mathematical analysis and design of 4-stage passive rc ... · gen. 4th order poly-phase filter...

Mathematical Analysis and Design of 4-Stage Passive RC Network in RF Front-End System

Minh Tri Trana, Nene Kushitab, Anna Kuwanac, and Haruo Kobayashid

Gunma

1. Research Background• Applications of RC Poly-phase Network• Analog Complex Signal Processing Concepts

2. Analysis of 4-Stage Passive RC Network• Frequency Responses• Image Rejection Ratio

3. Proposed Model of 4-Stage Passive RC Network• Model of 4-Stage Passive RC Network • Simulation Results

4. Conclusions

Outline

1. Research BackgroundTypical Applications of RC Poly-phase Network

Wi-Fi

Blue-tooth

Wi-FiBlue-tooth

Wi-Fi

Blue-tooth Blue-tooth

Blue-tooth

Applications: Wi-Fi, WiMax, UWB, GSM, WCDMA, LTE , 4G, Cordless Phones, RFID, ZigBee, Bluetooth, TV Set Top Box, Sensing, Radar…

1inV

2inV

4inV

3inV

1outV

2outV

3outV

4outV

1R

1C1R

1C1R

1C1R

1C

2R

2C2R

2C2R

2C2R

2C

3R

3C3R

3C3R

3C3R

3C

1AV

2AV

4AV

3AV

1BV

2BV

3BV

4BV

4R

4C4R

4C4R

4C4R

4C

1CV

2CV

3CV

4CV

RC Poly-phase Filter

Design of Image Rejection Filter for Blue-tooth Receiver:o Low image noiseo Simple Model of RC Network

o Derivation of Transfer Function of 4-Stage Passive RC Network Based on Superposition Principle

Approach

Objective

1. Research BackgroundResearch Objective & Design Achievements

o Mathematical Analysis of 4-Stage Passive RC Networko Image Rejection Ratio: -33dB

Achievements

1. Research BackgroundRFIC Design & Trade-offs

RF Design

Communication Theory

Random Signals

Transceiver Architectures

IC Design

CAD Tools

Wireless Standards

Multiple Access

Signal Propagation

Microwave Theory

Noise Power

Frequency

GainSupply Voltage

LinearityRF Design“trade-off”

Theoretical Concepts Practical Designs

1. Research BackgroundModern Receiver Architectures

LNA

AGC

ADCAGC

ADC

90

PLL

BPF

LPF

LPF

Mixer

VCO

Antenna

Direct-conversion Receiver

LNA 90

PLL

AGC ADC

BPF

Polyphase FilterMixer

VCO

Antenna

Low-IF Receiver

LO

IF IF

Wanted signal Image signal

frequency

SpectrumDC

offsetLO

IF IF

Wanted Signal Image signal

IF

frequency

Spectrum

Dc-offsets Image noise

1. Research BackgroundFrequency Plane of Low-IF Signals

1

anti-clockwise

2

3

4

4V jV

1V V3V V

2V jV

clockwise

3

1

2

4

1V V

2V jV

3V V

4V jV

Wanted Signals

Image Signals

MIXER

cos ω tLO

sin ω tLO

Vsig

-jV sig

+Vsig

+jVsig

-Vsig

-jV im

+jVim

-Vim

+Vim

Vim

ω ωLOωsig ωim

+

1

3

4

2

image signals

wanted signals

0 ffrequency

amplitude 2

dBH j f

Wanted Signals

Image Signals

1. Research BackgroundWireless Communication Specifications

Low Energy Bluetooth

Low Power (Wifi) (IEEE 802.11)

ZigBee (IEEE 802.15.4)

Frequency Range 2.402-2.482 GHz 2.40-2.50 GHz 2.402-2.482GHz

Discrete Channels 3 3 16

Max Channel Bandwidth 1~8 MHz 22 MHz 5MHz

Modulation GFSK QAM64 QPSK

Nominal Data-Rate 1 Mbps 1 Mbps 250 KbpsEstimated Max Potential

Data-Rate1 Mbps 54 Mbps 500 Kbps

Nominal Range (0 dBm) 10 m 25 m 75m

Average Power for ten 256-byte massages per day

50uW 570 uW 414uW

1. Research BackgroundSignal Bandwidth & Channel Bandwidth

Wifi

Meritso Low-cost o Small-size o High-integration

Demeritso Image Noises o Power Losso Noise Figure

1. Research BackgroundLow-IF Receiver System Architecture

LNA

MIXER

QuadGen .

4th order Poly-phase Filter

Limiting amplifier

ADC

VCO

RSSI

Antenna

PLL

BPF

This Work

Bandgap Reference .

