Lecture 14 RLC Resonators with TEM feed & TEM Resonators

1. Lecture 14RLC Resonators with TEM feed& TEM Resonators 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel

2. Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • Digital & Analog Communications • TEM transmission lines (cables and IC/PCB traces) • Digital communications (transients) • RF communications (matching loads to amplifiers) • RLC resonators (application: e.g. filters) • RLC resonators • Matching (Coupling to) RLC resonators • RLC resonators with TEM feed • Examples: cellphone channel selection filter. Notch filter. • TEM resonators • Half-wave and quarter-wave resonators • Quality factor of TEM resonators • Coupling to TEM resonators Today

3. Power Delivery (Coupling) to RLC Resonators I(w) Ri Power delivered into series resonator Ri : PDi() RS L + VS - C To maximize PDi choose: to maximize power delivery: drive at resonance frequency! 1 Dw Half-power bandwidth: 1/2 w wo If Rs is given, to maximize PDi choose Ri s.t.: Critically Matched! For critically matched resonator:

4. RLC resonators with TEM feed Ri + G L Zo =RS G Zo =RS L C - Ri C Behavior at resonance: 0 if critically matched (Ri=Z0) 1 if Ri=0/∞ (short/open) circuit Behavior away from resonance: Series resonance: Open circuit Parallel resonance: Short circuit

5. Example #1 – Cell Phone Filter (band-pass) RS + G L Zo =RS C Vs - Ri 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 RS = 100-ohm Filter solution: Parallel resonators look like short circuits far from o Small, hard to build, use TEM?

6. Zo L Zo R VTh C Example #2 – Notch Filter (i.e. interference suppression) Notch (band-stop) filter specs for suppression of interfering channel: Far from wo the load is matched (signal goes to amplifier R) At wo reflect all incident power; let G = -1 (short circuit) wo = 2 106 Dw = 2  104 (rejected band, notch filter) Zo = 100-ohm line Filter solution: Lossless series resonators look like short circuits at wo

7. Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • Digital & Analog Communications • TEM transmission lines (cables and IC/PCB traces) • Digital communications (transients) • RF communications (matching loads to amplifiers) • RLC resonators (application: e.g. filters) • RLC resonators • Matching (Coupling to) RLC resonators • RLC resonators with TEM feed • Examples: cellphone channel selection filter. Notch filter. • TEM resonators • Half-wave and quarter-wave resonators • Quality factor of TEM resonators • Coupling to TEM resonators Today

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

9. Open and Short Circuit “Half-Wave” TEM Resonators i + v Zo, n - D Short-Circuit Resonator: z i + v Zo, n - D v(z,t) Z0 i(t,z) case: n=2 z 0 D Open-Circuit Resonator: v(z,t) Z0 i(t,z) z D 0 case: n=2

10. “Quarter-Wave” TEM Resonators E.g. Short-Open Resonator: i + v Zo, n - D v(z,t) Z0 i(t,z) 1 2 z 0 D D = l/4, 3l/4, 5l/4,… = (ln/4)(2n - 1)