1. 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 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

2. 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 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

3. 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 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥

4. 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

5. 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,…

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

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

8. 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 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

9. 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× × ×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