This example calculates the phase shift across a compact metamaterial-inspired transmission line (TL) phase shifter composed of alternating sections of negative refractive index (NRI) metamaterials and printed coplanar waveguide (CPW) TLs. The results from FDTD and TL theory are compared for a 1,2, and 4-stage NRI-TL phase shifter sharing a representative unit-cell design.
Background
Whereas the phase lags in the direction of group velocity in conventional positive refractive index (PRI) TLs, (hence negative phase as energy propagates away from a source), the phase leads in the direction of positive group velocity in NRI-TL metamaterials. This allows the phase of a PRI medium to be compensated by the phase of a NRI medium cascaded after it, allowing for a zero-degree phase shift across the two medium at a designed frequency. This example closely follows the work of Anoniades et al. [1] and investigates a NRI-TL phase shifter implemented in coplanar waveguide (CPW) technologies that is loaded by discrete (lumped) capacitors and inductors.
The above figure presents the TL model of one-unit cell of the NRI phase shifter which consists of a CPW TL possessing a characteristic impedance Z0 and propagation constant βTL loaded in series by a discrete capacitor of value C0 and in shunt by an discrete inductor of value L0. Periodic (Bloch) analysis is used to find the structure’s dispersion relation. When the periodicity d0<0 (i.e., βblochd0<<1 and βTLd0<<1 , the dispersion equation can be simplified to the form \( \beta_{e f f}=\beta_{T L}-\frac{1}{\omega \sqrt{L_{0} C_{0}} d_{0}}\), where ω is the angular frequency [1]. The phase shift across one-unit cell is then given by ϕ0=βeffd0. Each unit cell is referred to as a “stage”, in which multiple (N) stages are often used to achieve a desired phase shift (Nϕ0). The CPW’s physical and electric parameters (in essence, βTL) and the inductance and capacitance can be engineered to achieve a zero-degree phase shift at a design frequency f0 (i.e., Nϕ0(f0)=0). It is clear that this zero-degree phase shift condition is independent on the number of stages.
The above figure presents the TL model of one-unit cell of the NRI phase shifter which consists of a CPW TL possessing a characteristic impedance Z0 and propagation constant βTL loaded in series by a discrete capacitor of value C0 and in shunt by an discrete inductor of value L0. Periodic (Bloch) analysis is used to find the structure’s dispersion relation. When the periodicity d0<0 (i.e., βblochd0<<1 and βTLd0<<1 , the dispersion equation can be simplified to the form \( \beta_{e f f}=\beta_{T L}-\frac{1}{\omega \sqrt{L_{0} C_{0}} d_{0}}\), where ω is the angular frequency [1]. The phase shift across one-unit cell is then given by ϕ0=βeffd0. Each unit cell is referred to as a “stage”, in which multiple (N) stages are often used to achieve a desired phase shift (Nϕ0). The CPW’s physical and electric parameters (in essence, βTL) and the inductance and capacitance can be engineered to achieve a zero-degree phase shift at a design frequency f0 (i.e., Nϕ0(f0)=0). It is clear that this zero-degree phase shift condition is independent on the number of stages.
The file mtm_phase_shifter.fsp contains the simulation model of the NRI-TL phase shifter. The figure above present the ungrounded CPW TL used in this example, which consists of an infinitesimally thin strip of width w=4 mm separated on both sides by a gap of width g=0.1 mm placed on top of a substrate of height h=0.5 mm with relative permittivity εr=2.2. The metallic traces are modeled using 2D rectangles assigned as Perfect Electrical Conductors (PEC). More details on designing the CPW TL can be found in Wadell [2].
The above figure presents different levels of magnification of the simulation setup used to calculate the transmission parameters of a 4-stage metamaterial phase shifter. It consists of 4 CPW unit cells of periodicity d0=8 mm loaded in series by capacitor C0=12 pF and in shunt by an inductors of value 2L0=100 nH (doubled with respect to TL theory to account for parallel connection between the signal line and the two ground planes). The capacitor shares the same width (w) as the signal line to avoid parasitic capacitance and have a length gc=0.2 mm, while the inductor shares the same length as the gap (g) and have a width wl=0.2 mm.
A mode source is used to calculate and inject the CPW-TL mode. A DFT monitor (T1) and mode expansion monitor are placed just in front of the source to measure the power injected by the mode source in to the fundamental mode. An output DFT monitor (T2) is placed precisely across the CPW TL such that the distance between the input and output monitors corresponds exactly to 4d0=32mm. It should be noted that the placement of the T1 and T2 monitors on the unit cell can be in multiple locations along its length as long as the correct periodicity is maintained. See the Microstrip with a Lumped RLC Element example for more details on setting up the RLC element and the monitors.
