Lecture 13 Matching TEM Lines & RLC Resonators 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel Lecture 13 Matching TEM Lines & RLC Resonators

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) Telegrapher equations in complex notation (frequency domain) Line Impedance and Reflection Coefficient along the line Smith Chart Voltage Standing Wave Ratio The Power Delivery Problem Matching (Coupling to) TEM lines RLC and TEM 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 Today 2

Power Delivery Problem Given ZS and ZL, design a connection that maximizes the average power delivered to ZL Zs + vs ZL - Problem 2: Zs Given source impedance ZS, find the best impedance ZA that maximizes the average power delivered to ZA + ZA =? vs -

Power Delivery Problem (matching TEM lines) Zs =RS+jXS Let’s connect the load with a TEM line Problem 1 + vs Z0 ZL - z Is there a length D s.t. the average delivered power is maximized? z=-D Im{} Xn=+j i.e. Z(-D)=RS-jXS toward z=-D Rn = 0 Not for all source-load combinations! Re{} Is there a length D s.t. we can match at least the real part of the source impedance? i.e. Re{Z(-D)}=RS No: only the resistive circles that are intersected. Rn=3 Xn = -j Rn=1 ZA=RS-jXS Z(-D)=RS+jXD Zs =RS+jXS Rn=1/3 jXM + vs XM= -XD-XS Z0 ZL - z=-D z

Today’s 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) Telegrapher equations in complex notation (frequency domain) Line Impedance and Reflection Coefficient along the line Smith Chart Voltage Standing Wave Ratio The Power Delivery Problem Matching (Coupling to) TEM lines RLC and TEM 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 5

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Today’s 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) Telegrapher equations in complex notation (frequency domain) Line Impedance and Reflection Coefficient along the line Smith Chart Voltage Standing Wave Ratio The Power Delivery Problem Matching (Coupling to) TEM lines RLC and TEM 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 10

terminated TEM lines, waveguides RLC Resonators Resonators trap energy: G C L V + - Parallel RLC resonator Also: terminated TEM lines, waveguides R L C I Series RLC resonator Circuit equations, series RLC resonator: KVL in frequency domain: i(t) t Series RLC resonator current i(t):

RLC Resonator Waveforms Series RLC resonator current i(t): i(t) t Stored Energy w(t): t wTo we(t) Dissipated Power Pd : Quality Factor Q: wTo/e Series resonator: Parallel resonator: Q radians, Q/o seconds

Power Delivery (Coupling) to RLC Resonators Representing the drivers (sources): For a series resonator: represent the source with a Thevenin equivalent For a parallel resonator: represent the source with a Norton equivalent Ri L C I(w) VS RS + - IS Ri C L V + - RS Quality Factors: Internal Qi = wowT/PDi (PDi is power dissipated internally, in Ri) External QE = wowT/PDE (PDE is power dissipated externally, in RS) Loaded QL = wowT/PdL (PdL is the total power dissipated, in Ri and RS) PDL = PDi + PDE for both series and parallel!

Power Delivery (Coupling) to RLC Resonators I(w) Power delivered into series resonator Ri : PDi() Ri 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: