Yunsong Xie, Xin Fan, and Yunpeng Chen

University of Delaware, Newark, Delaware

Jeffrey D. Wilson and Rainee N. Simons

Glenn Research Center, Cleveland, Ohio

John Q. Xiao

University of Delaware, Newark, Delaware

Characterizing the In-Phase Refl ection Bandwidth 

Theoretical Limit of Artifi cial Magnetic Conductors 

With a Transmission Line Model

NASA/TM—2013-217866

May 2013
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Yunsong Xie, Xin Fan, and Yunpeng Chen

University of Delaware, Newark, Delaware

Jeffrey D. Wilson and Rainee N. Simons

Glenn Research Center, Cleveland, Ohio

John Q. Xiao

University of Delaware, Newark, Delaware

Characterizing the In-Phase Refl ection Bandwidth 

Theoretical Limit of Artifi cial Magnetic Conductors 

With a Transmission Line Model

NASA/TM—2013-217866

May 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

International Microwave Symposium for 2013 (IMS2013)

cosponsored by the Institute of Electrical and Electronics Engineers, Microwave Theory and Techniques 

Society (MTT-S)

Seattle, Washington, June 2–7, 2013
Acknowledgments

The University of Delaware authors acknowledge the support that they have received from the NASA EPSCoR Program under 

contract number NNX11AQ29A. In addition, thank Prof. S.T. Chui and Prof. Dermott Mullan for their support and interests.

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Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2013-217866 1

Characterizing the In-Phase Reflection Bandwidth Theoretical Limit of 

Artificial Magnetic Conductors With a Transmission Line Model

Yunsong Xie, Xin Fan, and Yunpeng Chen

University of Delaware

Newark, Delaware 19716

Jeffrey D. Wilson and Rainee N. Simons

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

John Q. Xiao

University of Delaware

Newark, Delaware 19716

Abstract

We validate through simulation and experiment that 

artificial magnetic conductors (AMC’s) can be well 

characterized by a transmission line model. The theoretical 

bandwidth limit of the in-phase reflection can be expressed in 

terms of the effective RLC parameters from the surface patch 

and the properties of the substrate. It is found that the 

existence of effective inductive components will reduce the inphase reflection bandwidth of the AMC. Furthermore, we 

propose design strategies to optimize AMC structures with an 

in-phase reflection bandwidth closer to the theoretical limit.

Introduction

The evolution of modern wireless technology demands 

antennas with high gain, low profile, and broad bandwidth. 

Three ways to achieve high gain in patch antennas include the 

use of (1) a ground plate positioned a quarter wavelength 

away which limits the bandwidth and the size miniaturization; 

(2) an electromagnetic wave absorption material under the 

patch (Refs. 1 and 2) which wastes half of the electromagnetic 

(EM) wave energy; and (3) an artificial magnetic conductor 

(AMC) underneath the patch (Ref. 3).

An AMC, which is also known as a perfect magnetic 

conductor (PMC) and high impedance surface (HIS), is a 

device that is artificially designed with thickness usually much 

smaller than the EM wavelength. Among all the properties of 

AMCs, the reflection phase is the most interesting one. With a 

normal incident wave from air, the reflection phase of a 

perfect electric conductor (PEC) is 180°, whereas, for an 

AMC, the reflection phase can be between –90° and 90°

(Ref. 3), resulting in in-phase reflection. As a result, unlike the 

PEC, the close proximity of an AMC to the patch will add the 

incident and reflected waves to significantly increase the gain 

and bandwidth, as well as reduce the overall antenna size.

Recent research has been focusing on bandwidth 

enhancement. The AMC can be modeled as a parallel LC 

circuit, where the in-phase bandwidth is improved by 

increasing the permeability of the spacer layer (substrate) and 

the separation between the surface patch and the metal back 

plate (Refs. 4 to 7). However, due to Snoek’s limit, low loss 

magnetic materials with high permeability at microwave 

frequency are scarce (Ref. 8). Further miniaturization is 

limited by the allowable substrate thickness. In addition, a 

model with a parallel LC circuit is not applicable in the 

mushroom structure without vias, in which in-phase reflection 

remains (Refs. 6 and 9). In this manuscript, using a 

transmission line (TL) model for structures without vias, we 

will show that the in-phase reflection bandwidth has a 

theoretical limit. We explain why most AMC’s do not reach 

the theoretical bandwidth and will show a design procedure to 

enable us to approach the theoretical limit.

