A discretely loaded resistive vee dipole (RVD) is designed and realized for use in ground-penetrating radar (GPR) applications. The RVD is a good antenna for GPR applications because it can radiate a temporally short pulse into a small spot on the ground, and its low radar cross section mostly eliminates the multiple reflections between the surface of the ground and the antenna. The antenna presented in this paper is printed on a circuit board and is discretely loaded with off-the-shelf surface-mount chip resistors. The resulting structure is easy and inexpensive to manufacture and is mechanically stable. The realized antenna is measured in terms of the radiated field and the reflected voltage in the feeding transmission line. The results are compared with those of the antenna with the same resistive loading without substrate. The effects of the transmission line impedance on the performance of the antenna are also presented.
 A resistively loaded vee dipole (RVD) is two identically loaded monopoles placed in a vee shape. It is driven differentially at the junction of the two monopoles. The resistive profile considered in this paper for the loaded monopoles is the Wu-King profile [Wu and King, 1965]. The Wu-King profile can be represented as the resistance per unit length:
where R0 is the resistance per unit length at the drive point, r′ is the distance along the arms from the drive point, and h is the length of the monopole. The parameter R0 is chosen such that the antenna satisfies the performance requirement for a specific application. For example, Montoya and Smith selected R0 by a trade-off between the initial reflection from a perfect electric conductor (PEC) ground plane and the tail clutter for a ground penetrating radar (GPR) application [Montoya and Smith, 1999].
 The RVD with the Wu-King profile has many advantages for use in GPR applications. It can radiate a temporally short pulse into a small spot on the ground, while having a low radar cross section (RCS). In addition, it is geometrically simple and light. Some of the research on the RVD for use in GPRs or mine detection systems can be found in the work of Kim , Montoya , and Montoya and Smith [1996a, 1996b, 1999].
 Various methods have been used to realize the loaded monopole with the Wu-King profile. One method is depositing a resistive material of variable thickness on a dielectric rod [Lally and Rouch, 1970; Shen, 1967]. With this method, one can build a cylindrical antenna whose geometry is closely related to the theoretical model, but it is difficult to achieve an accurate resistive profile. Another method to implement the profile is tapering a resistive film of constant thickness [Esselle and Stuchly, 1991; Montoya, 1998; Montoya and Smith, 1999]. This method gives a better control over the resistive profile. However, the realized structure is mechanically weak, and the bonding between the resistive film and the metal at the drive point can be problematic. In a third method, one may use a series of discrete resistors that approximates the continuous profile. Maloney and Smith  successfully approximated a resistive profile by stacking special high-frequency resistors in series and soldering them end to end [Maloney, 1992; Maloney and Smith, 1993]. However, this structure is mechanically fragile to be used in a GPR applications, where the antennas are used in the field.
 As an alternative method to realize the Wu-King profile, we use standard off-the-shelf surface-mount chip resistors. The resistors are mounted on metal strips printed on a dielectric substrate using standard printed circuit board (PCB) manufacturing technology. The structure is schematically drawn in Figure 1. The resulting structure is easy and inexpensive to manufacture and mechanically stable. In this paper, the parameters used for the Wu-King profile in equation (1) are R0 = 1526 Ω/m and h = 30.6 cm. Other parameters needed for the realization of the RVD are discussed in the following sections.
2. Discretization of the Continuous Wu-King Profile
 To realize the resistive profile with standard chip resistors, equation (1) is discretized. The arm of length h is divided into a number of sections, and a chip resistor is placed in the center of each section so that the resistance of each section agrees closely with that of equation (1). The chip resistors used in this paper are 1.6 mm in length. Thus a maximum of 191 resistors can be used along each arm. However, to reduce the complexity and cost of the RVD, we will minimize the number of resistors used.
 The effects of the number of resistors are numerically investigated using the method of moments code in the electromagnetic interactions generalized (EIGER) code suite [Sharpe et al., 1997]. First, we generated the meshes for the numerical model by writing a MATLAB® script. The continuous profile is discretized using 11, 26, and 47 resistors per arm for the meshes. Figure 2 shows a portion of the mesh with 47 resistors per arm as an example of the generated meshes.
 In the numerical model, the metal strips are discretized using the rectangular cells whose widths are 0.8 mm, which is the same as the width of a chip resistor. The chip resistors are modeled using delta gap lumped impedance elements. The drive point is approximated by a delta gap voltage source placed between two triangle elements at the junctions of the two monopoles. The electric field integral equation with linear basis functions is used in the model.
Figure 3 shows the results from the numerical model when the antennas are driven by a Gaussian voltage pulse with tFWHM/τa = 0.15 which is incident in a 100 Ω transmission line. Here, τa = h/c is the time required by light to travel the length of a dipole arm, and tFWHM is the full-width half-maximum of the Gaussian pulse, which is expressed by
where V0 is the maximum amplitude of the pulse. In Figure 3, the graphs in the upper row show the reflected voltages in the feeding transmission line as functions of time, and the graphs in the lower row show the radiated fields as functions of retarded time (tr = t − r/c).
 In the graphs, the waveforms for the RVDs with 26 and 47 resistors for each arm look clean and simple; however, the waveforms for 11 resistors for each arm have ripples in the tails. Thus the resistive profile discretized by 11 resistors poorly approximates the continuous profile, which is well approximated by 26 and 47 resistors. Because we want to minimize the number of resistors, we choose the discretization using 26 resistors. Figure 4 shows the Wu-King profile discretized with 26 chip resistors. In the figure, the Wu-King profile is graphed in normalized conductance. The discrete profile is plotted by dots at the locations of the chip resistors.
3. Substrate Selection
 The substrate also affects the performance of the antenna. First, to investigate the effects of the substrate thickness, the radiated field of the RVD is obtained from the numerical model for a number of substrate thicknesses. Because an RVD is printed on a finite-size substrate, the numerical model should also include a finite-size substrate. However, EIGER cannot efficiently calculate the currents on the RVD on a finite-size substrate. This problem can be alleviated by assuming that the finite-size substrate can have a significant effect on the current distribution on the antenna, but it does not have a significant effect on the fields radiated by the currents. This allows the radiated fields to be calculated in a two step process. First, the current distribution on this antenna is obtained with the antenna on an infinite slab whose thickness is equal to that of the finite-size substrate. Second, the radiated fields from these currents are obtained in free space ignoring the dielectric slab.
Figure 5 shows the radiated fields of the RVDs when the antennas are driven by a Gaussian voltage pulse with tFWHM/τa = 0.15 incident in a 100 Ω transmission line. Note that the radiated waveforms as well as their amplitudes depend on the substrate thickness. The waveform is degraded with increasing substrate thickness. The reason for this is that the speed of current propagation on the antenna varies as a function of substrate thickness.
Figure 6 compares the currents of an RVD with and without a 1.45-mm-thick FR-4 substrate at 11 equally spaced points along the dipole arm. The currents are plotted as functions of time and vertically displaced according to the distance from the drive point when the antennas are driven by a Gaussian voltage pulse with tFWHM/τa = 0.15 incident in a 100 Ω transmission line. As the current pulses travel away, the current pulse of the RVD with 1.45-mm-thick FR-4 substrate appears later in time than that of the RVD without a substrate. Thus the existence of the substrate slows down the speed of current propagation. Note that the current on the RVD with the substrate propagates at approximately the same velocity along the entire length of the arm.
