1. ELECTROMAGNETICS AND APPLICATIONS Lecture 14 RLC Resonators with TEM feed & TEM Resonators Luca Daniel

2. L13-2 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 resonators (application: e.g. filters)  RLC resonators  Matching (Coupling to) RLC resonators  RLC resonators with TEM feed  Examples: cellphone channel selection filter. Notch filter. oTEM resonators  Half-wave and quarter-wave resonators  Quality factor of TEM resonators  Coupling to TEM resonators Outline Today

3. L13-3 Power Delivery (Coupling) to RLC Resonators Power delivered into series resonator R i : P Di (  ) Half-power bandwidth: oo   1 1/2 To maximize P Di choose: to maximize power delivery: drive at resonance frequency! RiRi L C I()I() VSVS RSRS + - If R s is given, to maximize P Di choose R i s.t.: Critically Matched! For critically matched resonator:

4. L13-4 RLC resonators with TEM feed RiRi L C Z o =R S  Behavior away from resonance: Series resonance: Open circuit Parallel resonance: Short circuit Behavior at resonance: 0 if critically matched (R i =Z 0 )  C L + - RiRi 1 if R i =0/∞ (short/open) circuit Z o =R S

5. L13-5 Example #1 – Cell Phone Filter (band-pass) Cell phone (band-pass filter) specifications for channel selection: - Looks like a short circuit far from channel frequency (reflects all power) - At channel freq reflect < 1/9 of the incident power (want max coupling) - channel frequency = 1 GHz - bandwidth = 10 MHz - available amplifier has R S = 100-ohm Filter solution: Small, hard to build, use TEM? Parallel resonators look like short circuits far from  o VsVs RSRS  C L + - RiRi Z o =R S

6. L13-6 Example #2 – Notch Filter (i.e. interference suppression) Notch (band-stop) filter specs for suppression of interfering channel: Far from  o the load is matched (signal goes to amplifier R) At  o reflect all incident power; let  = -1 (short circuit)  o = 2   10 6  = 2   10 4 (rejected band, notch filter) Z o = 100-ohm line Filter solution: Lossless series resonators look like short circuits at  o R L C V Th ZoZo ZoZo

7. L13-7 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 resonators (application: e.g. filters)  RLC resonators  Matching (Coupling to) RLC resonators  RLC resonators with TEM feed  Examples: cellphone channel selection filter. Notch filter. oTEM resonators  Half-wave and quarter-wave resonators  Quality factor of TEM resonators  Coupling to TEM resonators Outline Today

8. L13-8 Standing Waves on Shorted TEM Line c z c Every half wavelength the TEM line behaves like a short circuit!

9. L13-9 Open and Short Circuit “Half-Wave” TEM Resonators Open-Circuit Resonator: 0 z v(z,t) Z 0 i(t,z) Short-Circuit Resonator: i D v + - Z o, case: n=2 z D0 v(z,t) Z 0 i(t,z) i v + - Z o, z D D case: n=2

10. L13-10 “Quarter-Wave” TEM Resonators D = /4, 3 /4, 5 /4,… = ( n /4)(2n - 1) E.g. Short-Open Resonator: 0 z v(z,t) Z 0 i(t,z) 11 22 i D v + - Z o, D