstrip. Fig. 5 displays the measured differential TDR step responses. The positive-going step waveform travels along a short 50- coaxial cable labeled from 0.5 to 0 ns and experi-ences a short-circuit load, thus swinging from (250 mV) to (0 mV). On the contrary, the negative-going step swings from ( 250 mV) to (0 mV). Two in-teresting time-domain propagation characteristics merit further investigation. First, an exponentially decaying and oscillating waveform of resonant frequency near 5.5 GHz is observed be-tween 0–2.0 ns. As soon as the differential step voltage waves enter the microstrip and generate the decayed resonance close to point (point 11 at 5.52 GHz) implies that the leaky mode at summit is largely excited because its characteristic impedance is close to the 50- reference load. Second, the returned signals from the open end, starting from 2.0 ns, show that the frequen-cies within the signals exceed 5.5 GHz, more precisely above the onset frequency of 7.2 GHz. This phenomenon implies that the energy of the leaky mode is almost exhausted during the round-trip as the step voltage waves return to the source end. The qualitative description of the measured time-domain responses agrees well with expectations based on the theoretical data from Section II.
IV. TIME-DOMAINANALYSIS
The step responses in the time domain can be recovered from the dispersive characteristics reported in Section II, namely, the propagation constant and complex characteristic impedance. If the input step waveforms are truly differential, only the odd mode is excited in the microstrip with an open end as a load. Thus, the electrical wall may be placed at the center of the microstrip and the entire experimental setup can be modeled by a half-circuit, as illustrated in Fig. 6, where denotes the positive-going step voltage source of 17.5-ps rise time, and represents the internal referenced impedance (typically 50 ). The complex propagation constant, complex characteristic impedance, and length together model the microstrip that propagates the mode and is terminated by an open end. Herein, the effective positions of the junction of the SMA connector are located on the differential TDR measurement, as shown in Fig. 6. The measured maximum reflection coefficient ( ) is 0.020 and 0.012 for and , respectively. The different values appear to result from the soldering or variation
Fig. 6. Equivalent circuit representation of the TDR experiment on the step response for a microstrip leaky mode.V s is the positive-going waveform with 500-mV step voltage and 17.5-ps rise time.
of the separate SMA connector. Since the reflection coefficient is very small, the effect of the coaxial-to-microstrip transition can be neglected in our analyses of the TDR waveforms. The time–harmonic terminal voltage at (the source end) can be expressed as
(3) where denotes the reflection coefficient at the source end, assuming that the reference impedance is ; represents the time–harmonic component of the step voltage waveform, and is the input impedance looking into the leaky line. Applying the transmission-line equation, becomes
(4) Finally, the step response is obtained by the inverse discrete Fourier transformation of . As shown in Fig. 5, the the-oretical differential TDR responses of the microstrip are super-imposed onto the measured data. Excellent agreement exists be-tween the two sets of data, although a small discrepancy is ob-served for the returned signals near 2.0 ns. Since the theoret-ical TDR responses virtually duplicate the measured results, the characteristic impedance reported in Section II is verified. Con-sequently, further time-domain investigation of the -mode propagation becomes meaningful. Closely examining the TDR response reveals that in the echoed (returned) signals between 2.0–4.0 ns, the high-frequency signals arrive earlier than the lower ones. Fig. 7 validates this observation. By computing the normalized group velocity [ , is the speed of light] using the data shown in Fig. 2(a), Fig. 7 shows that the minimum group velocity occurs near 5.58 GHz, which is slightly higher than summit . Above 5.58 GHz, the group ve-locity increases from to the asymptotic limit of . Notably, the group velocity is below the speed of light. The figure also presents the phase constant , not normalized to , to better relate the phase constant and group velocity. Below 5.58 GHz, is small and nearly independent of frequency. How-ever, the group velocity cannot be approximated by taking the first-order derivative of the diagram [22] and, thus, the upper limit at which is equal to unity is imposed in Fig. 7. Time required for the voltage wave to travel from the source end back to the original position is
(5)
Fig. 7. Phase constant ( ) and normalized group velocity (V =c) of the first higher order microstrip mode (h = 30 mil, w = 16 mm, " = 2:55).
Fig. 8. Amplitude spectra for analyzing signals contained in the wavepacket. The gating signals are between 2.0–4.0 ns. V (!) is the time–harmonic component of the step waveform with 17.5-ps rise time and 500-mV step voltage.
