In this paper, a novel configuration of postvaractor loaded circular microstrip antenna is presented. A comprehensive cavity model analysis of such electrically thin antenna leads to very interesting results like compactness and high tunability. In earlier published works, varactor diode had been directly connected between the antenna and the ground plane. In this case, however, a metallic post is used to help connect the varactor diode in SOT-23 package to the ground. The metallic post acts as an inductance, and therefore the circuit changes from capacitive loading to series L-C loading resulting in vastly different results. The numerical results agree well with experimental results.
 In microstrip antennas (MSA) the impedance bandwidth is generally narrow with pattern bandwidth being usually many times larger. Therefore the general convention in MSA design is to evolve new techniques to enhance the impedance bandwidth. One of the many methods available for enhancing the useful frequency range of MSA is to load the structure with varactor diodes [Bhartia and Bahl, 1982]. By changing the bias level of varactor diodes, the resonant frequency is tuned. A tuning range of 20% was achieved with a 10V bias. Varactor diodes have also been used for design of polarization agile antenna [Haskins and Dahele, 1994, 1997]. In an earlier work [Waterhouse and Shuley, 1994] a varactor loaded rectangular MSA has been characterized using full-wave spectral domain integral technique. The dominant mode resonance is shown to have decreased by approximately 8% at 0V bias and the tuning range at 3V bias is about 16%.
 In Chakravarty and De , authors have presented a configuration of a circular patch loaded by a single shorting post. Analysis of such structure leads to the conclusion that asymmetric location of the shorting post leads to generation of dual band in TM11 mode. The analytical tool presented by Chakravarty and De  is further refined in Chakravarty and De  where the impedance formulation of the shorting post is altered to include a correction factor. Inclusion of the correction factor has lead to near accurate results for all the modes.
 In this paper we present a novel configuration of a postvaractor diode loaded circular patch. Here the varactor diode is connected between the upper patch and the bottom ground plane via a thin inductive post. In the previous articles [Chakravarty and De, 1999, 2002], it has been shown that accurate results can be obtained for electrically thin patch by considering the post as an inductor. In the present analysis, this method is further extended by including the equivalent circuit of the varactor diode in series with the shorting post. It is difficult to connect the varactor diodes in SOT-23 package directly across the thin patch and the ground plane. So for all practical purposes a pin will be required to connect the diode to the patch. It can be understood from the method that once the impedance of the post is removed from the load impedance, the resulting formulation leads to the case where only the varactor diode is connected to the patch directly. Such formulation is compared with the published results to verify the accuracy. In this paper the proposed structure is theoretically analyzed, and the computed results compare well with the experimental results.
2. Characteristics of Varactor Diode
 The operation of varactor diode is based on reverse-biased p-n junction. An increase in the reverse bias widens the depletion region between the p- and n-type substrates, and the junction capacitance is reduced. The junction capacitance as a function of reverse bias is given as [Chang, 2001]
where Cj(0) is the junction capacitance at zero bias, Φb is the contact potential (0.7V for silicon and 1.3V for GaAs) and M is a coefficient depending on junction doping profile. M = 0.5 for abrupt type varactor. For hyper-abrupt type varactor M varies from 0.5 to 5 depending on reverse bias.
 The equivalent circuit of the diode is modeled by a voltage dependent junction capacitance Cj(V) and a series resistance Rs(V), associated with the ohmic contact and the finite thickness of the epitaxial layer [Chang, 2001]. Since the depletion region expands as the bias is increased, the undepleted region becomes smaller which decreases the resistance of the structure. The series resistance should be as low as possible in order to keep the losses associated with the diode low. For a packaged diode (e.g., SOT-23), the package inductance Ls and the package capacitance Cp also affect the functioning of the diode. Usually, the effect of package capacitance on the characteristics of the diode is negligible and is therefore ignored. The impedance of a varactor diode can be obtained from the equivalent circuit as [Chang, 2001]
The varactor diode should be operated below a cutoff frequency. This cutoff frequency is given as
Since both Rs and Cj decrease as a function of the increased bias, the cutoff frequency is usually determined at zero-bias.
 In order to keep the diode in capacitive mode, the series resonant frequency of the diode should be much higher than the operating frequency. The series resonant frequency (SRF) can be obtained from the equivalent circuit and is given as
For low-loss designs such as antenna applications, fcutoff is usually much higher than the series resonant frequency (>10fr), which means that in case the series resonance frequency requirement is fulfilled, the cutoff frequency requirement is also fulfilled.
