Multiplexer scattering matrix definition

The multiplexer S-parameters are calculated considering the scattering matrices of the individually designed filters ($S^{F_{n}}$) and the junction (S ^M) as it is depicted in Fig. 2[Reference Cameron, Kudsia and Mansour6]. The scattering matrix of the overall multiplexer [S ^P] can be written as follows,

(1)$$S^{P}=S^{PP}+S^{PC}\lsqb L-S^{CC}\rsqb ^{-1}S^{CP}\comma\; $$

where the matrices S ^PP, S ^PC, S ^CC, and S ^CP represent the relation between external ports P and internal ports C of the network, and L is the connection matrix that relates the internal ports C. All these matrices are properly defined in [Reference Macchiarella and Tamiazzo2] and are linked to $S^{F_{n}}$ and S ^M. The external ports P define the scattering matrix of the multiplexer network.

Fig. 2.

N channel multiplexer.

Ideal star-junction matrix implementation

According to the selected topology of the multiplexer, Fig. 1 illustrates in dashed line, an ideal star-junction. This nodal idealization is useful as a representation of the node where incident power in port 1 is distributed to all filters. Taking into account its geometry, this network is reciprocal and ideally lossless.

Considering a star-junction of P (P = N + 1) ports, the circuit theory provides a direct scattering matrix solution. This solution is a first basic model to evaluate the ideal junction behavior of multiplexers with zero electrical length from the filters to the junction. It would be straightforward to introduce the electrical delay in the side of the filter or through a reference plane shift over the junction matrix elements.

To analyze the junction behavior, port 1 and internal ports C from Fig. 2 are loading with a reference impedance Z _0i, considering Z ₀₁ = Z ₀₂ = ...Z _P = Z ₀. The input impedance can be calculated as

(2)$$Z_{in} = Z_{02} \Vert Z_{03} \Vert ...\Vert Z_{P} = {Z_{0}\over P-1}\comma\; $$

resulting in a network that is not matched at any port since Z _in ≠ Z0. The reflection coefficient at each port (S _ii) is calculated substituting (2) in (3) and S _ii is obtained as

(3)$$S_{ii} = {Z_{in}-Z_0\over Z_{in}+Z_0}={2-P\over P}\comma\; $$

For a lossless and passive device, its scattering matrix is unitary and

(4)$$\sum_{m=1}^{P}\left\vert S_{m 1}\right\vert ^{2}=\left\vert S_{11}\right\vert ^{2}+\left\vert S_{21}\right\vert ^{2}+\cdots+\left\vert S_{P 1}\right\vert ^{2}=1$$

Considering the network symmetry, S _ij = S _ji; i ≠ j so

(5)$$\left\vert S_{i i}\right\vert ^{2}+\lpar P-1\rpar \left\vert S_{i j}\right\vert ^{2}=1\comma\; $$

Substituting (3) in (5), S _ij can be expressed as

(6)$$\left\vert S_{i j}\right\vert ={2\over P}.$$

Finally, since no electrical delay is considered due to the ideal nodal approach, no phase contribution is comprised. Then, the representing matrix of the P ports star-junction is defined as follows:

(7)$$S_{\lsqb PxP\rsqb }^{M}=\left[\matrix{{{2-P\over P}} & {{2\over P}} & {\cdots} & {{2\over P}} \cr {{2\over P}} & {{2-P\over P}} & {\cdots} & {{2\over P}} \cr {\vdots} & {\vdots} & {\ddots} & {\vdots} \cr {{2\over P}} & {{2\over P}} & {\cdots} & {{2-P\over P}}}\right].$$

Filter matrix implementation

Once the junction matrix is obtained, it is necessary to compute the scattering matrix of each filter $S^{F_n}$. The use of a systematic methodology approach that allows the evaluation of filter parameters in its first-tier yields good results for later optimization stages. As a consequence of the demanding specifications of the telecommunication standards, the use of filtering function with transmission zeros at finite frequencies is necessary.

