Design of coupled line diplexer

Figure 1 shows the parallel coupled line diplexer comprising of two second-order filters connected to 50 Ω line. The top section (Branch 1) in the diplexer is designed to operate at 2.4 GHz and having a pass band ranging from 2.28 to 2.55 GHz. The bottom section (Branch 2) is designed at a center frequency of 3.5 GHz having a pass band from 3.2 to 3.58 GHz.

Figure 1. Schematic of coupled line diplexer.

Figure 2. Smith chart showing matching of diplexer for branch 1 & branch 2. 2(a) Branch 1 - high impedance for higher band & 2(b) branch 2 - high impedance for lower band.

The even- and odd-mode capacitances of the individual filters have been calculated using equations (1[a] & 1[b]) respectively. Appropriate matching at the input reduces the interference between the two pass bands. This input matching can be done by using different lengths of a 50 Ω transmission line. The input line connecting to branch 1, which is operating at lower band should provide high impedance for band 2, thereby obstructing the passage of the signal and vice versa.

(1(a))\begin{equation}{Z_{0e}} = {Z_0}\left[ {1 + J{Z_0} + {{\left( {J{Z_0}} \right)}^2}} \right]\end{equation}

(1(b))\begin{equation}{Z_{0o}} = {Z_0}\left[ {1 - J{Z_0} + {{\left( {J{Z_0}} \right)}^2}} \right]\end{equation}

where, admittance inverter constants can be found using equations 2(a)–(c).

(2(a))\begin{equation}{Z_0}{J_1} = \sqrt {\frac{{\pi \Delta }}{{2{g_1}}}} \end{equation}

(2(b))\begin{equation}{Z_0}{J_n} = \frac{{\pi \Delta }}{{2\sqrt {{g_{n - 1}}{g_n}} }}\end{equation}

(2(c))\begin{equation}{Z_0}{J_{N + 1}} = \sqrt {\frac{{\pi \Delta }}{{2{g_N}{g_{N + 1}}}}} \end{equation}

The schematic of the diplexer circuit in Fig. 1 is simulated in advance design systems. Figure 2(a) and (b) shows S₁₁ curves in smith chart validating matching of the circuit at the input of branches 1 and 2, respectively.

The relation between the spacing “S” and height of substrate “h” is given [Reference Garg and Bahl34] as

(3)\begin{equation}0.05 \leqslant \frac{s}{h} \leqslant 2{ }for{ }{\in _r} \geqslant 1{\text{ }}\end{equation}

The lengths (W1, W2, W3, and W4) of coupled microstrip resonators are considered approximately as $\frac{{{\lambda _g}}}{2}$for bandpass regions of each filter section.

Theoretical calculation of coupled line section even- and odd-mode impedances for a given microstrip line configuration

Z_0o and Z_0e of second order coupled line filters are calculated using equations 4(a)–(a₁) [Reference Garg and Bahl34, Reference Kirschning and Jansen35] and are given in Table 1(a) and (b). The equivalent widths and lengths of microstrip lines calculated for this design are also given in Table 2.

Table 1. (a) Even- and odd-mode impedances of section 1. (b) Even- and odd-mode impedances of section 2

Table 2. Optimized dimensions of T-CLAD using MLG in mm

For a coupled microstrip line of width W, spacing “s” and height “h” fabricated on a substrate having ${\varepsilon _r}$ = 4.4, the following equations apply. For the case where,

\begin{equation*}0.1 \leqslant u\left( { = \frac{W}{h}} \right) \leqslant 10,{ }0.1 \leqslant g\left( { = \frac{s}{h}} \right) \leqslant 10,{ }1 \leqslant {\varepsilon _r} \leqslant 18\end{equation*}

${Z_0}_{{\text{odd}}}$ and ${Z_0}_{{\text{even}}}$ are calculated by following equations