1. Research BackgroundDesign Targets of Low-IF Receiver System

Parameter LNA Mixer PPF RSSIInput RF frequency 2.4GHz 2.4GHz 5MHz 5MHz

LO frequency 2.405GHzInput RF Power -85 dBm -68dBm -60dBm -60dBm

LO Power 0 dBmNF 2dB 10 dB 10dB 10dB

Gain 17 dB 8dB 0dB 60dBP1dB (input) >-30dBm > -10dBm

Input referent IP3 >-20dBm > 0 dBmCurrent Consumption < 5 mA <10 mA <5 mA <5 mA

Supply Voltage 1.8 V 1.8 V 1.8 V 1.8 VImage Rejection Ratio

IMRR< -30 dB

1. Research BackgroundSuperposition Principle

n n

i=1 i

i

i i=1A

1 =d d

(t)( EE t)

1d

2d3d

4d5d

6d7d

nd

…

1E (t)

2E (t)

3E (t)4E (t)

5E (t)

6E (t)

7E (t)nE (t)

AE (t)

1. Research BackgroundExample1 of Superposition Principle

R1

V1

R2

~ C1

VA

V3

V2~

~11

1CZ j C

31 2

1 1 2 1 1 2

1 1 1

C CA

V=R Z R R Z R

V VV

1 1 2 1 1

1 1 2

2

1 2

3

1

1C C

CA

C

VV VVR Z R R Z

=R Z R R Z R

1 2 31 2 1 2 1 1

1 1 2 1 1

C C

C CA

Z R R R R Z=R

VZ

VVR

VR Z

2 1 2 11 2

1 1 2

3 1

11AV R j R R C R=

j R R RVC

VV

1d

2d3d

4did…

1E (t)

2E (t)

3E (t)

nE (t)

AE (t)kd

2_1subd

2_2subd

Sub branches

1. Research BackgroundSuperposition Principle with Sub Branches

4E (t)

_ 1 _ 2

...

n n

i i ik

k sub k su

A=1

i

b

=1 iE (t)

d dd

d d

E (t)1 1 =11 1

1. Research BackgroundExample2 of Superposition Principle

R1V1 C3

R2~ C1

VA

R4

C2 R3

VC

VB1 2 3 1

1 1 1 AB

C C C

=Z Z R

VZ

V

2 3 4 2

1 1 1 A

CC =R Z R

VRV

1 11 2

2 3 3 4

11 1 1

1 1 11 1

1C

C C

A =R RZ R

Z

V

R

V

Z R

33

1CZ j C

11

1CZ j C

22

1CZ j C

1. Research BackgroundEnergy Propagation

Lumped circuits: resistors, capacitors, inductorsneglect time delays (phase)

account for time delays (phase change) Distributed circuit elements: transmission lines

0( , ) cos( )dE d t E e t d

Wavelength 2 m

Energy Propagation Function:

1. Research BackgroundSome Properties of Hilbert Transforms

o Negative Hilbert transform: -π/2 phase shift (-j) o Positive Hilbert transform: +π/2 phase shift (+j)

Hilbert Transforms Table1:

1. Research BackgroundComplex Signals

• Complex signal: a set of two (or four) sources with the same frequency and 90-degree separated phase

_ 21

coscos

;Neg complex V tSA

tt

AV t

j

clockwise

1V V

2V jV

Negative Hilbert transform

_ 21

coscos

;Pos complex V tA

Stt

AV t

j

anti-clockwise

1V V

2V jV

Positive Hilbert transform

Positive Complex Signal Negative Complex Signal

1. Research BackgroundPoly-phase Signals

• Poly-phase signal = Complex signal (four sources)• Poly-phase filter = Complex filter

clockwise

3

1

2

4

1V V

2V jV

3V V

4V jV

1

anti-clockwise

2

3

4

4V jV

1V V3V V

2V jV

Positive Hilbert transform Negative Hilbert transform

Negative Poly-phase SignalsPositive Poly-phase Signals

1 432_

s

; ;coss

i

osn

ci

;

n

Neg poly

AV t

A tt

A t

V t VtS V t

A t

1 432_

c

; ;sino

cosi

s

s

;

n

Pos poly

A

V V t

A

tA tA t

t

t

t

VS V t

1. Research BackgroundPhase Plane of Poly-phase Signals

clockwise

3

1

2

4

1V V

2V jV

3V V

4V jV

1

anti-clockwise

2

3

4

4V jV

1V V3V V

2V jV

Negative Poly-phase Signals Positive Poly-phase Signals

1 432_

s

; ;coss

i

osn

ci

;

n

Neg poly

AV t

A tt

A t

V t VtS V t

A t

1 432_

c

; ;sino

cosi

s

s

;

n

Pos poly

A

V V t

A

tA tA t

t

t

t

VS V t

0

negative positive|H(jω)|

je je phasor

1. Research BackgroundPhase Plane of Received Signals

Negative Poly-phase

Signal

Positive Poly-phase

Signal

0

|H(jω)|dB

je je Phasor

Wanted SignalsImage Signals

1 432_ ; ; ;

1 ;1; ;Ne

e

g ly

g

po

n

V t tVS t

j

V

t

V t

j V

3 41 2_

1

; ; ;

