RF circuits design
Grzegorz Beziuk
Microstrip and Stripline PCB
techniques
References
[1] Richard C. Li, RF circuits design, 2008, John-Wiley & Sons
[2] Jia-Sheng Hong, M. J. Lancaster, Microstrip Filters for RF/Microwave
Applications, 2001, John-Wiley & Sons
[3] Inder Bahl, Lumped elements for RF and microwave circuits, 2003,
Artech House
[4] Fabian Wai Lee Kung, RF/Microwave Circuit Design, 2008, Multimedia
University, (open source lectures: http://pesona.mmu.edu.my/~wlkung/ADS/ads.htm)
[5] R. Ludwig, Introduction to RF circuit design, 2007, Worcester, MA,
(open source lectures: https://ece.wpi.edu/~ludwig/ece3113/)
[6] Geoff Smithson, Practical RF printed circuit board design, 2000,
Plextek Co.
Printed Circuit Boards
Copper thikness
Dieelectric thikness
εr
Standard
material consist
of epoxy with glass
fiber reinforcement
Copper (Usually gold plated
to protect against oxidation)
Printed Circuit Boards
Typical dielectric thickness: 32mils (0.8mm), 62mils (1.56mm) for double sided
board. For multi-layer board the thickness can becustomized from 2 – 62 mils, in
1 mils step.
Copper thickness is usually expressed in terms of the mass of copper spread over
1 square foot.
Standard copper thickness are: 0.5, 1.0, 1.5 and 2.0 oz/foot2.
0.5 oz/foot2 ≅ 0.7mils thick.
1.0 oz/foot2 ≅ 1.4mils thick.
2.0 oz /foot2 ≅ 2.8mils thick.
Printed Circuit Boards
Examples of PCB materials
The above values are only rough approximation and depend on processes
* Taken from „Introduction to high-speed PCB design”, Kung [4]
Printed Circuit Boards
Summary of typical PCB process limitation for a range of cost options
* Taken from „ Practical RF printed circuit board design”, Smithson [6]
Printed Circuit Boards
Choices of Substrates depends on:
- operating frequency range
- electrical characteristics - e.g. nominal dielectric constant, anisotropy
loss tangent, dispersion of dielectric constant
- copper thickness (affect low frequency resistance)
- Tg, the glass transition temperature
- costs
- tolerance.
- manufacturing technology - thin or thick film technology.
- thermal requirements - e.g. thermal conductivity, coefficient of thermal
expansion (CTE) along x,y and z axis.
- mechanical requirements
Transmission lines concept
ZG
It
UG
Ut
It
ZL
Transmission lines concept
The Telegraphic Equations:
Fourier Transform
∂v
∂i
= − Ri − L
∂t
∂z
∂i
∂u
= −Gv − C
∂z
∂t
∂V
= −(R + jωL )I = − ZI
∂z
∂I
= −(G + jωC )U = −YU
∂z
Inverse Fourier
Transform
Transmission lines concept
Solutions:
Wave travelling in +z
direction
V (z ) = V e
+ − γz
0
Wave travelling in +z
direction
− γz
0
−V e
I ( z ) = I 0+ e −γz − I 0− eγz
Attenuation
factor
γ = α (ω ) + jβ (ω ) =
Propagation
constant
Phase
factor
(R + jωL )(G + jωC )
Transmission lines concept
Transmission lines parameters:
- characteristic impedance Z0
Z0 =
U 0+ e −γz U 0+ U 0− eγz U 0−
R + jω L
= + = − γz = − =
+ −γz
I0 e
I0
I0 e
I0
G + j ωC
- propagation velocity vp
ω
c
vp = =
β
ε err
- attenuation α
- dispersion (vp depends on frequency)
Transmission lines concept
The lossless (R=0, G=0) transmission lines parameters:
- characteristic impedance Z0
- propagation velocity vp
Z0 =
vp =
L
C
1
LC
- lack of attenuation and dispersion
- propagation constant
γ = jβ = jω LC
Transmission lines on PCB
Striplines:
- are planar-type transmission lines
- can be easily fabricated on a printed circuit board or
- the most common transmission line configurations using stripline technology
are: microstrip line, stripline and co-planar stripline.
Transmission lines on PCB
* Taken from „Introduction to high-speed PCB design”, Kung [4]
Transmission lines on PCB – paired
strips
W
Applications:
- toroidal power divider
εr, µr
h
with phase inverting
- directional branch filters
Transmission lines on PCB – paired
strips
T-line characteristic impedance Z0 for (W/h>1):
120π W ln (4 ) ε r + 1  πε r (W / h + 0.94 )  ε r + 1  ε rπ 2  
Z0 =
+
ln
 +
 + 2πε 2 ln  16  
2πε r 
2
π
εr  h


r
T-line characteristic impedance Z0 for (W/h<1):
2
ε − 1   π  ln(π / 4) 
120   4h 
W 
 ln  +

