1. Two-Plate Waveguide PEC plate Subject to b.c. Subject to b.c. 𝐷 𝑥 𝑧
𝐸 𝑥 = 𝐴 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
Subject to b.c.
𝐸 𝑦 = 𝐶 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
Subject to b.c. 𝐷
𝐸 𝑧 = −𝐵 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑥 𝑧
PEC plate
𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,…
Boundary conditions:
sin 𝛽 𝑧 𝑧 =0
𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2
𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2
𝑧=0, 𝐷

2. Two-Plate Waveguide Boundary conditions: PEC plate 𝐷 𝑥 PEC plate 𝑧
𝐸 𝑥 = 𝐴 𝑚 sin 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥+ 𝜋 2
𝐸 𝑦 = 𝐶 𝑚 sin 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥+ 𝜋 2 𝐷
𝐸 𝑧 = − 𝐵 𝑚 cos 𝑚𝜋 𝐷 𝑧 𝑒 −𝑗 𝛽 2 − 𝑚𝜋 𝐷 2 𝑥 𝑥
PEC plate 𝑧
𝛽 𝑧 = 𝑚𝜋 𝐷 , 𝑚=0,1,…
Boundary conditions:
𝛽 𝑥 = 𝛽 2 − 𝑚𝜋 𝐷 2
𝐸 𝑥 and 𝐸 𝑦 vanish on the PEC surface
sin 𝛽 𝑧 𝑧 =0, 𝑧=0, 𝐷
𝛽 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2
𝛽=𝜔 𝜇𝜀

3. Two-Plate Waveguide PEC plate 𝐷 𝑥 PEC plate 𝑧
𝐻 𝑥 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐷
𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐻 𝑧 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝑥
PEC plate 𝑧
𝛻× 𝐸 =− 𝜕 𝐵 𝜕𝑡
𝐻 = 𝛻× 𝐸 −𝑗𝜔𝜇
𝛻× 𝐸 =−𝑗𝜔𝜇 𝐻

4. Two-Plate Waveguide 𝐷 𝑥 𝑧 Field modes: standing wave patterns
𝐻 𝑥 =− 𝐶 𝑚 1 𝜂 𝛽 𝑧 𝛽 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐸 𝑦 = 𝐶 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝐷
𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐻 𝑧 = 𝐶 𝑚 1 𝜂 𝛽 𝑥 𝛽 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 𝑥 𝑧
TM: 𝐻 𝑥 = 𝐻 𝑧 =0
TE: 𝐸 𝑥 = 𝐸 𝑧 =0
Field modes: standing wave patterns
One m value corresponds to one mode
TM: 𝐶 𝑚 =0, 𝐸 𝑦 =0
TE: 𝐴 𝑚 = 𝐵 𝑚 =0, 𝐻 𝑦 =0
Field modes: TM mode and TE mode
TM: Transverse magnetic field → 𝐻 𝑥 = 𝐻 𝑧 =0
TE: Transverse electric field → 𝐸 𝑥 = 𝐸 𝑧 =0

5. Two-Plate Waveguide 𝐷 𝑥 𝑧 𝐸 𝑥 = 𝐴 𝑚 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2 TM modes
𝐻 𝑦 = 𝐴 𝑚 2 + 𝐵 𝑚 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐷
TM modes
𝐸 𝑧 = − 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑥 𝑧

6. 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

7. 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

8. 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