The methodology presented in [Reference Verdú, Evdokimova, de Paco, Bauer and Wagner7] addressed the filter synthesis with arbitrarily positioned attenuation poles at finite frequencies. The extracted-pole technique presented in [Reference Rhodes and Cameron8] and, more recently, the introduction of non-resonating nodes (NRNs) [Reference Amari and Macchiarella9] allows the synthesis of filters implemented by dangling resonant nodes (RN) and NRNs, suitably coupled to each other through an admittance inverter, J _r. This configuration results in higher modularity and flexibility in the synthesis, making it possible to create fully canonical networks without a direct source-load coupling. The low-pass nodal representation is depicted in Fig. 3 where the NRNs are implemented by a frequency invariant reactance B _i, and the RNs are represented by a unitary capacitor in parallel with a constant reactance b _i. The RN–NRN pairs are coupled by admittance inverters J _i.

Fig. 3.

Nodal diagram of a low-pass filter with dangling resonators.

For a defined set of transmission zeros (TZ), return loss level (RL), and filter degree (N), the scattering matrices of the filters individually designed $S^{F_n}$ are obtained following [Reference Giménez, Verdú and de Paco Sánchez4]. The reflection and transmission parameters are formulated as the ratio of two finite-degree polynomials and normalization constants ε and ε_r, where E(s), F(s), and P(s) are the characteristics polynomials of a generalized Chebyshev filtering function.

(8)$$\eqalign{ & {S_{11}\lpar s\rpar =e^{\,j \phi_{11}} {F\lpar s\rpar / \varepsilon_{r}\over E\lpar s\rpar }} \cr & {S_{21}\lpar s\rpar ={P\lpar s\rpar / \varepsilon\over E\lpar s\rpar }}. }$$

The phase term $e^{j \phi _{11}}$ introduces a shift in the original reflection coefficient phase. Since ϕ₁₁ is a real quantity, the magnitude of S ₁₁ is not affected. The proper calculation of this phase shift is essential to avoid the loading effect of the filters in duplexers and multiplexers, as will be seen later.

Multiplexer design approach

Most of the synthesis techniques related to multiplexers presented in the literature are based on iterative procedures or optimization techniques [Reference Wu and Meng10–Reference Yang, Wu, Yin and Yu12]. Both strategies demand significant computational time and resources. In general, designing multiplexers with an ideal star-junction configuration, where the interacting channel filters are directly connected in a common node, requires the proper combination of the imaginary part of the input admittance of all filters at the center frequency of each one as it is expressed in (9).

(9)$$\sum_{i=1}^{N} {\rm Im}\left(Y_{i n_{i}}\left(\,f c_{m}\right)\right)=0\comma\; \quad m=1\colon N\semicolon\; \ m \neq i.$$

Thus, the combined loading effect produced by other filters is minimized, and additional immittance compensation networks are avoided. The fulfillment of this condition depends strongly on the value of the reflection coefficient phase shift ϕ₁₁ of each filter.

The selection of the filtering function of each filter in multiplexers has been driven in the literature by the multiplexer classification as contiguous or not. Relying on this classification, there is a practical agreement for the use of doubly/singly terminated filters, respectively, in the early stages of the design for non-contiguous/contiguous channels [Reference Cameron, Kudsia and Mansour6].

For contiguous multiplexer designs based on singly terminated filters, dummy channels were required to imitate absent channels at the edges of the total multiplexer bandwidth, thus forming an additional annulling network for the compensation of the channel interactions. To avoid the addition of dummy channels or optimization techniques from the beginning, a direct approach is proposed in [Reference Silveira, Verdú and de Paco5] combining singly and doubly terminated filters independently designed under the MSN approach.

