L1-1 ELECTROMAGNETICS AND APPLICATIONS Lecture 8 TEM Transmission Lines Luca Daniel

L8-2 Course Outline and Motivations Electromagnetics: –How to transport signals and power on “guided systems” Applications –digital electronics: e.g. analyze transients when you send a signal from the CPU chip to the GPU chip, or from your keyboard to your iPad CPU RAM GPU A/D D/A keyboard iPad

L8-3 Course Outline and Motivations Electromagnetics: –How to transport signals and power on “guided systems” Applications –digital electronics: e.g. analyze transients when you send a signal from the CPU chip to the GPU chip, or from your keyboard to your iPad –analog and biomedical electronics: e.g. match load of RF cables bringing signal from power amplifier to MRI coil antennas to avoid reflections CPU RAM GPU A/D D/A PA

L8-4 Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces Digital & Analog Communications oTEM transmission lines (cables and IC/PCB traces) oDigital communications (transients) oRF communications (matching loads to amplifiers) oRLC and TEM resonators Microwave Communications Optical Communications Wireless Communications Acoustics Course Outline

L8-5 Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces Digital & Analog Communications oTEM transmission lines (cables and IC/PCB traces)  Parallel plate TEM transmission line  Characteristic Impedance  From Maxwell eqn. to Telegrapher eqn.  Finite difference - equivalent circuit interpretation  Examples of Practical TEM transmission lines (cables, traces…) oDigital communications (transients) oRF communications (matching loads to amplifiers) oRLC and TEM resonators Today’s Outline Today

L8-6 Parallel-Plate Transmission Line Boundary Conditions: Uniform Plane Wave x polarized wave propagating in +z direction in free space      z x y between perfect conductors plates     

L8-7 Currents and Voltages Corresponding to the Fields Currents in Plates: I(z) = H(z) W [A] Surface current density J s [A/m]: z W I(z) C Voltages between plates: V(z) is uniquely defined at fixed z Surface charge density  s (z) [C/m 2 ]: (Boundary condition; from ) 11 22 h      V(z) - + c x y z I(z) V(z) = E(z) h [V], independent of path

L8-8 Characteristic Impedance of the Transmission Line W      z x y z - + Electromagnetic Fields in the medium between the plates Corresponding Voltages and Currents on the Plates (induced by boundary conditions) h

L8-9 Maxwell Equations vs. Telegrapher’s Equations Electromagnetic Fields in the medium between the plates Corresponding Voltages and Currents on the Plates (induced by boundary conditions) Solutions: Note:

L8-10 Telegrapher’s Equations – Equivalent Circuit Interpretation Equivalent Circuit: i(z) i(z+  z) LzLz v(z) v(z+  z) zz LzLzLzLz CzCzCzCzCzCz Finite Difference Approximation: z h

L8-11 Examples of Practical TEM Transmission Lines microstrip Parallel wires Coaxial cable Parallel plates z Arbitrary cross-section 2) Compute the capacitance per unit length C (OR the inductance per unit length L) 1) Compute velocity from medium properties: 3) Compute Characteristic Impedance: General procedure to compute the Characteristic Impedance: