Rectangular Waveguide 𝑚=0,1,2,… 𝑛=0,1,2,… 𝑚=𝑛≠0 𝑚=1,2,… 𝑛=1,2, … 𝑇 𝐸 𝑚,𝑛 𝑧 𝑇 𝑀 𝑚,𝑛 𝑧 𝐸 𝑥 = 𝑗𝜔𝜇 𝛽 𝑦 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ cos 𝛽 𝑥 𝑥 sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑥 =− 𝑗 𝛽 𝑥 𝛽 𝑧 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴 cos ( 𝛽 𝑥 𝑥) sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ sin ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑦 =− 𝑗 𝛽 𝑦 𝛽 𝑧 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴 sin ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑧 =0 𝐸 𝑧 =𝐴 sin ( 𝛽 𝑥 𝑥) sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦 𝐻 𝑥 =− 𝜔𝜀 𝛽 𝑧 𝐸 𝑦 𝐻 𝑦 = 𝛽 𝑧 𝜔𝜇 𝐸 𝑥 𝐻 𝑦 = 𝜔𝜀 𝛽 𝑧 𝐸 𝑥 𝐻 𝑧 =𝐴′ cos ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑧 =0 𝛽 𝑧 = 𝛽 2 − 𝛽 𝑥 2 + 𝛽 𝑦 2 𝛽 𝑥 = 𝑚𝜋 𝑎 𝛽 𝑦 = 𝑛𝜋 𝑏 𝛽=𝜔 𝜇𝜀
Rectangular Waveguide Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏) 𝑚=1, 𝑛=0 𝛽 𝑥 = 𝜋 𝑎 , 𝛽 𝑦 =0 𝐸 𝑥 =0 𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦 𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑥 =𝑗 𝛽 𝑧 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑥 =𝑗 𝛽 2 − 𝑎 𝜋 2 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑦 =−𝑗𝜔𝜇 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑦 =0 𝐸 𝑧 =0 𝐻 𝑧 = 𝐴 ′ cos 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
Rectangular Waveguide Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏) 𝑚=1, 𝑛=0 𝛽 𝑥 = 𝜋 𝑎 , 𝛽 𝑦 =0 𝐸 𝑥 =0 𝐸 𝑦 =−𝑗𝜔𝜇 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐸 𝑧 =0 𝐻 𝑥 =𝑗 𝛽 2 − 𝑎 𝜋 2 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧 𝐻 𝑦 =0 𝐻 𝑧 = 𝐴 ′ cos 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
Rectangular Waveguide Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏) 𝑚=1, 𝑛=0 𝐸 𝑥 =0 𝐸 𝑦 =𝜔𝜇 𝑎 𝜋 𝐻 0 sin 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧− 𝜋 2 𝐸 𝑧 =0 𝐻 𝑥 = 𝛽 2 − 𝑎 𝜋 2 𝑎 𝜋 𝐻 0 sin 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧+ 𝜋 2 𝐻 𝑦 =0 𝐻 𝑧 = 𝐻 0 cos 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧
Rectangular Waveguide Cut-off frequency 𝛽 𝑧 = 𝛽 2 − 𝛽 𝑥 2 + 𝛽 𝑦 2 y 𝛽 𝑐 2 = 𝛽 𝑥 2 + 𝛽 𝑦 2 x 𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2 b 𝛽 𝑐 =2𝜋 𝑓 𝑐 𝜇𝜀 𝑓 𝑐 = 1 2𝜋 𝜇𝜀 𝛽 𝑥 2 + 𝛽 𝑦 2 = 1 2𝜋 𝜇𝜀 𝑚𝜋 𝑎 2 + 𝑛𝜋 𝑏 2 a z
Rectangular Waveguide Transverse E-field: Wave Impedance 𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2 𝛽=𝜔 𝜇𝜀 𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦 𝑍 𝑤 +𝑧 = 𝐸 𝑥 𝐻 𝑦 =− 𝐸 𝑦 𝐻 𝑥 𝛽 𝑧 =𝛽 1− 𝛽 𝑐 𝛽 2 𝛽 𝑧 =𝛽 1− 𝑓 𝑐 𝑓 2 𝐻 𝑦 = 𝛽 𝑧 𝜔𝜇 𝐸 𝑥 𝑍 𝑤 +𝑧 = 𝜔𝜇 𝛽 𝑧 𝛽 𝑐 = 𝛽 𝑥 2 + 𝛽 𝑦 2 𝑍 𝑤 +𝑧 = 𝜔𝜇 𝛽 1− 𝑓 𝑐 𝑓 2 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 /𝑓 2 𝑓> 𝑓 𝑐 ∞ 𝑓= 𝑓 𝑐 𝑍 0 𝑗 𝑓 𝑐 /𝑓 2 −1 𝑓< 𝑓 𝑐 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2 𝜔𝜇 𝛽 = 𝜇 𝜀 = 𝑍 0
Rectangular Waveguide Transverse H-field: Wave Impedance 𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2 𝛽=𝜔 𝜇𝜀 𝐻 𝑥 =− 𝜔𝜀 𝛽 𝑧 𝐸 𝑦 𝑍 𝑤 +𝑧 = 𝐸 𝑥 𝐻 𝑦 =− 𝐸 𝑦 𝐻 𝑥 𝛽 𝑐 = 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐻 𝑦 = 𝜔𝜀 𝛽 𝑧 𝐸 𝑥 𝑍 𝑤 +𝑧 = 𝛽 𝑧 𝜔𝜀 𝛽 𝑧 =𝛽 1− 𝛽 𝑐 𝛽 2 𝛽 𝑧 =𝛽 1− 𝑓 𝑐 𝑓 2 𝑍 𝑤 +𝑧 = 𝛽 𝜔𝜀 1− 𝑓 𝑐 𝑓 2 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 /𝑓 2 𝑓> 𝑓 𝑐 0 𝑓= 𝑓 𝑐 𝑗 𝑍 0 𝑓 𝑐 /𝑓 2 −1 𝑓< 𝑓 𝑐 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2 𝛽 𝜔𝜀 = 𝜇 𝜀 = 𝑍 0
Rectangular Waveguide Wave Impedances: 𝑇 𝐸 𝑧 and 𝑇 𝑀 𝑧 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2 𝑇 𝐸 𝑧 𝑍 0 = 𝜔𝜇 𝛽 𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2 𝑍 0 = 𝛽 𝜔𝜀 𝑇 𝑀 𝑧