1; ; ; pos

Pos poly tt

V t

tS t

j

V V

j

VV

Negative Poly-phase Signals Positive Poly-phase Signals




4. Conclusions

Outline

2. Analysis of 4-Stage Passive RC Network Design of 4-Stage Passive RC Network

1inV 4inV

2inV 3inV

1outV4outV

3outV2outV

1R 1C1AV

2AV 4AV

3AV

1R

1R

1R1C

1C

1C

2R

2R

2R

2R2C

2C

2C

2C3R 3C

3R

3C

3R3C

3R

3C

1BV2BV

3BV4BV

1CV

2CV

3CV

4CV

4R4C4R4C 4R

4C

4R 4C

Parameter PPFInput IF frequency 5MHz

Input IF Power -60dBmNF 10dB

Gain 0dBCurrent

Consumption<5mA

Supply Voltage 1.8VImage Rejection

Ratio IMRR< -30 dB

2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages on 1st Loop

1 12 2

2 23 3 3 3

3 4 4 3 4 4 3 4 4 3

1 1 1 1 2 2 3 3 3 3 4 4 3 3 4 4 1 1 2 2

_

1 1 2 2

1

4 4

2 2 2

1 1 1 11 1

1 1 1 1 1 11 1 1 1

1 2 1 2 1 2 1 2

Cto

C C C C C C C C C C C C

t

C

CC C

C C C C C

al A

C

R Z R Z

Z RR Z R Z

R Z R Z R Z R Z R Z R Z R Z R Z R Z R Z R Z R

Y

Z Z R R Z R Z Z R R Z

Z

R

R

1 2 2 3 3 3 3 4 4 2 2 3 3 4 42 2C C C C C C C CZ R Z R Z R Z R Z R Z R Z R Z

4

3

_1

22 _

1

2_

1

_1

34

3

1

1

1

1

4

1

1

1

total AC

inA total A

C

intot

in

i

i

inA

i

al AC

total AC

nA

n

n

n

A

i

YZ

VV YZVYZ

Y

V

R

R

Z

VVV

RVV

V

VR

V

1inV 4inV

2inV 3inV

1outV4outV

3outV2outV

1R 1C1AV

2AV 4AV

3AV

1R

1R

1R1C

1C

1C

2R

2R

2R

2R2C

2C

2C

2C3R 3C

3R

3C

3R3C

3R

3C

1BV2BV

3BV4BV

1CV

2CV

3CV

4CV

4R4C4R4C 4R

4C

4R 4C

Here:

2 2 3 3 3 3 4 4 2 2 3 3 4 4

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2 2

1

2

1 1

2 1

1

1

4

1

2

1

2 2

2 2

2

C C CC

A in Cin

in C C C C

C C C C C C

C C C C C C

A inin

in

RZ

V V

Z R Z R Z R Z R Z R Z R Z

R Z R Z R Z R Z R Z R Z


Z

R

RV

V

V

V VV

2 2 3 3 3 3 4 4 2 2 3 3 4 4

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2 2

2 2 3211

3 3 3 43

33

2 2

2 2

2

2

C C C C C C C

C C C C C C

C C C C C C

C C Cin

i

i

n

nCA

R Z R Z R Z R Z R Z R Z R Z



R ZVZ

R Z R Z RV

VRV

4 2 2 3 3 4 4

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2 2

2 2 3 3 3 34 4

4 4 2 2

4

3 3311

4

2

2 2

2

2 2

C C C C

C C C C C C

C C C C C C

C CC

C C C CA in

in

in

Z R Z R Z R Z



R Z R Z R Z R Z R Z R Z R ZVV VV

Z R

4

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2 2

2 2

2

C

C C C C C C

C C C C C C



Voltages on 1st loop

2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages on 1st Loop

2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages on 2nd Loop

4_

2 2

_2

3

3

2

_2

11

2

_2 2

22

44

3

1

2

total BC

total BC

total BC

total B

A

AB

A

AB

BA

A

C

A

B

A

YR Z

YR Z

YR Z

Y

VV

V

V

V

VV

VV

ZV

V

VR

2 2 3 3

_2 2

3 3

3 3

3 4 4 3 4

4 4 2 2 3 3 4 4

2 2 3 3

4

3 3 4 4

1 1 1 11 1

1 2

2

1 2

2

2

total BC

C

C C C C C C C

C

C

C

C C

C C

R Z R Z R

YR Z R Z

Z R Z R Z R Z R Z

R Z R Z R

Z Z R R R

Z R

Z

Z

Here:

