Two-Plate Waveguide 𝑥 𝑡=0 Plane wave: TEM mode 𝑚=0 TM 0 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 =0 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 𝑚 𝑒 −𝑗 𝛽 𝑥 𝑥 =− 𝐵 𝑚 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

Two-Plate Waveguide 𝑥 𝑡=0 𝑚=1 TM 1 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 = 𝐴 1 sin 𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 1 cos 𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

Two-Plate Waveguide 𝑥 𝑡=0 𝑚=2 TM 2 mode TM modes 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥 = 𝐴 1 sin 2𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2 𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐸 𝑧 =− 𝐵 1 cos 2𝜋 𝐷 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥 𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

Two-Plate Waveguide 𝐷 𝑥 𝑧 𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 TE modes 𝐻 𝑥 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐷 TE modes 𝐻 𝑧 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝑥 𝑧

Two-Plate Waveguide TE 0 mode does not exist 𝐷 𝑥 𝑧 𝑚=0 TE 0 mode 𝐸 𝑦 =0 TE modes 𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,…

Two-Plate Waveguide sin 𝜋 𝐷 𝑧 𝑚=1 TE 1 mode TE modes 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,…

Two-Plate Waveguide sin 2𝜋 𝐷 𝑧 𝑚=2 TE 2 mode TE modes 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,…

Two-Plate Waveguide TE modes TM modes Electric field 𝐸 𝑦,𝑚 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑥,𝑚 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐸 𝑧,𝑚 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 Magnetic field 𝐻 𝑥,𝑚 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐻 𝑧,𝑚 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐻 𝑦,𝑚 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑚=0 TE 0 does not exist TM 0 =TEM is a plane wave Possible modes TE 1 , TE 2 , … TM 0 , TM 1 , TM 2 , … 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,… A necessary condition for a mode to exist in a two-plate waveguide is 𝛽> 𝑚𝜋 𝐷 𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2 𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2

Two-Plate Waveguide Question: Given 𝐷 and 𝑓, what modes can exist? 𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2 𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,… 𝛽> 𝑚𝜋 𝐷 𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2 Question: Given 𝐷 and 𝑓, what modes can exist? 2𝜋𝑓 𝜇𝜀 > 𝑚𝜋 𝐷 Existing modes: 2𝑚+1 TE 1 , TE 2 , … TE 𝑚 , 𝑚 TE modes TM 0 , TM 1 , … TM 𝑚 , 𝑚+1 TM modes 𝑚<2𝜋𝑓 𝜇𝜀 𝐷 𝜋 Example: 𝐷=3 cm, 𝑓=10 GHz 𝑚<2𝜋𝑓 𝜇𝜀 𝐷 𝜋 = 2𝑓 𝑐 𝐷 = 2× 10 10 3× 10 8 ×0.03=2 If 𝐷=1.5 cm, 𝑚=0 1 mode No TE mode 1 TM mode: TM 0 If 𝐷=5 cm, 𝑚<3.333 𝑚=3 7 modes 3 TE modes 4 TM modes 𝑚=1 3 modes: TE 1 , TM 0 , TM 1