A number of mesh override regions are placed throughout the simulation in regions the fields are highly confined including over the signal traces and over the capacitor and inductor. This ensures we properly resolve the dimensions of the TL and loading elements and accurately capture the highly varying nature of the fields. Furthermore, since the measured phase shift will be strongly dependent on the distance between the T1 and T2 monitors, mesh override regions are used over them to lock the mesh grid points along the z direction such that they land near to the monitors z location
Parametrize Number of Unit Cells
Often it is useful to setup the simulation to compute the phase shift over a number of different stages, which requires a parametrization of the number of unit cells in the simulation. This is easily done using the built in scripting in the model analysis group. In this example, this script functions to set the unit-cell’s dimensions, duplicate the number of unit cells, generate the mesh override regions, position the source and monitors, and set the size of the FDTD region in the direction of propagation. The above figure presents the list of variables that control the number of stages and the physical and electric parameters of each unit cell.
A parameter sweep is setup on the number of unit cells (N) using the optimization and sweeps tool. The above figure shows the simulation setup for the 1-, 2-, and 4- stage NRI-TL phase shifter.
Source Setup - Single Stage
For the mtm_phase_shifter.fsp file (which will be used to simulate a single stage), the frequency range is set from 0.875-1.125GHz and the "optimize for short pulse" setting in the mode source is turned on. This allows for a quick simulation time which is set to 15ns in the FDTD settings.
Source Setup - Multiple Stages
For the mtm_phase_shifter_sweep.fsp file (which will be used to simulate multiple stages), the frequency range is set from 0.5-1.3GHz. Due to the pulse's increased bandwidth, a small but significant DC component is injected into the simulation which causes the autoshutoff to not be reached. In the RF methodology discussion, a number of suggestions are given to reduce or remove this problem. To begin, the "optimize for short pulse" setting in the mode source is turned off. While this requires us to significantly increases the simulation time to 40 ns in the FDTD settings, the DC component is significantly reduced and autoshutoff level will be reached.
Results and Analysis
Single Stage
After opening the file mtm_phase_shifter.fsp and running the simulation, the script mtm_phase_shifter.lsf is used to compare the simulated and theoretical phase shift (phase S21) for a single stage. The simulation takes approximately 5 minutes to run on a good workstation. The theory assumes d0=8 mm, C0=12 pF, L0=100 nH, and εeff of the TL is found from the mode expansion monitor. The theory and FDTD results are in agreement.
Multiple Stages
After opening the file mtm_phase_shifter_sweep.fsp, the parameter sweep Unit Cell Sweep is used to run the simulation for 1-, 2-, and 4- stages. The complete sweep takes roughly 20 minutes to complete on a good workstation.
After the sweep has ended, the script mtm_phase_shifter_sweep.lsf is used to calculate the phase shift (phase S21) for each stage in the sweep and generates a plot (figure below) that compares the simulated results to theory. These simulated results were obtained with a finer mesh over the inductors, capacitors, and strip width. Nevertheless, the coarser mesh in the simulation file gives comparable results.
It is clear that as the number of stages increases the slope of the phase increases and larger phase shifts can be achieved. The simulated and theoretical zero-degree phase shift occurs f=1.06 GHz and f=1GHz, respectively, which is within a 6% percent error and is independent on the number of stages. It is encouraging to see the zero-degree phase shift frequency does not vary with the number of stages, agreeing with theory.
Simulation results generated in the script prompt using the script file mtm_phase_shifter_sweep.lsf.
=======Maximum Difference in Simulated and Theoretical Phase======== 1 - Stage - Delta Phase = 2.0deg / stage 2 - Stage - Delta Phase = 2.4deg / stage 4 - Stage - Delta Phase = 1.7deg / stage
The script also displays the maximum difference between the simulated and theoretical phase shift. The small difference between simulation and theory is due to the assumptions used to derive the simplified theoretical dispersion relation (small argument approximations and dropping of higher order terms). Using a more advanced circuit model and decreasing the mesh size on the inductors and capacitors would further improve the match.
Short Pulse Vs Long Pulse
Whereas the previous results were obtained with the "optimize for short pulse" setting off and the autoshutoff level is reached (but at the cost of a much longer simulation time), they are now compared in the above figure against the approach in which the "optimize for short pulse" setting is on and the autoshutoff level isn't reached (with the benefit of a much quicker simulation time). Generally, this is not advised because any residual energy in the system would impact the FDTD results. However, it is clear that in this case the phase of S21 is not impacted by the autoshutoff level not being reached for the shorter pulse. This is because the majority of the residual energy in the simulation will be at DC. This energy cannot propagate and stays localized inside the mesh grid in the source's injection plane and, to a lesser extent, inside the inductors. By placing the source far enough behind the T1 and T2 monitors, the simulated results are not impacted. This residual energy can be observed in the figure below, which plots the electric field intensity at the end of the simulation.
Related references
[1] Antoniades, Marco A., and George V. Eleftheriades., Compact linear lead/lag metamaterial phase shifters for broadband applications., IEEE Antennas and Wireless Propagation Letters, pp.103-106, (2003).
[2] Brian C Wadell, Transmission Line Design Handbook, Artech House, pp. 79, (1991).