1outV4outV

3outV2outV

1AV

2AV 4AV

3AV

2R

2R

2R

2R 2C

2C

2C

2C3R 3C

3R

3C

3R3C

3R

3C

1BV2BV

3BV4BV

1CV

2CV

3CV

4CV

4R4C

4R

4C 4R4C

4R 4C

2

3 3 3 3 4 4

2 2 3 3 3 3 4 4

2 2 3 3 4 4

3 3 3 3 4 4

2 2 3 3 3 3 4 4

2 2

3

4

3 3 4 4

1 1

12

2 2

2 2

2

2

2

2

2

2

C C C

C C C C

C C C

C C C

C C C C

C C

B A

C

A

C AB A

C

B

VR Z R Z R Z

R Z R Z R Z R

V

Z

R Z R Z R

V

V

Z

R Z R Z R Z

R Z R Z R Z R Z

R Z R

V

V

Z R

Z R

Z R Z

V

4 4

3 3 3 3 4 43

2 2 3 3 3 3 4 4

2 2 3 3 4 4

3 3 3 3 4

2

43

2 2 3 3 3 3 4 4

2 2 3 3

2 2

2 2

4 4

2

2

2

2

2

2

C C CA

C C C C

C C C

C C CA

C

C

CBC C C

C C

A

A

C

Z R

Z

R Z R Z R ZV

R Z R Z R Z R Z

R Z R Z R Z

R Z R Z R ZV

R Z R ZV V

R Z R Z

R

V

Z R Z R Z

R

Voltages on 2nd loop


3

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1

14 4

1 2 11 1

1 1

2 3 3 3 3 44

41

2 2

2 2

2

2 2

2

in

C C C C C C

C C

inC C

in in

iiC C

B

C C C

Cn

Cn

i

C

in

n

B

V


R Z R Z R Z R Z R Z

V VV

R

R Z R R R Z R ZV

Z Z

ZV

VV

Z

R Z

V

V

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1

1

3

1 1

2 2 1 1 2 2

3

42 1 2 3 3 3 3 4

1 1

12 2

21

4

2 2

2

2i

C C C C C C

C C C C C

C Cin i

in in

i

nC C C

n

inC

C

inB

nC

i

R Z R Z R Z

R Z R R R ZZV

R Z R Z

R Z R Z R Z

R Z R Z R Z R Z R Z

V

V

V

Z

R Z

V

VV

VV

V

3 3

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2 2

2

44

3

11 2 1 2 3 3 3 31

1

4

4

24

1

2 2

2

2C C C Cin

Cin in

C C C C C C

C C C C C C

inB in

in

n

inC

V


R Z R Z R Z R Z R

VV V

R Z R R R ZV

ZV

Z

V

Z

R

Z

R

RV

Z Z

R

3

3

1 1 1 1 2 2 3 3 3 3 4 4

3 3 4 4 1 1 2

2

2 1

2 1 2 3 3 3 3

1 2

44

2

41

2 2

2

2in

in

C C C C C C

C

C C C Ci

C C C C C

in

nC

VZ R R R ZV


R Z R Z R Z R Z R

R Z

Z

R

Z

VV

R

ZZ

Voltages on 2nd loop


2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages on 3rd Loop

3

3 3 4

3 4 4 3

4_

3

3 44 3

21 1 2 C C C

Cto

C CCtal CY

Z R Z R R ZRR Z Z R Z Z R

3 3

3 3

3

3

_

_

_

3

2

3

3

4

_

11

4

23

1

4

2

3

total C

total C

total C

tota

BC

B

C

C

C

Cl C

B

C

B

B

C

C

B

B

B

R Z

R Z

R Z

R

V

V

V

V

VVZ

VV

V

Y

V

YV

V

Y

Y

Here:

