Planewave reflection at Metallic interfaces

Figure 3.5 illustrates reflection of a planewave that impinges on the interface of a perfectly conducting metallic halfspace (see Section 1.18 for a definition of halfspace). The incident planewave is represented by E_i, H_i, and k_i, which are the electric field, magnetic field, and propagation vector, respectively, of a planewave propagating in free space toward the right.

The wave is said to be normally incident on the conducting interface because the direc-tion of propagadirec-tion k_i is normal (perpendicular) to the metallic interface, which makes the Ei and Hi vectors tangential to this surface. Metal objects with perfect conductivity do not allow the wave to penetrate at all. They create a reflected wave that propagates in the opposite direction (in this case, to the left). The electric field vector of the reflected wave is inverted from the original at the interface; we say the reflected electric field is out of phase from the incident electric field. This means that the reflected field is 180 degrees out of phase from the incident field, and that the reflected electric field is the negative of the incident electric field at the metallic interface.

The fact that the field reflects off a metal surface is observed and used in a wide variety of applications. One obvious example that you have seen (literally) is light reflecting from a metallized mirror. Light is an electromagnetic wave, and you have undoubtedly seen light reflections from numerous metallic structures.

Another application where reflections are used is in antenna design. Antennas require both a positive and a negative “arm,” but many antennas (for example, your cell phone antenna) have only one arm. If a metal plate is placed under one arm of an antenna, the reflection from the plate makes it look to the antenna as if its other arm is “virtually”

there underneath this ground plane, just as a reflection would appear to be behind the sur-face of the mirror. Radio stations often create a very large ground plane for their transmit-ting antennas by burying wires in a radial pattern like the spokes of a wheel at the base of the antennas. The spokes are close enough that they act as if the plane they create is solid (just as the screen of a microwave oven door reflects electromagnetic fields, as described in Section 1.17). Cell phones and other handheld devices are not large enough to have large, perfect ground planes underneath them, but the tops of their cases are generally metalized (typically plastic coated inside with metal paint). This creates a small ground plane that creates a partial reflection, somewhat like a small handheld mirror that reflects only part of the face.

Shielded metal rooms are used to test electromagnetic devices because they reflect exter-nal fields and do not allow them to interfere with the sensitive tests occurring inside. If the fields being tested are high, they also keep people on the outside of the chamber from being exposed to high electromagnetic fields. Power line workers who repair live power lines wear clothing with metallic covering to reflect the external field and keep them safe.

The reason for this reflection can be explained in terms of the simple boundary condi-tions originally explained in Section 1.12. When the planewave impinges on the metal and begins to propagate into it, the perfect conductivity forces the E field to go to zero.

The E inside the metal must be zero because the conduction current density is given by Jc = σcE (σc is conductivity), and since σc is infinite, E must be zero or else the conduction current density would be infinite, which is not possible. Because the boundary condition (Equation 1.18) requires the tangential E field to be continuous at the metallic interface, and because the E field in the metal is zero, the tangential E field at the metallic interface must also be zero.

This requirement causes a reflected wave to be generated, with the E field of the reflected wave equal and opposite to that of the incident wave at the interface. The actual total field at the boundary (and everywhere outside the metal) is the sum of the incident and

H_s

A planewave normally incident on a planar metallic interface and the reflected planewave produced by the metal. The subscript i stands for the incident wave, and the subscript s stands for the scattered (reflected) wave.

The H fields are directed into the paper and are therefore represented by a cross in a circle.

reflected waves. The reflected wave, or scattered wave as it is often called, is represented in Figure 3.5 by E_s, H_s, and k_s, which are the electric field, magnetic field, and propagation vec-tor, respectively, of the scattered wave. At the boundary, the two equal and opposite elec-tric fields add to zero, thus satisfying the boundary condition. Note, however, that the incident and scattered magnetic fields are both equal and in the same direction. Thus, the total tangential magnetic field is twice the incident magnetic field on the boundary.