 The analysis is based on cavity model [Bahl and Bhartia, 1980] where it is assumed that the substrate is electrically thin (h ≪ λ0). The electric field within the substrate has only z-component and is nonvariant in z-direction. The magnetic field has essentially x and y components. The basic disk geometry with varactor loading is shown in Figure 1 The circular patch of radius r1 is loaded with a varactor diode at angular location 180° on the circumference of a concentric circle of radius r2 where r2 < r1. The varactor diode is connected between the ground plane and the upper patch via a conducting post of radius Δ. The coaxial probe is dc isolated from the varactor bias through a capacitor Cb. For the case of a probe-fed circular patch, this configuration may be assumed to be a series LC circuit. We assume that the inductive post along with varactor diode divides the patch into two concentric circles, namely, region I (0 < r < r2) and region II (r2 < r < r1).
 The expressions for electric field and magnetic field are obtained for the two regions and the solution is derived through the application of boundary conditions. The fields in region I can be written as
where Jn(X) is Bessel function of first kind of order n, ωnp and knp are the angular frequency and propagation constant for TMnp mode and C1 is a constant. Prime denotes a derivative with respect to its argument. The integer ‘n’ corresponds to the order of the Bessel function and ‘p’ denotes the pth zero of Jn′(knpr). The subscript ‘z’ is ignored for particular mode identification as the electric field is nonvariant along z direction.
 Similarly the expressions for the fields in region II can be re-written as
where Nn(X) is Bessel function of second kind.
 It is assumed that the diameter of the circular post is small. Such a thin post can be assumed to be replaced by a conductor in the form of a circular arc strip having arc length equal to diameter of the post coincidence with a circle of radius r2. For arc strip of small arc length the axial current may be assumed to be uniform along its width. This current is given as Ez/Zo where Zo is the impedance per unit length for the post. The impedance of such a post is given as [Chakravarty and De, 2002]
The disk radiator structure can be seen to be loaded with two reactance, namely, the post inductance Lp and the diode reactance Zd. These two reactive components are placed in series to each other and can be considered to be loading the disk resonator in shunt. The total impedance can be written as
The electric and magnetic fields given by equations (5)–(10) satisfy the following boundary conditions.
where α is the angle subtended by post at the centre of patch and we assume that complex wall impedance for the thin patch is negligibly small.
 Applying boundary conditions given by equations (13) and (14), two homogenous equations in Cn(1) and Cn(2) are obtained. For Cn(1) and Cn(2) to be nonvanishing the determinant of the equations so derived should be zero, which leads to
where t = r2/r1; T = sin(nα)/nα; x = knpr1 and ɛn = 1 for n = 0, ɛn = 2 for n ≠ 0. It is to be noted that XT represents the imaginary component of the total impedance as given in equation (12).
 The resonance frequency for a given mode ‘np’ is obtained by solving equation (15) where integer ‘n’ denotes the order of Bessel's function and ‘p’ corresponds to the pth zero of equation (15). The expression (15) is similar to the expressions given by Chakravarty and De [1999, 2002]. In this case, however, the results will be different due to the presence of varactor diode parameters in the load impedance represented as XT.
4. Evaluating the Varactor Diode
 The diode used for computing equation (15) is ZC836B (5% tolerance on capacitance values) from Zetex Semiconductor. This diode is packaged in SOT-23 style where total lead inductance Ls = 2.8 nH and stray capacitance is 0.08 pf [Chadderton, 1996].
 ZC836B silicon diode is a hyper-abrupt device with capacitance varying from 100 pf (2V) to approximately 17 pf (28V). For this diode the various parameters are as follows.
Table 1 lists the series resonant frequency (SRF) at different capacitance values. The series resonant frequency is measured using a vector network analyzer. The resonance is noted and using the known value of inductance, the effective capacitance is deduced. Since the proposed antenna structure shows reduction in dominant mode resonance, the various parameters of the loaded antenna are to be selected in such a way that the operating frequency is always less than the cutoff frequency.
