Notes 8 ECE 5317-6351 Microwave Engineering Waveguides Part 5: Fall 2012 Prof. David R. Jackson Dept. of ECE Notes 8 Waveguides Part 5: Transverse Equivalent Network (TEN)

Waveguide Transmission Line Model Our goal is to come up with a transmission line model for a waveguide mode. z a b x y The waveguide mode is not a TEM mode, but it can be modeled as a wave on a transmission line. + - z

Waveguide Transmission Line Model (cont.) For a waveguide mode, voltage and current are not uniquely defined. z a b x y x y a b A B TE10 Mode The voltage depends on x!

Waveguide Transmission Line Model (cont.) For a waveguide mode, voltage and current are not uniquely defined. TE10 Mode x y a b x1 x2 z a b x y Current on top wall: Note: If we integrate around the entire boundary, we get zero current. The current depends on the length of the interval!

Waveguide Transmission Line Model (cont.) Examine the transverse (x, y) fields: Modal amplitudes The minus sign arises from: Wave impedance

Waveguide Transmission Line Model (cont.) Introduce a defined voltage into field equations: We may use whatever definition of voltage we wish here. where or

Waveguide Transmission Line Model (cont.) Introduce a defined current and then from this define a characteristic impedance: We may use whatever definition of current we wish here. where

Waveguide Transmission Line Model (cont.) Summary:

Waveguide Transmission Line Model (cont.) Note on Z0: We can define voltage and current, and this will determine the value of Z0. Or, we can define voltage and Z0, and this will determine current.

Waveguide Transmission Line Model (cont.) The transmission-line model is called the transverse equivalent network (TEN) model of the waveguide + - z

Waveguide Transmission Line Model (cont.) Power flow down the waveguide:

Waveguide Transmission Line Model (cont.) Set Then

Waveguide Transmission Line Model (cont.) Summary of Constants (for equal power) Once we pick Z0, the constants are determined. The most common choice:

Waveguide Transmission Line Model (cont.) We have two constants (C1 and C2) Here are possible constraints we can choose to determine the constants: We can define the voltage We can define the current We can define the characteristic impedance We can impose the power equality condition Any two of these are sufficient to determine the constants.

Rectangular Waveguide Example: TE10 Mode of Rectangular Waveguide z a b x y Method 1: Define voltage Define current (This determines Z0) Method 2: Choose Z0 = ZTE Assume power equality

Example: TE10 Mode (cont.) z a b x y Method 1 Define:

Example: TE10 Mode (cont.) z a b x y Hence Since we have defined both voltage and current, the characteristic impedance is not arbitrary, but is determined.

Example: TE10 Mode (cont.) z a b x y

Example: TE10 Mode (cont.) z a b x y

Example: TE10 Mode (cont.) z a b x y Method 2

Example: TE10 Mode (cont.) z a b x y Take the conjugate of the second one and multiply the two together. Solution: The solution is unique to within a common phase term.

Example: TE10 Mode (cont.) z a b x y (as expected)

Example: Waveguide Discontinuity For a 1 [V/m] (field at the center of the guide) incident TE10 mode E-field in guide A, find the TE10 mode fields in both guides, and the reflected and transmitted powers. b a z z = 0 x y A B a = 2.2856 cm b = 1.016 cm r = 2.54 f = 10 GHz

Example (cont.) TEN Convention: Choose Z0 = ZTE Assume power equality

Example (cont.) TEN Equivalent reflection problem: Note: The above TL results come from enforcing the continuity of voltage and current at the junction, and hence the tangential electric and magnetic fields are continuous in the WG problem.

Example (cont.) TEN Recall that for the TE10 mode:

Example (cont.) TEN Hence, we have that

Example (cont.) Substituting in, we have

Example (cont.) Substituting in, we have (wave impedance) (our choice)

Example (cont.) Substituting in, we have

Example (cont.) Substituting in, we have (our choice)

Example (cont.) Summary of Fields

Example (cont.) Power Calculations: Alternative: Note: In this problem, Z0 and  are real. Alternative:

Example (cont.) For a 1 [V/m] incident TE10 mode E-field in guide A (field at the center of the guide) , find the TE10 mode fields in both guide, and the reflected and transmitted powers. Final Results: b a z z = 0 x y A B a = 2.2856 cm b = 1.016 cm r = 2.54 f = 10 GHz

Discontinuities in Waveguide Note: Planar discontinuities are modeled as purely shunt elements. Rectangular Waveguide (end view) The equivalent circuit gives us the correct reflection and transmission of the TE10 mode.

Discontinuities in Waveguide (cont.) Inductive iris in air-filled waveguide Top view: z x TE10 Higher-order mode region 1 T  Because the element is a shunt discontinuity, we have TEN Model Z0TE Lp

Discontinuities in Waveguide (cont.) Much more information can be found in the following reference: N. Marcuvitz, Waveguide Handbook, Peter Perigrinus, Ltd. (on behalf of the Institute of Electrical Engineers), 1986. Equivalent circuits for many types of discontinuities Accurate CAD formula for many of the discontinuities Graphical results for many of the cases Sometimes, measured results