1outV4outV

3outV2outV

3R 3C

3R

3C

3R3C

3R

3C

1BV2BV

3BV 4BV

1CV

2CV

3CV

4CV

4R4C

4R

4C 4R4C

4R 4C

4 4

3 3 4 4 3 3

4 4

3 3 4 4 3 3

4 43

3 3 4 4 3 3

4 4

3 3

3 3

3 3

3

4

4 4

1 1

2 2

33 3 4 4 3

1

3

23

3

2

2

2

2

C B

C

C C C

C

C C C

CB

C C C

CB

C

C

C

C

C

C

B

C B

B

B

C

C

C

B

Z R

Z R

Z

Z RZ R Z R R Z

Z RZ R Z R R Z

Z RV

Z R Z R R Z

Z RV

Z R Z

V

V

V

R R

V

VV V

V R

ZZ RV

V

Voltages on 3rd loop


4 4 4

4 4 41 2 1 2 3 1 2

3 3 3

3

1 1 1 1 2 2 3 3 3 3 4

21 1 1 1 2 3 41

1 1 11 1

4

2 2

in in iC C

in in in

in

C C C

in in in

in

C C

C C Cn

in in inC C

n

C

C i

V V VV

V V V VZ Z Z ZR Z R R Z R

V V VV

R

V V V

Z R Z R Z R Z R Z

R

V

R Z

R

Z R R

4

3 3 4 4 1

1 1 1 12 2 2

2 2

1 2

41 1 1

2

1 2 1 2 3 1

1 1 2 2

3

1 1 12 1

4 42

4

2C

C C C C C C

Cin in ini

Cn in in

in iC C C

in in inC i

n i n

in

n

Cin

CZ Z

Z

R Z R Z R Z R Z R Z R

V V VV V V

R Z R R Z ZV V V

V V

Z R Z RV

VV

RV

Z

V

3 4 4

1

1 1 1 1 2 2 3 3 3 3 4 4

3

2 2

3 4 4 1 1 2 2 1 1 2

2

2

3 33

11 2

32

1 32

1

2 2

2C C C C C C

C C C C C C

Cin in i

i

in in iniC C

in

nC C

n

C


R Z R Z R Z R Z R Z R

V V

Z

V V VV

R Z R

VZ

V

R RZV

ZV

R Z

3

1 1 1 1 2 2

1 12 2

1 11 2 1 2

3

3

3 3 3 4 4

3 3 4 4 1 1 2 2 1 1 2

4

44 4

1

4

2

3

41

2 2

2

n

C C C C C C

C C C C C

in

inC i

C Cin in

C

in inC

i

n

C

n

n

C

i

Z ZV V


R Z R Z R Z R Z R Z

V

VV V

R Z

V VR Z Z

ZV

VR R R R

3 3

3 3 3

1

1 2 1 2 3 1 2 1 2 3 4 42 2 2

1 1 12

1 1 1 2 2 3 3 3 3 4 4

3 3 4

4 4

1

2 2

2

inin in

in in in

C C C C C C

C

in in inC C C

inC

iC

iC

nC

n

RV V

V V V

R Z R Z R Z R Z R Z

V V VZ Z ZZ R R Z R Z R R R Z

V V

R Z

R Z R

VR

V

4 1 1 2 2 1 1 2 2C C C C CZ R Z R Z R Z R Z

Voltages on 3rd loop


2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages on 4th Loop

4

44 44

4_

1 1tot

Cl D

Ca

C

R ZR ZR

YZ

4

4

4

3

4

4_

_

_

44

4

2

4

2

4

4

3

1

1

1

3_

2

Ctotal D

total D

t

C

C

C

C

C

Cot

ou

al

t

D

Cout total D

Cou

C

ou

C

Ct

t

VY

Y

Y

VV

ZV

Z

RV

VR

V

RV

V

V

Y

VZ

Z

V

R

Here:

1outV4outV

3outV2outV

1CV

2CV

3CV

4CV

4R4C

4R

4C 4R4C

4R 4C

4

4

44

2 44

4

1 1

1

44

44

44

3

3 4

4

44

3 2

4

2

4

1

1

1

1

C

out C

CC

C CC

o

ou

ut C CC

C

out

C

C

C

C

C

t

V

V

V

RR

RR

RR

R

ZZ

V

V

V V

V

V

V Z

R

Z

VZ

ZZ

V Z

Voltages on 4th loop


33

3 3

1 1 1 1

1 1 1 1

1 1

1

1

24 4

4

4

4

4 4

1

2 23

2

23

i

in inC

Cn

in in

iC

in

i

n

C

in

in i

in

in

C C

iC C

o

n

nn

ut

n i

Z R ZV V

VZ

V

VV

V V

VV

VV V

R

Z R Z R VR R

Z

Z

Z

VV

R

1 1 1 13 24 4

3 32

2

2

2 2

3

31 2

4

31

4

4 313

1

1

in in

in

in

in

in

in

C

C

in

in i

in

inin

in

CC

Cin

C

C

V

V

Z RZ R

R

V VVR

V

V VV

V

Z R

Z R ZV V

VR Z

V

Z

V R

1 1 2 2 1 1 3 3 3 3 4

3

4 3 3 4 4 1

4

1 2 2

2

2

2

2

1

2 2

1

1

2

1

11

2 2 2C C C C C C C C C C C C

C

C

in

inC

in

i

u

C

o t

n

n i

R Z Z R Z R R Z Z R Z R Z R Z R Z R R

Z

Z Z R R Z

Z

Z R

V

VV

ZV

V

V

4

4

4

1 1 1 1

1 1

2

2

2

2

3

1

1 1

4

4

1

2

1

32 1

2 11

3in

in

in

n

in in

in

in in

C C

inC

in

in in

i

C

C

C

Cn

Z R Z R

Z R Z

V V V

VV

V

RV V V

RV

VRZV

R

V

Z

R

Z

2

4 4 31 1

1 1 1 1

4

4 3

1 1

1 343

2

32

42

Cin

in in

i

in in

in

in

in

in in

in

n

nC C

i

i nC

RVV V

VZ

V

V R

Z R

Z R Z

VV

VRV

RVV

VV

2

1 1 2 2 1 1 3 3 3 3 4 4 3 3 4 4 1 1 2 2 2 2 1 1

31 1 1

4

213

1

3

2

3

23 2

2 2 2C C C C C C C C C C C C

in inC

inC

in

iin nC

out

C

C

in

V

R Z Z R Z R R Z Z R Z R Z R Z R Z R R Z Z R R Z

V VZ R ZZ

V

R

Z

VR

VVV

Z

1 1 1 14

4

4

2 22

23

2 2

2

3

1

2 21 1 1 1

13

1

21 1

3 C Cin in

in i

in inin

in

in i

in

in

in

in

C

Ci

Cn

C

Cn n

V VV

V RV V

ZV V

Z RV

V VV

Z

R

R Z R

Z R Z

RVR

V

V

V

Z

4 4

4

3

1 1 2 2 1 1 3 3 3

1

1

1 13

2

3

2 1 121 1

2 2

inC

C C C

in

in

in i

C

in in

inC

n

C

iC

n

C

V

R Z Z R Z

Z

R R Z

RRV

Z R Z

V VR Z RV

ZVV V

3 24 4

4 4 3 3 4 4 1

2

2

4

1 2 2 2 2 1 1

3 3

3

4

2

23

1 1 1 1

1 14

3

3

4

2 C C C C C C

in inC

in in

in C

i

in

in

i

C

Cn

in

i

u

n

n

o

C

C

t

C

R Z R Z R Z R R Z Z R R Z

V VVV V

V Z

V

Z RZZ RV

VR

Z R

Z Z R

V

VV

12 2

2

2 2

3

44

1 1 1 1

1

33

1

2 2

3 32

3

2 231

1

in

in inin

in i

C

C

C C

Cin

in in

in

Cin

n

C

inin

in

V VZ R Z RZ R

Z

Z

R Z

VV

V V R

V VV

R VV

R

ZV

V V

1 1 2 2 1 1 3 3 3 3 4 4 3 3 4 4 1 1 2 2

1

2

1

12 2

2

4

1 1

1 12 1

2 2 2

in in

inin in

C C C C C C C C

i

C

Cn

C C CR Z Z R Z R R Z Z R Z R Z R Z R Z R R Z Z R

R VV

R Z

R Z RVV V V

Voltages on 4th loop


2. Analysis of 4-Stage Passive RC Network Model of Input Wanted Signals

0

|H(jω)|

1je 2je 3je 4je phasor

phase plane

_Pos BS_Pos AS _Pos CS

_Pos DS

4_

32 2

1

1

2 3; ; ;

1; ; ; 1; s; c1 o;n

p

Pos p

os k k k

oly

k

j jj

S t

j j

tt

V t Ae e

V t

e

V

t

VV

2

1

41

4

3

32

i in

i

in

i

n

in

n

n

i

in n

VV

VV

jV

VVV

Positive poly-phase signals

Wanted Signals

1 1 2 2 1 1 3 3 3 3 4 4

3 3 4 4 1 1 2 2

4 3 2 1

4 3

2 2 1 1

1 1 2 2 1

2 1

4 3 2 1

4 3 2 12

1

32

1

12 2

2

2 2

C C C C C C

C C C

ou

ouC C C C

C C C C

i

t in

n

C C

C C C

t

C

R R Rj j j j

j j j

R Z Z R Z R R Z Z R Z R

Z R Z R Z R R Z Z R R Z

R Z

Z Z Z Z

Z Z Z Z

R

R R R RV

Z R Z R

V

R

j

V

V

3 3 3 4 4

3 3 4 4 1 1 2 2 2 2 1 1

31 1 2 2 1 1 3 3 3 3 4 4

3 3 4 4 1 1 2 2 2 2

4 34 3 2

1 1

2 13

1

2

2 2

2

C C C

C C C C C C

inC C C C C C

C C C C C C

C C Cout

Cj j j

Z Z R Z R

Z R Z R Z R R Z Z R R Z

VR Z Z R Z R R Z Z R Z R

Z

jZ Z Z Z

R Z R Z R R Z Z Z

R R R

R R

VR

4 4 3 2 1

1 1 2 2 1 1 3 3 3 3 4 4

3 3 4 4 1 1 2 2 2 2 1

2

1

43 1

42 2

2C C C C C C

out

C C

C C C

C C C

inC

C

R Z Z R Z R R Z Z R Z R

Z R Z R Z R R Z

R RZ Z

Z R R Z

RZ ZjV

jRV

j j

Output voltages

2. Analysis of 4-Stage Passive RC Network Derivation of Output Voltages (Wanted Signal)

Transfer Function

3

1 1 2 2 1 1 3

1 3

4 3 2

3 3 3 4 4

3 3 4 4 1

2

4 3

1 2 2 2 2 1 1

41

2

42

1 1

2 2

2

in

C C C

i

C C C C

C C C

C

o ou out

inn

ouut t

C

n

C

t

C C

i

C

VR R R R

V

R Z Z R Z R R Z Z R Z

VH

j

R

Z R Z R Z

j j j

R R Z Z R

VV V

R Z

Z Z Z ZV

V

1 2 21

3 31 32

4 41

1 1 2 2

3 3

4 4

2 1

1 23 3

442 4

13

4 2 4 3

2 2 2

2 2 2

1 1 1; ; ;

1 1 1; ; ;

1 1 1 1;2 2 2

;2

; ;

R

R R

R

C

C C

C

f f f

f f f

R C R C

f f fC C

R C

R RC Rf

1 21 2

33

44

1 1;

1

2 2

21

2

C C

C

C

Z ZC C

ZC

Z

j f j f

j C

j f

f

2. Analysis of 4-Stage Passive RC Network Derivation of Transfer Function (Wanted Signal)

Transfer Function

4 3 2

4 3

4 3 2 1 4 3 2 1 3 2 1 2 1

4 3 2 1 2 1 3 2 1 4 3 2 1

4 3 2 1 43 42 41 32 31 21

4 3 2 1 3 2 1 2 1

4 32 31 21 3 42 4143 2

2

14

2

2

f f f f f f f f f f f f f

f f f f f f f f f f f f f

f f f f f f f f f f

f f f f f f f f f

f f f f

s s

f f ff

s

js

s

sf

j

Hs

21

2 43 41 31 1 43 42 32

4 3 21 2 31 1 324 3 2 1 2 1 3 2 1

3 2 41 1 42 2 1 43

4 3 2 1

2

f

f f f f f f f f

f f f f f f ff f f f f f f f f

f f f f f f f f

f

s

f f f


1 4 3 2 1

2 4 3 2 1 3 2 1 2 1

3 4 3 2 1 2 1 3 2 1

4 4 3 2 1

5 1 43 42 41 32 31 21

4 32 31 21 3 42 41 216 2 43 21

2 43 41 31 1 43 42 32

7 3 4

2

2

2

4

a f f f fa f f f f f f f f f

a f f f f f f f f f

a f f f f

a a f f f f f f

f f f f f f f fa a f f

f f f f f f f f

a a f

3 21 2 31 1 32 3 2 41 1 42 2 1 43f f f f f f f f f f f f f f

4 3 2

1 2 3 4

54 3 2

6 7 4

2Posa a as s s s

H j fs s

j j aa as a sa

Transfer function


2. Analysis of 4-Stage Passive RC Network Model of Input Image Signals

3_

32 2

1 2 4; ; ;

