rf circuits design 1 - ue.pwr.wroc.pl€¦ · rf circuitsdesign grzegorz beziuk introduction. basic...

RF circuits designGrzegorz BeziukGrzegorz Beziuk

Introduction. Basic definitions and parameters

References[1] Tietze U., Schenk C., Electronic circuits : handbook for design and application, Springer

2008

[2] Golio M., RF and microwave passive and active technologies in: RF and Microwave

handbook, 2008, CRC Press Taylor and Francis Group

[3] Golio M., RF and microwave applications and systems in: RF and Microwave handbook,

2008, CRC Press Taylor and Francis Group

[4] Maxim, Application Note 742, Impedance Matching and the Smith Chart: The

Fundamentals, AN 742, 2002, Maxim

Introduction

* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]

Introduction


Component dimensions relative to signal wavelenght.

l < λ/10 – phase shift is neglected,

lumped element

l > λ/10 – phase shift can not be

neglected, distributed circuit

description

Introduction


Several common guided wave

structures: coaxial cable,

rectangular waveguide, stripline,

microstrip, coplanar waveguide

Introduction


Microwave and RF

frequency industrial

and IEEE band

designation

Introduction


U. S. Military

frequency band

designation

Introduction

* Taken from „RF and microwave applications and systems in : RF and Microwave handbook” Golio M. [3]

Wlan RF ISM Bands

(Industrial, Scientific

and Medical band).

Operating channels

for direct sequence

Introduction


Attenuation of

electromagnetic signals

in atmosfere

Transmission lines

* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]

Cross section and

field lines of the

coaxial and the

symmetric cables

Transmission lines

[ ]Ω====

=

r

for

r

r

r

r

W

r

H

EZ

εε

µπ

εε

µµµ

377120

1

0

0

Wave impedance of transmission line:

r

for

rr

p

ccv

r

εµε

µ

===1

Velocity of the wave propagation in transmission line:

Where µ, ε are magnetic and electric permitivities, respectively, c is the light

velocity in the free space c = 3*108 [m/s].

Transmission lines

Characteristic impedance of transmission line:

−

===

1ln1

ln2

1

20

d

a

d

a

d

d

ZkZI

UZ

i

a

WgW

π

π

Ω

−

Ω

=

][1ln120

][ln60

20

d

a

d

a

d

d

Z

r

i

a

r

ε

ε

Transmission lines

Equivalent circuit for an incremental length of transmission line. A finite length of

transmission line can be modelled as a series concatenation of sections of this form.

Transmission linesWe can write the following equations:

( )( ) 112

112

''

''

UdzCjdzGII

IdzLjdzRUU

ω

ω

+−=

+−=

When we replace:dUUU += 12

dIII += 12

Then we divide equations by dz, and assuming that length

of the line section tends to 0:

IIIUUUdz =→=→→ 2121 ,,0

Transmission linesWe get the following equations:

( )

( )UCjGdz

dI

ILjRdz

dU

''

''

ω

ω

+−=

+−=

When we differentiate first equation by z and then put second

equation into obtained differential equation we will get

transmission line equation:

( )( ) UUCjGLjRdz

Udγωω =++= ''''

2

2

Transmission lines

The solution of this equation is

( ) z

r

z

f eUeUzU γγ += −

Where γ is a propagation constant:

( )( )'''' CjGLjR ωωγ ++=

For the low loss lines the propagation constant is described by

the equation:

βα

ωγ '''

'

2

'

'

'

2

'CLj

C

LG

L

CR++≈

Transmission lines

α – it is elementary attenuation constant

β – it is elementary phase shift constant

In the case of lossless transmision lines (R’ = G’ = 0)

attenuation is equal to zero.

When we write:

( ) ( ) ( ) ( )

( ) ( )ztuztu

zteUzteU

eUeUezUztu

rf

wavereflected

f

z

r

waveincident

f

z

f

ztj

r

ztj

f

tj

,,

coscos

ReRe,

__

+=

=++++−=

=+==

−

+−

ϕβωϕβω αα

γωγωω

Transmission lines

Incident wave in

transmission line in

the time To and ¼ of

the period later.

Reflected wave in

transmission line in

the time To and ¼ of

the period later.


Transmission lines

For incident wave:

''

10

CLdt

dzvzt pf ===⇒=+−

β

ωϕβω

Wave length:

f

v

CLf

p===⇒=

''

122

β

πλπβλ

For electromagnetic wave in the free space wave lenght is

given by equation:

f

c=λ

Transmission lines

Characteristic impedance of the line is given by expression:

''

''0

CjG

LjRZ

ω

ω

+

+=

The incident and reflected current wave in the line are given

by equation:z

r

z

f

zrzfeIeIe

Z

Ue

Z

UI γγγγ −=−= −−

00

For lossless lines:

'

'0

C

LZ =

Transmission lines

Attenuation of a typical coaxial

50Ω cable in the versus of

frequency


Transmission lines

Four-pole representation of the transmission line

U1

U2

I2

I1

Z0

0 lz

γ = α + jβ

Transmission linesVoltages on the line terminals:

l

r

l

f

rf

eUeUU

UUU

γγ +=

+=

−2

1

Currents on the line terminals:

lrlfl

r

l

f

rf

rf

eZ

Ue

Z

UeIeII

Z

U

Z

UIII

γγγγ

00

2

00

1

−=−=

−=−=

−−

Transmission lines

Now we get the following expressions:

l

r

l

f

eUIZU

eUIZU

γ

γ

2

2

202

202

=−

=+ −

Reflected wave depends on a load resistance of the line. If we

connect to the end of the line resistance R = Z0 reflected wave

is equal to zero.

Transmission lines

Then we obtain expressions:

( ) ( )

( ) ( )llll

llll

eeI

eeZ

UI

eeIZ

eeU

U

γγγγ

γγγγ

−−

−−

−++=

−++=

22

22

2

0

21

2021

for ( )ll eel γγγ −+=2

1)cosh(

( )ll eel γγγ −−=2

1)sinh(

Transmission lines

For the transmission line loaded by arbitrary impedance Z2

its input impedance is given by expression:

( )

( ) 10

2

02

+

+=

ltghZ

Z

ltghZZZ I*

γ

γ

We obtain four pole equation of a transmission line:

( ) ( )

( ) ( )

=

2

2

0

0

1

1

coshcosh1

sinhcosh

I

U

llZ

lZl

I

U

γγ

γγ

U1

U2

I2

I1

Z0

0 lz

γ = α + jβZ2

ZIN

Transmission lines


Transmission lines

Input impedance of the line opened at the end (for l < λ/8):

CjlCjZ I*

ωω

1

'

1==

In the case of the line short at the end (for l < λ/8):

LjlLjZ I* ωω == '

In the case of line electrically short (l<10λ) ZIN = Z2.

For the line of lenght l=λ/4:

2

2

0

Z

ZZ I* =

Reflection coefficient

Relationships between voltages, currents and characteristic

impedance of the transmission line:

ff IZU 0= rr IZU 0=

Therefore waves parameters of the line are given by

expressions:

Both, incident and reflected wave could be described by one

parameter.

0

0

0

0

ZIZ

Ub

ZIZ

Ua

rr

f

f

==

== Incident wave

Reflected wave


[ ] [ ] WVAba ===

‘a’ and ‘b’ describes power of incident and reflected waves:

For transmitted power:

2*

2*

0

0

Re

Re

bIUP

aIUP

realZ

rrr

realZ

fff

==

==

When we take into consideration voltages and currents:

rf

rf

III

UUU

−=

+=


Then:

( )

( )baZ

I

baZU

−=

+=

0

0

1

And finally:

−=

+=

0

0

0

0

2

1

2

1

ZlZ

Ub

ZlZ

Ua

U

I

=

a

b


Definition of the reflectrion coefficient:

( )a

b

U

U

waveincident

wavereflectedntcoeffieciereflection

f

r ===Γ_

___

0

0

ZZ

ZZ

a

b

U

U

f

r

+

−===Γ

Or:

Γ−

Γ+=

1

10ZZ


jImZ

ReZZ0

jImΓ

ReΓ

j

-j

-1

1

00 =Γ⇒= ZZ

10 −=Γ⇒=Z


Matching – Z = Z0, b = 0, Γ = 0

Shorted end of the line – Z = 0, b = -a, Γ = -1, Pr = Pf

Open end of the line – Z = ∝, b = a, Γ = 1, Pr = Pf

Resistive load – Z = R, 0 < R < ∝, -1 < Γ < 1

Inductive load – Re(Z) = 0 and Im(Z) > 0, |Γ| = 1 and

0 < arg(Γ) < π.

Capacitive load – Re(Z) = 0 and Im(Z) < 0, |Γ| = 1 and

-π < arg(Γ) < 0.

Reflection coefficientjImΓ

ReΓ

j

-j

-1 1(short)

inductive

capacitive

resistive-inductive

resistive-capacitive

matching

open

resistive

01 =⇒−=Γ Z

LjZ ω=

CjZ

ω

1=

RZ =

0

0

1,

ZCjZZj

ω=−=⇒−=Γ

ωjZLjZZj 00 , ==⇒=Γ

∞=⇒=Γ 01 Z

00 ZZ =⇒=Γ

0→L

0→R

0→C

∞→L

∞→R

∞→C


For passive circuit the power emmision of a load resistance

always is positive or equal to zero:

( ) 012222

≥Γ−=−=−= abaPPP rf

We can define the Power Transmission Factor:

21 Γ−=Pk

U

I

=I

UZ =0Z

a

b=Γ0Z

a

b


Magnitude of the

reflection

coeffiecient and

power

transmision

factor for different

load resistances



2Z0Z 2a

2b2

2

Z

Γ

βαγ j+=1

1

Z

Γ 1a

1b

0

z

l

ljll eee βαγ 22

2

2

21

−−− ⋅Γ=Γ=Γ

Transmission line influences on reflection coeffiecient – it causes its attenuation

and phase shift like a transformer.

