Home Networking Basis: Transmission Environments and Wired/Wireless Protocols

As shown by Maxwell's equation, radio waves travel at the speed of light which has been measured at 186,281.7 miles/second or, equivalently, 299,792.8 kilometers/second in open space. For optimal performance, the dimensions of radio transceiver antenna depend on the radio wavelength. In the normal condition the wavelength is obtained by dividing the speed of light by the frequency of the radio wave as in

Equation 5.1

where c is the speed of light and f is the frequency of the radio wave. These wavelengths are between 0.32 and 0.33 m, 0.12 and 0.125 m, and 0.051 and 0.053 m for the 902 928 MHz, 2.4 2.4835 GHz, and 5.725 5.850 GHz frequency bands, respectively.

Similar to light waves, radio waves at these wavelengths in a home environment experience reflection when bounced off a wall or any household object and attenuation when passing through household objects. The received radio wave is usually a combination of direct, reflected, and attenuated forms of the original radio wave. Because of the phase differences of these direct, reflected, and attenuated radio waves, the received radio wave strength might vary rapidly with distance. This is usually called fast fading. The received radio wave strength can be estimated using the ray-tracing method for a particular home structure according to known or estimated dielectric properties of wall and household objects. Table 5.3 [2] shows typical dielectric properties of some popular building materials. These dielectric properties along with the physical location and size of these household objects can be used to calculate the strength of reflected and diffracted radio waves.

5.2.1 Directly Received Radio Wave

The power of a directly received radio wave is calculated according to

Equation 5.2

where P_t is the transmit power, P_r(d) is the distance-dependent received power, G_t is the transmitter antenna gain, G_r is the receiver antenna gain, d is the separation distance, and l is the wavelength. Both separation distance and wavelength should use the same measurement unit, meters for example. The received power is proportional to the transmit power, the transmit antenna gain, and the receiver antenna gain. The received power is also proportional to the square of the wave length and to the inverse of the square of distance.

The permitted transmit power at these ISM bands is 0.25 or 1 W. The antenna gain is usually between 0 to 6 dB. The wavelengths are approximately 0.32, 0.12, and 0.052 m for the 902- to 928-MHz, 2.4- to 2.4835-GHz, and 5.725- to 5.850-GHz frequency bands, respectively. The received power to transmit power ratios are 40 to 52, 48.5 to 60.5, and 55.8 to 67.8 dB for these ISM frequency bands, respectively, at a distance of 10 m. For a transmit power of 1 W, the directly received powers at a distance of 10 m are 6 100, 0.9 14, and 0.16 2.8 microwatts (µW), respectively.

5.2.2 Reflected and Penetrated Radio Wave

When a radio wave hits a wall, some of the energy is reflected, and the remaining energy passes through the wall. To be more specific, there are two sides, therefore two air interfaces, for a particular wall. The reflected wave consists of the first reflection when the radio wave reaches the wall as well as the radio waves reflected at the other side of the wall and between both sides of the wall and escaped in the reflected wave direction. The radio wave that passed through the wall also consists of the directly passed wave and waves reflected between both sides of the wall. We can calculate the reflected and transmitted waves by using Maxwell's equation as well as boundary conditions. The wave energy reflection and transmission coefficients can also be similarly calculated.

Figure 5.2 shows the relationship between the incident, reflected, and transmitted waves. A wave arrives with an angle of q_i and reflected with an angle of q_r. In general, the incident wave angle and the reflected wave angle are the same (i.e., q_i = q_r). Incident and transmitted wave angles are related by their dielectric properties as shown by

Equation 5.3

The reflected electric field is related to the incident electric field through a constant , i.e., E_r. = E_i. More specifically, an electric field can be divided into a parallel component and a perpendicular component with respect to the observation plane, which is in parallel with the page surface of Figure 5.2. In other words, we have E_{r|| ||}E_i|| and E_r = E_i, specifically, for

Equation 5.4

where

Equation 5.5

At the boundary interface, the transmitted wave equals the sum of the incident and the reflected waves; that is, E_t = E_i + E_r = (1 = )E_i = TE_i. For parallel and perpendicular components, we have E_t = T_||E_i and E_t = TE_ifor

Equation 5.6

Equation 5.7

At an air-to-wall interface, we have m₁ = m₂ = e₁ = 1 and e₂ = 9. For q_i = 30°, we found q_t = 73.22°. We then calculate _|| and as shown by

Equation 5.8

Equation 5.9

We also calculate T_|| and T as shown by

Equation 5.10

Equation 5.11

We find that about 22% of the parallel components and 70% of the perpendicular components of the incident wave are reflected, while 78% of the parallel components and 30% of the perpendicular components are transmitted through the air-to-wall interface. When a radio wave travels through one medium to another with a higher permittivity, through air to a wall for this case, the reflected wave has an 180° phase change, while the phase of the transmitted wave is the same as that of the incident wave. Also the magnitude of the wave at the interface boundary is reduced.

