Post on 20-May-2020
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
* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]
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
* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]
Several common guided wave
structures: coaxial cable,
rectangular waveguide, stripline,
microstrip, coplanar waveguide
Introduction
* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]
Microwave and RF
frequency industrial
and IEEE band
designation
Introduction
* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]
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
* Taken from „ RF and microwave passive and active technologies in: RF and Microwave handbook” Golio M. [2]
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.
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
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
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
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
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
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
Reflection coefficient
[ ] [ ] 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
−=
+=
Reflection coefficient
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
Reflection coefficient
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
Reflection coefficient
jImZ
ReZZ0
jImΓ
ReΓ
j
-j
-1
1
00 =Γ⇒= ZZ
10 −=Γ⇒=Z
Reflection coefficient
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
Reflection coefficient
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
Reflection coefficient
Magnitude of the
reflection
coeffiecient and
power
transmision
factor for different
load resistances
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
Reflection coefficient
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
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
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
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
Bipolar transistor hybryd π model,
without case
Bipolar transistor hybryd π model,
with taking into consideration a case
parameters
S-parameters of a bipolar transistor
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]
S-parameters of a bipolar transistor
* Taken from „Electronic circuits : handbook for design and application” Tietze U., Schenk C. [1]