ECE 5350/6350 2015 Assignment 4 Solution

1.

Electric Field

! Magnetic Field

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These are the expected field patterns for a microstrip waveguide mode. Since an alternating potential drop has been imposed between the top trace and the ground plane, there is an associated charge difference between these conductors, that takes the form of charges of equal magnitude and opposite sign. We expect to see (and do see) an electric field originating on one of these conductors and terminating on the other. At

the instant (and cross-section) shown, the top trace has a positive charge and the ground plane has a negative charge.

As the mode propagates down the guide, these charges flow, back and forth, along the guide direction (into and out of the plane of this page). For the top trace, the mode field pattern encompasses the top trace, so we expect to see (and do see) magnetic field circulating around the top trace. The ground plane is broader than the modes strongest field region, so circulating fields are not apparent, but we do see a discontinuity of the tangential component of the field associated with a surface current.

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For the instant (and cross-section) shown, the top trace is conducting current out of the page plane (by the right hand rule). For the ground plane, the current is flowing into the page plane. (For the formula above, assume region 1 is inside the substrate, region 2 is below it and the normal points from region 1 into region 2, i.e. down.)

We can use a symmetry plane, and we should, since it decreases simulation time by half. We cannot use a z symmetry plane since we are exciting the guide from one side, a non-symmetric excitation, with respect to the z direction. Also, the microstrip guide does not possess mirror symmetry in the y direction. The guide is, however, mirror symmetric in the x direction. We see that magnetic field is everywhere normal to the x = 0 plane (what CST calls the YZ plane). So we should choose a magnetic symmetry plane for preserving this mode.

2.

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The transmission is near unity (0dB) across the entire frequency range which is exactly what you would like in a waveguide. The transmission characteristics can be described as low (insertion) loss and broadband. This kind of bandwidth is normally only possible for TEM or quasi-TEM waveguides which have two (or more) conductors.

n̂× H2 −H1( ) = Js

3.

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Each of the resonator features are spanned by a single mesh interval. This is the bare minimum for quantitative results. One could argue this is adequate for a class project. If one wants accuracy at the few percent level or better, however, this is not adequate.

4. Electric field y-component (left) and z-component (right).

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One can clearly see the wave fronts (iso-phase surfaces) propagating down the waveguide in the y-component plot animation. We expect this from the E-field mode plots, which are dominated by y-oriented field.

In the z-component plot, the wavefronts appear to be moving in approximately the broadside (y) direction. This is as expected. Consider a magnetic dipole moment arising from the Babinet equivalent resonator. This dipole is oriented across the “capacitive” gap, which for us is the x-direction. (This orientation is normal to the page in the above-right figure.) The electric field of a magnetic dipole moment circulates around the dipole, so is oriented in the y- and z-direction. In the direction of radiation, which is close to broadside (i.e. y-directed), the radiation is predominately z-polarized.

The figure below shows the circulating magnetic field of an electric dipole. The fields are swapped for our case, a magnetic dipole.

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5. Phase (top) and magnitude (bottom) of the the 𝜃-component of the electric far-field versus the permittivity of the liquid crystal.

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We see from both the phase and magnitude of the radiation, that the radiating element has its radiation condition centered at a permittivity of about 3.95. The range of permittivity to achieve the available range of phase is from 3.7 to 4.2 (in my data).

Though not part of the question, we see that to cover this range of permittivity, we need to set the average permittivity, epsLC, to 3.95, and the sinusoidal amplitude, epsLC1 to 0.25 (for my data).

6. Starting from the familiar statement of Nyquist sampling theorem

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we use the relationship between frequency and period, and angular and cyclic frequency

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to find

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Moving to the spatial domain from the temporal domain, we replace the temporal phase constant with the spatial phase constant. We also replace the temporal sampling period with the spatial sampling period, which for us is just the metamterial periodicity, a.

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Using the equality sign for the case of minimal unaliased sampling rate, we then obtain

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The spatial dependence of the permittivity, in the spcript is

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To represent a half oscillation, the minimum spatial phase constant times the length of the device must be π

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The length could be classified as either maximum distance between “samples” or the total length of the “sample” intervals

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We then find the minimum spatil phase constant to be

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fs > 2 fmax

fs =1Ts

and ωmax = 2π fmax

1Ts

> 2ωmax

2π ⇒ ωmax <

πTs

ωmax ⇔ βLCmax and Ts ⇔ a

βLCmax =πa! 2.827

ε z( ) = εLC + εLC1 sin βLCz( ) where z = ia

βLC minΔz = π

Δz = N −1( )a or Na

βLC min =π

N −1( )a! 0.149 or π

Na! 0.141

7.

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8. Plot of directivities for the four configurations highlighted in the previous plot. The plots are normalized to the most directive configuration, the one shown for with a beam around 30 degrees.

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3max [degree]0 30 60 90 120 150 180

Dm

ax

0

1

2

3

4

5

6

7

8

9

10

11

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

9. I plotted all of the configurations, though I asked for just the ones plotted in the previous plot. In theory, these points should lie below the sine curve, which accounts for the projection of the physical length as seen from the direction of radiation.

! 3[degree]20 40 60 80 100 120 140

l e, l

0.2

0.4

0.6

0.8

1

1.2

1.4