 To find substrate-related design parameters such that the substrate minimally slows down the current speed, we developed a simple model. First, note that the speed of current propagation (v) can be related to the medium surrounding the current as
where εre is the effective relative permittivity of the surrounding medium. Thus we have to find substrate-related parameters such that εre becomes as close to 1 as possible. The effective relative permittivity can be obtained by noting that the wave front is approximately spherical as it originates at the drive point (Figure 7). For this wave front, the geometry of the RVD looks approximately like a pair of coplanar strips. Thus the effective relative permittivity experienced by the wave front after it propagates a certain distance away from the drive point can be estimated using the formulas developed for the coplanar strip geometry:
where a is the distance from the symmetry plane to the inner edge of the strip line, w is the width of the metal strip, d is the thickness of the substrate, εr is the relative permittivity of the substrate material [Gupta et al., 1996], and K is the complete elliptic integral of the first kind [Abramowitz and Stegun, 1972].
Figure 8 shows the effective relative permittivities obtained using equation (4) for two RVDs, i.e., (Figure 8a) 2α = 60° on a substrate with εr = 4.2 and (Figure 8b) 2α = 44° on a substrate with εr = 3.4. The effective relative permittivity is plotted as a function of substrate thickness for a number of metal strip widths. For the graphs, the parameter a is obtained by first drawing a circle of radius h/10 centered at the drive point of the RVD and then taking the arc length from the symmetry plane to the inner edge of the metal strip.
 Assuming the current speed is approximately constant through out the arm, we can check the accuracy of the simple model. The current speed is obtained from the numerical model by first drawing a line along the locations of the pulse maximums in Figure 6 and measuring the slope of the line (v = r′/t). The current speed can be related to εre using equation (3). The two dots in Figure 8a mark εre's predicted by the numerical model and the simple model for the RVD in Figure 6. The difference is small (Δεre = 0.04) considering the simplicity of the simple model.
 The two graphs in Figure 8 show that the effective relative permittivity decreases with decreasing substrate thickness d and increasing metal strip width w. It is also evident from equation (4) that a smaller εr results in a smaller εre. Thus, in order to decrease εre, one has to print wide metal strips on a thin substrate with a small εr. However, note that the metal strip width must be chosen such that it is comparable with the width of chip resistors.
4. Performance of the Realized RVD
 A resistive vee dipole has been designed and realized based on the discussions in the previous sections and in the work of Kim . The interior angle has been chosen to be 44°, the substrate has been chosen to be a 0.05-mm-thick Kapton® (εr = 3.4), and the metal strip width has been chosen to be 3 mm. For this design, εre is approximately 1.02 at r′/h = 0.1 as marked by a dot in Figure 8.
Figure 9a shows the realized RVD. The arms are printed 3 mm wide on a vee-shaped 0.05-mm-thick Kapton® substrate with an interior angle 44°. Each arm is loaded with 26 surface-mount chip resistors. Because the 0.05-mm-thick Kapton substrate is thin and fragile, it is attached to a blank FR-4 substrate of thickness 1.45 mm to enhance the mechanical strength. To minimize the effect of the FR-4 substrate, the FR-4 substrate is cut out underneath the RVD. The picture of the FR-4 substrate is shown in Figure 9b.
 To see the performance of the realized RVD, the reflected voltage in the transmission line and the radiated fields at a number of observation angles have been measured. The measured results are then compared with the results from a numerical model. In this numerical model, the geometry of the RVD is the same as the realized RVD, but the modeled RVD radiates in free space without a dielectric substrate. Thus the comparisons show how the realized RVD functions closely to a theoretical RVD, which does not have a substrate.
Figure 10 shows the reflected voltages in a 100 Ω transmission line as functions of time for a Gaussian voltage pulse with tFWHM/τa = 0.15 incident in the transmission line. The figure compares the voltage obtained from the measurement with the voltage obtained from the numerical model for the realized RVD. The figure shows that the agreement is good, and therefore, the realized RVD on the substrate works as well in terms of the reflected voltage as the theoretical RVD without a substrate.
