II. MATCHING NETWORK DESIGN USING REAL FREQUENCY DIRECT COMPUTATIONAL TECHNIQUE (RFDCT)

Matching network design is preferred to be done using “RFDCT” in which transducer power gain (TPG) of the double matched system seen in Fig. 2 is described in terms of the driving point immitances of the generator [Z _G or Y _G ], equalizer (antenna matching network, BPMN) [Z _B or Y _B ] and the load (V-SPMA antenna) [Z _L or Y _L ]. TPG for the V-SPMA antenna matching problem is given as [Reference Yarman5]

(1) $$T=\displaystyle{{1 - {{\left\vert {{G_{22}}} \right\vert }^2}} \over {{{\left\vert {1 - {G_{22}}{S_{in}}} \right\vert }^2}}}{T_{EL}}.$$

Fig. 2.

(a) Double matching problem, (b) cascaded connection of two lossless two-ports [G] and [EL].

In terms of the standard normalization resistor R ₀ = 50 Ω, unit normalized generator reflectance G ₂₂ and input reflectance S _in of [EL] structure are given, respectively, by

(2) $${G_{22}}=\displaystyle{{{Z_G} - {R_0}} \over {{Z_G}+{R_0}}}\comma \;$$

(3) $${S_{in}}={\eta _B}\displaystyle{{{Z_L}\lpar j\omega \rpar - {Z_B}\lpar - j\omega \rpar } \over {{Z_L}\lpar j\omega \rpar +{Z_B}\lpar - j\omega \rpar }}\comma \;$$

where Z _L (jω) is the input impedance data of the V-SPMA antenna to be computed by using its given input reflectance data S ₁₁(jω) over the frequency band of interest as

(4) $${Z_L}\lpar j\omega \rpar ={R_0}\displaystyle{{1+{S_{11}}\lpar j\omega \rpar } \over {1 - {S_{11}}\lpar j\omega \rpar }}.$$

Z _B (jω) is the back-end impedance of the [E] equalizer (BPMN) to be computed by

(5) $${Z_B}\lpar p\rpar =\displaystyle{{{a_1}{p^n}+{a_2}{p^{n - 1}}+\cdots+{a_n}p+{a_{n+1}}} \over {{b_1}{p^n}+{b_2}{p^{n - 1}}+\cdots+{b_n}p+{b_{n+1}}}}\comma \; \; \quad p=j\omega\comma \;$$

where p = jω is the Laplace variable. Z _B (p) is a positive real (PR) impedance function that is to be determined in the optimization process. The all pass function η _B (p) is given as

(6) $${\eta _B}\lpar p\rpar ={\lpar \!-\! 1\rpar ^{{n_{dc}}}}\displaystyle{{b\lpar \!-\! p\rpar } \over {b\lpar p\rpar }}\comma \;$$

where n _dc is the number of transmission zeros at DC. The TPG T _EL of the lossless two port [EL], which is composed of matching network and the antenna is given by [Reference Yarman5]

(7) $${T_{EL}}\lpar \omega \rpar =1\! -\! {\left\vert {{S_{in}}\lpar j\omega \rpar } \right\vert ^2}=\displaystyle{{4{R_B}\lpar p\rpar {R_L}\lpar j\omega \rpar } \over {{{\left\vert {{Z_B}\lpar p\rpar +{Z_L}\lpar j\omega \rpar } \right\vert }^2}}}\comma \; \quad p=j\omega\comma \;$$

where R _L and X _L are real and imaginary parts of the antenna input impedance Z _L (jω), respectively. R _B is the real or resistive part of back-end impedance function Z _B (p) and it is given by [Reference Yarman5]

(8) $$\eqalign{{R_B}\lpar {p^2}\rpar &=Even\left\{{{Z_B}\lpar p\rpar } \right\}\cr &=\displaystyle{{{A_0}{p^{2{n_{dc}}}}} \over {{B_1}{p^{2n}}+{B_2}{p^{2\left({n - 1} \right)}}+\cdots+{B_n}{p^2}+1}} \cr &=\displaystyle{{A\lpar {p^2}\rpar } \over {B\lpar {p^2}\rpar }} \geq 0\comma \; \forall \omega\comma \; }$$

where the denominator is an even polynomial given by B(p ²) = 1/2[c(p)² + c(−p)²], which is formed with an auxiliary polynomial $c\left(p \right)={c_1}{p^n}+{c_2}{p^{n - 1}}+\cdots+{c_n}p+1$ . Once the auxiliary polynomial coefficients {c _i ; i = 1, 2, …, n} and ${A_0}=a_0^2 \geq 0$ of the real part R _B (ω) are initialized, we can generate the error function ε as follows [Reference Yarman5]:

(9.i) $$\eqalign{{\varepsilon _i} &=T\lpar {p_i}\rpar - {T_t}\lpar {p_i}\rpar \cr & =\displaystyle{{1 - {{\left\vert {{G_{22\comma i}}} \right\vert }^2}} \over {{{\left\vert {1 \!-\! {G_{22\comma i}}{S_{in\comma i}}} \right\vert }^2}}}{T_{EL}} - {T_t}\lpar {p_i}\rpar \cr &=\displaystyle{{1 - {{\left\vert {{G_{22\comma i}}} \right\vert }^2}} \over {{{\left\vert {1 - {G_{22\comma i}}{S_{in\comma i}}} \right\vert }^2}}}\left\{{\displaystyle{{4{R_B}\lpar {p_i}\rpar {R_{L\comma i}}} \over {{{\left\vert {{Z_B}\lpar {p_i}\rpar +{Z_{L\comma i}}} \right\vert }^2}}}} \right\}- {T_t}\lpar {p_i}\rpar \comma \cr &{\rm for}\left\{{i=1\comma \; 2\comma \; {\rm \ldots }\comma \; nd\, \, {\rm and}\, \, {p_i}=j{\omega _i}} \right\}\comma \; }$$

where nd is the number of frequency data uniformly distributed within the optimization band. T _t (p) is a mathematically generated Butterworth or Chebyshev-type bandpass gain function to be used as the objective or target function curve to be tracked by the optimization algorithm such that

(9.ii) $$\eqalign{&{T_t}\lpar {p^2}\rpar = \cr &\quad \left\{\matrix{ {T_{bw}}\lpar {p^2}\rpar \comma \; {\rm Butterworth}\, {\rm type}\, {\rm bandpass}\, {\rm target}\, {\rm gain}\, {\rm function} \cr T_{ch}\lpar {p^2}\rpar \comma \; {\rm Chebyshev}\, {\rm type}\, {\rm bandpass}\, {\rm target}\, {\rm gain}\, {\rm function} \cr} \right..}$$

The sole purpose of the above formulated process is to determine the Z _B (p) = a(p)/b(p) impedance function as a realizable PR function belonging to the equalizer [E], i.e. the BPMN, by minimizing the error function (9.i) in a non-linear optimization process. Then, the corresponding LC lossless ladder network with resistive termination is obtained by applying the very well-known long division process to the Z _B (p) function. Resistive termination is replaced by an ideal transformer with transformer ratio [R = n ²:1] which completes the design [Reference Yarman5].

Design steps in Matlab code:

• • Enter the element number n of the BPMN to be designed.

• • Enter normalized frequencies ω _{s 1}, ω _{c 1}, ω _{c 2}, ω _{s 2} (lower stopband, lower corner, upper corner, upper stopband frequencies) to be used in constructing the Butterworth (or Chebyshev) template or target gain function of order 2n, T _t (p ²) as given in (9.ii).

• • Set $c=\left\{{{c_1} {c_2} \cdots {c_n} 1} \right\}$ and a ₀ with arbitrary real numbers and construct the optimization initial vector by combination of both as x ₀ = [ca ₀].

• • Execute an optimization function equipped by a suitable nonlinear optimization algorithm such that x = OptAlgorithm(‘OptFunc.m’, x ₀, options); OptAlgorithm could be chosen as lsqnonlin or fminsearch. Corresponding line is then typed as x = lsqnonlin(‘OptFunc.m’, x₀ , options); OR x = fminsearch(‘OptFunc.m’, x ₀, options);

• • Execute OptFunc.m, to minimize the error function given in (9.i).

• • After obtaining the solution vector x = [ca ₀] via a successful optimization, construct A(p ²) and B(p ²) even polynomials as in (8) using the scalar a ₀ and the vector [c].

• • Determine Z _B (p) = a(p)/b(p) input impedance of the antenna matching network (equalizer [E]), i.e. BPMN, and synthesize it to obtain the resulting LC lossless ladder network.

III. SIMULATION RESULTS

Designed BPMN matching network together with V-SPMA antenna seen in Fig. 3 is simulated in MWO [10] environment. Simulated gain curve of this BPMN–V-SPMA structure is seen in Fig. 4 in solid blue color. As understood from this curve, the new structure, i.e. the V-SPMA antenna equipped with the BPMN matching network, has the operating band ranging from 800 to 5200 MHz, a very wide frequency band in which many communication standards could be covered such as GSM, 3 G and Wi-Fi. Even though a large frequency band portion between 1.22–5.2 GHz has a very small gain deviation of ~0.66 dB, a relatively small frequency band between 800–1220 MHz has a large gain deviation of ~2.6 dB. This case is thought to be caused by the geometry of the antenna that is essentially designed for 2.4 GHz RFID applications and could be improved in a future work yielding a gain flatness alongside the whole 800–5200 MHz band.