; ; ; ; ; ;

Neg poly

j jj

S

j j e eV t V t

VV t

V t V

t

t V t V tV t V t

V

e

tV t

2

1

41

4

3

32

i in

i

in

i

n

in

n

n

i

in n

VV

VV

jV

VVV

Negative poly-phase signals

0

|H(jω)|

1je 2je 3je 4je phasor

phase plane

_Neg BS_Neg AS_Neg CS

_Neg DSImage Signals

Transfer Function

1 1 2 2 1 1 3 3 3 3 4 4

3 3 4 4 1 1

4 3 2 1

2 2 2 2

4 2

1

3 1

1

2 2

2C C C C C C

C C C C C

C

C

C C C

R Z Z R Z R R Z Z R

Z Z Z Z

Z R

Z R Z R

j j jR R

Z R R Z Z R

RjH

R

R

Z

2. Analysis of 4-Stage Passive RC Network Derivation of Transfer Function (Image Signal)

4 3 2

1 2 3 4

54 3 2

6 7 4

2Nega a as s s s

H j fs s

aa a a as

js

j

4 3 2 1 4 3 2 1 3 2 1 2 1 4 3 2 1 2 1 3 2 1 4 3 2 1

4 3 2 1 3 2 1 2 1

4 32 31 21 3 42 41 214 3 2 1 43 42 41 32 31 2143 21

2 43

4 3 2

4 3 2

42

2

f f f f f f f f f f f f f f f f f f f f f f f f f f

f f f f f f f f f

f f f f f f f ff f f f

s s s jsH

s s sf f f

f

j

f f ff f

f

41 31 1 43 42 32

4 3 21 2 31 1 324 3 2 1 2 1 3 2 1 4 3 2 1

3 2 41 1 42 2 1 43

2

f f f f f f

f f f fs

f f ff f f f f f f f f f f f f

f f f f f f f f

2. Analysis of 4-Stage Passive RC Network Composed Transfer Function of 4-Stage Passive RC Network

1 2 3 4

4 6 7 5

4 3 2

2 24 2 3;

f f f fH f f R

f

a a a a

a a a faf f

Composed transfer function

4 3 2

1 2 3 4

54 3

62

7 4

2 ; 0Negj ja a a aa

s s s sH j f f

s s a a as s

4 3 2

41 2 3

34

5 6 72

4

2 ; 0Poss s s s

H j f fs

a a a aa a a asj

s sj

Wanted Signals

Image Signals

2. Analysis of 4-Stage Passive RC Network Analysis of Transfer Function (Positive Frequency)

1 24 3

3 4

4 6 7 5

2

2 24 2 3; 0

f f f fH f f

f

a a a a

a a a faf f

4 3 2

2 20 0 4 2 3

1 2 3 4

4 6 7 5

lim lim 1f f

a a af f f fH f

f f f f

a

a a a a

4 3 2

2

1 2 3 4

4 6

24 2 37 5

lim lim 1f f

a a af f f fH f

f f

a

a a a f a f

4 3 24

1min f f ffH as ff

2 2 2 2, 0, 2 ; min 2 " "a b a b ab a b ab as a b

2 2

4 6 7 54 2 3

4 6 7 54 2 32a a a a a a a af f f f f f f f

min14 2 3

m6 5

in4 7

2

max1...... max 2

;H ff

f f f fa af H

a af

Applying Cauchy-Schwarz inequality theorem:

2. Analysis of 4-Stage Passive RC Network Analysis of Transfer Function (Negative Frequency)

4 3 2

2

1 2

24 2

3 4

4 6 7 53

; 0a a a af f f f

Ha a a a

f ff f f f

4 3 2

2 20 0 4 2 3

1 2 3 4

4 6 7 5

lim lim 1f f

a a af f f fH f

f f f f

a

a a a a

1 2 3 4

4 6 7 5

4 3 2

2 24 2 3lim lim 1f f

a af f f fH f

f f f f

a a

a a a a

1 2 3 4min ;dB

H f a f f fs f f f f f

3 41 2 2 3max1 ;max 2 ;max 3H f as f H f as f H f as ff f f f f f

2. Analysis of 4-Stage Passive RC Network Frequency Responses

f0

2dB

H j f

max1fmax 2f

314

2 4f ff f

1f2f3f4f

32f f

43f f21f f

1 2 3 4

4 6 7 5

4 3 2

2 24 2 3;