The group velocity at 7.0 and 15.0 GHz is and , re-spectively. Meanwhile, the corresponding delay times are 2.967 and 2.01 ns, as clearly displayed in Fig. 5. The echoed signals that return from the microstrip open end are spread out. The 15.0-GHz signal component travels at a higher group velocity and returns to the source end approximately 2 ns after the step voltage waveform is inflicted. Meanwhile, the 7.0-GHz signal component, which is in the leaky-mode region, travels at a lower group velocity, thus appearing approximately 1 ns later than the 15.0-GHz signal.
V. FREQUENCY-DOMAINANALYSIS
The signals in the wavepacket (from 2.0 to 4.0 ns) are an-alyzed using experimental and theoretical data by the discrete fast Fourier transformation (DFFT). Fig. 8 illustrates the rela-tionship between the amplitude and frequency of the computed spectra. Amplitudes below 5.2 GHz are close to zero, clearly indicating that signals below 5.2 GHz (near point 9) are hard to propagate. This phenomenon agrees with the implication in Fig. 3 that signals below point 7 (4.6 GHz) can be disregarded. Such signals can be disregarded because the complex charac-teristic impedance is nearly zero, thus reflecting most signals below point 8 back to the source end. Fig. 8 also plots , the Fourier transform of the step waveform (500 mV in ampli-tude, 17.5 ps in rise time). Above 7.2 GHz, which is the onset frequency of the leaky mode (Fig. 3, point 16), both ex-perimental and theoretical values closely follow the curve with little discrepancy. Therefore, by only propagating the
Fig. 9. Experimental and theoretical voltage gain(V (!)=V (!)) defined as the time–harmonic component of the echoed signals to that of the step voltage.
mode, the microstrip line acts as a high-pass filter with corner frequency at the onset frequency. Lying between 7.2–5.2 GHz is the leaky region excited by the differential step voltage waves. Notably, the DFFT gating process causes a residual value at zero frequency (dc). For an open microstrip line, the highly leaky region of the third higher order ( ) mode lies at approxi-mately the triple frequency of the mode [23]–[25]. Mean-while, the mode (also an odd mode) can be excited by the differential TDR apparatus. The loss displayed in Fig. 8 be-tween 16.0–20.0 GHz [figures that are nearly three times 5.46 (point 10) and 6.5 GHz (point 14)] is closely associated with the leakage of the mode. The voltage gain is defined as the ratio of the time–harmonic components of the echoes to
written as
Echoed Signals
(6) Fig. 9 plots , and clearly reveals three propagating zones for the mode. Below 4.7 GHz, the theoretical (ex-perimental) data show a reduced gain approximately 50-dB ( 40-dB) gain reduction, confirming that only a very small portion of the signal can propagate below point 8 in Fig. 3 This region is thus designated as a reflection zone. Above 7.2 GHz, both the theoretical and experimental gain curves approach 0 dB, indicating that a high-pass propagating zone exits. The gain plot between 5.2–7.2 GHz quantifies how the leaky mode gets excited and survives the round-trip leakage. Near 7.2 GHz (point 16 of Fig. 3), the leaky mode exhibits nearly zero attenuation constant [see Fig. 2(a)] and, thus, a gain reduction of nearly 0 dB is observed in Fig. 9. Meanwhile, at 6.0 GHz (near point 9 of Fig. 3), the leaky mode has an attenuation constant of 0.05 and experiences a reduction of 20 dB in the round-trip, closely approaching the predicated value of 19.5 dB, computed by
leaky zone (7) Fig. 10 compares the computed attenuation constant using (7) and the theoretical data shown in Fig. 2. Excellent agreement is obtained with small values [from 0 0.007 (1/mm)] for the at-tenuation constant between 5.8–7.2 GHz. The sensitivity of the TDR scope prohibits accurate assessment of the larger attenu-ation constant and, thus, Fig. 10 shows a growing discrepancy in the attenuation constant below 5.8 GHz. Shortening the leaky
Fig. 10. Comparison of the attenuation constants obtained by differential TDR experiment and the 2-D field theory. Noticed that is not normalized to k . The experimental data is obtained from the gain of Fig. 9 by (9).
line used to measure large attenuation constants is one way to enhance the accuracy of the measurement.
