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The entire microwave theory is based on Maxwell’s equations, whereas the entire electronic circuit theory is based on Kirchhoff’s electrical current and voltage laws. In this paper, we show that the traditional microwave design methodology can be simplified based on a broadside-coupled microstrip line as a low loss metamaterial. That is, Kirchhoff’s laws are still valid in the microwave spectrum for narrowband signals around various designated frequencies. The invented low loss metamaterial has been theoretically analyzed, simulated, and experimentally verified in both time and frequency domains. It is shown that the phase velocity of a sinusoidal wave propagating on the low loss metamaterial can approach infinity, resulting in time-space shrink to a singularity as seen from the propagating wave perspective.

Microwave theory and techniques, based on Maxwell’s equations, and electronic circuit theory and techniques, based on Kirchhoff’ electrical current and voltage laws, are traditionally two distinguishable disciplines in both university education and industrial research and development. It is known that Kirchhoff’s electrical current and voltage laws can be derived from Maxwell’s equations under a time-invariant or slowly changing field condition [

Now a challenging question is, can we find a way to accurately use Kirchhoff’s laws in the microwave and even mm-wave spectrums? If the answer is yes, microwave designs that are difficult for many electronic engineers can be simplified. Moreover, new circuitry architectures and topologies may be worked out for future communication technologies. Theoretically, it is possible to converge Maxwell’s equations and Kirchhoff’s laws for microwave and even mm-wave circuit designs, as shown in the next section. The precondition is to let both permittivity (

Metamaterial has become a research interest since early 2000s, when its existence was experimentally verified with man-made structures [_{0} = 8.854 x 10^{−12} F/m and_{0}^{−7} H/m, whereas the permittivity and permeability values in a medium are larger than these values. To date, various metamaterial structures in one, two, and three dimensions have been proposed and studied [

In this paper we present a low loss metamaterial structure based on broadside-coupled microstrip lines, which avoids the aforementioned problems. Moreover, it is compatible with traditional transmission line design, e.g., microstrip lines on a planar structure. Based on this low loss metamaterial structure, a detailed study around the transition region from left- to right-handed wave propagations on the metamaterial has been done. It is shown that at various designated frequencies, the phase velocity of a sinusoidal wave approaches infinity, leading to time-space shrink to a singularity and resulting in zero permittivity and zero permeability of the metamaterial. Consequently, a simplified design methodology based on Kirchhoff’s laws can be utilized for microwave circuitry analyses, when the low loss metamaterial is utilized at various designated frequencies above 1 GHz.

We start from the first principle, i.e., from Maxwell’s equations, and let both permittivity and permeability be zero, and then we study our proposed metamaterial structure with broadside-coupled microstrip lines.

The four Maxwell’s equations in differential form under the condition of null magnetic current source are described below [

Similarly, from (

It is apparent that Kirchhoff’s current law, i.e., (_{r}_{o}_{r}_{o}_{r}_{o} = 0, indicating no electric charge accumulation. When

Note that in the above derivations,

Figures _{1} = 0.17 mm and_{2} = 1.52 mm, respectively. The substrate material used is Rogers RO4350B with a dielectric constant

Unit dimensions.

| | | | |
---|---|---|---|---|

1 | 3.67 | 9.80 | 0.25 | 12.3 |

2 | 3.67 | 4.67 | 0.25 | 7.05 |

3 | 3.67 | 3.20 | 0.25 | 4.46 |

Metamaterial with broadside-coupled microstrip lines and short-circuit stubs with multiple vias. (a) Oblique-top view of a unit cell. (b) Side view of the unit cell. (c) Two or more unit cells are cascaded.

Figure _{p} is a series capacitance of the broadside-coupled line segment, either to the left or to the right of the middle short-circuit stab having a shunt inductance of_{s}. The shunt capacitance of the broadside-coupled line segment to ground is_{pg}. To simplify this model for analysis, we can choose the middle unit section between two cascaded unit cells with a length of

Lossless equivalent model of (a) the unit cell shown in Figure

Using the model shown in Figure _{L} and_{L}, instead of unit length capacitance and unit length inductance. The reason is that, unlike _{L} and_{L} are not linearly scalable with

It should be pointed out that, in a conventional transmission line theory [_{a} in Figure _{b}. As seen in Table _{b} = 12.3 mm, the unit cell length_{a}

The propagation constant

For a microstrip line, the quasi-TEM (transverse electromagnetic wave) mode can be utilized [

Substituting (

Under the following designated condition,

The phase velocity

The above theoretical derivations indicate that the phase velocity of a sinusoidal wave on the metamaterial shown in Figure

In order to make a detailed analysis around the transition frequency, i.e., _{d} become zero, independent of

The group velocity on the metamaterial described by (

The above derivations indicate that for a TEM-mode or sinusoidal wave with a constant frequency, e.g., a single frequency carrier for communication transceivers, the singularity of time-space, i.e., vanishing of time-space, is realized when the phase velocity

To verify the above analytical results, simulations in both time and frequency domains are done. The simulator used is Advanced Design System (ADS) version 2017 from Keysight. Parameters listed in Table ^{−8} Ωm.

