Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 9 ECE 6340 Intermediate EM Waves 1

Fields of a Guided Wave Assume z Then The “ t ” subscript denotes transverse (to z ) Theorem Guided wave 2

Fields of a Guided Wave (cont.) Proof (for E y ) or Now solve for H x : so 3

Substituting this into the equation for E y yields the following result: Next, multiply by Fields of a Guided Wave (cont.) 4

so The other components may be found similarly. Fields of a Guided Wave (cont.) 5

Summary of Fields 6

Types of Guided Waves TEM mode: 7 TM mode: TE mode: Hybrid mode: Transmission line Waveguide Fiber -optic guide

Wavenumber Property of TEM Wave To avoid having a completely zero field, Assume a TEM wave: We then have Note: The plus sign is chosen to give a decaying wave: 8

Wavenumber Property (cont.) Wavenumber notation: Note that k z is called the “propagation wavenumber” of the mode. TEM mode: 9 Note: A TEM mode can propagate on a lossless transmission line at any frequency.

Wavenumber Property (cont.) so Lossless TL: The phase velocity is equal to the speed of light in the dielectric. 10 The field on a lossless transmission line is a TEM mode (proven later).

Wavenumber Property (cont.) Note: The TEM z assumption requires that R = 0. Otherwise, E z  0 (from Ohm's law). Lossy TL (dielectric but no conductor loss): The mode is still a TEM mode (proven later). Real part: Imaginary part: 11 Dividing these two equations gives us:

and are 2D static field functions. Static Property of TEM Wave The fields of a TEM mode may be written as: Theorem 12

Therefore, only a z component of the curl exists. We next prove that this must be zero. Proof Static Property of TEM Wave (cont.) 13

Also, Use Static Property of TEM Wave (cont.) 14

Hence Therefore, Static Property of TEM Wave (cont.) 15

Also, ( No charge density in the time-harmonic steady state, for a homogeneous medium) Therefore, Hence, Static Property of TEM Wave (cont.) 16

Static Property of TEM Wave (cont.) 17

The potential function is unique (because of the uniqueness theorem of statics), and hence is the same as a static potential function (which also obeys the Laplace equation and the same BCs.). A B Static Property of TEM Wave (cont.) 18

The static property shows us why a TEM z wave can exist on a transmission line (two parallel conductors). Static Property of TEM Wave (cont.) A B V0V0 A nonzero field can exist at DC. Transmission line 19

The static property also tells us why a TEM z wave cannot exist inside of a waveguide (hollow conducting pipe). Static Property of TEM Wave (cont.) Waveguide (This would violate Faraday's law: at DC the voltage drop around a closed path must be zero.) No field can exist inside at DC. C 20

so Similarly, C A B Static Property of TEM Wave (cont.) 21

TEM Mode: Magnetic Field so 22

TEM Magnetic Field (cont.) Also, so This can be written as 23

TEM Mode: Charge Density TEM mode x y A E B 24

TEM Charge Density (cont.) so Hence Note: 25 In general

TEM Mode: Homogeneous Substrate A TEM z mode requires a homogeneous substrate. Contradiction! x y Assume a TEM mode Coax partially filled with dielectric 26

Example: Microstrip Line Assume a TEM mode: w h x y (requires a homogeneous space of material) w x y h 27 Homogeneous model

Example (cont.) Strip in free space (or homogeneous space) with a static charge density (no ground plane): In this result, I 0 is the total current [ Amps] on the strip at z = 0. (This was first derived by Maxwell using conformal mapping.) Hence: This is accurate for a narrow strip (since we ignored the ground plane). 28 w x y

Example: Coaxial Cable Find E, H a b We first find E t0 and H t0 29

Example (cont.) Boundary conditions: so Hence Therefore 30

Example (cont.) Hence 31

Example (cont.) E This result is valid at any frequency. The three-dimensional fields are then as follows: 32

Example (cont.) E x y JsJs Find the characteristic impedance 33

Example (cont.) (assume  =  0 ) Result: 34 Find ( L, C) for lossless coax

Example (cont.) Result: 35 Find ( L, C, G) for lossy coax

TEM Mode: Telegrapher’s Eqs. x y TEM mode (lossless conductors) 36 A E B i H CvCv A B

Telegrapher’s Eqs. (cont.) Note: v is path independent in the ( x, y ) plane: 37

Hence, we have Use Telegrapher’s Eqs. (cont.) Take x and y components: 38

But so Hence Telegrapher’s Eqs. (cont.) Note: L is the magnetostatic (DC) value (a fixed number). CvCv dx dy  39 (flux per meter)

Telegrapher’s Eqs. (cont.) If we add R into the equation: This is justifiable if the mode is approximately a TEM mode (small conductor loss). See the derivation on the next slide. 40

Faraday's law: Telegrapher’s Eqs. (cont.) Include R Assume that current still flows in the z direction only, and R is unique in the time domain. Hence: 41  l = flux/meter

Telegrapher’s Eqs. (cont.) A B CiCi so Ampere’s law: The contour C i hugs the A conductor. Note: There is no displacement current through the surface, since E z = 0. Now use this path: x y 42

Now use Telegrapher’s Eqs. (cont.) Take x and y components: 43

Hence Telegrapher’s Eqs. (cont.) 44

Telegrapher’s Eqs. (cont.) Note: C and G are the static (DC) values. CiCi dx dy But 45

Hence or Telegrapher’s Eqs. (cont.) 46

Telegrapher’s Eqs.: Alternate Derivation Ampere's law: Alternate derivation of second Telegrapher’s equation 47 Hence