Microwave Engineering

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Microwave Engineering ECE 5317-6351 Microwave Engineering Adapted from notes by Prof. Jeffery T. Williams Fall 2011 Fall 2018 Prof. David R. Jackson Dept. of ECE Notes 15 Transverse Resonance Method

This leads to a “Transverse Resonance Equation (TRE).” Transverse Resonance Method This is a general method that can be used to help us calculate various important quantities: Wavenumbers for complicated waveguiding structures (dielectric- loaded waveguides, surface waves, etc.) Resonance frequencies of resonant cavities (resonators) The transverse resonance method involves establishing a reference plane and enforcing the KVL and KCL. This leads to a “Transverse Resonance Equation (TRE).”

Transverse Resonance Method To illustrate the method, consider a lossless resonator formed by a lossless transmission line with reactive loads at the ends. We wish to find the resonance frequencies of this transmission-line resonator. (Here we develop the method. We will do the actual algebra for this structure a little later.) A resonator can have nonzero fields at a resonance frequency, when there is no source.

Transverse Resonance Method (cont.) We start by selecting an (arbitrary) reference plane R. R = reference plane at arbitrary x = x0 Note: Although the location of the reference plane is arbitrary, a good choice will often simplify the derivation of the TRE and the complexity of the final TRE.

Transverse Resonance Method (cont.) Examine the voltages and currents at the reference plane:

Transverse Resonance Method (cont.) Define impedances: Boundary conditions: Hence: TRE

Summary TRE: or

Derive the resonance frequency of a parallel RLC resonator. Example: Derive the resonance frequency of a parallel RLC resonator. Lossless: At the resonance frequency, voltages and currents exist with no sources.

A reference plane is chosen. RLC Resonator (cont.) A reference plane is chosen.

RLC Resonator (cont.) The TRE is obtained.

Solve the quadratic equation. RLC Resonator (cont.) Solve the quadratic equation.

RLC Resonator (cont.) Hence, the plus sign is the correct choice. Factor out 4LC from the square root. For the lossless limit, G  0: (must be a positive real number) Hence, the plus sign is the correct choice.

complex resonance frequency RLC Resonator (cont.) Hence, we have complex resonance frequency We can write this as where

quality factor of RLC resonator RLC Resonator (cont.) Ratio of imaginary and real parts of complex frequency: so or where quality factor of RLC resonator

In the time domain we have: RLC Resonator (cont.) In the time domain we have: In the phasor domain:

RLC Resonator (cont.)

Q of a General Resonator The quality factor (Q) for a general resonator is defined as: Note: is often denoted simply as 0 in this equation.

Q for RLC Resonator For the RLC resonator we have: Phasor domain Hence

Q for RLC Resonator (cont.) For the RLC resonator we have: Similarly: Phasor domain Hence so

General Q Formulas These formulas hold for any resonator:

Transmission Line Resonator Example: Derive a transcendental equation for the resonance frequency of this lossless transmission-line resonator. We choose a reference plane at x = 0+. Note: The load reactances may be functions of frequency.

Transmission Line Resonator (cont.) Apply TRE:

Transmission Line Resonator (cont.)

Transmission Line Resonator (cont.) After simplifying, we have Special cases:

Transmission Line Resonator (cont.) For the resonance frequencies, we have We then have

Rectangular Resonator Example: Derive a transcendental equation for the resonance frequencies of a rectangular resonator. Orient the structure so that b < a < h The structure is thought of as supporting rectangular waveguide modes bouncing back and forth in the z direction. The index p describes the variation in the z direction. We have TMmnp and TEmnp modes.

(Choose Z0 to be the wave impedance.) Rectangular Resonator (cont.) We use a Transverse Equivalent Network (TEN) to model any one of the waveguide modes: We choose a reference plane at z = 0+: (Choose Z0 to be the wave impedance.) Hence

Rectangular Resonator (cont.) Hence

Rectangular Resonator (cont.) Also, we have Hence, we have

The TMz and TEz modes have the same resonance frequency. Rectangular Resonator (cont.) Solving for the wavenumber k we have: Hence Note: The TMz and TEz modes have the same resonance frequency. or TEmnp mode: The lowest mode is the TE101 mode.

The other field components, Ey and Hx, can be found from Hz. Rectangular Resonator (cont.) TE101 mode: Note: The sin is used to ensure the boundary condition on the PEC top and bottom plates: The other field components, Ey and Hx, can be found from Hz.

Rectangular Resonator (cont.) Here we examine the Q of the resonator: (dielectric and conductor loss) We now allow for dielectric and conductor loss in the resonator.

Rectangular Resonator (cont.) Q of TE101 mode

(This holds for any mode.) Rectangular Resonator (cont.) Q of TE101 mode Results (from Pozar book): This gives: (This holds for any mode.)

Excitation of Resonator h Practical excitation by a coaxial probe Circuit model: (probe inductance) Tank (RLC) circuit Note: The value of the circuit elements will depend on where the resonator is fed, and also the size of the probe. The values of Q and 0 do not depend on the feed.

Excitation of Resonator (cont.) (from circuit theory – see next slide) where Approximate form (see slide 35):

Excitation of Resonator (cont.) Derivation of ZRLC formula: Note: Red color highlights where equivalent substitutions have been made. Blue color highlights where terms combine to unity.

Excitation of Resonator (cont.) Derivation of approximate form:

Excitation of Resonator (cont.) Note: A larger Q of the resonator (less loss), means a more sharply peaked response, and a larger R value. Lossless resonator:

Excitation of Resonator (cont.) High Q resonator Low Q resonator Note: A larger Q of the resonator (less loss), means a more sharply peaked response, and a larger R value.

Grounded Dielectric Slab Derive a transcendental equation for wavenumber of the TMx surface waves by using the TRE. Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction.

Grounded Dielectric Slab (cont.) We think of the transmission lines in the TEN running in the x direction.

TMx Surface-Wave Solution TEN: The reference plane R is chosen at the interface.

TMx Surface-Wave Solution (cont.) TRE:

TMx Surface-Wave Solution (cont.) Letting we have or Note: This method (TRE) is a lot simpler than doing the EM analysis and applying the boundary conditions!

The modes are hybrid in the z direction (not TEz or TMz). Waveguide With Slab Choose this representation: TExmn modes TMxmn modes Note: The modes are hybrid in the z direction (not TEz or TMz). TEN:

Waveguide With Slab (cont.) TEN: TRE:

Waveguide With Slab (cont.) Choose TEx: Separation equations: so

Waveguide With Slab (cont.) Final transcendental equation for the unknown wavenumber kz: TEx: with Note: The integer n is arbitrary but fixed. The equation has an infinite number of solutions for kz for a given n: m = 1, 2, 3, …

Waveguide With Slab (cont.) Limiting case: w  0: This mode becomes the usual TE10 mode of the hollow waveguide. Hence, we have

Waveguide With Slab (cont.) Longitudinal section plane (yz plane) Alternative notation: Another notation that is common in the literature for slab-loaded waveguides is the designation of “LSE” and “LSM” modes. LSE mode: “longitudinal section electric” mode LSM mode: “longitudinal section magnetic” mode LSE: TExmn (The electric field vector stays in the “longitudinal section”.) LSM: TMxmn (The magnetic field vector stays in the “longitudinal section”.)