The time-domain behavior of the planewave in front of a metal surface is particularly interesting. The total EM fields in the space to the left of the conducting halfspace in Figure 3.5 are the sums of the incident and scattered fields. Figure 3.6 shows the incident and reflected E field waves and their sum at nine different instants of time. The sum is zero at the metallic interface for all instants of time (because tangential electric fields are always zero on perfectly conducting metal). If you look closely at Figure 3.6, you will see that at certain positions in front of the metallic interface, the sum of the incident and reflected waves is zero for all nine instants of time. This is shown more clearly in Figure 3.7(a), which shows just the sum of the incident and reflected wave electric fields as a function of distance z at all nine instants of time superposed on the same graph. At points that lie at half-wavelength intervals in front of the metallic surface, the total E is zero at all nine instants of time. It turns out that at these points the E is zero at all instants of time, not just the nine shown in the figure. These zero values of E are called nulls. The nulls occur

t = t₁ t = t₂ t = t₃

t = t₄ t = t₅ t = t₆

t = t₉ t = t₈

t = t₇

z

z

z

z

z

z

z

z

z Incident

Reflected Sum

Figure 3.6

The electric fields of the incident wave (light solid lines) of Figure 3.5, the scattered (reflected) wave (dashed lines), and the sum of the incident and scattered waves (bold lines) as a function of distance, at nine different instants of time. At t1 and t9, the scattered wave lies on top of the incident wave. At t5 the sum of the waves is zero. The gray rectangle represents the position of the planar metallic interface.

because the incident and scattered E fields are equal in magnitude and opposite in sign at those points, so that they cancel each other out at all times. This cancellation occurs because the incident and scattered waves propagate with the same phase velocity, but in opposite directions, and because the incident and scattered E must add to zero at the metallic interface.

If you imagine the electric field patterns at different times in Figure 3.7(a) appearing sequentially as a “movie,” you can see why the wave in front of a metal surface is called a standing wave. The total electric field (sum of the incident and reflected waves) moves up and down at any given point (except the nulls, which remain at zero) in a sinusoidal fash-ion. If you measured the electric field at any individual point in front of the planewave, you would find that its amplitude is a sinusoidal function of time. Figure 3.7(b) shows the sinusoidal time variation of the amplitude at one point, z = z1.

Figure 3.8 shows a similar standing wave pattern for the total H fields. Because of the mutually orthogonal relations between E, H, and k in each of the waves, the incident and scattered H fields add at the metallic interface instead of cancelling, and the first null in H occurs a quarter-wavelength back from the metallic interface, and then at half-wavelength intervals thereafter.

(a) The sums of the electric fields of the incident and scattered waves of Figure 3.6 for nine instants of time all plotted on the same graph as a function of distance in front of the planar metallic object. This is an illustration of a standing wave. The E at t1 is shown in bold to illustrate the typical pattern. (b) The electric field at position z1 as a function of time.

Standing wave patterns are often represented by just the envelope of the pattern, as shown in Figure 3.9. The envelopes of the E and H standing waves clearly show the positions of the nulls and the maximum values that the E and H attain.

When a planewave is incident on a metal object at an angle other than normal (or head on, where θi = 0°), even more interesting field patterns occur. This is referred to as being obliquely incident and is shown in Figure 3.10. The angle θi that the propagation vector (ki) makes with a normal to the metallic surface is called the angle of incidence. As with normal incidence, the boundary conditions at the metallic interface require the tangential electric field to be zero there, again producing a scattered wave. The field reflects off the metal surface at an angle θs that is equal to the angle of incidence θi. This means that the propaga-tion vector of the scattered wave ks makes an angle θs with the normal to the surface that is equal to the angle of incidence. Once again, the sum of the electric fields of the incident and scattered waves is a standing wave, but for oblique incidence the nulls are farther apart than one half wavelength. The greater the angle of incidence, the farther apart the nulls.

Figure 3.11 shows the incident and scattered waves at one instant of time in terms of the peaks and troughs—the peaks as black lines and the troughs as gray lines. At points where black and gray lines intersect, the sum of the E in the two waves is zero. Although inter-mediate values of E between the peaks and troughs are not explicitly shown in Figure 3.11, the sum of the incident and scattered E is zero everywhere along the dashed lines at every instant of time. Thus, a standing wave pattern occurs for oblique incidence just as it does for normal incidence, but the nulls in the standing wave for oblique incidence are farther apart than for normal incidence, as mentioned above.

Another way of illustrating the electric fields of the incident and reflected waves and their sum is shown in Figures 3.12 to 3.14. In these figures, the amplitude of the E is represented by shades of gray, with white being maximum and black being minimum.

Figure 3.12 shows the E of the incident wave at one instant of time, Figure 3.13 the E of the scattered wave, and Figure 3.14 the sum of the two. As time progresses, the patterns move according to the propagation directions of the waves shown by the two arrows in Figure 3.11, but the nulls in the total E always occur at positions along the dashed lines in Figures 3.11 and 3.14.

The sums of the magnetic fields of the incident and scattered waves of Figure 3.6 for nine instants of time all plotted on the same graph as a function of distance in front of the planar metallic object. This is an illustration of a standing wave. The H at t1 is shown in bold to illustrate the typical pattern.