Table 1. Series Resonant Frequencies for ZC836B at Increasing Reverse-Bias Voltages
Reverse-Bias Voltage, V
Typ. Cj(V) in pf (From Datasheet)
Series Resonant Frequency in MHz (Using Datsheet Values)
Measured SRF in MHz
Reconstructed Cj(V) in pf (From Measurement)
 The microstrip patch under consideration has the following dimensions: (1) dielectric constant = 2.2, (2) height of substrate = 0.79 mm, and (3) patch radius = 33.1 mm.
 To measure the tunability of different modes for a circular patch the feed location is to be chosen in such a way that all the modes are excited with different levels of efficiency. The patch radiators are excited by SMA coaxial connectors from underneath. The probe location determines the resonant resistance seen for a given mode. Resonant resistance for each mode will be different. The probe location should be selected in such a way so that the modes of interest are excited [Kishk and Shafai, 1986]. The resonance frequency for TMnp mode is given by [Bahl and Bhartia, 1980]
where χnp is the eigenvalue for TMnp mode, fnp is measured in GHz and re is the effective radius of the patch due to fringing fields.
Table 2 lists the results for the above patch when loaded with ZC836B. The feed point is dc isolated using a blocking capacitor of 220 pf (SMD). The measured return loss values are also given.
Table 2. Computed and Measured Resonance Frequencies of TM01 Mode for Circular Patch When Loaded With ZC836B Varactor Diodea
Resonant Frequency, MHz r2 = 30 mm, Δ = 0.95 mm
Resonant Frequency, MHz r2 = 30.65 mm, Δ = 0.3 mm
Resonant Frequency, MHz r2 = 20 mm, Δ = 0.95 mm
Here ɛr = 2.2, h = 0.79 mm, r1 = 33.1 mm.
 It is seen that with increasing reverse-bias, the resonant frequency for TM01 mode increases. Maximum reduction in frequency (over dominant mode for unperturbed patch) takes place when the post is located near the edge of the patch (t ∼ 1). Frequency tunability for this mode is wide, nearly 105% (as measured). However, useful frequency range is narrower considering the fact the return loss figure varies at different bias levels. For 2:1 VSWR boundary the useful range is 213 MHz. TM11 mode (1.762 GHz for unperturbed patch) does not display any tunability for this case. This agrees excellently with theoretical prediction. For this particular varactor diode the cutoff frequency varies from approximately 442 MHz (@0V) to 3.2 GHz (@−15V). For all practical purposes, the higher order modes are not affected by the varactor diode as these resonances are well above the cutoff. To observe the tunability for TM11 mode we use another varactor diode BB833 (from Siemen's) which is usable upto 2.5 GHz. This diode is available in package SOD-323, for whom Ls ≈ 1.4 nH. Table 3 lists the results. The capacitance value as reconstructed from measurements is seen to vary from 18 pf (@0V) to 3.35 pf (@15V). The blocking capacitor value is 47 pf.
Table 3. Computed and Measured Resonance Frequencies of TM11 Mode for Circular Patch When Loaded With BB833 Varactor Diodea
Cutoff Frequency, GHz
Resonant Frequency as Measured, GHz
Resonant Frequency as Computed, GHz
Here ɛr = 2.2, h = 1.59 mm, r1 = 33.1 mm, r2 = 30 mm, Δ = 0.15 mm, Ls = 1.4 nH.
 The tunability for this mode is 10% for a capacitance ratio of 5.37. Theoretical prediction using (22) and (23) shows the possibility of existence of (1, X) and (2, X) modes at frequencies lower than TM01 and TM11 modes for loaded patch. These modes can be theoretically generated at frequencies which are lower than the series resonant frequency for the shorting post in series with varactor diode. Figure 2 shows the variation of these modes with increasing reverse bias and compares it with measured resonance. In Figure 2 the series resonant frequency (SRF) for the post inductance in series with varactor reactance is also plotted. SRF separates the net loading effect into two zones, namely, capacitive and inductive. It is seen that TM01 is excited only when the net reactive loading is inductive in nature. The measured resonance closely follows both (1, X) and (2, X) modes and since these modes are closely spaced, possibility exists that the measured resonance is influenced by either or both the modes. The tunability for this mode is also very high, of the order of 78%. However, like the previous case the useful range is limited because of constraint in acceptable return loss levels. The useful range for a given feed location is of the order of 13.7% only. For Δ = 0.95 mm the tunability for lowest order mode in this case increases to 88.5% and that of TM11 mode is nearly 16.6%.
