Poc of microstrip circuit design

The considered PoC is a two-port distributed passive circuit. As illustrated by the 2-D design shown by Fig. 1(a), the circuit is constituted by two microstrip circular lines presenting physical inner radius, R ₁, outer radius, R ₂ and width, w. The two circular lines are connected in parallel. The access lines have physical length, l_port and width, w_port.

Fig. 1.

(a) 2-D design and (b) photo of the ring-stub circuit under study.

The top circular line is connected to an open-ended straight stub having physical length, l ₁, and width, w ₁, at its mid-point. The physical parameters of this PoC prototype are indicated in Table 1. The fabricated circuit prototype photographed in Fig. 1(b) is implemented on Cu-metalized FR4 dielectric substrate in microstrip technology. The substrate physical characteristics are addressed by Table 2.

Table 1.

Geometrical parameters of the circuit PoC

Table 2.

Physical parameters of the PoC circuit substrate

BP NGD function specification

The analysis of the ring-stub passive symmetric circuit is based on the 2-D frequency-dependent S-matrix:

(1)$$[ {S( j\omega ) } ] = \left[{\matrix{ {S_{11}( j\omega ) } & {S_{21}( j\omega ) } \cr {S_{21}( j\omega ) } & {S_{11}( j\omega ) } \cr } } \right]$$

with the angular frequency, ω. The main parameters are basically the transmission coefficient magnitude, and the reflection coefficient, respectively:

(2)$$S_{21}( \omega ) = \vert {S_{21}( j\omega ) } \vert $$

(3)$$S_{11}( \omega ) = \vert {S_{11}( j\omega ) } \vert $$

Then, the essential parameter is the GD defined by:

(4)$$GD( \omega ) = {-}\partial \arg [ {S_{21}( j\omega ) } ] /\partial \omega $$

Figure 2(a) represents the GD response of BP NGD ideal circuit. This BP NGD function is characterized by the cut-off angular frequencies, ω1<ω2, with:

(5)$$GD( \omega _1 < \omega < \omega _2) < 0$$

Fig. 2.

BP NGD circuit ideal responses: (a) GD, (b) S ₂₁ and (c) S ₁₁.

The associated NGD bandwidth (BW) is given by:

(6)$$\Delta \omega = \omega _2-\omega _1$$

The response is also characterized by a center angular frequency ω ₁ < ω ₀ < ω ₂ which can be defined by:

(7)$$GD( \omega _0) = GD_0 < 0$$

The behavior of the transmission coefficient characterized by attenuation limit:

(8)$$S_{21}( \omega _0) = A$$

is presented by Fig. 2(b). Then, the access matching can be limited with respect to the diagram of Fig. 2(c) by:

(9)$$S_{11}( \omega _0) = B$$

These specifications will serve the BP NGD electrothermal characterization in the rest of the paper.

To investigate the electrothermal NGD behavior of the circuit, an innovative experimental setup described in the following section was considered.

Principle of the experimental setup for the NGD electrothermal analysis

The electrothermal empirical study principle of microwave circuit is fundamentally aimed to determine the transmission and reflection coefficients of S-matrix proposed in equation (1) with the evolution of ambient temperature, T:

(10)$$[ {S( jf, \;T) } ] = \left[{\matrix{ {S_{11}( jf, \;T) } & {S_{21}( jf, \;T) } \cr {S_{21}( jf, \;T) } & {S_{11}( jf, \;T) } \cr } } \right]$$

It means that the empirical S-parameter data should be expressed in the function of a couple (jω = j2πf, T). To measure the S-parameters, we proposed the innovative experimental setup highlighted by an illustrative diagram in Fig. 3. The vector network analyzer (VNA) is facing the side of the test box. After calibration, two ports are respectively connected with a 1 m-long cable. The cable enters the test box through the hole on the side of the test box. At the same time, the two cables are relatively parallel and do not cross for a certain distance to avoid mutual interference. After putting it into the test chamber, fix both ends of the cable on the iron frame inside the test chamber with adhesive tape, and fix the thermometer. After placing, close the door of the test chamber to start the test. Of course, the VNA apparatus is needed to record the S-parameters. The thermal room specifications will be described in the following subsection.

