6. 2 ray propagation model part 2

1. WIRELESS COMMUNICATION
SERIES
2 Ray Propagation Model
[Part – 2]

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3. 𝑬 𝒕𝒐𝒕 =
𝑬 𝒐 𝒅 𝒐
𝒅′
. 𝑪𝒐𝒔 𝝎 𝒄(𝒕 −
𝒅′
𝒄
) −
𝑬 𝒐 𝒅 𝒐
𝒅′′
. 𝑪𝒐𝒔 𝝎 𝒄(𝒕 −
𝒅′′
𝒄
)
Path Difference (Δ) ≈
𝟐. 𝒉 𝒕 𝒉 𝒓
𝒅
𝑷𝒉𝒂𝒔𝒆 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 (𝜽∆) =
𝝎 𝒄.∆
𝒄
𝑻𝒊𝒎𝒆 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 (𝝉 𝒅) =
𝜽∆
𝟐𝝅𝒇
It should be noted that when d becomes large, the difference between the distances d’ and d’’ becomes very small, and the
amplitudes Elos and Eg are virtually identical and differ only in phase
𝐸 𝑜 𝑑 𝑜
𝑑
=
𝐸 𝑜 𝑑 𝑜
𝑑′ =
𝐸 𝑜 𝑑 𝑜
𝑑′′
If the received E – Field is evaluated at some TIME , say t = d’’ / c, then E Field Eqn. can be written as
𝐸𝑡𝑜𝑡(𝑑) =
𝐸 𝑜 𝑑 𝑜
𝑑
cos
𝜔𝑐. ∆
𝑐
− cos(0)
𝐸𝑡𝑜𝑡(𝑑, 𝑡 =
𝑑′′
𝑐
) =
𝐸 𝑜 𝑑 𝑜
𝑑′ . 𝐶𝑜𝑠 𝜔𝑐(
𝑑′′
𝑐
−
𝑑′
𝑐
) -
𝐸 𝑜 𝑑 𝑜
𝑑′′ . 𝐶𝑜𝑠 𝜔𝑐(
𝑑′′
𝑐
−
𝑑′′
𝑐
)
𝐸𝑡𝑜𝑡(𝑑) =
𝐸 𝑜 𝑑 𝑜
𝑑
cos 𝜃∆ − cos(0)

4. 𝑐𝑜𝑠 𝜔𝑐 𝑡 − 𝑐𝑜𝑠 𝜔𝑐 𝑡 + 𝜃∆ = 2 𝑠𝑖𝑛
𝜃∆
2
𝑠𝑖𝑛 𝜔𝑐 𝑡 +
𝜃∆
2
𝐸𝑡𝑜𝑡(𝑑) =
𝐸 𝑜 𝑑 𝑜
𝑑
cos 𝜃∆ − cos(0)
𝐸𝑡𝑜𝑡(𝑑) = 2.
𝐸 𝑜 𝑑 𝑜
𝑑
𝑠𝑖𝑛
1
2
𝜔𝑐∆
𝑐
𝐸𝑡𝑜𝑡(𝑑) = 2.
𝐸 𝑜 𝑑 𝑜
𝑑
𝑠𝑖𝑛
1
2
.
1
𝑐
. 2𝜋𝑓.
2ℎ 𝑡ℎ 𝑟
𝑑
𝑬 𝒕𝒐𝒕(𝒅) = 𝟐.
𝑬 𝒐 𝒅 𝒐
𝒅
𝒔𝒊𝒏
𝟐𝝅𝒉 𝒕 𝒉 𝒓
𝝀𝒅
𝑠𝑖𝑛
𝜃∆
2
=
𝜃∆
2
for θ < 0.3 𝑟𝑎𝑑
𝐸𝑡𝑜𝑡(𝑑) = 2.
𝐸 𝑜 𝑑 𝑜
𝑑
2𝜋ℎ 𝑡ℎ 𝑟
λ𝑑
𝑬 𝒕𝒐𝒕(𝒅) = 𝑬 𝒐 𝒅 𝒐
𝟒𝝅𝒉 𝒕 𝒉 𝒓
𝝀𝒅 𝟐
By using the identity,
By using the Trigonometric identity,
Relating Field to Power by using the simple relation
(Field)2 = Power
𝑷 𝒓 = 𝑬 𝒕𝒐𝒕
𝟐
. 𝑨 𝒆 = 𝑬 𝒐 𝒅 𝒐
𝟐
. (𝟒𝝅) 𝟐
𝒉 𝒕 𝒉 𝒓
𝝀𝒅 𝟐
𝟐
.
𝑮 𝒓 𝝀 𝟐
𝟒𝝅
𝑷 𝒓 = 𝑬 𝒕𝒐𝒕
𝟐
. 𝑨 𝒆 = 𝑷 𝒕 𝑮 𝒕 .
𝒉 𝒕 𝒉 𝒓
𝝀𝒅 𝟐
𝟐
. 𝑮 𝒓 𝝀 𝟐
𝑷 𝒓 =
𝑷 𝒕 𝑮 𝒕 𝑮 𝒓 𝒉 𝒕 𝒉 𝒓
𝟐
𝒅 𝟒