Finally, an attempt is made to extract complex characteristic impedance from time-domain differential TDR waveforms. The terminal voltage of (3) is expressed in terms of multiple reflec-tions in the transmission line as [26]
(8) where and denote the reflection coefficients at the source and load terminals, respectively. The terminal voltage be-fore the first round-trip reflection wave arrives at the source end can be reduced to
(9) Equation (9) simulates the case of an infinitely long trans-mission line. In practice, as long as the returned signals do not interfere with the transient signals that have sufficient time to settle. Fig. 5 shows that the transient signals decay to become negligible before the returned signals arrive at the source end at 2.0 ns. The reflection coefficient and the characteristic impedance can thus be obtained from the terminal voltage by
(10) (11) where is the discrete Fourier transform, using the gating transient voltage waveform between 0–2.0 ns and then padding with zero above 2.0 ns. Thus, the reflection coefficient is ob-tained. Fig. 11 plots the input reflection coefficient in decibels, including both its real and imaginary parts. The figure also con-tains the numeral points of Fig. 3. Notably, points 10 (summit ), 11 (point ), and 12 (summit ) show that and
are near zero, displaying good input matching to 50 between points 10 and 13. Detailed observation reveals that the minimum of is approximately 15.6 dB at 5.75 GHz. Between points 11 and 13, is below 10 dB, indicating
Fig. 11. Reflection coefficientS obtained from the TDR data by (10) using the gating experimental data from 0 to 2.0 ns. The numerals shown in the plot follow the inset of Fig. 3.
Fig. 12. Comparison of the complex characteristic impedance obtained both from theory and experiment that are computed by (1) and (11).
that this region is easily excited by the present apparatus. Points 1–7 are outside the curve, as shown in Fig. 3. Notably, applying estimates that significant electromagnetic energy enters the transmission line when using the present ex-perimental apparatus, specifically, 6.5% at 3.0 GHz (point 5) and 21% at 4.6 GHz (point 7). The reflection zone reported in Fig. 9 actually comprises a transitional region between the re-flection and leaky zones. Fig. 12 compares the directly extracted complex characteristic impedance of the mode using (11) and the theoretical data obtained by (1), clearly showing excel-lent agreement between the theoretical and experimental data. The differential TDR experiment, through (11), confirms that the complex characteristic impedance of the mode has an inductive (resistive) component that peaks at summit ( ) or point 10 (12) of Fig. 2(a). The discrepant peak values at sum-mits and are caused by truncating the infinite oscillation.
VI. CONCLUSIONS
This study has experimentally and theoretically confirmed the definition proposed by Das [4] for obtaining the complex characteristic impedance of a leaky line by a case study of the first higher order ( ) microstrip mode. The microstrip mode may propagate in three types of zones in the broad-band spectra, namely, the propagation, leaky, and reflection zones. In the propagation zone, where the slow wave propagates with zero attenuation, the propagation constant and characteristic impedance are both real. The leaky zone enters below the
Fig. 13. Typical transition region of the spectral gap. The normalized dispersion curves are divided into three regions. The right of pointC, where the real spectral solution touches one, is the bound-wave region and is physical. The left of pointA, which is the complex solution, is the leaky-wave region and is physical. Between the leaky- and bound-wave regions is the so-call spectral gap, where solutions are nonphysical.
propagation zone, showing the complex propagation constant and the complex characteristic impedance. The complex characteristic impedance is close to 50 , together with a large inductive part, implying that the leaky-mode antenna of this type is usually easy to match to 50 . The group velocity is lowest in the leaky region, falling to approximately ( : the speed of light) at 5.58 GHz in this particular case study. Although the phase velocity exceeds in the leaky zone, the group velocity is smaller than .
Below the leaky zone is the reflection zone, where complex characteristic impedance falls into two regions. The first re-gion contains complex characteristic impedance of nearly zero (near 3.0 GHz, for the particular case study), repsenting a short-circuit phenomenon. Meanwhile, the second re-gion is characterized by characteristic impedance with a small real part, but a growing imaginary (inductive) part, with the imaginary part growing with frequency. Such inductive circuit loading closely reflects the mode launched from the differ-ential TDR heads, thus creating a transitional region (between 3.0–4.6 GHz in the particular case study) transforming the pure reflection zone into a leaky zone.