In [Reference Silveira, Verdú and de Paco5], it is established that in star-junction multiplexer configuration with N filters, inner filters (2:(N − 1)) are designed as MSN and outer filters (1 and N) are calculated as doubly non-MSN with the proper input phase adjustment. In case of MSN filters, ϕ₁₁ = 0^°. The phase calculation for non-MSN filters is made for channels 1 and N evaluating at the central frequency Ω_C of the adjacent filters 2 and N − 1, respectively, as follows

(10)$$\eqalign{ e^{\,j \phi_{11}} & =\varepsilon_{r} E\lpar s\rpar /\left.F\lpar s\rpar \right\vert _{s=j \Omega_{C 2}} \cr e^{\,j \phi_{1\comma N}} & =\varepsilon_{r} E\lpar s\rpar /\left.F\lpar s\rpar \right\vert _{s=j \Omega_{C\lpar N-1\rpar }}. }$$

Triplexer example

For a non-contiguous band triplexer example, the inner filter is designed as an MSN doubly terminated filter. The outer filters are calculated as non-MSN doubly terminated too, with the proper input phase adjustment. A triplexer compound of fully canonical 5th-order filters was designed. The synthesis input parameters are shown in Table 1.

Table 1.

Triplexer design specifications

Applying the theory developed before, S ^M can be obtained with (7), and $S^{F_{n}}$ is obtained with the synthesis methodology. Finally, the scattering matrix of the multiplexer S ^P is calculated applying (1). In the triplexer case, P = 4 and C = N = 3. The dimensions of each matrix are related to the quoted parameters above. The involved matrices in the matrix system for this example are expressed as

$$\eqalign{& S_{\lsqb {\rm PxP}\rsqb }^{M}=\left[\matrix{{-0.5} & {0.5} & {0.5} & {0.5} \cr {0.5} & {-0.5} & {0.5} & {0.5} \cr {0.5} & {0.5} & {-0.5} & {0.5} \cr {0.5} & {0.5} & {0.5} & {-0.5} }\right]\cr & S_{\lsqb {\rm PxP}\rsqb }^{PP}=\left[\matrix{{S_{11}^{M}} & {0} & {0} \cr {} & {S_{22}^{F_{1}}} & {0} & {} \cr {0} & {} & {S_{22}^{F_{2}}} & {0} \cr {} & {0} & {} & {S_{22}^{F_{3}}}}\right]\cr & S_{\lsqb 2 {\rm CxP}\rsqb }^{C P}=\left[\matrix{{S_{21}^{M}} & {0} & {} & {0} \cr {S_{31}^{M}} & {} & {0} & {} \cr {S_{41}^{M}} & {0} & {} & {0} \cr {0} & {S_{12}^{F_{1}}} & {0} & {} \cr {} & {} & {S_{12}^{F_{2}}} \cr {0} & {0} & {} & {S_{12}^{F_{3}}}}\right]\cr & S_{\lsqb P \times 2 C\rsqb }^{PC}=\left[\matrix{{S_{12}^{M}} & {S_{13}^{M}} & {S_{14}^{M}} & {0} & {} & {0} \cr {0} & {} & {0} & {S_{21}^{F_{1}}} & {0} & {} \cr {} & {0} & {} & {0} & {S_{21}^{F_{2}}} & {0} \cr {0} & {} & {0} & {0} & {} & {S_{21}^{F_{3}}}}\right]\cr & S_{\lsqb 2C\times 2C\rsqb }^{CC}\left[\matrix{{S_{22}^{M}} & {S_{23}^{M}} & {S_{24}^{M}} & {0} & {} & {0} \cr {S_{32}^{M}} & {S_{33}^{M}} & {S_{34}^{M}} & {} & {0} & {} \cr {S_{42}^{M}} & {S_{43}^{M}} & {S_{44}^{M}} & {0} & {} & {0} \cr {} & {0} & {} & {S_{11}^{F_{1}}} & {0} & {}\cr {0} & {} & {0} & {} & {S_{11}^{F_{2}}} & {0}\cr {} & {0} & {} & {0} & {0} & {S_{11}^{F_{3}}}}\right]\cr & L_{\lsqb 2C\times 2C\rsqb }\left[\matrix{{0} & {} & {0} & {1} & {} & {0} \cr {} & {0} & {} & {0} & {1} & {} \cr {0} & {} & {0} & {} & {0} & {1} \cr {1} & {0} & {} & {0} & {} & {0}\cr {} & {1} & {0} & {} & {0} & {}\cr {0} & {} & {1} & {0} & {} & {0}}\right]}$$