f f f fH f f R

f

a a a a

a a a faf f

Transfer function

Frequency Responses

2. Analysis of 4-Stage Passive RC Network Image Rejection Ratio

Notch Band Pass Band

Rejection Ratio

f0

2dB

H j f

1 2 3 4

1 2 3 4

22N

Pos

eg

H j f f f f fIR

f f f ff f f

RH j f f ff f f

Image Rejection Ratio = ----------------Pass Band

Notch Band




4. Conclusions

Outline

3. Proposed Model of 4-Stage Passive RC NetworkModel of 4-Stage Passive RC Network

1

2 21

3 31 32

4 41 42 43

0.7 ;1.5 ; 0.7 ;4 ; 0.7 ; 1.5 ;10 ; 0.7 ; 1.5 ; 4 ;

f MHzf MHz f MHzf MHz f MHz f MHzf MHz f MHz f MHz f MHz

1inV

2inV

4inV

3inV

1outV

2outV

3outV

4outV

1R

1C1R

1C1R

1C1R

1C

2R

2C2R

2C2R

2C2R

2C

3R

3C3R

3C3R

3C3R

3C

1AV

2AV

4AV

3AV

1BV

2BV

3BV

4BV

4R

4C4R

4C4R

4C4R

4C

1CV

2CV

3CV

4CV

1 1

2 2

3 3

4 4

1 ; 227 ;1 ; 106 ;1 ; 39.8 ;1 ; 15.9 ;

R k C pFR k C pFR k C pFR k C pF

3. Proposed Model of 4-Stage Passive RC NetworkSimplified Model of 4-Stage Passive RC Network

1 4 3 27

1 1.62 10a f f f f

2 4 3 2 1 313

2 1 2 1 7.19 10a f f f f f f f f f

3 4 3 2 1 2 1 3 12

201.03 10a f f f f f f f f f

4 4 3 2 1254.2 10a f f f f

5 1 43 42 41 32 31 2172 3.44 10a a f f f f f f

4 32 31 21 3 42 41 216 2 43 21

2 43 41 31

14

1 43 42 32

2 1.9 14 0f f f f f f f f

a a f ff f f f f f f f

7 3 4 3 21 2 31 1 32 3 2 41 1 42 2 1 43202 2.26 10a a f f f f f f f f f f f f f f f

7 13 20 24 3 2

2 24

5

25 14 202 7 3

1.62 10 7.19 10 1.03 10 4.2 10

4.2 10 1.9 10 2.26 10 3.44 10dB

f f f fH f

f f f f

Transfer function

5

5

5

0 ;1.33 ; 1.55 100.55 ; 25.5 10

0

20l1.4

o2 ; 405 10

0 ;

g

f

H

f

dBdB fdB fdB fdB

4 3 2

2 24

7 13 20 25

25 14 20 72 3

1.62 10 7.19 10 1.03 10 4.2 10

4.2 10 1.9 10 2.26 10 3.; 0

44 10

f f f fH ff f f f

5

5

5

1;1.16; 1.55 100.938; 25.5 101.18; 405 10

1

0

;

ff

f

ff

H

3. Proposed Model of 4-Stage Passive RC NetworkAnalysis of System Model (Positive Frequency)

7 13 204 3 2

2 24 2 3

25

25 14 20 7

1.62 10 7.19 10 1.03 10 4.2 10

4.2 10 1.9 10 2.26 10 3.44 10; 0

dB

f f f fH ff f f f

5

5

5

5

5

5

5

1;0; *10

0.016; *100; *10

0.018; *100;

0710.2

1524.5

4063.2

1

*100.024;

00*10

0; *101;

ffff

H fff

ff

5

5

5

5

5

5

5

0 ;; *10

35.6 ; *10; *10

34.8 ; *

0710.2

1520log 24.5

4063.2

100

10; *10

32.1 ; *10; *10

0 ;

dB

dB

dB

dB

fff

fH f

ff

ffdB

3. Proposed Model of 4-Stage Passive RC NetworkAnalysis of System Model (Negative Frequency)

3. Proposed Model of 4-Stage Passive RC NetworkSimulation Results of 4-Stage Passive RC Network

Frequency Responses

33dB

Rejection Ratio

Parameter PPFInput IF frequency 5MHz

Input IF Power -60dBmNF -

Gain -0.3dBCurrent

Consumption-

Supply Voltage -Image Rejection

Ratio IMRR-33dB

-33dB




4. Conclusions

Outline

This Work:• Derivation of transfer function of poly-phase network

based on superposition principle • Mathematical analysis and model of 4-stage passive

RC poly-phase filter• Image rejection ratio: -33dB

Future of Work• Analysis of IQ mismatches of poly-phase signals• Analysis of Parasitic of RC components

4. Conclusions

Thanks for your kind attention!

Questions & Answers1) Will the variations of R and C components change

the characteristics of this filter?Yes, they will.(The variations of R and C components will cause the IQ mismatches. Therefore, the image rejection ratio will be changed.)

2) Are the simulation results of RC poly-phase filter in this research best? No, they aren’t.(There are some design trade-offs in this research. So, these simulation results are acceptable for the design targets.)

mathematical analysis and design of 4-stage passive rc ... · gen. 4th order poly-phase filter...

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