Figure

Illustration of time domain simulations, when a layout component (

Figure _{d} is negative, indicating (see (_{d} is zero and thus_{0} according to (_{d} is positive, indicating (see (_{0}, i.e., the phase velocity (see (

Time domain simulation when the section number

Obviously, the results shown in Figures _{0}, the phase delay time shown in (

Figure _{m} = 19.6, 29.4, and 39.2 mm, are still much shorter than a wavelength of 300 mm in free-space or 157 mm in the substrate with a relative permittivity of_{r} = 3.66.

Time domain simulation of cascaded sections at 1 GHz, when

Simulation results similar to Figure

Due to symmetry of the line structure (see Figure

Simulation of S-parameters when_{21} phase response; zero phase delay at 1 GHz is seen.

Figure

Simulation of S-parameters, _{11} and S_{21} magnitudes (dB), and (b) S_{21} phase response.

Figure _{0}_{0 }

Simulation of S-parameters,_{11} and S_{21} magnitudes (dB), and (b) S_{21} phase response.

Figure _{0 }

Simulation of S-parameters,_{11} and S_{21} magnitudes (dB), and (b) S_{21} phase response.

For verification purposes, experimental samples were designed and fabricated in our printed circuit laboratory at Linköping University. Measurements in the time domain were done in our test and measurement laboratory with an oscilloscope WaveMaster/SDA/DDA 8 Zi-B from LeCroy, whereas measurements in the frequency domain were done with a vector network analyzer ZVM 20 GHz from Rohde & Schwarz.

Samples with the structure shown in Figure

Photo of fabricated samples.

Figure

Time domain measurement when

Figure

Measurement of S-parameters, _{11} and S_{21} magnitudes (dB), and (b) S_{21} phase response.

Measurement of S-parameters,_{11} and S_{21} magnitudes (dB), and (b) S_{21} phase response.

Both the simulation and experimental results have verified that the phase velocity of a sinusoidal wave can approach infinity at a designated angular frequency

Equation (

One probable doubt can be that the observed time-space singularity at a designated frequency in this study is just standing waves between the input and output ports such that the phase difference is zero between the two ports. However, the simulated S21 curves in Figures _{0}. All those results also support the analytical results of (

This study has started from the first principle of Maxwell’s equations and the transmission line theory. The only two approximations made are, first, the minimum unit cell length _{0} when both permittivity and permeability are near zero. This can be analyzed further with (_{0}. Therefore, a certain bandwidth can be utilized, in which both permittivity and permeability are near zero, but not exactly zero.

With a near-zero permittivity and permeability realized with the presented low loss metamaterial, microwave designs can be simplified. According to (

Illustration of time-space reduces to a singularity of the metamaterial shown in Figure

It is commonly understood that in a homogenous and right-handed material, a harmonic electromagnetic wave propagates with a phase velocity, but a modulated wave from a harmonic electromagnetic wave propagates with a group velocity. However, one must notice that phase, group, and propagation velocities are different concepts; the propagation velocity of a wave can be different from either a phase or a group velocity.

This study has shown that a phase velocity on our presented metamaterial can approach infinity at a designated frequency of_{0}, but the group velocity is only half of the light velocity in the same substrate of the metamaterial. This result agrees well with that from an electromagnetic field study [

Within a small bandwidth_{0}, a phase velocity on our presented metamaterial still approaches infinity. Thus, narrowband signals, e.g., a frequency-modulated signal with a bandwidth of_{0} or a phase-modulated signal of_{0}, can be transferred with near-zero time delay between the input and output ports, according to the principle illustrated in Figure _{0} propagates through the metamaterial with a phase velocity approaching infinity.

Using our invented low loss metamaterial of broadside-coupled transmission lines with short-circuit stubs, it is shown that a phase velocity of a traveling sinusoidal wave can approach infinity at various designated frequencies, resulting in zero permittivity and zero permeability. This means that the traveling sinusoidal wave experiences time stop and space shrink to a singularity, independent of its size. This property has been derived from theory, analyzed with simulation, and verified with experimental results.

With this low loss metamaterial, the traditional microwave theory and techniques can be simplified for narrowband signals around various designated frequencies where the phase delay is near zero. That is, the traditional electrical circuit theory based on Kirchhoff’s laws is still valid in the microwave or mm-wave spectrum for narrowband signals, utilizing a low loss metamaterial. Consequently, the microwave design methodology in terms of impedance matching, utilization of Smith Chart, and consideration of phase delay is not needed. Instead, the design methodology for analog circuits in terms of voltage loop and current divergence can be used directly. Moreover, some new circuit topology with zero phase delay of interconnects of arbitrary lengths can be worked out.

The simulation and experimental data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors thank Gustav Knutsson at Linköping University for fabrication of experimental samples. This work has been financially supported by the Faculty of Science and Engineering at Linköping University, Sweden. The second author also acknowledges the support by the China Scholarship Council (File No. 201706075057).