The proposed filter design methodology and the numerical analysis that has been conducted are technologically independent. Therefore, the experimental validation has been addressed using equivalent lumped reactance elements and microstrip technology for manufacturing simplicity. The dangling resonators in the low-pass model shown in Fig. 3 are transformed into bandpass domain, resulting in a ladder-type filter with resonators based on the Butterworth van Dyke (BvD) model. Figure 4 shows the equivalent electrical model of each channel. The input and output access lines of each filter were designed as 50 Ω lines. Its lengths were a priori fixed following a geometrical criterion and later integrated into the filter synthesis process due to its behavior as a phase shifter, in order to meet (9).

Fig. 4.

Circuital elements of 5th-order bandpass ladder-type filter based on the BvD model.

The synthesis procedure described in [Reference Giménez, Verdú and de Paco Sánchez4] provides the elements of the BvD model in the bandpass domain, allowing the fabrication of the prototype. Table 2 displays the synthesized values of the elements for each triplexer filter. Considering the available commercial values, most of the elements were split into two series inductors or parallel capacitors. The external parallel inductors (L _S, L _L) of filter F1 and F2 were dismissed because of its behavior as an open circuit at the triplexer frequency. The prototype manufactured in a single 1.52 mm-thick FR4 substrate is shown in Fig. 5. The triplexer has been accomplished using wirewound high-Q chip inductors and multi-layer high-Q capacitors from Johanson Technology. The inductors and capacitors Q-factor depends on the value of each element and the selected frequency; ranging from 35 to 45 for Q _L (Q-factor of inductors) and 500 to 3000 for Q _C (Q-factor of capacitors) for our design specifications.

Fig. 5.

Fabricated triplexer with lumped elements.

Table 2.

Bandpass elements of the Butterworth van-Dyke model for the triplexer filters

Electromagnetic simulation of the star-junction was carried out to evaluate the accuracy of (7). Figure 6 shows the excellent agreement between the solution obtained using the matrix approach and the simulation results. As expected, the higher the frequency, the higher the deviation from the theoretical value, but keeping very acceptable values to give an accurate result in the multiplexer response evaluation. The simulated and measured transmission, reflection, and cross-isolation responses are shown in Figs. 7, 8, and 9 respectively. As a consequence of the low Q-factor, especially Q _L, the F1, F2, and F3 insertion loss are kind of high: − 2.89, − 4.05, and − 4.19 dB, respectively. The triplexer performance is not good, but still demonstrates the theory presented in this work. The technology and frequency range were chosen to help with a fast and easy manufactured proof of concept, but the developed theory can be applied at higher frequencies and other technologies where other effects should be considered.

Fig. 6.

S-parameters comparison between the star-junction electromagnetic simulation and the matrix approach.

Fig. 7.

Simulated and measured transmission triplexer response.

Fig. 8.

Simulated and measured reflection triplexer response.

Fig. 9.

Simulated and measured cross-isolation response.

In order to demonstrate the minimization of the loading effect in the filter passband with the proposed methodology, filter F1 was measured detached from the triplexer. Figure 10 shows the excellent agreement between the measurements when the filter is part of the triplexer and when it is considered as stand-alone. In spite of the technology effects in the outcome, the experimental validation evidences the methodology accuracy to minimize the loading effect through the proper choice of ϕ₁₁.

Fig. 10.

Comparison between the filter F1 transmission response when it is measured as stand-alone and connected as part of the triplexer.