Transmission Line Theory Shahid Bahonar University of Kerman Faculty of Engineering Electrical Engineering Department Transmission Line Theory By: Kambiz Afrooz

𝑉(𝑧)=𝑎 𝑒 −𝛾𝑧 +𝑏 𝑒 𝛾𝑧 =𝑎 𝑒 −𝛼𝑧 𝑒 −𝑗𝛽𝑧 +𝑏 𝑒 𝛼𝑧 𝑒 𝑗𝛽𝑧 𝜕 𝑉(𝑧 𝜕 𝑧 =−(𝑅+𝑗𝐿𝜔) 𝐼(𝑧)  𝜕 𝐼(𝑧 𝜕 𝑧 =−(𝐺+𝑗𝐶𝜔) 𝑉(𝑧)  𝜕 2  𝑉(𝑧 𝜕  𝑧 2 =−(𝑅+𝑗𝐿𝜔)  𝜕 𝐼(𝑧 𝜕 𝑧 =(𝑅+𝑗𝐿𝜔)(𝐺+𝑗𝐶𝜔)𝑉(𝑧)  𝜕 2  𝑉(𝑧 𝜕  𝑧 2 = 𝛾 2 𝑉(𝑧) 𝛾= 𝑅+𝑗𝐿𝜔)(𝐺+𝑗𝐶𝜔 =𝛼+𝑗𝛽 𝑆 2 = 𝛾 2        ⇒𝑆=±𝛾 𝑉(𝑧)=𝑎  𝑒 −𝛾𝑧 +𝑏  𝑒 𝛾𝑧 =𝑎  𝑒 −𝛼𝑧 𝑒 −𝑗𝛽𝑧 +𝑏  𝑒 𝛼𝑧 𝑒 𝑗𝛽𝑧 𝑣(𝑧,𝑡)=𝑎  𝑒 −𝛼𝑧  cos(𝜔𝑡−𝛽𝑧)+𝑏  𝑒 𝛼𝑧  cos(𝜔𝑡+𝛽𝑧) 

𝑍 𝐿 = 𝑉(0) 𝐼(0) = 𝑍 ∘ 𝑉++𝑉− 𝑉+−𝑉− 𝑉− 𝑉+ = 𝑍 𝐿 − 𝑍 ∘ 𝑍 𝐿 + 𝑍 ∘ In lossless case: 𝑉 𝑧 =𝑉+  𝑒 −𝑗𝛽𝑧 +𝑉− 𝑒 𝑗𝛽𝑧 𝐼(𝑧)= 𝑉 + 𝑍 ∘ 𝑒 −𝑗𝛽𝑧 − 𝑉 − 𝑍 ∘ 𝑒 𝑗𝛽𝑧 𝑉 0 =𝑉++𝑉− 𝑍 𝐿 = 𝑉(0) 𝐼(0) = 𝑍 ∘ 𝑉++𝑉− 𝑉+−𝑉− 𝑉− 𝑉+ = 𝑍 𝐿 − 𝑍 ∘ 𝑍 𝐿 + 𝑍 ∘ 𝐼(0)= 𝑉 + 𝑍 ∘ − 𝑉 − 𝑍 ∘

𝐼(𝑧)= 𝑉 + 𝑍 ∘ 𝑒 −𝑗𝛽𝑧 − 𝑉+ГL 𝑍 ∘ 𝑒 𝑗𝛽𝑧 = 𝑉+(z)−𝑉−(z) 𝑍 ∘ 𝑍 𝐿 − 𝑍 ∘ 𝑍 𝐿 + 𝑍 ∘ =Г𝐿 1+Г𝐿 1−Г𝐿 = 𝑍 𝐿 𝑍 ∘ =𝑧𝐿 𝑉−= 𝑍 𝐿 − 𝑍 ∘ 𝑍 𝐿 + 𝑍 ∘ 𝑉+=Г𝐿𝑉+ 𝑉 𝑧 =𝑉+  𝑒 −𝑗𝛽𝑧 +𝑉+ГL 𝑒 𝑗𝛽𝑧 = 𝑉+(z)+ 𝑉−(z) V(𝑧+λ)=V(z) 𝐼(𝑧)= 𝑉 + 𝑍 ∘ 𝑒 −𝑗𝛽𝑧 − 𝑉+ГL 𝑍 ∘ 𝑒 𝑗𝛽𝑧 = 𝑉+(z)−𝑉−(z) 𝑍 ∘ I(𝑧+λ)=I(z) Г(z) = 𝑉−(z) 𝑉+(z) = 𝑉+ГL 𝑒 𝑗𝛽𝑧 𝑉+  𝑒 −𝑗𝛽𝑧 = ГL 𝑒 𝑗2𝛽𝑧 Z(z)= 𝑉(𝑧) 𝐼(𝑧) = 𝑍 ∘ 𝑍 𝐿 −𝑗 𝑍 ∘ tan⁡(β𝑧) 𝑍 ∘ −𝑗 𝑍 𝐿 tan⁡(β𝑧)