(4(a))\begin{equation}{Z_0}_{{\text{odd}}} = {Z_0}_{{\text{surf}}}.\left[ {\frac{{\sqrt {\frac{{{\varepsilon _r}_{{\text{eff}}}}}{{{\varepsilon _r}_{{\text{eff}},o}}}} }}{{1 - \left( {\frac{{{Z_0}_{{\text{surf}}}}}{{{\eta _0}}}.{ }{q_{10}}.\sqrt {{\varepsilon _r}_{{\text{eff}}}} { }} \right)}}} \right]\end{equation}

(4(b))\begin{equation}{Z_0}_{{\text{even}}} = {Z_0}_{{\text{surf}}}.\left[ {\frac{{\sqrt {\frac{{{\varepsilon _r}_{{\text{eff}}}}}{{{\varepsilon _r}_{{\text{eff}},\,e}}}} }}{{1 - \frac{{{Z_0}_{{\text{surf}}}}}{{{\eta _0}}}.{ }{q_4}.\sqrt {{\varepsilon _r}_{{\text{eff}}}} { }}}} \right]\end{equation}

Where ${Z_0}_{{\text{odd}}},{Z_0}_{{\text{even}}}$are the odd- and even-mode impedances of the coupled microstrip lines. And ${ }{Z_0}_{{\text{surf}}}$ is surface impedance, ${\varepsilon _r}_{{\text{eff}}}$ is the effective dielectric constant, ${\eta _0}{\text{ }}$.is free space impedance, ${\varepsilon _r}_{{\text{eff}},\,o\,}$.is static odd-mode effective dielectric constant, ${\varepsilon _r}_{{\text{eff}},e}$ is static even-mode effective dielectric constant. The variables ${q_4}$ and ${q_{10}}$ are constants given by equations 4(n) and 4(o).

For $\frac{W}{h} \leqslant 1$ the effective dielectric constant is calculated as

(4(c))\begin{equation}{\varepsilon _r}_{{\text{eff}}} = \frac{{{\varepsilon _r} + 1}}{2} + \frac{{{\varepsilon _r} - 1}}{2}.\left( {\sqrt {\frac{W}{{W + 12h}}} + 0.04{{\left( {1 - \frac{W}{h}} \right)}^2}} \right)\end{equation}

When $\frac{W}{h} \geqslant 1$ the effective dielectric constant is calculated as

(4(d))\begin{equation}{\varepsilon _r}_{{\text{eff}}} = \frac{{{\varepsilon _r} + 1}}{2} + \frac{{{\varepsilon _r} - 1}}{2}.\left( {\sqrt {\frac{W}{{W + 12h}}} } \right)\end{equation}

In our design $\frac{W}{h} \geqslant 1$.

To calculate surface impedance ${Z_0}_{{\text{surf}}}$ we use the following equations from 4(e) to 4(g)

(4(e))\begin{align}{Z_0}_{{\text{surf}}} &= { }\frac{{{\eta _0}}}{{2\pi \sqrt 2 \sqrt {{\varepsilon _r}_{{\text{eff}}} + 1} }}.{\text{ln}}\left( 1 + \left( {4.\frac{h}{{{W_{{\text{eff}}}}}}} \right) \right.\nonumber\\ &\quad \left. .\left( \left( {4.\frac{h}{{{W_{{\text{eff}}}}}}} \right).{ }\frac{{14.{\varepsilon _r}_{{\text{eff}}} + 8{ }}}{{11.{\varepsilon _r}_{{\text{eff}}}}} \right) + temp \right)\end{align}

Where ${W_{{\text{eff}}}}$ is effective width of the line

(4(f))\begin{equation}{W_{{\text{eff}}}} = W + \frac{t}{\pi }.{\text{ln}}\left( {\frac{{4e}}{{\sqrt {{{\left( {\frac{t}{h}} \right)}^2} + {{\left( {\frac{t}{{W\pi + 1.1\pi }}} \right)}^2}} }}} \right).\frac{{{\varepsilon _r}_{{\text{eff}}} + 1{ }}}{{2.{\varepsilon _r}_{{\text{eff}}}}}\end{equation}