Fig. 3.

Schematic diagram of NGD electrothermal test device.

Electrothermal test and the thermal room specification

Figure 4 shows the photograph of the electrothermal test setup. The employed VNA is referenced Agilent® AV3672B-S, which operates from 10 to 26.5 GHz. We use two identical SMA cables 1meter-length. Then, the main innovative part of this research work is the consideration of the thermal room with volume 1 m × 1 m × 1 m. This later one is referenced ESPEC® [35]. During the experimentation, the circuit under test (CUT) is placed in the room as indicated in Fig. 4. Then, it is connected to the VNA as previously depicted in Fig. 3. Prior to running the test, the VNA was calibrated in SOLT configuration at the ambient temperature, T_a = 20°C. We underline that the present study is limited to the case of temperature, T, higher than T_a.

Fig. 4.

Photo of the considered electrothermal experimental setup.

Then, we realize that such calibration remains valid to measure the circuit with the variation of the room temperature, ΔT = T−T_a<100°C. Moreover, the influence of the room temperature on the interconnection cables and CUT connectors is negligible.

Guideline of the thermal room test operation

The thermal room test operation can be summarized by the following steps:

• • Step A: Installation with respect to the diagram shown by Fig. 3. This step requires the familiarization to the use of VNA and the control of the temperature (alternating humidity and heat) with the thermal test chamber.

• • Step B: Connect the extension cable and put it into the high and low temperature (alternating humidity and heat) test chamber without a test circuit to ensure that the two cables entering the temperature chamber are equal in length and parallel. Close the thermal room and connect the NGD circuit to be tested on the no-load cable.

• • Step C: VNA calibration with respect to the range of the frequency of the CUT.

• • Step D: Measure the S-parameters of the cable with the temperature change when no load.

• • Step E: Adjust the room temperature and wait for the temperature stabilization.

• • Step F: Adjust the temperature box, the range of each rise is 4°C within the time of 5 min. After reaching a rise, keep the temperature stable during 1 min. Then, derive the S-parameter data from the VNA.

• • Step G: The previous step must be repeated until the temperature variation reaches 40°C. For the present case of the study, the users must save 11 groups of ring-stub circuit S-parameter data families.

To verify the feasibility of the robustness analysis of the NGD circuit under ambient temperature stress, a practical investigation will be presented in the next section.

Principle of the experimental setup for the NGD electrothermal analysis

To validate the BP NGD behavior of the CUT, comparisons between the simulated (“Sim.”) and measured (“Meas.”) results were carried out in the frequency band from 2.35 to 2.5 GHz. The simulated results were obtained from the commercial tool of electronic and RF/microwave circuit simulator ADS® from Keysight Technologies®. The ring-stub circuit operates as a dual-band NGD function. The simulated, and measured GDs and magnitudes of the reflection and transmission coefficients are plotted in Fig. 5(a) and in Fig. 5(b), respectively. It can be pointed out that these comparative results present a very good agreement. In addition, the simulated and measured GD responses are plotted in Fig. 5(c). These results confirm that the ring stub CUT behaves as a BP NGD function. Table 3 presents the comparison between the simulated and experimented results of BP NGD specifications. As expected, the CUT exhibits NGD center frequency of approximately f ₀ = 2.42 GHz and NGD value, GD(f_o), of −1.9 ns. It can be seen from Fig. 5(b) that in the test frequency band, the attenuation remains better than −4 dB and the reflection coefficient is better than −11 dB. The slight frequency shifts of NGD center frequency are mainly due to the fabrication inaccuracies, substrate effective permittivity tolerance, and losses versus the numerical computation accuracy.

Fig. 5.

(a) Reflection and (b) transmission coefficients and (c) GD responses of the ring-stub CUT prototype shown by Figs. 1 in the second bandwidth at ΔT = 0.

Table 3.

Comparison of BP NGD specifications from the ring-stub CUT

The electrothermal test results of the ring-stub circuit were generated with respect to the guideline indicated in Subsection C of section III. The experimented data results obtained after the post-processing will be analyzed in the two following subsections.

Interconnection cable calibration including the temperature stress effect

Before the NGD characterization of the ring-stub circuit, the interconnection cable was separately calibrated at each sample of the room temperature variation, ΔT, was progressively increased from 0 to 40°C step 1°C. During the test, the VNA was calibrated from 2.3 to 2.6 GHz to generate the thermal and frequency-dependent S-parameters of the employed interconnection cable which can be denoted:

(11)$$[ {S_{cable}( jf, \;T) } ] = \left[{\matrix{ {S_{11cable}( jf, \;T) } & {S_{21cable}( jf, \;T) } \cr {S_{21cable}( jf, \;T) } & {S_{11cable}( jf, \;T) } \cr } } \right]$$

The associated GD determined from the transmission coefficient is given by:

(12)$$GD_{cable}( f, \;T) = \displaystyle{{-\partial \arg [ {S_{21cable}( jf, \;T) } ] } \over {2\pi \;\partial f}}$$

Consequently, the obtained electrothermal mappings of the employed interconnection cable GD, transmission and reflection coefficients are displayed in Figs 6(a), 6(b) and 6(c), respectively. Significant variations of the interconnection cable S-parameter magnitudes can be observed. The NGD center frequency, f ₀, of the ring-stub circuit under temperature stress is considered to carry out the calibration. Accordingly, we consider the GD(f ₀), S ₂₁(f ₀), and S ₁₁(f ₀) empirical data of the tested cable summarized in Table 4. The influence of the interconnection cable can be quantified from this table. It can be emphasized from this empirical result that the cable response presents the variations of:

• • Interconnect cable GD presenting medium value of about mean(GD_cable(f_o))≈9.1 ns and variation max(GD_cable(f_o))-min(GD_cable(f_o))<0.1 ns,

• • Transmission coefficient having absolute variations of about max(S _21cable(f_o))-min(S _21cable(f_o))<0.2 dB,

• • And reflection coefficient with maximal value max(S _11cable(f_o)) lower than −23 dB.

Fig. 6.

Mappings of (a) GD, (b) S ₂₁ and (c) S ₁₁ for the interconnection cable versus frequency and temperature effect.

Table 4.

GD, transmission and reflection coefficients at the NGD center frequency of the interconnected cable

By taking into account the results of the cable calibration, the BP NGD characterization of the ring-stub circuit including the electrothermal robustness analysis will be discussed in the following paragraph (Fig. 7).

Fig. 7.

(a) GD, (b) S ₂₁ and (c) S ₁₁ of the interconnection cable at NGD center frequencies versus temperature effect.

BP NGD characterization after processing at ambient temperature on the ring-stub circuit

The heart of the thermal stress robustness investigation is explored in the present subsection. The robustness study of the BP NGD CUT is focused around the NGD frequency band. We choose the operation frequency band between 2.35 and 2.5 GHz. The following paragraph explains the calibration approach by considering the cable electrothermal effect.

NGD circuit calibration approach by taking into account the cable electrothermal effect response

The electrothermal observable data processing is elaborated from the measurement of the total circuit composed of the interconnection cable characterized in the previous subsection connected to the CUT. Accordingly, the total circuit S-parameters can be denoted by:

(13)$$[ {S_{total}( jf, \;T) } ] = \left[{\matrix{ {S_{11total}( jf, \;T) } & {S_{21total}( jf, \;T) } \cr {S_{21total}( jf, \;T) } & {S_{11total}( jf, \;T) } \cr } } \right]$$

As the interconnection cable with S-parameters given by equation (11) and the tested circuit with S-parameters given by equation (10) are very well matched in the NGD frequency band, we can deduce the CUT transmission coefficient from the following relation:

(14)$$S_{11}( jf, \;T) \approx S_{11total}( jf, \;T) $$

(15)$$S_{21}( jf, \;T) \approx \displaystyle{{S_{21total}( jf, \;T) } \over {S_{21cable}( jf, \;T) }}$$

It can be derived from equation (14) that the ring-stub CUT measured GD and transmission coefficient magnitude in decibel are respectively expressed by:

(16)$$GD( f, \;T) \approx GD_{total}( f, \;T) -GD_{cable}( f, \;T) $$

(17)$$S_{{21}_{dB}}( f, \;T) \approx S_{21total_{dB}}( f, \;T) -S_{21cable_{dB}}( f, \;T) $$

Based on the proposed approach, the empirical results are examined in the following paragraph.