Г(𝑙) = 𝑉−(𝑙) 𝑉+(𝑙) = 𝑉+ГL 𝑒 −𝑗𝛽𝑙 𝑉+ 𝑒 𝑗𝛽𝑙 = ГL 𝑒 −𝑗2𝛽𝑙 𝛤(𝑙)=| 𝛤 𝐿 |  𝑒 𝑗 ∠  𝛤 𝐿   𝑒 −2𝑗𝛽𝑙 | 𝛤(𝑙 |=| 𝛤 𝐿 |  𝑒 𝑗  𝜑 𝐿   𝑒 −2𝑗𝛽𝑙 ∠ 𝛤(𝑙)= 𝜑 𝐿 −2𝛽𝑙 Z(𝑙)= 𝑉(𝑙) 𝐼(𝑙) = 𝑍 ∘ 𝑍 𝐿 +𝑗 𝑍 ∘ tan⁡(β𝑙) 𝑍 ∘ +𝑗 𝑍 𝐿 tan⁡(β𝑙) V(𝑙+0.5λ)=−V(𝑙) Z(𝑙+0.5λ)=Z(𝑙) Г(𝑙+0.5λ)=Г(𝑙) I(𝑙+0.5λ)=−I(𝑙)

Smith Chart: 𝛤(𝑧)= 𝑍(𝑧)− 𝑍 ∘ 𝑍(𝑧)+ 𝑍 ∘ = 𝑧(𝑧)−1 𝑧(𝑧)+1 𝑧(𝑧)= 1+𝛤(𝑧 1−𝛤(𝑧 𝑧(𝑧)=𝑟+𝑗𝑥,𝛤(𝑧)= 𝛤 𝑟 +𝑗 𝛤 𝑖 𝑟+𝑗𝑥= 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 = 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1− 𝛤 𝑟 +𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 +𝑗2 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 𝑟= 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 ⇒ 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 = 1 𝑟 − 𝛤 𝑟 2 𝑟 − 𝛤 𝑖 2 𝑟 ⇒𝑟−2𝑟 𝛤 𝑟 +𝑟 𝛤 𝑟 2 +𝑟 𝛤 𝑖 2 =1− 𝛤 𝑟 2 − 𝛤 𝑖 2 1+𝑟) 𝛤 𝑟 2 −2𝑟 𝛤 𝑟 +(1+𝑟) 𝛤 𝑖 2 =1−𝑟⇒ 𝛤 𝑟 2 −2 𝑟 𝑟+1 𝛤 𝑟 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 𝛤 𝑟 − 𝑟 𝑟+1 2 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 + 𝑟 2 1+𝑟 2 = 1 1+𝑟 2

Smith Chart: 𝛤(𝑧)= 𝑍(𝑧)− 𝑍 ∘ 𝑍(𝑧)+ 𝑍 ∘ = 𝑧(𝑧)−1 𝑧(𝑧)+1 𝑧(𝑧)= 1+𝛤(𝑧 1−𝛤(𝑧 𝑧(𝑧)=𝑟+𝑗𝑥,𝛤(𝑧)= 𝛤 𝑟 +𝑗 𝛤 𝑖 𝑟+𝑗𝑥= 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 = 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 +𝑗 𝛤 𝑖 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1− 𝛤 𝑟 +𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 +𝑗2 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 𝑥= 2 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 = 2 𝑥 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 − 2 𝑥 𝛤 𝑖 =0 1− 𝛤 𝑟 2 + 𝛤 𝑖 − 1 𝑥 2 = 1 𝑥 2

