Chapter 2: Transmission lines and waveguides 2.1 Generation solution for TEM, TE and TM waves 2.2 Parallel plate waveguide 2.3 Rectangular waveguide 2.4 Circular waveguide 2.5 Coaxial line 2.6 Surface waves on a grounded dielectric slab 2.7 Stripline 2.8 Microstrip 2.9 Wave velocities and dispersion 2.10 Summary of transmission lines and waveguids

Transmission lines and waveguides

2.1 Generation solution for TEM, TE and TM waves Two-conductor TL Closed waveguide Electromagnetic fields (time harmonic eiωt and propagate along z axis): where e(x,y) and h(x,y) represent the transverse (x,y) E and H components, while ez and hz are the longitudinal E and H components.

In the case without source, the harmonic Maxwell’s equations can be written as: With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz: Wavevector in xy plane: (kc2=kx2 + ky2)

(1) TEM waves (Ez = Hz = 0) Propagation constant: (1) TEM waves (Ez = Hz = 0) Propagation constant: (kc = 0, no cutoff) The Helmholtz equation for Ex: For a e-jz dependence, and the above equation can be simplified Laplace equations, equal to static fields Similarly, Transverse magnetic field:

Characteristic impedance of a transmission line, Z0: Wave impedance: Note: Wave impedance, Z relates transverse field components and is dependent only on the material constant for TEM wave. For TEM wave Characteristic impedance of a transmission line, Z0: relates an incident voltage and current and is a function of the line geometry as well as the material filling the line. For TEM wave: Z0= V/I (V incident wave voltage I incident wave current)

(2) TE waves (Ez = 0 and Hz  0) The field components can be simplified as:  is a function of frequency and TL/WG structure Solve Hz from the Helmholtz equation Because , then where . Boundaries conditions will be used to solve the above equation. Solve Hz first and then obtain Hx, Hy, Ex, Ey TE wave impedance:

(3) TM waves (Hz = 0 and Ez  0) The field components can simplified:  is a function of frequency and TL/WG structure Solve Ez from Helmholtz equation: Because , then where . Boundaries conditions will be used to solve the above equation. Solve Ez first and then obtain Hx, Hy, Ex, Ey TM wave impedance:

(4) Attenuation due to dielectric loss Total attenuation constant in TL or WG = c + d. c: due to conductive loss; calculated using the perturbation method; must be evaluated separately for each type. d: due to the dielectric loss; calculated from the propagation constant. Taylor expansion (tan << 1)

Mode Definition Propagation constant Wave impedance Solve the fields TEM Ez = Hz = 0 TE (H wave) Hz  0 and Ez = 0 TM (E wave) Ez  0 and Hz = 0 Ez Hx, Hy, Ex, Ey Hz

2.2 Parallel plate waveguide w >> d (fringing fields and any x variation could be ignored) Formed from two flat plates or strips Probably the simplest type of guide Support TEM, TE and TM modes Important for practical reasons.

(a) TEM modes (Ez = Hz = 0) Laplace equation for the electric potential  (x,y) for Boundary conditions: The transverse field , so that we have Characteristic impedance: Phase velocity:

(b) TMn modes (Hz = 0) The transverse ez(x,y) satisfies General solutions: Boundary condition 1: Bn = 0 Boundary condition 2: Solutions of TMn modes:

The components of TMn: Example: TM1

(b) TMn mode (Hz = 0) Propagation constant: k > kc Traveling wave k = kc Tunneling? k < kc Evanescent wave Cutoff frequency: @ (k = kc) The TMn mode cannot propagate at f < fc! Power flow: Frequency and geometry dependent

(c) TEn mode (Ez = 0) The transverse hz(x,y) satisfies And boundary Ex(x,y) = 0 at y = 0, d. An = 0 TE1 Cutoff frequency : where the propagation constant Wave impedance:

Parallel plate waveguide TEM TM1 TE1

Require: start from the first two Maxwell’s equations Homework: 1. Derive the field solutions of TE1 mode for a parallel-plate metallic waveguide and plot the field pattern of each component roughly if possible. Require: start from the first two Maxwell’s equations

Substitute Eq. (1) into (2) by eliminating H and we have (no source, isotropic)

In the case without source, the harmonic Maxwell’s equations can be written as: With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz: Wavevector in xy plane: (kc2=kx2 + ky2)