Simulation and Measurement Set Up

Four AMC samples with different structures are considered. 

The sample, labeled as AMC_Via, has a via at the center 

connecting the surface patch to the ground plate with 

structural parameters shown in Figure 1 of a = 8 mm, h =

1.57 mm, d = 0.4 mm, and g = 1 mm. The other three samples 

do not have vias (d = 0 mm) and are labeled as AMC_0.2, 

AMC_0.5, and AMC_1.0 with the number referring to the 

value of g in mm. The metal is copper with conductivity of 

5.8×107 S/m, and the substrate is made of dielectric material 

FR-4 with μ = 1 and ε = 4.4 – j0.088.

The TL model for an AMC is shown in Figure 1(c). The

series-connected RLC circuit is used to characterize the surface 

patch. A TLS component is used to represent the existence of the 

substrate which includes the parameters of ɛ, μ, and thickness. 

The metal back plate is modeled as a shorted line.

HFSS software (Ref. 10) is used to obtain the S-parameters 

of the structure involving the surface patch, via, and substrate, 

which are designated as the front components. With known ɛ, 

μ and the thickness of the substrate, the values of R, L, and C 

can be found by fitting the TL model shown in Figure 1(d) to 
NASA/TM—2013-217866 2

the S-parameters of the front components (surface patch and 

substrate). To validate the TL model, we compare Sparameters (transmission and reflection) of the AMC front 

components with those of an HFSS simulation. The ideal 

situation arises when both results match throughout the 

frequency range of interest.

Figure 1.—Schematics and dimensions of the 

AMC structure in both (a) top view and (b) side 

view. The TL model of (c) an AMC structure, 

and (d) the surface patch for achieving the RLC 

circuit by fitting.

In HFSS simulations, only one unit is used. Several pairs of 

master and slave boundaries are set on the lateral sides of the 

unit to mimic the infinite periodic distribution of the AMC 

units (Ref. 10). In studying the properties of the surface patch

without the ground plate, two Floquet excitation ports are set 

on the front and back sides of the structure to obtain both 

reflection (S11) and transmission (S21). The distance between 

the port and the structure is as far as 20 mm to mimic the 

plane wave propagation condition. Only one Floquet 

excitation port is used since the reflection is the only 

parameter of interest.

In the experimental measurements, two antennas are 

connected to the Agilent 8712ES network analyzer and are 

positioned in front of the sample. The distance between the 

antenna and sample is larger than 2D2

/λ to produce the plane 

wave propagation condition, where D is the diameter of the 

smallest sphere containing the horn antenna and λ is the 

wavelength of the microwave. The reflection of the AMC 

sample is calculated by Meas

Metal

Meas RAMC =SAMC S , where Meas SAMC

and Meas SMetal are the measured transmission between the two 

antennas when the AMC sample and a thick circular metal 

plate of area 2 A = r are respectively located in front of the 

antennas. 

Calculation and Measurement Result

Table I shows the extracted RLC parameters by fitting the 

S-parameter of the front component simulated using HFSS. 

The fitted results are excellent as shown in Figure 2. It should

be pointed out that the LC resonance frequencies based on the 

parameters shown in Table I are higher than 100 GHz for all 

four structures. In the considered frequency range (2 to

10 GHz), the impedance of the AMC structure is dominated 

by the effective capacitive component C in the RLC group. 

For example, at 12 GHz, the impedance of the L and C

components of the AMC_Via sample are j5.93 and –

j88.89 , respectively. It is also interesting that the RLC 

parameters are very similar for AMC_1 and AMC_Via 

samples, which confirms that the function of the via in the 

structure may not be important to the in-phase reflection 

performance (Refs. 6 and 9).

TABLE I.—THE RLC PARAMETERS OF AMC SAMPLES 

OBTAINED BY FITTING THE S-PARAMETERS OF

THE FRONT COMPONENT

AMC_0.2 AMC_0.5 AMC_1 AMC_Via

R (Ohm) 1.3466 2.1094 3.5812 3.5574

L (10–9 H) 0.0182 0.0425 0.1027 0.0957

C (10–12 F) 0.3964 0.2725 0.1811 0.1817

Figure 3 shows the phase of the reflection obtained from the 

HFSS simulation (red circles) and the TL model (blue lines) of 

each AMC structure. The results from the TL model match 

well with the HFSS simulations, showing that the frequency of

0° reflection phase decreases with decreasing g. It can be
NASA/TM—2013-217866 3

Figure 2.—S-parameters of the front components of 

the samples (a) AMC_0.2, (b) AMC_0.5, (c) AMC_1 

and (d) AMC_Via from HFSS simulation (squares 

and circles) and calculated by using the extracted 

RLC parameters in Table I.

Figure 3.—The experimental data (black squares), 

and HFSS simulated results (red circles) and TL 

model calculated results (blue lines) of the 

samples (a) AMC_0.2, (b) AMC_0.5, (c) AMC_1 

and (d) AMC_Via.
NASA/TM—2013-217866 4

observed that the discrepancy between the model results and 

experiment increases as g increases. The RLC parameters of 

the surface patch are affected by the addition of the metal 

ground plate due to the change of the electric field 

distribution, which influences the C component. Samples with 

smaller g (closer patch) have less effect on the C component 

from the ground plate due to confinement of the electric field 

between the patches.

Figure 3 also shows the experimental measurement of the 

four AMC samples (black dots). In order to satisfy the infinite 

periodic boundary condition, each sample has a size of 190 by

203 mm, which is more than four times the wavelength of the 

electromagnetic wave in vacuum at the frequency with 0°

phase reflection. Due to the limitation of the horn antenna 

working frequency, we are only able to measure the reflection 

from 4.5 to 10 GHz. It is clear that the experimental results 

agree very well with simulations for all samples. Since the TL 

model calculation is very similar to HFSS simulation and 

experimental results, it is reasonable to believe that the TL 

model will be able to characterize and quantitatively calculate 

the behavior of other AMC configurations. 

Analysis of the TL Model

By analyzing the TL model in Figure 1(c), the existence of 

the L component of the surface patch will reduce the 

bandwidth of the in-phase reflection. Since we are only 

interested in the reflection phase behavior of the AMC 

structure, we assume a lossless AMC with a negligible 

inductive component and the thickness of the substrate is 

much smaller than the EM wavelength. Then the center 

frequency f0,center and normalized bandwidth NBW0 of the inphase reflection in this baseline model can be derived to be:

0

0

0, center 4

4 1

C Z

h

cCZ

f

÷ + ÷

= (1)

÷ + ÷

= =

h

f cCZ0 0, center

0

4 1

Bandwidth 2 NBW

(2)

Where c is the speed of light in vacuum, C is the capacitive 

component of the surface patch, Z0 is the intrinsic impedance 

of vacuum, and h and are the thickness and the permeability 

of the substrate, respectively. Using a Taylor expansion to 

include the effect of a small but not negligible inductive 

component, the normalized bandwidth becomes

( )

( )

L

4

4 3 NBW NBW 3 / 2

0 0

0

L 0

Z h h cCZ

c h cCZ

+

+ = (3)

Equation (3) shows how the inductive component in the 

transmission line affects the in-phase reflection bandwidth. 

Since the coefficient of L is negative, NBWL will have the 

maximum value NBW0 when L = 0. Furthermore, considering 

the property of the coefficient of L, three conclusions referring 

to the strategy of broadening the in-phase reflection bandwidth 

can be drawn:

1. The NBW of the AMC structures has a theoretical 

limitation which is expressed by Equation (2) using a thin 

substrate approximation.

