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

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)
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

3. 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 -

4. 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

5. 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

6. Course Outline and Motivations
Electromagnetics:
How to analyze, design and couple energy to/from resonators
Applications
e.g. in cellphone receivers: electrical (RLC) resonator filters
and MEMs resonators filters
LNA
ADC I Q LO
Micron Technology, Inc

7. Course Outline and Motivations
Electromagnetics:
How to analyze, design and couple energy to/from resonators
Applications
e.g. couple energy to MRI coils driven at resonance
CPU
RAM
GPU
A/D
D/A PA

8. Course Outline and Motivations
Electromagnetics:
How to analyze, design and couple energy to/from resonators
Applications
Cavity/Optical resonators (e.g. lasers)
Prof. Ippen, MIT

9. Course Outline and Motivations
Electromagnetics:
How to analyze, design and couple energy to/from resonators
Applications
acoustical resonators (e.g. musical instruments and vocal chords, and... your own shower “room”) d
vocal chords

10. 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

11. 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):

12. 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

13. 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!

14. 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: