ELECTROMAGNETICS AND APPLICATIONS

Lecture 13Matching TEM Lines& RLC Resonators

Luca Daniel

L13-2

• Review of Fundamental Electromagnetic Laws• Electromagnetic Waves in Media and Interfaces• Digital & Analog Communications

o TEM transmission lines (cables and IC/PCB traces)o Digital communications (transients)o 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

o 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

Outline

TodayToday

L13-3

Power Delivery Problem

vs

+

-

Given ZS and ZL,design a connection thatmaximizes the averagepower delivered to ZL

ZL

Zs

Problem 1:

Problem 2:

vs

+

-

Zs Given source impedance ZS, find the best impedance ZA

that maximizes the averagepower delivered to ZA

ZA =?

1

2*

A A AP Re V I

1

2*

A A ARe Z I I

21

2 A AR I2

1

2s

AS A

VR

Z Z

2

2 2

1

2

A S

S A S A

R V

R R X X

A SX X

2

3

10

2

s AA

SA S A

R RdPV

dR R R

A SR R

2

2

1

2 2 S

A,MAX S

S

RP V

R

2

8 S

S

V

R

L13-4

Power Delivery Problem (matching TEM lines)

vs

+

-

ZL

Zs =RS+jXS Problem 1Let’s connect the load with a TEM line

Z0

i.e. Z(-D)=RS-jXS

0z

z=-D Im{}

Re{}

L

Xn = -j

Xn=+j

Rn=1

Rn = 0

Rn=3

Is there a length D s.t.the average delivered power is maximized?

Not for all source-load combinations!

i.e. Re{Z(-D)}=RS

Is there a length D s.t. we can match at least the real part of the source impedance?

No: only the resistive circles that are intersected.

vs

+

-

ZL

Zs =RS+jXS

Z0

0 zz=-D

toward z=-D

Z(-D)=RS+jXD

XM= -XD-XS

jXM

Rn=1/3ZA=RS-jXS

L13-5

• Review of Fundamental Electromagnetic Laws• Electromagnetic Waves in Media and Interfaces• Digital & Analog Communications

o TEM transmission lines (cables and IC/PCB traces)o Digital communications (transients)o 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

o 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’s Outline

L13-6

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

ADC

I

Q

LO

Micron Technology, Inc

L13-7

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/DD/A PA

L13-8

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

L13-9

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

L13-10

• Review of Fundamental Electromagnetic Laws• Electromagnetic Waves in Media and Interfaces• Digital & Analog Communications

o TEM transmission lines (cables and IC/PCB traces)o Digital communications (transients)o 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

o 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’s Outline

L13-11

RLC Resonators

Resonators trap energy:

RL

CI

Series RLC resonator

GC

L

V+

-

Parallel RLC resonator

Also:

terminated TEM lines,

waveguides

Circuit equations, series RLC resonator:

21 Rfor 2LLC

oR

j j2L

o1ω LC

1 j L I + R I + I = 0j C

KVL in frequency domain:

Series RLC resonator current i(t):

R t2L

o oi(t) I e cos( t)

R t2L

0I e

i(t)

0 t

0I

0 o

1 2T 2 LCf

L13-12

RLC Resonator Waveforms

Series RLC resonator current i(t):

Q radians,Q/o seconds

wTo/e

t0

wTo

we(t)

Stored Energy w(t):R t2 2 L

m o

R t2 2 Le o

1w (t) Li cos ( t)e2

1w (t) Cv sin ( t)e2

e,max m,maxw w

R t2L

o oi(t) I e cos( t)R t2L

0I e

i(t)

0 t

0I

0 o

1 2T 2 LCf

max maxLv iC

t

T m e T0w (t) w (t) w (t) w e

Tw

0CQ RC RL

T0

D

wQ

PT

Ddw

Pdt

0

Q

0

RC

Dissipated Power Pd : Quality Factor Q:

Series resonator:

Parallel resonator:

LR

0L 1 LQR R C

L13-13

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

Quality Factors:

Internal Qi = wT/PDi (PDi is power dissipated internally, in Ri)

External QE = wT/PDE (PDE is power dissipated externally, in RS)

Loaded QL = wT/PdL (PdL is the total power dissipated, in Ri and RS)

PDL = PDi + PDE L i E

1 1 1Q Q Q

Ri

LC

I()

VS

RS

+-

IS Ri CL

V+

-RS

L i S

1 1 1Note :R R R

L i SNote :R R R

for both series and parallel!

0 0L S i

LS i

0 L 0S i

L L seriesR (R R )

QR R

R C C parallelR R

L13-14

Power Delivery (Coupling) to RLC Resonators

Power delivered into series resonator Ri : PDi()

Half-power bandwidth:

o

1

1/2

2S

Di iV1P ( ) R

2 Z

2

0 21

2S

Di iS i

VP ( ) R

R R

2

Thi 2

2S i

V1R2 1R R L

C

1LC

0

1LC

To maximize PDi choose: to maximize power delivery: drive at resonance frequency!

Ri

LC

I()

VS

RS

+-

0

Di

Di

P ( )

P ( )

0 L

1Q

If Rs is given, to maximize PDi choose Ri s.t.:

2

3

10

2s i

Si S i

R RdPV

dR R R

i SR RCriticallyMatched!

i EQ Q

For critically matched resonator:

0 L i S

1 2 2Q Q Q

2

2

1

2 2S

Di,max S

S

RP V

R

2

8S

S

V

R