COMM (603) Lecture #7 1 - GUC · 2016. 4. 10. · © Dr. Hany Hammad, German University in Cairo

GUC (Dr. Hany Hammad) 4/10/2016

COMM (603) Lecture #7 1

© Dr. Hany Hammad, German University in Cairo

Lecture # 8

• Series & Parallel Resonant Circuits

– Series Resonant Circuits.

– Parallel Resonant Circuits.

– Loaded and unloaded Q.

• Transmission line resonators

– Shorted-circuited /2 line.

– Shorted-circuited /4 line.

– Open-circuited /2 line.


Resonators

• Applications:

– Filters.

– Oscillators.

– Frequency Meters.

– Tuned Amplifiers.




Series Resonant Circuits

CjLjRZin

1

2

2*

2

1

2

1

2

1

in

inininZ

VZIZVIP

RIPloss2

2

1

LIWm2

4

1

ωCjLjRIPin

1

2

1 2

The complex power delivered to the resonator

The power dissipated by the resistor

The average magnetic energy stored in the inductor

The average electric energy stored in the capacitor C

ICVW ce 222 1

4

1

4

1

emlossin WWjPP 2

Microwave Engineering, 3rd Edition by David M. Pozar Copyright © 2004 John Wiley & Sons



2

21

2

22

I

WWjP

I

PZ emlossinin

Resonance em WW

At resonance RI

PZ lossin 2

21

(pure real)

LC

10

Quality Factor l

em

P

WWQ

dloss/secon energy

stored energy average

At resonance RCωR

Lω

P

WωQ

o

o

loss

mo

12


CILI

o

2

22 1

4

1

4

1

0





Near resonance

LCLjRZin 2

11

2

22

oLjR

o

LCo

1

222 222 ooo0

LjRLjRZin

2

22

o

RQjR

2

Note 2

BW

o

BW=Bandwidth Ratio

o

in QjRZ

21

L U

R

LωQ o

Note:

Zin as function of BW



o

o

o

o

R

RCR

Lj

RCR

Lj

CLjR

R

V

V

1

1

11

1

1

RCωR

LωQ

o

o 1

2

21

1

o

o

R

QV

V

For half power

2

1

1

1

2

2

o

o

Q

2

212

o

o

Q


o

o

R

jQV

V

1

1

Relation Between Q & BW ?





1

2

2

o

o

Q 1

o

o

Q

1

L

o

o

LQ

1

U

o

o

UQ

U

o

o

U

L

o

o

L

To find a relation

U

oU

L

oL

22

UL

ULo

UL

oLU

22

11ULo

2

QL

o

o

L 1

Q

o

L

oL

2

Q

oUL

LU

oQ

UoL



Qoo

LU 12

QBW

1

o

in

RQjRZ

2

707.02

RRZin


jRjRR 1




Parallel Resonant Circuits

1

11

Cj

LjRZin

*

22* 1

2

1

2

1

2

1

in

ininZ

VIZVIP

Cj

L

j

RVPin

1

2

1 2

R

VPloss

2

2

1

The complex power delivered to the resonator

The power dissipated by the resistor

The average magnetic energy stored in the inductor

The average electric energy stored in the capacitor

2

4

1VCWe

LVLIW Le 2

22 1

4

1

4

1




emlossin WWjPP 2

2

21

2

22

I

WWjP

I

PZ emlossinin

Resonance em WW

At resonance RI

PZ lossin 2

21

LC

10

Quality Factor l

em

P

WWQ

dloss/secon energy

stored energy average

At resonance RCωLω

R

P

WωQ o

oloss

mo

2




Parallel Resonant Circuits 1

11

Cj

LjRZin

.11

1

x

x

1

11

Cj

LjRZ o

o

in

o 01

CjLj

o

o

Let

1

1

11

CjCj

LjR

Z o

o

o

in

o

o

1

1

1

1

11

CjCj

LjRZ o

o

oin

1

2

1

Cj

Lω

Δωj

RZ

o

in

1

21

Cj

RZin

LCo

1

CRjΔ

RZin

21

o

in

jQ

RZ

21

Note:



