Equivalent Circuit Analysis and Matching Network

The impedance characteristics of the dielectric-loaded QHA can be observed from simple equivalent circuits. Because the configuration of BHA is constructed as a twisted pair line, the BHA model consists of a differential transmission line and resistor (Figure 2.8(a)). The characteristic impedance of the transmission line in the model is approximately 65 Ω. This value is similar to the simulated characteristic impedance of the twisted line in a BHA. The resistor includes the effects of radiation resistance and ohm resistance, whose value equals 1.3 Ω (obtained by curve fitting compared with full- wave simulation.) The high permittivity of the ceramic rod makes the antenna very small as compared with the wavelength in free space, and decreases radiation resistance dramatically. The small radiation resistance means that the loss tangent of the ceramic rod becomes very critical to antenna efficiency.

L₁

L₂ R_A

L₁

Antenna feed

Antenna feed

R_A R_A

(a) (b)

Figure 2.8 (a)The equivalent circuit of the proposed BHA. (b)The equivalent circuit of the proposed QHA.

1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62 1.63

Frequency (GHz)

-5 0 5 10

(i) Solid line: Impedance of QHA from equivalent circuit simulation (ii) Dashed line: Impedances of BHAs from equivalent circuit simulation (iii) : Impedance of QHA from full wave simulation

Im p ed an ce (o h m )

Real part

Imaginary part

Figure 2.9 Impedances from full-wave simulation and equivalent circuit simulation.

A resonant QHA can be considered as two resonant BHAs arranged orthogonally with quadrature-phase excitation. To simplify the feeding network, the phase quadrature is obtained using the self-phasing method such that only one set of feeding lines is required. By this method, two BHAs are fed in parallel (Figure 2.8 (b)), with one BHA slightly larger than the other. In the proposed design, the electrical lengths L

and L

of the equivalent transmission line of the two BHAs are 180.8° and 179.2°

at the QHA resonant frequency, what are inductive and capacitive, respectively, and cancel each other at the center frequency. The very small difference in length required to achieve circular polarization is based on the small input resistance of the antenna.

To compare full-wave simulation result with the equivalent circuit, the input impedance of the un-matched antenna was de-embedded to eliminate parasitic inductance from feeding lines. Figure 2.9 plots the calculated input impedances of the BHA and QHA from equivalent circuits, and the input impedance of QHA from full-wave simulation, where the reference plane is set as reference plane 1 (Figure 2.7). The calculated result for the circuit model agrees with the full-wave simulation, and the impedance of the QHA is the impedance of two BHAs connected in parallel.

Near the center frequency of the QHA, the impedance curve of an imaginary part

becomes flat and the curve of the real part reaches a maximum. The corresponding

Smith chart has a tip on the impedance curve around center frequency (Figure 2.10).

Sweep

: QHA with the parasitic inductance : QHA matched by adding a SMD capacitor : QHA with the parasitic inductance : QHA matched by adding a SMD capacitor L_s

Figure 2.10 (a) Schematic diagram of the matching circuit. (b)The simulated input impedance of the antenna at reference plane 2. The solid line represents the antenna without the capacitor. The dashed line represents the antenna matched by adding the capacitor.

The small input resistance is a critical problem when feeding the proposed antenna. To transform a very small resistance to 100Ω, the efficiency of the matching circuit should be considered. Furthermore, the matching circuit should also be small such that the entire antenna is compact. Figure 2.10(a) shows the proposed matching circuit, in which two inductors and a capacitor are used to match antenna resistance RA of 1.3 Ω at the center frequency. To transform the small resistance to 100 Ω, the required value of the inductors and capacitor are 0.6 nH and 8.5 pF, respectively.