H_s

E_s k_s

H_i E_i

k_i

Planar conductor θ_i

θ_s

Conducting halfspace Scattered planewave

Incident planewave The angle of

reflection equals the angle of incidence

Empty space

Figure 3.10

A planewave obliquely incident on a perfectly conducting planar metallic interface and the reflected (scattered) planewave produced by the metal. The subscript i stands for the incident wave, and the subscript s stands for the scattered (reflected) wave. The angle of reflection θs is equal to θi, the angle of incidence.

λ/2

z

λ/2

z E envelope

H envelope

Conducting halfspace Conducting halfspace

Figure 3.9

Envelopes of the E and H standing waves of Figures 3.7 and 3.8.

Conducting halfspace T

T

T T

P P

P P

Plane of cancellation

P is peak T is trough

Figure 3.11

Wavefronts of the incident and scattered planewaves of Figure 3.10 at one instant of time. P stands for peak, and T stands for trough. The dashed lines show where the peaks and troughs add to zero. The respective wavefronts propagate in the directions of the arrows as time progresses.

Conducting halfspace

Figure 3.12

The amplitude of the E of the incident wave of Figure 3.10 at one instant of time. The peaks are white, and the troughs are black. Values in between peaks and troughs are shown in various shades of gray.

grOunDIng AnD SHIELDIng In ELECTrOMAgnETICS

Grounding is an important concept in electrical engineering. Voltages are measured relative to a ground, which is usually treated as zero voltage. Currents flow from a positive voltage down to a ground. Properly grounding equipment is important to prevent shocks and hazard.

At low frequencies (Chapter 2), one electrode has a positive voltage and the other is treated as the ground. Most often the ground electrode is attached on the body at some distance from the first electrode. The current will pass from the positive electrode to the ground electrode. At low frequencies, this current flows along the path of least resistance. It will tend to flow and concentrate in the high-water-content, high-conductivity tissues. At low frequencies, a ground can be created in many dif-ferent ways; any metallic contact will create a ground. A rod, wire, or clip can all be used to create a sufficient ground for low frequencies.

At middle and higher frequencies, the situation is quite different. The current fol-lows the path of least inductance, not least resistance. A single wire often will not be sufficient to provide a ground, and the current will choose (continued on next page)

Conducting halfspace

Figure 3.13

The amplitude of the E of the scattered wave of Figure 3.10 at one instant of time. The peaks are white, and the troughs are black. Values in between peaks and troughs are shown in various shades of gray.

Conducting halfspace

Figure 3.14

The amplitude of the sum of the E of the incident wave and the E of the scattered wave of Figure 3.10 at one instant of time. The maxima are white, and the minima are black. Values in between the maxima and minima are shown in various shades of gray. The dashed lines indicate planes of cancellation of the inci-dent and scattered waves.

many other paths to reach the ground. The current is no longer constrained to just the metallic contact points. Ground plates are commonly used, and multiple parallel connections are attached to them. Sometimes this creates another problem—ground loops. The current can pass up some of the parallel connections and down others, creating loops that cause delays in parts of the signal and make it spread out over time. Pulsed electromagnetic fields have both high- and low-frequency components, and they should be treated as high-frequency fields when considering grounding.

Shielding is another important aspect of high-frequency devices. Since high-fre-quency fields are no longer contained in the metal and can be transmitted through space, shielding is important to provide protection from electromagnetic interference (EMI). Since electric fields cannot penetrate metal, putting the device to be shielded in a completely enclosed metal container or can (with the seams electrically sealed) will prevent the device from receiving any signals from outside of the can. Of course, we usually want to get a signal in or out of the device in the can, but as soon as we make a hole in the can to run a wire, the can is now “leaky” with respect to elec-tromagnetic fields. As with the microwave oven door described in Section 1.17, the size of the hole determines which frequencies will pass through and which will be rejected. So typically we use a shielded cable such as a coax to reach the device inside the can. The outer shield of the cable is connected (with no holes) to the body of the can, with the inner conductor of the coax reaching the device inside the can. We also have to be careful to do the same on the other end of the cable.

Full shielding is often important for very sensitive electromagnetic measurements done in a laboratory. In this case, the shielding is provided by a metal mesh or plates completely surrounding the entire room: a shielded room. The measurement equip-ment and power supply are often left outside the room, and the receiving antenna and device being measured are inside the room. This protects the measurements from corrupting RF signals such as radio stations and cell phone signals, and makes it possible to receive very tiny signals without excessive outside noise.