 We have denoted the lowest order mode as TM1X mode instead of denoting as TM11 mode. The reason behind this is the observation that with postvaractor loading TM11 and TM12 modes (as defined for unperturbed patch) still resonates nearly at the same frequency range as that for a unperturbed patch. In Figure 3, the computed values for first three TM1p modes are shown for different post locations. For this test case the various parameters chosen are as follows: ɛr = 2.2, r1 = 33.1 mm, Δ = 0.15 mm, h = 0.79 mm, Ls = 1.4 nH, Ctest = 10 pf.
 For P = 0, the resonant frequencies for TM11 and TM12 modes are 1.763 GHz and 5.103 GHz, respectively. The shorting post location r2 is altered from 2.0 mm to 32.0 mm and eigen-frequency computed using equation (15). Again, SRF divides the region into two reactive zones, capacitive and inductive, respectively. It is seen that in the region where net reactive loading is inductive in nature, TM11 and TM12 modes behave in identical fashion as they would behave for post loading only. However, in the capacitive zone, another mode can be excited for n = 1 whose resonance actually decreases with post being pushed towards the periphery of the patch.
 In Figure 4 the effect of variation in post location is compared for TM01 mode for two cases: one, with only post loading and two, with post in series with packaged varactor diode. For this case, Ls = 2.8 nH, Ctest = 100 pf. For both these cases, the resonant frequency is seen to decrease monotonically as the post moves towards the edge of the patch. However, for case I, the variation is much larger than case II. This is due to the fact that the varactor reactance now dominates the inductive impedance of the shorting post.
 It would be interesting to study the effect of varactor diode on resonance of microstrip patch without using the shorting post. To study this effect the diode used by Waterhouse and Shuley  is taken for analysis. The diode chosen is Alpha Schottky barrier diode DMF 3078 which has the following parameters: Cj(0) = 0.8 pf, Rs = 7Ω, Ls = 0.5 nH. In reference [Waterhouse and Shuley, 1994] a rectangular patch of following dimensions was loaded with the diode by drilling a small hole through the substrate and soldered into position. ɛr = 2.17, h = 0.7874, L = 13.6 mm, W = 16.4 mm, xp1 = 3.0 mm, yp1 = 0. For the unloaded patch the fundamental resonance is at 7.125 GHz. To compare, a circular patch of suitable dimensions giving a dominant mode resonance (unloaded) at 7.125 GHz was chosen and subsequently analyzed by loading it with the above diode parameters. These parameters are: ɛr = 2.17, h = 0.7874 mm, r1 = 7.9 mm, r2 = 1.742 mm.
Figure 5 compares the results. It is seen that the range of operation for both the patch shapes when loaded with same diode is nearly same. For the circular patch, however, the reduction in resonance and the tunable range is marginally higher. For the circular patch the dominant mode is TM11 mode. For this diode the SRF is at 7.95 GHz (@0V) and higher. Therefore, a significantly lower mode either (1, X) or (0, 1) is not seen.
 Package parasitic plays a crucial role in determining the mode structure of the loaded patch. For purely capacitive loading, only the conventional eigen-modes will exist at a reduced frequency. This is shown in Table 4.
Table 4. Computed Resonance Frequencies of TMnp Modes for Purely Capacitive Loading of a Circular Patcha
Resonant Frequency for (1, 1) Mode, GHz
Resonant Frequency for (2, 1) Mode, GHz
Resonant Frequency for (0, 2) Mode, GHz
Resonant Frequency for (1, 2) Mode, GHz
Here ɛr = 2.17, h = 0.7874 mm, r1 = 7.9 mm, r2 = 1.742 mm.
P = 0, unloaded
P = 1, Cj = 0.8 pf
P = 1, Cj = 0.348 pf
 If package inductance is now included, the effect changes sharply. The effects are shown in Table 5.
Table 5. Computed Resonance Frequencies of TMnp Modes for Varactor Loading of a Circular Patcha
Resonant Frequency for (1, 1) Mode, GHz
Resonant Frequency for (1, X) Mode, GHz
Resonant Frequency for (0, 2) Mode, GHz
Resonant Frequency for (1, 2) Mode, GHz
Here ɛr = 2.17, h = 0.7874 mm, r1 = 7.9 mm, r2 = 1.742 mm, Ls = 0.5 nH.