2. Decreasing the inductive component L is an efficient way

to push the NBW of an AMC closer to its theoretical 

limit.

3. Increasing the permeability of the substrate decreases the 

coefficient of L but enhances the L component. It is 

unclear that the use of a magnetic substrate will increase 

the NBW closer to the theoretical limit. However, it is 

certain that, according to Equation (3) the reduction of 

NBW caused by the existence of the inductive component 

can be minimized through increasing either the thickness 

of the substrate h or the capacitive component of the 

surface patch.

Conclusion

We have demonstrated in both HFSS simulation and 

experimental measurements that our TL model is able to 

describe the reflection phase behavior of AMC structures. A 

theoretical limit of the in-phase reflection bandwidth is 

derived using the TL model. The reduction of the in-phase 

reflection is caused by the existence of the inductive 

component of the surface patch, which can be minimized by 

using a thicker substrate or increasing the capacitive 

component of the surface patch. 

References

1. F. Costa, et al., “Analysis and design of ultrathin

electromagnetic absorbers comprising resistively loaded 

high impedance surfaces,” IEEE Transactions on 

Antennas and Propagation, vol. 58, pp. 1551–1558,

May 2010.

2. F. Costa and A. Monorchio, “A frequency selective 

radome with wideband absorbing properties,” IEEE 

Transactions on Antennas and Propagation, vol. 60,

pp. 2740–2747, Jun 2012.

3. D. Sievenpiper, et al., “High-impedance electromagnetic 

surfaces with a forbidden frequency band,” IEEE 

Transactions on Microwave Theory and Techniques,

vol. 47, pp. 2059–2074, Nov 1999.

4. A.P. Daigle, et al., “Numeric simulations of a novel 

wideband electromagnetic band gap metamaterial 

utilizing oriented cobalt-substituted z-type barium 
NASA/TM—2013-217866 5

hexaferrites,” Magnetics Letters, IEEE, vol. 2,

pp. 0500104-0500104, 2011.

5. D.J. Kern and D.H. Werner, “Magnetic loading of EBG 

AMC ground planes and ultrathin absorbers for 

improved bandwidth performance and reduced size,” 

Microwave and Optical Technology Letters, vol. 48,

pp. 2468–2471, Dec 2006.

6. A.P. Feresidis, et al., “Artificial magnetic conductor 

surfaces and their application to low-profile high-gain 

planar antennas,” IEEE Transactions on Antennas and 

Propagation, vol. 53, pp. 209–215, Jan 2005.

7. L. Yousefi, et al., “Enhanced bandwidth artificial 

magnetic ground plane for low-profile antennas,” IEEE 

Antennas and Wireless Propagation Letters,

vol. 6, pp. 289–292, 2007.

8. X.K. Zhang, et al., “High frequency properties of 

polymer composites consisting of aligned Fe flakes,” 

Journal of Applied Physics, vol. 99, Apr 15 2006.

9. J. Joubert, et al., “CPW-fed cavity-backed slot radiator 

loaded with an AMC reflector,” IEEE Transactions on 

Antennas and Propagation, vol. 60, pp. 735–742,

Feb 2012.

10. H.S. Wu and C.K.C. Tzuang, “Artificially integrated 

synthetic rectangular waveguide,” IEEE Transactions on 

Antennas and Propagation, vol. 53, pp. 2872–2881,

Sep 2005.
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Characterizing the In-Phase Reflection Bandwidth Theoretical Limit of Artificial Magnetic 

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We validate through simulation and experiment that artificial magnetic conductors (AMC’s) can be well characterized by a transmission line 

model. The theoretical bandwidth limit of the in-phase reflection can be expressed in terms of the effective RLC parameters from the surface 

patch and the properties of the substrate. It is found that the existence of effective inductive components will reduce the in-phase reflection 

bandwidth of the AMC. Furthermore, we propose design strategies to optimize AMC structures with an in-phase reflection bandwidth closer 

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