 The radiated fields are measured by a small dipole probe shown in Figure 11, which is placed at an angle in the plane of the RVD at r = 2.72 m away from the drive point of the RVD. The dipole probe is 4 cm long and connected to a 100 Ω balanced transmission line. In Figure 12, the voltage (VL) across the dipole probe is plotted as a function of time and vertically displaced according to the angular location of the dipole probe. The solid lines represent the results from the numerical model where the RVD does not have a substrate, and the dotted lines represent the results from the measurement. The agreement is good, and therefore, the realized RVD works well in terms of the radiated field. The amplitudes of the radiated fields are slightly larger. This is believed to be from a small error in the calibration procedure.
 Because the RVD has a symmetrical geometry, a balun structure is required. Two balun structures have been tested for use with the RVD. The RVDs with identical baluns are placed r = 2.72 m apart facing each other (Figure 13a), and the voltage at the balun input is measured. Figure 14 compares the measured results with the numerical results. In the numerical model, the RVDs have ideal baluns.
Figure 14a shows the results with a balun assembly whose schematic diagram is shown in Figure 13b. The balun assembly is made using a Picosecond Pulse Labs Model 5315A Balun and a pair of 60-cm-long 50 Ω semirigid coaxial cables. The coaxial cables are connected to the two output ports of the balun, which are 180° different in phase. The outer conductors of these cables are connected together forming a 100 Ω balanced transmission line. The 60-cm-long cable pair provide a time window of approximately 5.8ns in which no multiple reflections exist between the antenna and the balun. Figure 14b shows the results with double-Y baluns with characteristic impedance Z0 = 188 Ω, which are designed and analyzed by Venkatesan and Scott .
 The graphs show that the RVD works well with the double-Y balun. The measured amplitudes are lower than the ideal cases due to the balun insertion losses. In Figure 14b, a bump seen at tr/τa ≃ 0.6 is believed to be from a reflection inside the balun.
 In addition, the RVD is matched better to the double-Y balun than to the balun assembly. Figure 15 shows the voltage standing wave ratios (VSWRs) in four feeding transmission lines, whose characteristic impedances are Z0 = 100, 200, 300, and 400 Ω. The figure shows that VSWR is large for Z0 = 100 Ω at low frequencies and for Z0 = 400 Ω at high frequencies. The antenna is matched better for Z0 = 200 or 300 Ω.
 In order to minimize multiple reflections between the antenna and the ground, the antenna must have a low RCS. The multiple reflections complicate the radar signals making it more difficult to detect buried objects. The RCS is obtained from the numerical model by first sending a plane wave toward the opening of the RVD and then taking the power reflected back in the opposite direction. The RCS is affected significantly by the impedance matching. Figure 16 shows the RCSs of the RVD for three feeding transmission line impedances. The figure also shows the RCS for an ideal feeding transmission line, whose characteristic impedance is the complex conjugate of the input impedance of the antenna at each frequency (Z0 = Z*in). As expected, the RCS is seen to be the lowest for the conjugate match. The RCS is degraded about 13 dB for Z0 = 200 or 300 Ω.
 A practical way to build an RVD was proposed and discussed. In the proposed method, the RVD is printed on a circuit board and loaded with off-the-shelf surface-mount chip resistors. To minimize the effect of the substrate, we printed wide metal strips on a thin low-permittivity substrate. The thin substrate was backed by a thick FR-4 substrate to enhance the mechanical strength. The RVD was made easily and inexpensively with this method, and the resulting structure was mechanically stable.
 The performance of the realized RVD was investigated through experiments. The results showed that the performance of the RVD on the PCB was as good as the RVD without a substrate. The realized RVD was tested with two types of baluns, and the waveforms are compared with those of the RVDs with ideal baluns. The test showed that the RVD with the double-Y balun worked better than the RVD with the balun assembly.
 The RVD is seen to work well with a transmission line whose characteristic impedance is around 200 or 300 Ω in terms of VSWR and RCS. However, the mismatch is still significant, and the RCS is increased by about 13 dB due to the mismatch. Thus the future research should be focused on developing a matching network or optimizing the structure of the RVD to improve the match.
 This work is supported in part by the U.S. Army RDECOM CERDEC Night Vision and Electronic Sensors Directorate, Countermine Division.