Voltage standing wave ratio


Voltage standing wave ratio

Voltage standing wave ratio:

Γ−

Γ+==1

1

min

max

U

UVSWR

For the full matching VSWR is equal to 1 and P = Pmax

Real power transmision:

VSWR

PP max=

Wave source matching

0Z

βαγ j+=gΓ

gb

ga

0

z

l

Ug

gZ

U

=

gΓgb

ga

Ug

gZ

U 0Zb

a

Γ

Wave source matching

Line input voltage:

( )g

g

g

gg

g

g U

ZZ

ZZU

ZZ

ZUU Γ−=

+

−−=

+= 1

21

2 0

0

0

0

( )g

g

gZ

U

Z

Ub Γ−== 1

2 00

0

In the case of full matching the generator and the line:

0

02 Z

Ub

g

g =

Impedance matching – Smith Chart

S-parameters

=

2

1

2221

1211

2

1

U

U

yy

yy

I

I

1I 2I

1U 2U

=

2

1

2221

1211

2

1

a

a

ss

ss

b

b=1a

1b 2a

2b

+=

+=

02

0

22

01

0

11

2

1

2

1

ZIZ

Ua

ZIZ

Ua

−=

−=

02

0

22

01

0

11

2

1

2

1

ZIZ

Ub

ZIZ

Ub

s11....s22 - scatering parameters

S-parametersS11 – the input reflection coeffiecient

[ ]0ZS111 s=Γ

1a

1111 asb =

02 =a

1212 asb =

0=ΓL 0ZRL =

0

2

101

01

111 ZR

aLLa

bs

==Γ

=

Γ=Γ==

S11 we can use to determine the input impedance of circuit:

11

110

1

10

1

1

1

1

1

1

00

0 s

sZZ

I

UZ

ZRZR

ZRI*

LL

L −

+=

Γ−

Γ+==

==

=

S-parametersS12 – return transmission coeffiecient

[ ]0ZS0=Γg

01 =a

2121 asb =

2a

2222 asb =

222 s=Γ0ZRg =

02

112

1=

=a

a

bs

S-parametersS21 – forward transmission coeffiecient

[ ]0ZS

0

02 Z

Ub

g

g =

1a

1111 asb =

02 =a

1212 asb =

0ZRL =2U1U

Ug

0ZRg =

01

221

2 =

=a

a

bs

0

22 2

21 ZRRu

gLg

AU

Us

====

S-parametersS22 – output reflection coeffiecient

[ ]0ZS0=Γg

01 =a

2121 asb =

2a

2222 asb =

222 s=Γ0ZRg =

0

1

202

02

222 ZR

agga

bs

==Γ

=

Γ=Γ==

S22 we can use to determine the output impedance of circuit:

22

220

2

20

2

2

1

1

1

1

00

0 s

sZZ

I

UZ

ZRZR

ZROUT

gg

g −

+=

Γ−

Γ+==

==

=

Y and S-parameters

( )( )

( )

( )

( )( )

21122211

2

002211

2

002211

22

2

002211

02121

2

002211

01221

2

002211

2

001122

11

1

1

1

2

1

2

1

1

yyyy

ZZyy

ZZyys

ZZyy

Zys

ZZyy

Zys

ZZyy

ZZyys

y

y

y

y

y

y

y

−=∆

∆+−+

∆−−+=

∆+++

−=

∆+++

−=

∆+−+

∆−−+=

21122211

2211

2211

0

22

2211

21

0

21

2211

12

0

12

2211

2211

0

11

1

11

1

21

1

21

1

11

ssss

ss

ss

Zy

ss

s

Zy

ss

s

Zy

ss

ss

Zy

s

s

s

s

s

s

s

−=∆

∆+++

∆−+−=

∆+++

−=

∆+++

−=

∆++−

∆−+−=

S-parameters of a bipolar transistor


Bipolar transistor hybryd π model,

without case

Bipolar transistor hybryd π model,

with taking into consideration a case

parameters

Noise Figure

Noise factor:

out

in

S*R

S*RF =

Noise figure (for temparature T0 = 290K):

( ) dBoutdBin

out

in S*RS*RS*R

S*RF*F ,,log10log10 −=

==

rf circuits design 1 - ue.pwr.wroc.pl€¦ · rf circuitsdesign grzegorz beziuk introduction. basic...

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