Notably, a small inductor is utilized, which helps reduce loss and thereby increase the efficiency of the matching network. In practice, the two small inductors are implemented using two parasitic short sections of transmission lines. The large capacitor used is a surface mount device (SMD) component. The capacitor is shunted with feeding lines. Figure 2.7 presents the layout of the proposed matching configuration. The circuit size is very small and only one SMD component is used. By appropriately positioning the capacitor, the required length of the short matching transmission-line sections and, thus, the matching inductance can be obtained.

Figure 2.10(b) shows the simulated input impedances of the antenna, with and

matching capacitor, the input impedance is near 1.3 Ω with a small inductive reactance provided by the short transmission line sections from reference plan 1 to 2 (Figure 2.7). After adding the capacitor, the impedance can be matched to 100 Ω at the center frequency. Matching circuit efficiency depends on the loss of components.

Since the loss of a short transmission line section is quite low, it can be ignored. The parasitic resistor of the SMD capacitor is of greatest concern. According to the capacitor model from the Murata library [55], the parasitic resistance associated with an 8.5 pF capacitor is 0.33 Ω. The matching efficiency is at least 80%. To increase efficiency further, the single 8.5 pF capacitor can be replaced by two shunt 4.25 pF capacitors.

2.2.3 Experimental Results

In this work, a good RHCP pattern in the upper half space is designed. Figure 2.11 presents the method used to attain self-phased quadrature excitation for RHCP.

Two feed points are connected to the feeding lines on a circuit substrate for a differential input signal. The circuit substrate is rotated by a small degree, θ, relative to the symmetric plane. After rotation, the lengths of two BHAs become different; this difference proportional to the rotational angle (Figure 2.11). However, average length of two BHAs remains the same, implying that resonant frequency will not change because the resonant frequency of a QHA depends on average length of the two BHAs. Rotation makes the length of one BHA longer than the average length and the other shorter. The longer BHA exhibits an inductive impedance at the center frequency and the shorter BHA shows capacitive impedance, which provides the required quadrature-phase excitation for circular polarization. To obtain the RHCP in the upper half space, θ should be positive; otherwise, a negative θ causes LHCP in the upper half space and RHCP in the lower half space.

Antenna feed

: Rotational angle

θ

Symmetric plane

Circuit substrate

Figure 2.11 Rotation method rotates the circuit substrate for achieving circular polarization.

Frequency (GHz)

1.560 1.565 1.570 1.575 1.580 1.585 1.590

Axial Ratio (dB)

0 1 2 3 4 5 6 7 8

θ^{~ 1o} θ~ 2^o θ~ 5^o θ^{~ 8o} θ^{~ 1o} θ~ 2^o θ~ 5^o θ^{~ 8o}

Figure 2.12 The simulated axial ratio compared with different rotational angles.

A set of simulations for different rotational angles was conducted to identify a suitable θ for good circular polarization. Figure 2.12 presents simulation results; the frequency responses of the axis ratio for θ from 1–8° are shown. The circular polarization improves when the rotational angle approaches 5°, and then worsens as the angle is increased further. Although not shown, the simulated axial ratios for θ = 0° are large (30–40dB), meaning that the radiation field is primarily linearly polarized.

Notably, the 3-dB axial ratio bandwidth when θ = 8° is wider than that when θ = 5°;

however, θ = 5° has the best axial ratio at the center frequency. Figure 2.13 shows the variation of impedance for different rotational angles. Since the impedance before matching is quite small, the simulations include the matching circuit for enhanced observation. When θ is around 5°, the minimum axial ratio is achieved, and the circular polarization occurs at the tip (1.572 GHz) of the Smith chart. When exceeds 5

°, a circle instead of a tip occurs on the impedance curve. The formation of the tip is

Sweep

Figure 2.13 The simulated impedances with different rotational angles.

-30 -25 -20 -15 -10 -5 0

Figure 2.14 The simulated radiation pattern when rotational angle is 5^o.

Figure 2.15 The current distribution of the proposed antenna with input phases of (a)0^o and (b)90^o.

The input impedance bandwidth of the proposed antenna is determined by the matching structure. By using the matching circuit, the simulated 10-dB return-loss bandwidth for θ = 5° is 12 MHz, which is wider than the corresponding 3-dB axial-ratio bandwidth (7 MHz). Both the small impedance and axial-ratio bandwidths result from the small radiation resistance. Nevertheless, both the bandwidths are sufficient for application on the GPS L1 band. Figure 2.14 shows the normalized simulation radiation pattern at the center frequency. A good semi-spherical circular polarized pattern is obtained, which has a 3-dB beamwidth as wide as 130°. The good cardioid-shaped radiation pattern demonstrates that the overlapping region of two BHAs with a top metal ring does not influence radiation performance that is similar to that of the conventional resonant QHA. It also shows that the RHCP gain is 4 dB higher than the LHCP gain. This result agrees with the helix design of left-handed orientation. Figure 2.15 plots the simulated current distribution of the QHA at 1.572 GHz. The current distribution of each BHA is one-wavelength resonance, just like a loop antenna. At the center frequency, where circular polarization occurs, the two BHAs resonate in phase quadrature as predicted. The closed loop resonance for each BHA is accomplished using the top metal ring.

The proposed antenna was fabricated for experiments. The antenna is fixed to the circuit board via the protrusion. A microstrip Balun with 2:1 impedance transformation provides the differential signal. The SMD capacitor used for impedance matching is a Murata GRM1885C1H6R8DZ01¬. As mentioned, the

whose lengths are determined by the shunt capacitor location. By properly tuning the capacitor position, the return loss of the antenna matched better than 10 dB at the center frequency. Figure 2.16 presents the measured input impedances with rotational angles of 0°, 4°, and 8°. The condition of θ = 0° indicates that the two BHAs are the same length and the corresponding impedance curve changes monotonously, as predicted in the simulation. The antenna with the best circular polarization, where the axial ratio is 0.8 dB, is measured when the rotational angle is 4°, where a tip at the center frequency of 1.592 GHz forms in the impedance curve. The shift of resonant frequency and the required rotational angle is mainly due to manufacturing tolerance of ceramic permittivity and metal printing. When the rotational angle is increased to 8°, a circle rather than a tip occurs in the curve near the center frequency. Under this condition, the impedance bandwidth widens at the expense of degraded circular polarization; this is tradeoff between circular polarization quality and impedance bandwidth. Additionally, the center frequency remains the same with different rotating angles (Figure 2.16). The self-phasing method used is convenient for tuning circular polarization without changing the center frequency.

Figure 2.17(a) presents a photograph of the finished antenna, and Figure 2.17(b) depicts the measured radiation pattern at 1.592GHz with a 4° rotational angle. The radiation pattern is measured using an RHCP standard antenna. The semi-spherical pattern is omni-directional in the horizontal plane and agrees with that simulated. The corresponding 3-dB beamwidth is 150°, which is sufficiently wide for practical applications. The measured peak gain is −4.3 dBic and the 3-dB axial ratio bandwidth is 8 MHz.

0 1.01.0-1.0 10.0

Figure 2.16 The measured impedances with different rotational angles.

-35 -30 -25 -20 -15 -10 -5 0

Figure 2.17 (a)Photograph of the realized antenna. (b)Measured radiation pattern of the proposed antenna at 1.592 GHz.

2.2.4 Summary

A semi-spherical circular-polarized radiation pattern and reasonable gain for proposed QHA were achieved. The proposed self-phasing method for achieving circular polarization was flexible when tuning the axial ratio while retaining the same resonant frequency. Experimental results agree well with simulation results. The differential signal for antenna feeding can be provided by a LC Balun or a differential amplifier. The small input resistance of the miniaturized QHA has been matched successfully via a simple structure without occupying additional area. Both the feeding and matching structures are quite small, low cost and simple to manufacture, and easily integrated into circuit boards.

Chapter 3 Miniaturized Antennas with Circuit