Consider a two-port network characterized by its ABCD parameters, as shown in Figure 1. Image impedance Z_i1 is the input impedance at port 1 when port 2 is terminated with Z_i2, and image impedance Z_i2 is the input impedance at port 2 when port 1 is terminated with Z_i1.

Figure 1: 
Two-port network terminated in its image impedance

In a two-port network, the two ports are matched when terminated with their image impedance. In terms of the ABCD parameters of the network, the image impedances of ports 1 and 2 are expressed in [14] as follows:

If the network is symmetric, A = D and Z_i1 = Z_i2.

Consider a periodic loading line of infinite length, as shown in Figure 2. Depending on the frequency and normalized susceptance values, the periodic line represents passbands or stopbands and thus can be considered as a type of filter. Each unit cell of this line consists of a transmission line of length d with shunt susceptance b that is normalized to characteristic impedance Z₀; k is the propagation constant. Here, I_n and I_n+1 represent the currents on both sides of the nth and n+1th unit cells, respectively. Similarly, V_n and V_n+1 represent the voltages on both sides of the nth and n+1th unit cells, respectively.

Figure 2: 
Periodically loaded transmission line’s equivalent circuit

If the finite line comprises a cascade connection of identical two-port networks, we determine the relationship between the voltages and currents on both sides of the nth unit cell using the ABCD matrix as follows:

The characteristic impedance at the end of the unit cell is called the Bloch impedance that is expressed in [14] as follows:

For symmetric unit cells, A = D; thus, Equation (4) can be simply expressed as follows:

where ± solutions imply the characteristic impedance of traveling waves in positive and negative directions. Except for the sign, these impedances are the same for symmetric networks. However, (5) can be used for periodic structures; but the structure need not have infinite unit cells.

Among the various radiating element shapes, the simplest rectangular patch was selected as the radiating element in this study. Because the electromagnetic field is concentrated between the signal pad and the ground pad, a lumped port, which is more suitable than a wave port, was used for the analysis. As shown in Figure 3, two lumped ports were attached to both sides of the radiating element, and the design parameters of the radiating element were optimized to resonate at an operating frequency of 79 GHz and maximize the antenna gain of a single radiating element. The optimized design parameters a, b, A, and B were 1.06 mm, 0.79 mm, 2.37 mm, and 3.18 mm, respectively.

Figure 3: 
Simulation setup for S parameters of a rectangular radiating element

To obtain the ABCD parameters, we assumed the radiating element to be a unit cell and converted the S parameters of the optimized radiating element, as shown in Figure 4, to the ABCD parameters.

Figure 4: 
Simulated reflection coefficient of the unit cell

By substituting the complex values of the ABCD parameters in Table 1 into Equations (1) and (4), we obtained the image and Bloch impedances. The result of the image impedance was the same as that of the Bloch impedance, and the absolute and real values of the calculated optimum impedance were 119 Ω and 90 Ω, respectively.

A conventional microstrip array antenna is required to verify the effect of the feed line on the performance of a series-fed microstrip. As shown in Figure 5, the rectangular radiating elements were arranged on the dielectric substrate at half-wavelength intervals (λ_g/2) and directly connected to the feed line. Therefore, all the radiating elements had the same phase characteristics. The end radiating element was connected to the inset-type feed line to match the feed line to the input impedance of the radiating element. Therefore, the inset-related parameters, w_i and depth, depended on the feed line. The series-fed microstrip array antenna consists of 16 identical rectangular radiating elements and a single inset-fed radiating element.

Figure 5: 
Geometry of a series-fed microstrip array antenna

The feed line width (W) of 119 Ω, which is the absolute value of the result obtained from the image parameter method and Bloch theory, was 0.054 mm on the Astra MT77 substrate at 79 GHz. The design parameters of the corresponding end radiating element were as follows: a_i = 0.56 mm, b_i = 1.071 mm, depth = 0.485 mm, and w_i = 0.137 mm. For the feed line of 90 Ω, which is the real value of the optimum impedance obtained from Equations (1) and (5), the line width (W) was 0.11 mm, and the matched end radiating elements had the following design parameters: a_i = 0.562 mm, b_i = 1.07 mm, depth = 0.487 mm, and w_i = 0.15 mm. When the characteristic impedance was 100 Ω, which is a common width in the literature for series-fed microstrip array antennas, the width of the feed line (W) was 0.08 mm. The design parameters for a series-fed microstrip array antenna with a characteristic impedance of 100 Ω were as follows: a_i = 0.69 mm, b_i = 1.0595 mm, depth = 0.5 mm, and w_i = 0.16 mm.

The simulated reflection coefficients of these series-fed microstrip array antennas are shown in Figure 6. The -10 dB reflection coefficient bandwidth of the simulation was 5.74 GHz at 119 Ω, 4.02 GHz at 90 Ω, and 5.52 GHz at 100 Ω. Notably, the feed line with an optimum characteristic impedance of 119 Ω increased the bandwidth by approximately 200 MHz compared with the common characteristic impedance of 100 Ω. Moreover, we note that the absolute value, rather than the real value, of the impedance obtained from the Bloch theory and image impedance method should be used as the impedance of the feed line.

Figure 6: 
Simulated reflection coefficients of the antenna with respect to the operating frequencies

The radiation patterns of the E- and H-planes simulated at 79 GHz are shown in Figure 7. The peak gains in the E- and H-planes were 17.33 dB for the characteristic impedance of 119 Ω with a feed-line width of 0.054 mm, 14.75 dB for the characteristic impedance of 90 Ω with a feed-line width of 0.11 mm, and 16.02 dB for the characteristic impedance of 100 Ω with a feed-line width of 0.08 mm.

Figure 7: 
Simulated radiation patterns at 79 GHz: (a) E-plane; (b) H-plane

The simulated antenna efficiency is shown in Figure 8. The antenna efficiencies were 97.55%, 96.13%, and 94.98% at 119 Ω, 100 Ω, and 90 Ω, respectively. The highest antenna efficiency was obtained using a 119 Ω feed line.

Figure 8: 
Simulated antenna efficiency at the operating frequency

For a feed line with a characteristic impedance of 119 Ω, the antenna gain was approximately 1.31 dB higher than that of a series-fed microstrip array antenna with a feed line of 100 Ω. Furthermore, the gain when using a feed line with a characteristic impedance of 119 Ω was approximately 2.58 dB higher than that when using a feed line with a characteristic impedance of 90 Ω. That is, when the absolute values of the optimal characteristic impedances obtained from the image parameter method and Bloch theory are applied, a larger antenna gain can be obtained than when the real values are applied.

The difference in gain between the characteristic impedances of 119 Ω and 100 Ω was linearly approximately 1.35 times; this is an effect that can theoretically reduce the number of radiating elements by 1.35 times compared with the conventional method. Therefore, the feed line of 100 Ω required 23 radiating elements to have the same gain as the series-fed microstrip array antenna with a feed line of 119 Ω and 17 radiating elements.