P = 0, unloaded
P = 1, Cj = 0.8 pf
P = 1, Cj = 0.348 pf
 Now, it is seen that (1, X) mode is excited in between the resonances for (1, 1) and (1, 2) mode. The asterisk is given because for this value of varactor parameters, the SRF(7.95 GHz) is very close to the eigen-value. Hence this mode may not be excitable at all for the particular capacitance value. TM01 mode is not found.
 Though the present paper considers the mode properties of the excited resonance of a postvaractor diode loaded antenna, the radiation pattern is of considerable interest. One of the highlights of the proposed configuration is the generation of TM01 mode at a frequency lower than the fundamental TM11 mode. This results in compact antenna. Also this particular mode is widely tunable with applied voltage. Thus it can be used as a compact and electronically programmable antenna. In Figures 6 and 7, the measured radiation pattern for this mode is presented. As predicted by theory, this mode has a null in the bore-sight of the elevation pattern. The azimuth pattern shows omni-directional properties. It is suggested that this mode is useful for indoor applications like WLAN where the antenna is made to look at the ceiling.
 The measured gain for the two lowest modes, namely, TM01 mode and TM11 modes are presented (Table 6). It is seen that the gain of TM01 is quite low. However, it can be improved by better matching through change in probe location. Proper impedance matching improves the efficiency of radiation [Kishk and Shafai, 1986]. The efficiency for the upper mode is approximately 48%. The efficiency is measured by comparing the measured gains of an unloaded patch and patch loaded with varactor diode.
Table 6. Gain Characteristics of Postvaractor Loaded Antennaa
Here ɛr = 2.2, h = 0.79 mm, Δ = 0.3 mm, r1 = 33.1 mm, r2 = 30.65 mm, V = −2.0V.
 This novel configuration of postvaractor loading of a circular microstrip patch antenna results in a compact reduced size antenna which is electronically tunable over wide range of frequency. The salient features of the proposed design can be listed as follows.
 1. The shorting post offers an inductive impedance. When placed in series with a varactor diode, it loads the patch with a series tuned circuit.
 2. The series resonant frequency of the load plays a crucial role in determining the modes that are to be excited. Below the SRF the net reactance of the load is capacitive in nature and in this zone of operation there exists a possibility of exciting TM1X mode which can have significantly reduced resonance as compared to TM11 mode for the unperturbed patch. For example, loading the patch of radius of 33.1 mm and h = 1.59 mm (dominant mode resonance = 1.738 GHz) with Cj(2) = 8.7 pf leads to dominant mode reduction of the order of 49.5% over the unperturbed value.
 3. Above SRF of the load, the net effect is inductive in nature and TM01 mode with resonant frequency lower than TM11 mode is excited. For example, when a patch with radius of 33.1 mm and h = 0.79 mm (dominant mode resonance = 1.762 GHz) is loaded with Cj(0) = 176 pf, the reduction in dominant mode resonance is of the order of 79.2% over the unperturbed value. It is to be noted that for all such cases (1, X) mode exists theoretically at a lower frequency as compared to (0, 1) mode, but whether this mode can be excited depends on whether its resonance is well separated from SRF or not.
 4. Both these lower order modes (1, X) or (0, 1) are widely tunable and depends on capacitance ratio C(0)/C(15). For a ratio of 9, the dominant mode is seen to be tuned by 105%. The useful range, determined by 10 dB return loss value for a given feed location, however, is less. For this case the useful range or the effective bandwidth is 213 MHz or 62.2% as compared to the lowest frequency.
 5. TM11 mode is also tunable albeit over a smaller range. For a capacitance ratio of 5.7 the tunable range is 10%.
 6. For both (1, X) and (0, 1) modes the eigen-frequency decrease monotonically as the post is moved towards the edge of the patch. Maximum reduction in dominant mode resonance therefore takes place when the post with varactor diode is located near the periphery of the patch.
 7. Careful selection of varactor diode parameters like package parasitics can eliminate the lower order modes altogether. If the SRF of the load is selected to be much higher than the dominant mode resonance of the unperturbed patch, the lower order modes (1, X) and (0, 1) modes are suppressed. For such case, TM11 is still the dominant mode at a reduced resonance frequency.
 The proposed antenna system is a compact, tunable antenna. Being electronically tunable, it is possible to program this antenna using a microcontroller. As pointed out by Waterhouse and Shuley , this antenna system can be used to avoid threat frequencies by altering the bias voltage.