5. 𝑷 𝒓 =
𝑷 𝒕 𝑮 𝒕 𝑮 𝒓 𝒉 𝒕 𝒉 𝒓
𝟐
𝒅 𝟒
𝑃𝑟
𝑃𝑡
=
𝑮 𝒕 𝑮 𝒓 𝒉 𝒕 𝒉 𝒓
𝟐
𝑑4
Calculating the Path Loss (PL) = Pt / Pr
𝑃𝑡
𝑃𝑟
=
𝑑4
𝑮 𝒕 𝑮 𝒓 𝒉 𝒕 𝒉 𝒓
𝟐
𝑷𝑳 𝒅𝑩 = 𝟒𝟎 𝐥𝐨𝐠 𝒅 − (𝟏𝟎 𝒍𝒐𝒈 𝑮 𝒕 + (𝟏𝟎 𝒍𝒐𝒈 𝑮 𝒓 + (𝟐𝟎 𝒍𝒐𝒈 𝒉 𝒕 + 𝟐𝟎𝒍𝒐𝒈(𝒉 𝒓)
Path Loss for 2 Ray model is given by:

6. 𝑬 𝒕𝒐𝒕 =
𝑬 𝒐 𝒅 𝒐
𝒅′
. 𝑪𝒐𝒔 𝝎 𝒄(𝒕 −
𝒅′
𝒄
) −
𝑬 𝒐 𝒅 𝒐
𝒅′′
. 𝑪𝒐𝒔 𝝎 𝒄(𝒕 −
𝒅′′
𝒄
)
Path Difference (Δ) ≈
𝟐. 𝒉 𝒕 𝒉 𝒓
𝒅
𝑷𝒉𝒂𝒔𝒆 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 (𝜽∆) =
𝝎 𝒄.∆
𝒄
𝑻𝒊𝒎𝒆 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 (𝝉 𝒅) =
𝜽∆
𝟐𝝅𝒇
𝐸 𝑑, 𝑡 =
𝐸 𝑜 𝑑 𝑜
𝑑
. 𝐶𝑜𝑠 𝜔𝑐(𝑡 −
𝑑
𝑐
)
𝐸𝑙𝑜𝑠 𝑑′, 𝑡 =
𝐸 𝑜 𝑑 𝑜
𝑑′
. 𝐶𝑜𝑠 𝜔𝑐(𝑡 −
𝑑′
𝑐
)
𝐸𝑔 𝑑′′, 𝑡 = 𝜞
𝐸 𝑜 𝑑 𝑜
𝑑′′
. 𝐶𝑜𝑠 𝜔𝑐(𝑡 −
𝑑′′
𝑐
)
𝑬 𝒕𝒐𝒕 = 𝑬𝒍𝒐𝒔 + 𝑬 𝒈
Δ = d’’ – d’
𝒅′ = (𝒉 𝒕 − 𝒉 𝒓) 𝟐 + 𝒅 𝟐
𝒅′′
= (𝒉 𝒕 + 𝒉 𝒓) 𝟐 + 𝒅 𝟐
𝐸 𝑜 𝑑 𝑜
𝑑
=
𝐸 𝑜 𝑑 𝑜
𝑑′ =
𝐸 𝑜 𝑑 𝑜
𝑑′′
𝑬 𝒕𝒐𝒕(𝒅) = 𝑬 𝒐 𝒅 𝒐
𝟒𝝅𝒉 𝒕 𝒉 𝒓
𝝀𝒅 𝟐
Relating Field to Power by using the simple relation
(Field)2 = Power
𝑷 𝒓 =
𝑷 𝒕 𝑮 𝒕 𝑮 𝒓 𝒉 𝒕 𝒉 𝒓
𝟐
𝒅 𝟒
𝑷𝑳 𝒅𝑩 = 𝟒𝟎 𝐥𝐨𝐠 𝒅 − (𝟏𝟎 𝒍𝒐𝒈 𝑮 𝒕 + (𝟏𝟎 𝒍𝒐𝒈 𝑮 𝒓 + (𝟐𝟎 𝒍𝒐𝒈 𝒉 𝒕 + 𝟐𝟎𝒍𝒐𝒈(𝒉 𝒓)
Path Loss for 2 Ray model is given by: