Two-Plate Waveguide Wave impedance TM mode 𝑍= 𝐸 𝐻 if 𝐸 × 𝐻 has the same direction as propagation direction 𝑍=− 𝐸 𝐻 if 𝐸 × 𝐻 has a direction opposite to propagation direction TM mode 𝐸 𝑧,𝑚 =− 𝐵 𝑚 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝐻 𝑦,𝑚 = 𝐴 𝑚 2 + 𝐵 𝑚 2 1 𝜂 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥 𝑍 𝑇𝑀 +𝑥 =− 𝐸 𝑧 𝐻 𝑦 𝐵 𝐴 2 + 𝐵 2 = sin 𝜃 𝑖 = 𝛽 𝑥 𝛽 𝑍 𝑇𝑀 +𝑥 =𝜂 𝐵 𝐴 2 + 𝐵 2 𝑍 𝑇𝑀 +𝑥 =𝜂 1− 𝑓 𝑐 /𝑓 2

Two-Plate Waveguide Wave impedance 𝑍 𝑇𝐸 +𝑥 = 𝜂 1− 𝑓 𝑐 /𝑓 2 𝑍 𝑇𝐸 +𝑥 = 𝜂 1− 𝑓 𝑐 /𝑓 2 𝑍 𝑇𝑀 +𝑥 =𝜂 1− 𝑓 𝑐 /𝑓 2 𝑍 𝑇 𝑀 0 +𝑥 =𝜂 𝑍 𝑇𝐸 𝑍 𝑇𝑀 = 𝜂 2 𝜂= 𝜇/𝜀

Two-Plate Waveguide Waveguide wavelength 𝜆 𝑥 /2 𝜆 𝑥 𝜆 𝑤𝑔 = 2𝜋 𝛽 𝑥 𝜆 𝑤𝑔 = 2𝜋 𝛽 𝑥 𝜆 𝑥 /2 𝛽 𝑥 = 𝛽 2 − 𝛽 𝑐 2 𝜆 𝑥 =𝛽 1− 𝑓 𝑐 /𝑓 2 𝜆 𝑤𝑔 = 2𝜋 𝛽 1 1− 𝑓 𝑐 /𝑓 2 𝜆 𝑤𝑔 = 𝜆 1− 𝑓 𝑐 /𝑓 2 𝜆= 2𝜋 𝛽 𝜆 𝑤𝑔 ≥𝜆

Two-Plate Waveguide Phase velocity 𝑣 𝑝,𝑤𝑔 ≥ 𝑣 𝑝 𝛽 𝑥 = 𝛽 2 − 𝛽 𝑐 2 =𝛽 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑝,𝑤𝑔 = 𝜔 𝛽 𝑥 𝑣 𝑝,𝑤𝑔 = 𝜔 𝛽 1 1− 𝑓 𝑐 /𝑓 2 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑝 = 𝜔 𝛽 𝑣 𝑝,𝑤𝑔 ≥ 𝑣 𝑝

Two-Plate Waveguide Phase velocity Group velocity 𝑣 𝑝,𝑤𝑔 = 𝜔 𝛽 𝑥 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 Group velocity 𝑣 𝑔 = 𝑑𝜔 𝑑 𝛽 𝑥 𝛽 𝑥 =𝛽 1− 𝜔 𝑐 /𝜔 2 = 𝛽 𝜔 𝜔 2 − 𝜔 𝑐 2 = 1 𝑣 𝑝 𝜔 2 − 𝜔 𝑐 2 𝑑 𝛽 𝑥 𝑑𝜔 = 1 𝑣 𝑝 𝜔 𝜔 2 − 𝜔 𝑐 2 = 1 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑔 = 𝑑𝜔 𝑑 𝛽 𝑥 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑔 ≤ 𝑣 𝑝

Two-Plate Waveguide Phase velocity Group velocity 𝑣 𝑝,𝑤𝑔 = 𝜔 𝛽 𝑥 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 Group velocity 𝑣 𝑝,𝑤𝑔 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑔 = 𝑑𝜔 𝑑 𝛽 𝑥 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑝,𝑤𝑔 𝑣 𝑝 𝑣 𝑔 = 𝑣 𝑝 1− 𝑓 𝑐 /𝑓 2 𝑣 𝑝,𝑤𝑔 𝑣 𝑔 = 𝑣 𝑝 2