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Phase and group propagation in metamaterial-based microwave components has always been intellectually challenging for students and engineers new to the area of periodic structures and metamaterials. This paper aims in tackling this important topic by studying the wave propagation in a metamaterial-based microwave device. Hence, the contribution of this paper is twofold. First, design of a novel metamaterial ring hybrid (or rate-race) is presented which has a large rejection band so that the second and third harmonics are effectively suppressed. Second, the electromagnetic phase and the group propagation in the ring hybrid are investigated by numerically exciting the input ports with band-limited Gaussian pulses and then finding their responses at various locations in the device.

Metamaterial microwave components have gained increased popularity in the last decade [

Here, the minus (−) sign indicates the traditional hybrid and the plus (+) sign indicates the metamaterial hybrid. Assuming an operational frequency of 0.75 GHz and a system impedance of 50 Ω, a metamaterial hybrid can be readily obtained by replacing the quarter-wavelength lines with left-handed unit cells having a combined phase shift of +90˚. Following the methodology of Ref. [

The underlying wave propagation in the metamaterial hybrid differs from its traditional counterpart on the account of its unique dispersion characteristics [

The dispersion equation of the periodic structure which constitutes the metamaterial hybrid can be obtained by applying the Bloch-Floquet theorem to the metamaterial unit cell (

where A and D are the elements of the overall transmission matrix of the metamaterial unit cell which can be calculated by multiplying the matrices of individual elements. The Bloch impedance is given by:

The corresponding dispersion diagram along with the impedance plot of the metamaterial unit cell is plotted as a function of frequency in _{21} and S_{31} (_{11} and S_{41} plots shown in

The phase velocity of the propagated wave in the ring hybrid can be obtained by simply dividing the radian frequency ω by the Bloch phase constant k (v_{p} = ω/k). The phase velocity is plotted in solid line in _{g} = dω/dk) and is obtained by differentiating (2):

where L_{o} and C_{o} are the distributed inductance of the host transmission. ω_{H} and ω_{L} are given by:

As shown in

rected phase and group velocities will be further clarified in the Section 4 by observing pulse propagation at various locations in the hybrid.

To understand the signal propagation in the hybrid, a Gaussian excitation signal is applied at the input port and the output is detected at various locations. Numerically, the response to a modulated Gaussian input pulse is given by the following integral:

where α is the standard deviation of the pulse, ω_{o} is the modulating frequency, and V(ω) and θ(ω) are the mag-

nitude and phase of the frequency response. First consider the in-phase hybrid mechanism when an input Gaussian pulse with a variance of 5ns and 0.75 GHz modulating frequency is applied at port 1. As depicted in

To show the contra-directed phase and group wave propagation in the ring hybrid, the progression of the Gaussian pulse travels between ports 1 and 2 is analyzed. The Gaussian pulse is excited at port 1 and the output responses are calculated at two points which are separated by two and three unit cells from the input port. Since each unit cell is 4.6mm in length (assuming zero length for the lumped elements), the total separation from the input port is 9.2 mm and 13.8 mm respectively. The points of determination of the output response are schematically shown in

Consider the Gaussian pulse responses at the two designated points shown in

the direction which is opposite to that of the envelope, a signature behavior that is observed in the backward wave regime or the regime identified by the negative phase velocity. The propagation times for the 9.2 mm and 13.8 mm distances on the hybrid are approximately equal to 0.34 ns and 0.5 ns respectively. Hence the approx

imate group velocities from Equation (7) are given by 0.092c and 0.92c respectively which are very close to the values obtained from the dispersion relations (2) and (4) and are shown in

Two important properties of metamaterial-based microwave devices i.e. the harmonic suppression and the contra-directional phase and group velocities are studied. In this numerical study, a novel metamaterial 180˚ ring hybrid is designed and simulated at 0.75 GHz. The design procedure exploits the inherent filtering properties of the metamaterial transmission lines by placing a stop band in such a way that the first and second harmonics are effectively suppressed. The unique wave propagation in the hybrid is studied by exciting the input port with a band-limited Gaussian pulse of standard deviation 5 ns. The particular relation between the phase and group propagation can be observed both in the frequency-domain dispersion diagrams and the time domain simulations. In particular, the Gaussian pulse is shown to propagate at a group velocity of 0.095 c through the hybrid at the design frequency.

Omar Siddiqui, (2015) Numerical Investigation of Phase and Group Propagation of Time-Domain Signals in a Novel Band-Reject Metamaterial Ring Hybrid. Journal of Computer and Communications,03,10-17. doi: 10.4236/jcc.2015.36002