QBW

o

12

j

R

jQ

RZ

o

in

1/21

2

RZin





Loaded and unloaded Q

QQQ eL

111

L

R

R

L

Q

o

L

L

o

e

for series circuits

for parallel circuits

LQ

eQ

Q Unloaded Quality Factor

Loaded Quality Factor

External Quality Factor



Short-Circuited /2 Transmission line Resonator

llj

ljlZZ oin

tanhtan1

tantanh

ljZZ oin tanh

Attenuation Constant Propagation Constant

ll tanh1l

ov

l

v

l

v

ll o

0 ljZZ oin tanLossless Case

small) is (

Where v is the phase velocity if the transmission line


yx

yxyx

tanhtanh1

tanhtanhtanh

Note:

&

xjjx tantanh

llj

ljlZZ oin

tanhtan1

tanhtanh

Assume a lossy transmission line.

Low loss





o

vl

2o Since for

o

l

o

l

tantanThen

ljjl

Zllj

ljlZZ

o

oooin

1tanhtan1

tantanh1 ol Since

o

oin jlZZ

jLRZin 2 (Series RLC)

lZR oo

oZL

2

LC

o

2

1

Accordingly

o

o

o

tantan1

tantantan

= 0

= 0

Note:

oo

tan

o

oin jlZZ



0

Resonance occurs when

2

l

RlZZ oin At resonance

2

nl

Resonance also occurs at

,3,2,1n


lZR o

o

oZL

2

R

LQ o

llZ

Z

R

LQ

oo

oo

o

2

1

2

l

2

2Note:

2Q





llj

ljlZZ oin

tanhtan1

tantanh

ljZZ oin tanh

o

o

v

l

v

l

v

ll

22

o

vl

24 o Since for let o

o

ooin

jl

ljZZ

2

21

12

o

l

o

o

o

2tan2

tan1

2tan2

tan

2tan

o

o

o

2tan2

tan1

2tan2tan1

2tan

= 0

= 0

o

oo

2

2tan

1

2tan

o

o

oin lj

jl

ZZ2

1

2

ll tan

oo

jl

Z

2

Assume lossy line

Note:

Note:



CjRZin

21

1

oo

injl

ZZ

2

l

ZR o

ooZC

4

CL

o

2

1

0Resonance occurs when 4

l

l

ZRZ oin

At resonance

24

lRCQ o

Parallel Resonance




Open-Circuited /2 Transmission line Resonator

ljl

lljZZ oin

tantanh

tanhtan1

ljZZ oin coth

Resonance occurs when 2

l at o

o

l

And so ooo

l

tantantan ll tan

CjRZin

21

1

so

L

ZR o

ooZC

2

CL

o

2

1

o

o

o

tantan1

tantantan

= 0

= 0

&

o

o

o

o

oin

jl

Z

jl

lj

ZZ

1

Note:


Open-Circuited /2 Transmission line Resonator

22

lRCQ o

2

nl ,3,2,1n





Example: Half-wave length microstrip resonator

Consider a microstrip resonator constructed from a /2 length of 50 open-circuited microstrip line. The substrate is Teflon (r=2.08, tan = 0.0004), with a thickness of 0.159 cm. The conductors are copper. Compute the length of the line for resonance at 5 GHz, and the Q of the resonator. Ignore fringing fields at the end of the line.

answer

From microstrip design equations cmW 508.0

The effective permittivity 8.1e

The resonant length is

cmf

c

f

vl

e

24.280.11052

103

222 9

8

The Quality factor is

2Q


Example: Half-wave length microstrip resonator

The propagation constant is 49.1401024.22

222

rad/m

0724.0)00508.0(50

1084.1 2

WZ

R

o

sc Np/m

024.008.180.12

0004.08.008.27.104

12

tan1

re

erod

k

Np/m

68.728

024.00724.02

49.140

2

Q

0184274.0

10813.52

1041052

2 7

79

osR




Filters

• Filter Design By The Insertion Loss Method

– Characterization by Power Loss Ratio

• Maximally Flat

• Equal Ripple

– Design Steps

• Low-pass prototype design.

• Scaling and conversion.

• Implementation.

– Using Stubs.

– Using High-Low Impedance Sections.


