(b)
Fig. 1. (a) Geometry of basic (ATL) building (b) Lumped element equivalent circuit model of the design
The lumped equivalent circuit in Fig. 2 (d) to the impedance matrix [Z] 90o electrical lengths and certain characteristic impedance were used in the calculation for the first step of an ATL design.
[Z] = \(\par \begin{bmatrix}\frac{Z_{2\ }\left(Z_{3}+Z_{2}\right)}{{2Z}_{2}+\ Z_{3}}\ Z_{1}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{Z_{2}^{2}}{{2Z}_{2}+\ Z_{3}}\\ \frac{Z_{2}^{2}}{{2Z}_{2}+\ Z_{3}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{Z_{2\ }\left(Z_{3}+Z_{2}\right)}{{2Z}_{2}+\ Z_{3}}\ Z_{1}\\ \end{bmatrix}\) (3)
Where Z1 = jxL1, Z2 = -j/(wCb), Z3 = jwLc. L1,L2,L3 and w were the lumped element values in Fig. 2 (b) and angular frequency, respectivaly.The image impedance of the [Z], Z0, for the characteristic impedance of the ATL could then be calculated.
Z0 = \(\sqrt{z_{11\ }^{2}-z_{22}^{2}}\) (4)
From S22 was given as Eq.(5) and the component of S21 should be zero for any λ/4 transmission line of θ = 90⁰, we could easily obtain the relation of Eq.(6) for a 90⁰ artificial model transmission lines (ATL).
S22 =\(\frac{2Z_{22\ }Z_{T}}{{(Z}_{11}+\ Z_{T)\ }{(Z}_{22}+\ Z_{T)-\ }Z_{22\ }Z_{21}}\)= \(\left|S_{21}\right|e^{\text{jθ}}\) (5)
Z1Z3 + (2Z1 + Z3) Z2 + Z23 = 0 (6)
How to design an ATL with a Z0 characteristic impedance an electrical length of 90⁰. For the first step, we chose L and WTL x (x = 1, 2, and 3), considering the circuit size we needed, and obtained the LTL value using Eq. (1).
Branch line coupler
The proposed design BLC in this paper was a compact design based on microstrip artificial transmission lines (ATLs) with triangular slot patch and T-shape, as shown in Fig. 2, i.e., the coupler fabricated prototype. The horizontal and vertical TL’s respective impedances were Z0/\(\sqrt{2}\) and Z0, which were 35Ω and 50 Ω. The width(W) and length(L) arms of the BLC could be calculated by using the equations in [31]. In our design, the chosen centre frequency of the proposed design was 3.5GHz. The optimised dimensions of the triangular slot ATL with 50Ω and 90o electrical length were optimised as Wp = 14.00mm, W1=1.56 mm, W2= 1.31 mm, Wf = 4.0 mm, Wt = 0.40 mm, Wts = 0.30 mm, Wg = 0.21 mm, Wm = 1.2 mm, L1 = 1.78 mm, L2 = 1.66 mm, Lm = 13.8 mm, Lp = 4.23 mm, Lf = 1.5 mm, Lt = 4.80 mm, Lts = 4.10 mm, Lg = 0.75 mm, respectively. The dimensions for vertical and horizontal of the unit-call were optimised to achieve the BLC centre frequency of 3.5 GHz. The main component considered the BLC for the BM design. The proposed design coupler was simulated by )CST) microwave studio. Using the proposed design of the impedances, 3 dB BLC Eq.(7) and (8) were computed through the outputs phase difference ϕ = 90o with Z0 = 50 Ω, the value of the horizontal and vertical arms, which were Z1 = Z0 = 50 Ω, Z2 = Z3 = 35.36 Ω. Z1 = Z0 from Eq.12, which can be simplified as:
Z1 = Z0d │sin ⏀ │ (7)
Z2 = Z3 =\(\frac{Z_{o}d\ sin}{\sqrt{1+\ d^{2}\ \sin^{2\ }}}\) (8)\(\theta_{1}\)=\(\frac{\text{π\ }}{2}\) (9)
\(\theta_{2}\)=tan-1\(\left(\frac{\left(Z_{0}\tan\right)}{Z_{1}}\right)\)(10)
\(\theta_{3}\)=\(\pi\)–tan–1\(\left(\frac{\left(Z_{0}\tan\right)}{Z_{1}}\right)\)(11)
\(\text{\ \ }\theta_{2}\) = tan-1 (tan⏀) ⏀ (12)
Where, ϕ is the phase difference at 3.5 GHz frequency and (d2) is the power division ration for the coupler. From Eq (7), for a given phase difference of BLC as 90⁰, the electrical lengths were \(\theta_{2}\) =\(\theta_{3}\) = 90⁰. The simulated and measured result for the return loss and phase difference to BLC are summarised in Fig. 2. The simulated in Fig. 2(a) to BLC operated between two bands for scattering parameter and isolation loss of 2.91GHz to 4.02GHz and 2.72GHz to 4.16 GHz. The results were below -10dB. The phase difference to the simulated between the output ports was found to be 90⁰, as shown in Fig. 2(c).However, the measured result to BLC is as shown in Fig. 2(b) to the scattering parameter loss and isolation loss which operated between 2.62GHz to 2.5GHz. The phase difference to the measured between the output ports was found to be 87.5⁰, as shown in Fig. 2(c). This small loss error at 3.6GHz due the SMA and coaxial cables. The insertion loss to the simulated and measured was also indicated in Fig.2 (a-b) -2±0.1dB and -3±0.2dB, respectively. There small difference between simulated and measured results for the proposed design BLC due fabricated and tools used. All of these measurements were carried out using Keysight (AgilentTechnologies) FieldFox N9925A vector network analyser (VNA). The λ/4 for the artificial transmission lines (ATLs) triangular slot patch provided good wide bandwidth and size reduction. The performance parameters such as return loss, insertion loss, isolation loss, and phase difference between output ports could be analysed with VNA. The BLC with dimension is as shown in Fig. 3(a), while the unused ports were terminated with 50Ω SMA terminator ports when the measurement took place, are illustrated in Fig. 3(b). Table 2 summarized to comparison the proposed design BLC with the previously published work.