Z Chart: 𝛤 𝑟 − 𝑟 𝑟+1 2 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 + 𝑟 2 1+𝑟 2 = 1 1+𝑟 2 𝛤 𝑟 − 𝑟 𝑟+1 2 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 + 𝑟 2 1+𝑟 2 = 1 1+𝑟 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 − 1 𝑥 2 = 1 𝑥 2

Y Chart: 𝛤(𝑧)= 𝑍(𝑧)− 𝑍 ∘ 𝑍(𝑧)+ 𝑍 ∘ = 𝑧(𝑧)−1 𝑧(𝑧)+1 𝑦(𝑧)= 1−𝛤(𝑧 1+𝛤(𝑧 𝑦(𝑧)=𝑔+𝑗𝑏,𝛤(𝑧)= 𝛤 𝑟 +𝑗 𝛤 𝑖 𝑔+𝑗𝑏= 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1+ 𝛤 𝑟 −𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 −𝑗2 𝛤 𝑖 1+ 𝛤 𝑟 2 + 𝛤 𝑖 2 𝑔= 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 1+ 𝛤 𝑟 2 + 𝛤 𝑖 2 ⇒ 1+ 𝛤 𝑟 2 + 𝛤 𝑖 2 = 1 𝑔 − 𝛤 𝑟 2 𝑔 − 𝛤 𝑖 2 𝑔 ⇒𝑔+2𝑔 𝛤 𝑟 +𝑔 𝛤 𝑟 2 +𝑔 𝛤 𝑖 2 =1− 𝛤 𝑟 2 − 𝛤 𝑖 2 1+𝑔) 𝛤 𝑟 2 +2𝑔 𝛤 𝑟 +(1+𝑔) 𝛤 𝑖 2 =1−𝑔⇒ 𝛤 𝑟 2 +2𝑔 𝑔 𝑟+1 𝛤 𝑟 + 𝛤 𝑖 2 = 1−𝑔 1+𝑔 𝛤 𝑟 + 𝑔 𝑔+1 2 + 𝛤 𝑖 2 = 1−𝑔 1+𝑔 + 𝑔 2 1+𝑔 2 = 1 1+𝑔 2

Y Chart: 𝛤(𝑧)= 𝑍(𝑧)− 𝑍 ∘ 𝑍(𝑧)+ 𝑍 ∘ = 𝑧(𝑧)−1 𝑧(𝑧)+1 𝑦(𝑧)= 1−𝛤(𝑧 1+𝛤(𝑧 𝑦(𝑧)=𝑔+𝑗𝑏,𝛤(𝑧)= 𝛤 𝑟 +𝑗 𝛤 𝑖 𝑔+𝑗𝑏= 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 −𝑗 𝛤 𝑖 1+ 𝛤 𝑟 +𝑗 𝛤 𝑖 1+ 𝛤 𝑟 −𝑗 𝛤 𝑖 = 1− 𝛤 𝑟 2 − 𝛤 𝑖 2 −𝑗2 𝛤 𝑖 1+ 𝛤 𝑟 2 + 𝛤 𝑖 2 𝑥= 2 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 = 2 𝑏 𝛤 𝑖 1− 𝛤 𝑟 2 + 𝛤 𝑖 2 − 2 𝑏 𝛤 𝑖 =0 1− 𝛤 𝑟 2 + 𝛤 𝑖 − 1 𝑏 2 = 1 𝑏 2

ZY Chart: 𝛤 𝑟 − 𝑟 𝑟+1 2 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 + 𝑟 2 1+𝑟 2 = 1 1+𝑟 2 𝛤 𝑟 − 𝑟 𝑟+1 2 + 𝛤 𝑖 2 = 1−𝑟 1+𝑟 + 𝑟 2 1+𝑟 2 = 1 1+𝑟 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 − 1 𝑥 2 = 1 𝑥 2 1− 𝛤 𝑟 2 + 𝛤 𝑖 − 1 𝑏 2 = 1 𝑏 2 𝛤 𝑟 + 𝑔 𝑔+1 2 + 𝛤 𝑖 2 = 1−𝑔 1+𝑔 + 𝑔 2 1+𝑔 2 = 1 1+𝑔 2