Electrothermal data mapping processing of the ring-stub CUT

The thermal stress measurement of the ring-stub circuit was carried out by including the interconnect cable. The experimented data represented by S-parameter equation (10) were recorded after 11 min of the temperature change. After the thermal stress, the BP NGD characterization of the ring NGD circuit is performed by using previously suggested equations (13), (15), and (16). Similar to the analysis of the previous subsection, the analyses were based on the mappings of the ring-stub CUT GD, transmission and reflection coefficients displayed by Figs 8(a), 8(b) and 8(c), respectively. As expected, an interesting electrothermal effect is curiously observed. It can be seen in Fig. 8(a) that the ring-stub circuit conserve the BP NGD behavior despite the thermal stress. However, the NGD value and the NGD center frequency are decreasing inversely to the temperature variation. Moreover, it can be underlined from Fig. 8(a) that the NGD bandwidth is slightly increasing. Furthermore, in the NGD frequency range, despite the temperature stress, at the NGD center frequency, the attenuation depicted by Fig. 8(b) remains better than 3 dB and the ring-stub circuit access matching illustrated by Fig. 8(c) remains better than −11 dB.

Fig. 8.

Mappings of experimented (a) GD, (b) S ₂₁ and (c) S ₁₁ of ring-stub circuit versus (f,T).

In addition, similar to the normal operation of BP NGD circuits available in the literature, the transmission coefficient magnitude presents an electrothermal behavioral effect shown by Fig. 8(b) similar to the GD of Fig. 8(a). Based on the explored raw data, the robustness study related to the BP NGD specifications of the ring-stub NGD circuit will be examined in the following paragraph.

Robustness related to the BP NGD specifications

In the area of electronic and microwave engineering, the robustness study must be referenced by observable data. For the present study, we are presenting a BP NGD behavior of a microstrip circuit against thermal stress. For the best of the authors' knowledge, this is the first study of electrothermal BP characterization applied to microstrip circuit.

As aforementioned in Fig. 2, the BP NGD specifications are quantified by the NGD center frequency, NGD value, NGD bandwidth, attenuation and reflection coefficient of the CUT. Based on the data depicted in Fig. 8, we extracted the NGD center frequency, NGD value, and NGD bandwidth versus stress temperature which are plotted in Figs 9(a), 9(b) and 9(c), respectively. Then, the variations of the transmission coefficient, reflection coefficient, and insertion loss flatness versus stress temperature are shown in Figs 10(a), 10(b) and 10(c).

Fig. 9.

Plots of NGD (a) center frequency, (b) value and (c) bandwidth versus temperature parameter after test processing of ring-stub circuit.

Fig. 10.

(a) Transmission and (b) reflection coefficients at the NGD center frequency, and (c) attenuation flatness versus temperature of the ring-stub circuit after test processing value.

The overall BP NGD specifications are characterized by the NGD center frequency, f ₀, BW, Δf = f ₂ − f ₁, value, GD(f ₀), attenuation, $S_{{21}_{dB}}( f_0)$, matching, $S_{{21}_{dB}}( f_0)$, and insertion loss flatness, $S_{{21}_{dB}}( f_1) -S_{{21}_{dB}}( f_0)$ and $S_{{21}_{dB}}( f_2) -S_{{21}_{dB}}( f_0)$. Table 5 recapitulates the quantified BP NGD characteristics of the ring-stub circuit under temperature variation from 0 to 40°C with a step of 4°C.

Table 5.

BP NGD specifications of the ring-stub CUT versus temperature

It is noteworthy from Table 5 that the attenuation and access matching are still better than 4 dB and 11 dB despite the stress. However, we can point out that:

• • The NGD center frequency presents a shift of about: max (f ₀) − min (f ₀) = 18 MHz,

• • And the GD maximal variation is of about:max [GD(f ₀)] − min [GD(f ₀)] = 1.26 ns.