Fig. 3.

MWO simulation setup for unmatched and matched antenna cases. (a) Unmatched antenna case: V-SPMA antenna without BPMN. (b) Matched antenna case: designed matching network (BPMN) together with V-SPMA antenna; BPMN–V-SPMA structure.

Fig. 4.

MWO simulation results: comparison of matched antenna gain (solid blue) and S ₁₁ (dashed blue) to unmatched antenna gain (solid red) and S ₁₁ (dashed red).

IV. DESIGN OF TRANSFORMERLESS MATCHING NETWORK VIA REOPTIMIZATION

Designed BPMN (of Fig. 3(b).) can be reoptimized to be able to obtain a new BPMN having 50 Ω resistive termination. To discriminate the BPMN of Fig. 3(b) from the newer reoptimized 50 Ω terminated version, we can name the former as BPMN_TR. Designed BPMN_TR has a transformer X ₁ with a turn ratio of 1.464. It is a challenging engineering issue to construct an UWB “real-world” transformer that is to be able to operate alongside a very wide frequency band ranging from 0.8 to 5.2 GHz. Therefore, to avoid from the hard issues of constructing such a transformer, we should prefer to re-design a 50 Ω terminated (i.e. transformerless termination) BPMN, via a method that may be called as reoptimization.

When the analytic form of the normalized immitance (impedance Z(p) or admittance Y(p)) [Reference Yarman5], is synthesized as a lossless two-port in resistive termination, one may end up with a transformer depending on the value of the termination resistance. “transformerless” or 50 Ω terminated network is always desired in the final synthesis, however, this can be achieved for low-pass design problems only (see p. 361, [Reference Yarman5] and also [Reference Köprü, Kuntman and Yarman11]). Indeed, the back-end impedance of the designed BPMN_TR given by

(10) $${Z_B}\lpar p\rpar ={R_0}\displaystyle{{1+{\Gamma _B}\lpar p\rpar } \over {1 - {\Gamma _B}\lpar p\rpar }}={R_0}\displaystyle{{g\lpar p\rpar +h\lpar p\rpar } \over {g\lpar p\rpar - h\lpar p\rpar }}$$

becomes Z _B (p = 0) = R ₀(g _{n +1} + h _{n +1})/(g _{n +1} −h _{n +1}) at DC, i.e. when p = jω = 0; where, the back-end reflectance is given as the ratio of “arbitrary” polynomial h(p) and “strictly Hurwitz” polynomial g(p) such that

(11) $${\Gamma _B}\lpar p\rpar =\displaystyle{{h\lpar p\rpar } \over {g\lpar p\rpar }}=\displaystyle{{{h_1}{p^n}+{h_2}{p^{n - 1}}+\cdots+{h_n}p+{h_{n+1}}} \over {{g_1}{p^n}+{g_2}{p^{n - 1}}+\cdots+{g_n}p+{g_{n+1}}}}.$$

By substituting h_n+1 = 0, (10) results Z _B (p = 0) = R ₀ g _{n +1}/g _{n +1} = R ₀ = 50 Ω. This means that, for low-pass designs only, a matching network would have a 50 Ω standard termination resistance, i.e. a “transformerless” design [Reference Yarman5, Reference Köprü, Kuntman and Yarman11].

To the best of our knowledge, for bandpass design problems, it is a challenging task to obtain 50 Ω termination via a procedure similar to the method mentioned above for low-pass designs. On the other hand, reoptimization would not be so easy if the optimum topology of the designed BPMN_TR is not available. This is owed to the RFTs such as RFDCT and simplified real frequency technique (SRFT) [Reference Yarman5, Reference Yarman6, Reference Köprü, Kuntman and Yarman11] that helps one to determine the optimum topology (suited for the considered problem) of the non-50 Ω BPMN_TR as the initial design to be able to pass to the reoptimization stage if desired [Reference Köprü, Kuntman and Yarman11].

A) Reoptimization of BPMN_TR via simulation tool in MWO environment

Once the RFDCT determines the topology of the BPMN_TR as in Fig. 3(b), the reoptimization is performed via the simulation tool in MWO or ADS (Advanced System Design, Agilent Inc.) [10, 12].

After removing the transformer X ₁ from the BPMN_TR network seen in Fig. 3(b) and preserving its topology as the same and keeping the termination resistance (which is the resistance of Port 1) constant at 50 Ω, simulation tool in MWO environment can try to reach and track a preset gain curve trajectory of around 0 dB along the desired band of interest, i.e. from 0.8 to 5.2 GHz. Running this simulation tool optimizes the element values of BPMN_TR (Fig. 3(b)), i.e. ideal lumped component values (cval_ideal), ideal microstrip elements (mval_ideal) as microstrip lines MLINs, microstrip tee junctions MTEEs, microstrip tapered lines MTAPERs, and vias VIAs, in a such a way that the resulting gain curve (solid pink in Fig. 5(d) if the simulation runs with tanδ = 0 meaning that FR4 laminate is assumed as if it is lossless; or solid blue if the simulation runs with tanδ = 0.015 meaning that a lossy FR4 laminate is used) could track as much precise as the desired target gain curve (solid blue in Fig. 4) and, more importantly, not to cause the upper corner frequency to decrease much less than the desired value of 5.2 GHz. The values of ideal components and the dimensions of the ideal microstrip elements of the resulting optimized BPMN circuit schematics (shown in Fig. 5(a)) are given in Table 1. The material used in the MWO simulation is an FR4 laminate having the specs of dielectric constant ε _r  = 4.33, dielectric thickness h = 1 mm, copper thickness T = 43 µm, loss tangent tanδ = 0.015. Furthermore, by replacing the ideal lumped components of Fig. 5(a) with 402 case chip inductor and chip capacitor models from Murata LQP15MN and GJM15 series [13], a new schematics in the cosimulation tool of ADS as seen in Fig. 5(b) is formed. The pseudo-physical layout-like final pcb design is also seen in Fig. 5(c) drawn with 402 case lumped element footprints together with the routed microstrip element traces. Running the cosimulation with Fig. 5(b) schematics (with tanδ = 0.015), has resulted gain (solid red) and S ₁₁ (dashed red) performances (seen in Fig. 5(d)) which are almost close to those (solid and dashed blue) of Fig. 5(a) (ideal) BPMN circuit.

Fig. 5.

Comparison of BPMN with ideal elements and BPMN with Murata lumped component models. (a) MWO sim.: reoptimized BPMN with ideal lumped and microstrip elements. (b) ADS cosim. : BPMN with lumped component models from Murata. (c) ADS layout with lumped component footprints (402 case) and microstrip traces. (d) Comparison of gain and S₁₁ performances: BPMN with ideal elements (with lossless and lossy FR4) versus BPMN with Murata lumped component models (with lossy FR4).

Table 1.

Values of ideal elements in BPMN in Fig. 5(a).

Note that the values of the lumped components used in the new reoptimized BPMN whose ideal, cosimulation and layout schematics are given in Figs 5(a), 5(b) and 5(c)), respectively, are standard values available in Murata 402 case chip inductor and capacitor series of LQP15MN and GJM15, respectively. Thus, we can produce the BPMN using Murata's standard microwave grade lumped components [13].

V. BPMN PRODUCTION AND ITS MEASUREMENT

The BPMN prototype board production effort consists of

• • processing the layout pattern in Fig. 5(c) on a FR4 PCB laminate using a prototyping machine with model name “Elevenlab” from MITS Electronics Inc. [14] available in the circuit laboratory [15] of Istanbul University;

• • soldering the Murata chip inductors and capacitors on the manufactured PCB; and

• • soldering wires through the vias.

Resulting assembled board that measures 23 mm × 10 mm = 0.90 inch × 0.40 inch, ready for vector network analyzer (VNA: 10 MHz–14 GHz Rohde-Schwarz [16] available in the microwave and antennas laboratory [17]) measurements, is shown in Fig. 6(a). The SMA connector on the left edge of the board is connected to Port-1 of the VNA and the right edge connector is connected to the V-SPMA antenna's input port as seen in Fig. 6(b).

Fig. 6.

Prototype BPMN–V-SPMA setup and its VNA measurement results. (a) Assembled BPMN board ready for VNA measurements. (b) Final VNA measurement setup: BPMN board connected to V-SPMA antenna (BPMN–V-SPMA structure). (c) S ₁₁ measurement result of BPMN–V-SPMA setup via VNA. (d) Gain and S₁₁ performance comparison: BPMN_cosim versus BPMN–V-SPMA (measured via VNA).

Note that the gain performance (solid black in Fig. 6(d)) of the BPMN–V-SPMA setup is computed via an MWO equation that uses the s1p touchstone file belonging to the measured S ₁₁ data (Fig. 6(c)) such that

(12) $$\eqalignno{ T\lpar {\omega _i}\rpar &=1 - {\left\vert {{S_{11}}\lpar {\omega _i}\rpar } \right\vert ^2}\comma \; {\omega _b} \leq {\omega _i} \leq {\omega _e}\comma \; \cr i&=\lcub 1\comma \; 2\comma \; ...\comma \; nd\rcub \comma \;} $$

where ω _i is the ith frequency at which the gain T(ω _i ) is evaluated. nd is the number of data uniformly distributed in the frequency axis ranging from f _b  = 100 MHz to f _e  = 14 GHz, where f _b and f _e are the start and the stop frequencies, respectively, in the measurement range of VNA.