Filter Design by The Insertion Loss Method

The perfect filter would have zero insertion loss in the pass-band, infinite attenuation in the stop-band, and a linear phase response in the pass-band.

21

1

load todeliverdPower

source from availablePower Ratio LossPower

load

incLR

P

PP

If both load and source are matched.

LRPIL log10

Characterization by Power Loss Ratio

refinc PP 2)(

The insertion loss (IL) is dB is

2

12

1

SPLR





22

22

)(

NM

M

Where M and N are real polynomials in 2.

2

2

1

N

MPLR

The magnitude of the voltage gain

o

o

ZZ

ZZ

)(

)()(

)()()(

)()()( *

jXZR

jXZR

o

o

2*2 )()()()()()( (Even functions of )

21

1

LRP

)()(

)()(

jXZR

jXZR

o

o

)()()( jXRZ


Proof

)()(

)()()(

jXZR

jXZR

o

o

)()(

)()()(*

jXZR

jXZR

o

o

)()()( *2

)()(

)()(

)()(

)()()()( *

jXZR

jXZR

jXZR

jXZR

o

o

o

o

)()(

)()()()(

22

22

*

XZR

XZR

o

o

2222

22

*

)()()()(

)()()()(

ooo

o

ZRZRXZR

XZR

oo

o

ZRXZR

XZR

)(4)()(

)()()()(

22

22

*





2

2

1

N

MPLR

Maximally Flat

Equal Ripple

Elliptic Function

Linear Phase

N

c

LR kP

2

21

c

NLR TkP

221

N

c

pA

2

1)(


Maximally Flat

– This characteristic is also called the binomial or Butterworth response and is optimum in the sense that provides the flattest possible pass-band response for a given filter complexity, or order.

– For a low-pass filter, it is specified by

N

c

LR kP

2

21

N

c

LR

kP

G2

21

11

Filter Order

Cutoff Frequency 1

0 c

G

Passband Stopband

Ideal Low Pass Filter




Maximally Flat

21 k

N

c

LR kP

2

21

0 5.0 0.1 5.1

LRP

c1

1kIf 211,1 ckLRP

2

1G dB 3

N

c

LR kP

2

2

For >c, the attenuation increases monotonically with frequency at a rate if 20N dB/decade.


Maximally Flat

N

c

LR kPIL

2

2

10 1log10log10

N

c

kIL

2

2

10 1log10

110 102

2

ILN

c

k

110102 C

ILk

c

IL kN

10

2

10

10/

10

log

log)110(log

2

1

)( cc ILIL

Designing For specific Insertion Loss given in dB.

Need to find filter order (N)

c




Equal Ripple

• If a Chebyshev polynomial is used to specify the insertion loss of an N-order law pass filter as Equal Ripple.

c

NLR TkP

221

c

N

LR

Tk

G

PG

221

1

1

N

cc

NT

2

2

1

c

N

c

LR

kP

22 2

4

4

22NGreater than the binomial response. Chebyshev case is


Equal Ripple

c

NTkIL

2210 1log10

c

c

ω

ωNk

ω

ωNk

IL

122

10

122

10

coshcosh1log10

coscos1log10 c 0

c

1101.02 rGk Gr is the ripple amplitude in decibels

c

GIL r

N1

1.01.01

cosh

110110cosh

Where IL is required insertion loss in decibels at a specified frequency

Designing For specific Insertion Loss given in dB.

Need to find filter order (N)

nTn coscos

cosx

xnxTn 1coscos

21 kGr




Elliptic Function

• They have equal-responses in the pass-band as well as the stop-band.

• Maximum attenuation in the pass-band Amax.

• Minimum attenuation in the stop-band is Amin.


Linear Phase

• In some applications it is very important to have a linear phase response in the pass-band to avoid signal distortion.

• Phase of the voltage transfer function of the filter

N

c

pA

2

1)(

N

c

d NpAd

d2

)12(1

The group delay is defined as




The process of filter design using IL method

COMM (603) Lecture #7 1 - GUC · 2016. 4. 10. · © Dr. Hany Hammad, German University in Cairo

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Transcript of COMM (603) Lecture #7 1 - GUC · 2016. 4. 10. · © Dr. Hany Hammad, German University in Cairo