Where $'{\text{temp}}'$ is a constant and given by,

(4(g))\begin{equation}{\text{temp}} = \sqrt {16{{\left( {\frac{h}{{{W_{{\text{eff}}}}}}} \right)}^2}.{{\left( {\frac{{14.{\in _r}_{{\text{eff}}} + 8}}{{11.{\in _r}_{{\text{eff}}}}}} \right)}^2} + \left( {\frac{{{\in _r}_{{\text{eff}}} + 1}}{{2.{\in _r}_{{\text{eff}}}}}} \right).{\pi ^2}} \end{equation}

To evaluate ${\varepsilon _r}_{{\text{eff}},\,o\,}$ and ${\varepsilon _r}_{{\text{eff}},\,e}$we use equations 4(h) to 4(q)

(4(h))\begin{equation}{\varepsilon _r}_{{\text{eff}},o} = \left( {0.5\left( {{\varepsilon _r} + 1} \right) + {a_0} - {\varepsilon _r}_{{\text{eff}}}.{e^{ - {c_0}.{g^{{d_0}}}}}} \right) + {\varepsilon _r}_{{\text{eff}}}\end{equation}

(4(i))\begin{equation}{a_0} = 0.7287\left( {{\varepsilon _r}_{{\text{eff}}} - \frac{{{\varepsilon _r} + 1}}{2}} \right).\left( {\sqrt {1 - {e^{ - 0.179u}}} } \right)\end{equation}

(4(j))\begin{equation}{b_0} = \frac{{0.747.{\varepsilon _r}}}{{0.15 + {\varepsilon _r}}}\end{equation}

(4(k))\begin{equation}{c_0} = {b_0} - \left( {{b_0} - 0.207} \right).{e^{ - 0.414u}}\end{equation}

(4(l))\begin{equation}{d_0} = 0.593 + 0.694{e^{ - 0.562u}}\end{equation}

(4(m))\begin{equation}g = \frac{s}{h}\end{equation}

(4(n))\begin{equation}{\varepsilon _r}_{{\text{eff}},e} = \frac{{{\varepsilon _r} + 1}}{2} + \frac{{{\varepsilon _r} - 1}}{2}.{\left( {1 + \frac{{10}}{v}} \right)^{ - ae\left( v \right).{b_e}\left( {{\varepsilon _r}} \right)}}\end{equation}

Where,

(4(o))\begin{equation}v = \frac{{u.\left( {20 + {g^2}} \right)}}{{10 + {g^2}}} + g{e^{ - g}}\end{equation}

(4(p))\begin{equation}ae\left( v \right) = 1 + \frac{{{\text{ln}}\left( {\frac{{{v^4} + {{\left( {\frac{v}{{52}}} \right)}^2}}}{{{v^4} + 0.432}}} \right)}}{{49}} + \frac{{{\text{ln}}\left( {1 + {{\left( {\frac{v}{{18.1}}} \right)}^3}} \right)}}{{18.7}}\end{equation}

(4(q))\begin{equation}{b_e}\left( {{\varepsilon _r}} \right) = 0.564{\left( {\frac{{{\varepsilon _r} - 0.9}}{{{\varepsilon _r} + 3}}} \right)^{0.053}}\end{equation}

To evaluate constants ${q_4}$, ${q_{10}}$ given in equation 4(a) & 4(b) the following equations are used

(4(r))\begin{equation}{q_4} = \frac{{2*{q_1}}}{{{q_2}*\left( {{e^{ - g}}*{u^{ - {q_3}}} + \left( {2 - {e^{ - g}}} \right)*{u^{ - {q_3}}}} \right)}}\end{equation}

(4(s))\begin{equation}{q_{10}} = \left( {\frac{1}{{q2}}} \right)*\left( {{q_2}*{q_4} - {q_5}*{e^{\left( {\ln \left( u \right)*{q_6}*{u^{ - {q_9}}}} \right)}}} \right)\end{equation}

The constants ${q_1}$, ${q_9}$ required to evaluate ${q_4}$ and ${q_{10}}$ are given by

(4(t))\begin{equation}{q_1} = 0.8695*{\text{ }}{u^{0.194}}\end{equation}

(4(u))\begin{equation}{q_2} = 1 + 0.7519*g + 1.89*{g^{2.31}}\end{equation}

(4(v))\begin{equation}{q_3} = 0.1975 + {\left( {16.6 + {{\left( {\frac{{8.4}}{g}} \right)}^6}} \right)^{ - 0.387}} + \frac{1}{{241}}{\text{ln}}\left( {\frac{{{g^{10}}}}{{1 + {{\left( {\frac{g}{{3.4}}} \right)}^{10}}}}} \right)\end{equation}

(4(w))\begin{equation}{q_5} = 1.794 + 1.14*{\text{ln}}\left( {1 + \left( {\frac{{0.638}}{{g + 0.517*{g^{2.43}}}}} \right)} \right)\end{equation}

(4(x))\begin{align}{q_6} &= 0.2305 + \frac{1}{{281.3}}*{\text{ln}}\left( {\frac{{{g^{10}}}}{{1 + {{\left( {\frac{g}{{5.8}}} \right)}^{10}}}}} \right) \nonumber \\ &\quad + \frac{1}{{5.1}}*{\text{ln}}(1 + 0.598)*{g^{1.154}}\end{align}

(4(y))\begin{equation}{q_7} = \frac{{10 + 190*{g^2}}}{{1 + 82.3*{g^3}}}\end{equation}

(4(z))\begin{equation}{q_8} = {e^{\left( { - 6.5 - 0.95*\ln \left( g \right) - {{\left( {\frac{g}{{0.15}}} \right)}^2}} \right)}}\end{equation}

(4(a₁))\begin{equation}{q_9} = \ln \left( {{q_7}} \right)*\left( {{q_8} + \frac{1}{{16.5}}} \right)\end{equation}

In this design, considering the thickness of the line as t = 0.035 mm, the height of substrate h = 1.58 mm, ${\varepsilon _r}$ = 4.4, the following values are obtained for odd- and even-mode impedances are obtained for coupled line sections 1 & 2 designed for ${f_1}$ and ${f_2}$ frequencies for a given spacing “s” and width W of the lines.

Based on the impedances obtained, using microstrip line theory [Reference Garg and Bahl34] the width of the lines have been calculated. The design parameters have been optimized for better results and the optimized dimensions of the circuit are given in the following table.

Implementation of the design in 3D EM software

Figure 3. Proposed tunable attenuating diplexer using MLG pads. (a) Schematic with graphene pads & (b) simulated and measured S-parameters of proposed diplexer without graphene.

The proposed tunable coupled line attenuating diplexer (T-CLAD) structure simulated in 3D Electro Magnetic (EM) solver Ansys HFSS is shown in Fig. 3(a). After optimization, the design is realized on FR-4 substrate of thickness 1.58 mm and having electrical properties ε_r = 4.4 and tanδ = 0.025. Dimensions of the circuit labelled in Fig. 3(a) are given in Table 1. The measurements are carried out using Keysight network analyzer PNA N5224B. Simulated and measured results of circuit without graphene pads are plotted in Fig. 3(b). Later Graphene pads are placed at locations shown in Fig. 3(a). As Graphene is lossy material with high dielectric constant and high loss tangent [Reference Pereira, Hardt, Fantineli, MV and LE31–Reference Rubrice, Castel, Himdi and Parneix33], it is effecting the transmission parameters slightly. Due to the standard losses of dielectric, there exist some transmission losses initially in both pass bands.

Characterization of MLG pads

Preparation and testing of MLG pads

Commercially available high-purity multilayer graphene (MLG) of flake size 1–5 µm has been chosen for making graphene pads. Initially, this MLG is mixed with isopropyl alcohol (IPA) at 10 mg/ml. This mixture is sonicated for 20 minutes to get the graphene particles dispersed uniformly in the liquid. Now this dispersed graphene liquid is transferred to the desired position on the circuit carefully using a dropper as shown in Fig. 4(a). The IPA in the mixture gets evaporated after some time leaving the pure graphene as a resistive sheet. The resistance of the graphene pad depends on the thickness and area of the pad. Controlled dropping of graphene liquid is required based on the requirement. Applying bias to these pads varies the resistance. Figure 4(b) shows the plot of current vs applied voltage and resistance vs applied voltage of the pads placed in the circuit. Further increase in bias voltage (>6 V) leads to the breakdown of graphene pads.

Figure 4. (a) Preparation of Graphene pads & (b) Measured resistance of MLG pad and voltage vs current plot.

Results & discussion

The MLG pads are placed at the edges of the center coupled-line section of the design for both branches as shown in Fig. 5(a) (circled). Initial resistance of graphene pads is around 250 Ω. As graphene is a resistive sheet, the coupling between the sections is slightly disturbed and the insertion loss in the higher band has slightly increased. Bias is applied to each of these pads and the resistance of pads is varied in branch 1 and branch 2 separately. The fabricated prototype of proposed T-CLAD and corresponding Measured and simulated S-parameters of the circuit after placing the graphene pads are shown in Fig. 5(a) and (b), respectively.

Figure 5. Proposed T-CLAD and its S-parameters. (a) Fabricated prototype of T-CLAD with MLG pads & (b) S-parameters without biasing the graphene.

The size of graphene pads placed for branch 1 is 0.65 × 1 mm and for branch 2 it is 1.2 × 1 mm. A variable three-channel DC multiple power supply (Model Number: PSD3304) from Scientific company is used to apply DC bias to the graphene pads as shown in Fig. 6 (a). As the coupler sections inherently block DC, there is no need for a bias tee for additional protection while carrying out this experiment. When the graphene pad in branch 1 is biased with voltage varying from 0 to 6 V, there is a corresponding change in resistance of the graphene pad from 250 to 54 Ω. Due to this attenuation in branch 1 is varied from 4.6 to 10.3 dB. Figure 6(b) shows the simulated and measured S-parameters corresponding to the biasing of the graphene pad in branch 1. The graphene pad in branch 2 is also biased similarly, resulting in variable attenuation from 4.8 to 14.1 dB in the second operating band. Measured and simulated results obtained by biasing graphene pad in branch 2 are shown in Fig. 6(c). Figure 6(a) shows the test setup of the circuit and biasing of graphene pads in the fabricated circuit.

Figure 6. (a) Test setup of T-CLAD with graphene pads, (b) variation in lower pass band, & (c) variation in higher pass band.

Table 3. Comparison of proposed attenuating diplexer with existing literature

This circuit can be used to control the power levels of signals appearing at the two branches of the diplexer. By biasing the graphene pads in branch 1 and 2, the power levels of the two operating bands can be controlled simultaneously. This circuit has applications in reconfigurable Bluetooth, WLAN systems. This concept can be applied to any other desired bands of interest. Multiple graphene pads can also be placed to obtain additional attenuation. However, insertion loss of circuit might degrade by using more graphene pads. Table 3 gives the comparison of proposed attenuating diplexer with existing literature.

In references [Reference Xu and Zhu5, Reference Xu6] tunable diplexers are realized using varactor diodes. In reference [Reference Xu and Zhu5], a total of nearly 18 varactor diodes are used to obtain the desired tuning characteristics making the circuit very complex and power-hungry. In reference [Reference Xu6], a total of 10 varactor diodes are used achieve the desired characteristics. In both these circuits, the use of several diodes makes these circuits very cumbersome. In this paper, by using two very tiny graphene pads we are able to achieve considerable attenuation levels in the pass band with tolerable insertion loss using extremely low powers (around 6 mW). In reference [Reference Chen, Zhu, Liang and Fan9] the size of the two graphene pads is almost 10 times larger than the pads used in our design. The use of such large pads will result in larger surface resistance leading to higher insertion losses. This work has the advantage of using a very simple diplexer design and a relatively easier bias mechanism which consumes very low power for its tunable operation.