The complex characteristic impedance, combined with the complex propagation constant of the mode, explicitly ex-plains the propagation characteristics of the three zones associ-ated with the differential step excitation of a long leaky line. The proposed differential TDR test method consists of time-domain signals in two parts. The first part involves the transient wave-forms instantly reflecting at source end. Meanwhile, the second part is focused on the signals returning from the open end. The transient signals are shown to be associated with the input reflec-tion coefficient ( ) and complex characteristic impedance of the leaky line. On the other hand, the echoed signals are related to the leaky and propagation zones, from which the attenuation (leaky) constant can be accurately extracted.
APPENDIX A
2-D DYADICGREEN’SFUNCTION
Assume that the time–harmonic dependence is and the guided-wave -dependant solution is . What follows is the 2-D dyadic Green’s function applied in the integral-equation formulation for obtaining the leaky-mode solution, as shown at the bottom of the following page.
APPENDIX B
ESTIMATION OFSPECTRAL-GAPBANDWIDTH
The bandwidth of the spectral gap can be related to the leaky constant at point and the slope of line segment by esti-mating the phase constant at point in terms of the leaky con-stant at point and assuming that the wavenumber between points and is linear. What follows is parallel to the work of [17], except the explicit mathematical expression relating the bandwidth of the spectral gap to the leakage attenuation con-stant.
According to Fig. 13, the propagation constant is near unity in the spectral gap. The leaky field propagates in the parallel-plate form. Thus, we can set , and the Helmholtz relation is reduced to
(A-1)
The direction of is parallel to the parallel plates. Herein, and are defined in terms of their real and imaginary parts as follows:
(A-2) (A-3) By normalizing and with respect to , substituting (A-2) and (A-3), into (A-1), and separating the real and imagi-nary parts, we obtain
(A-4) (A-5)
TE mode TM mode
(A-7) (A-8) As pointed out in [17], from points to , is constant while becomes smaller. At point , , , and are nonzero, and from (A-5). We solve (A-4) and get
(A-9) Since , (A-9) is reduced to
(A-10) Assuming that the wavenumber between points and de-pict a straight line, is linear in this range. Thus, the expres-sion relating to the bandwidth of the spectral gap and the leakage constant at point is obtained immediately
(A-11) where is the slope of the line segment , as shown Fig. 13. In practice, the slope of the line segment is the same as that of , where the solutions are usually easier to obtain. Assume that the spectral width of ( ) equals that of ( ) in (A-11).
This relation is verified using the data in [17, Figs. 4 and 5]. We get meter from [17, Fig. 5] and calculate the normalized leakage constant at point as
. Applying (A-10) to estimate at point , we obtain
, where correlates well with that shown in [17, Fig. 4].
The data from [17] are listed as follows: GHz
GHz
GHz
GHz
The spectral gap of [17, Fig. 4] is 14 MHz. Applying (A-11) allows us to calculate the bandwidth of the spectral gap
GHz MHz
kHz
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Shyue-Dar Chen (S’96–M’99) was born in Taiwan,
R.O.C., on August 11, 1958. He received the B.S. degree in physics from the National Taiwan Normal University, Taipei, Taiwan, R.O.C., in 1981, the M.S. degree from Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1986, and is currently working toward the Ph.D. degree in communication engineering at the National Chiao Tuang University, Hsinchu, Taiwan, R.O.C.
From 1986 to 1996, he was with the Chung Shang Institute of Science and Technology, Lung-Tan, Taiwan, R.O.C., where he was involved with the design and development of hardware for microwave systems. His research interests are mainly in the area of computational electromagnetics and propagation of guiding structures.
Ching-Kuang C. Tzuang (S’80–M’83–SM’84–
F’97) received the B.S. degree in electronic engi-neering from the National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1977, the M.S. degree from the University of California at Los Angeles, in 1980, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1986.
From 1981 to 1984, he was with TRW, Redondo Beach, CA, where he was involved with analog and digital monolithic microwave integrated circuits. Since 1986, he has been with the Institute of Communication Engineering, National Chiao Tung University. His research activities involve the design and development of millimeter-wave and mi-crowave active and passive circuits and the field theory analysis and design of various complex waveguiding structures and large array antennas. To date, 53 M.S. and 14 Ph.D. degree students have graduated under his supervision.
Dr. Tzuang helped form the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), Taipei Chapter, and served as its secretary, vice chairman, and chairman in 1988, 1989, and 1990, respectively. He has served on the Asia–Pa-cific Microwave Conference International Steering Committee, where he has served as the international liaison officer representing the Taipei Chapter since 1994.