At the wall-to-air interface, we have m₁ = m₂ = e₂ and e₁ = 9. The angle values are reversed (i.e., we have q_i = 73.22° and q_i = 30°). We calculate another set of reflection factors, '_|| and ', as shown by

Equation 5.12

Equation 5.13

We then calculate T_|| and T as shown by

Equation 5.14

Equation 5.15

When a radio wave travels from a medium with a higher permittivity to another, through a wall to air for this case, the reflected, transmitted, and incident waves all have the same phase. Also the magnitude of the wave at the interface boundary is enlarged.

The energy transport rate of a radio wave in a direction normal to the boundary interface is Esinq. More specifically, we have energy transport rates E_i||sinq_i, E_isinq_i, E_r||sinq_r, E_rsinq_r, E_T||sinq_t, and E_Tsinq_r for parallel and perpendicular components of incident, reflected, and transmitted waves, respectively. Because the incident wave is divided into a reflected and a transmitted wave, the total energy of the reflected and the transmitted waves should equal that of the incident wave. Therefore, we have R + T = 1, where energy reflection coefficients, R_|| for parallel component and R for perpendicular component, and transmission coefficients, T_|| for parallel component and T for perpendicular component, are defined by the following expressions:

Equation 5.16

Equation 5.17

Equation 5.18

Equation 5.19

At the first air-to-wall interface, we calculate energy reflection coefficients R_|| and R as shown by the following expressions for m₁ = m₂ = e₁ = 1, e₂ = 9, q_i = 30°, and q_i = 73.22°.

Equation 5.20

Equation 5.21

We then calculate energy transmission coefficients T_|| and T as shown by

Equation 5.22

Equation 5.23

At the wall-to-air interface, we have m_i = m₂ = e₂ = 1 and e₁ = 9. The angle values are reversed (i.e., we have q_i = 73.22° and q_t = 30°) and so are relative permittivities. This double reversal results in the same values for energy reflection and transmission coefficients. We have

Equation 5.24

Equation 5.25

Equation 5.26

Equation 5.27

Values of reflection and transmit coefficients differ according to relative permittivity of material as well as incident angle. Figure 5.3 shows the relationship between incident and transmit angles for an air-to-wall interface where the relative permittivity of the wall is 9.

The minimum transmit angle is about 70.5°. In fact, when the incident angle approaches zero, the reflection coefficient approaches one and the transmit coefficient approaches zero for both parallel and perpendicular components of the wave. Figure 5.4 shows parallel components of reflection and transmit coefficients as variables of the air-to-wall incident angle. At an incident angle of about 19°, the reflection coefficient is about zero while the transmit coefficient is about one for the parallel component of the wave.

Figure 5.5 shows perpendicular components of reflection and transmit coefficients as variables of the air-to-wall incident angle. The perpendicular component of the reflection coefficient decreases as the incident angle and the perpendicular component of the transmit coefficient increase.

The relationship between the wall-to-air incident and transmit angles and the reflection and transmit coefficients is still determined by relative permittivity of material and incident angle. However, they look different because the wave travels from a high relative permittivity material to one with less relative permittivity; therefore, the existence of a minimum incident angle exists. The value of the minimum incident angle is the same as that of the minimum transmit angle when the wave goes from the air to the wall. Below the minimum incident angle, the wave travels inside the wall. Figure 5.6 shows the relationship between incident and transmit angles for a wall-to-air interface.

Figure 5.7 shows parallel components of reflection and transmit coefficients as variables of the incident angle from inside the wall-to-air interface. Reflection and transmit coefficients are one and zero, respectively, for incident angles that are less than the minimum. At an incident angle of about 72° corresponding to the 19° transmit angle or incident angle for the air-to-wall example, the reflection coefficient is about zero while the transmit coefficient is about one for the parallel component of the wave. Consequently, when a radio wave with an incident angle of about 19° hits the wall, the parallel component will pass the wall without reflection.

Figure 5.8 shows perpendicular components of reflection and transmit coefficients as variables of the wall-to-air incident angle. Above the minimum incident angle of about 70°, the perpendicular component of the reflection coefficient decreases as the incident angle and the perpendicular component of the transmit coefficient increase.

The reflected wave from a wall includes the first reflection at the air-to-wall interface and those reflected between two sides of the wall and escaped from the wall to the air. For example, at an incident angle of 45°, the strengths of the first reflected wave are 0.13 and 0.37 for parallel and perpendicular components, respectively. The strengths of the second reflected wave, which first pass the air-to-wall interface and then reflect at the wall-to-air interface and finally pass the wall-to-air interface, are estimated for parallel and perpendicular components as follows. We have the second reflected wave strength for the parallel component of 0.87 x 0.1 x 0.9 = 0.0783. We also have the second reflected wave strength for the perpendicular component of 0.63 x 0.4 x 0.6 = 0.1512. Strengths for the third reflected wave are 0.87 x 0.1 x 0.1 x 0.1 x 0.9 = 0.000783 and 0.63 x 0.4 x 0.4 x 0.4 x 0.6 = 0.024192 for parallel and perpendicular components, respectively. We observe that the strengths of the second reflections are much weaker than those of the first reflections especially for the parallel component. The effects of the third reflection can be ignored for their weaker strength.