(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.