Z0 =
ln  + 0.125  − r
ε r 
ε r   W 
 h  2(ε r + 1)   2 
−1
Transmission lines on PCB –
coplanar strips
b
a
εr, µr
h
The coplanar strips transmission line is one of possible the
complementary structure of waveguide.
Transmission lines on PCB –
coplanar strips
T-line characteristic impedance Z0 :
where:
a
k=
b
ε reff = 1 +
k1 ' = 1 − k12
ε r − 1 K (k ')K (k1 )
2
Z0 =
120π K (k )
ereff K (k ')
K (k )K (k1 ')
k'= 1− k 2
 πa 
sinh  
 4h 
k1 =
 πb 
sinh  
 4h 
K(k) is a first kind eliptic integral. K(k’) is an associated integral with K(k).
Transmission lines on PCB –
microstrip line
The most common and
analysed transmission line
structure. It is easy to use,
low cost and has a good
range of practical
impedances.
W
t
h
εr, µr
a > 10W
Transmission lines on PCB –
microstrip line
T-line characteristic impedance Z0 for (W/h < 1):
Z0 =
 8
120
W (t ) 
ln 
+ 0.125

h 
ε reff (t ) W (t ) / h
T-line characteristic impedance Z0 for (W/h > 1):
Z0 =
120π W (t )
 W (t )

+ 1.393 + 0.667 ln
+ 1.444 

ε reff (t )  h
 h

−1
Transmission lines on PCB –
microstrip line
Where:
W 1.25 t 
4πW 
 h + π h 1 + ln t  for (W / h ≤ 0.5π )
W (t ) 


=
h
W + 1.25 t 1 + ln 2h  for (W / h ≥ 0.5π )
 h
π h
t 
ε reff (t ) = ε reff −
ε r −1 t / h
4. 6
W /h
ε reff =
ε r +1 ε r −1 
2
+
h
1 + 12 
2 
W
−0.5
Transmission lines on PCB –
stripline
W
The most common and
analysed triplate
t
(sandwitched) transmission
line structure.
εr, µr
a > 10W
b
Transmission lines on PCB –
stripline
T-line characteristic impedance Z0:
Z0 =
2


8b
 8b 
ln 1 + 0.5
+ 
+
6
.
27


πw'
εr 
 πw' 


60
where:
∆W
w' = W +
t
t
6
m=
2t
3+
b


e
∆W 1 
= ln 
2
m
π
t
  1 + 1 +  1 / 4π 
  2b / t   W / t + 1.1 
Transmission lines on PCB –
coplanar waveguide
a > 10W
W
a > 10W
t
εr, µr
h
Main advantage of coplanar waveguide is its single-sided
nature. Grounding components does not require plated
through-holes plane on the other side of substrate. This
makes it ideal for use with surface mounted components.






Transmission lines on PCB –
coplanar waveguide
T-line characteristic impedance Z0 :
Z0 =
30π K (kt ')
ereff ,t K (kt )
where:
ε reff ,t = ε reff −
ε reff − 1
(b − a ) / 2 K (k ) + 1
0.7t K ' (k )
ε reff = 1 +
ε r − 1 K (k ')K (k1 )
2
K (k )K (k1 ')
Transmission lines on PCB –
coplanar waveguide
where:
 πa 
sinh  
a
 4h 
a
kt = t k =
k1 ' = 1 − k12 k ' = 1 − k 2 kt ' = 1 − kt2 k1 =
 πb 
bt
b
sinh  

1.25t 
4
π
a


 4h 
at = a +
1 + ln


K(k) is a first kind eliptic integral.
π 
 t 
K(k’) is an associated integral
with K(k).
bt = b −
1.25t 
 4πa 
1 + ln


π 
 t 
Transmission lines on PCB –
coplanar waveguide with ground
a > 10W
W
a > 10W
t
εr, µr
h
The transmission line occurring in the case of top and
bottom grounded PCB.
Transmission lines on PCB –
coplanar waveguide with ground
T-line characteristic impedance Z0 :
where:
a
k=
k1 ' = 1 − k12 k ' = 1 − k 2 k1 =
b
K (k ') K (k1 )
K (k ) K (k1 ')
=
K (k ') K (k1 )
1+
K (k ) K (k1 ')
1+ ε r
ε reff
Z0 =
 πa 
tanh 
 4h 
 πb 
tanh 
 4h 
60π
1
(
)
K
k
K (k1 )
ereff
+
K (k ') K (k1 ')
K(k) is a first kind eliptic integral. K(k’) is an
associated integral with K(k).
Transmission lines discontinuities
W
W
1.42W
b
b = 0.57W
Appropriate T-line banding
It is needed for frequencies above 300 MHz.
Via holes
Designing a PCB one can use three types of through hole
vias:
- blind via
- buried via
- through hole via
Via holes
PCB cross-section showing different plated through hole
types
* Taken from „ Practical RF printed circuit board design”, Smithson [6]
Via holes
Via hole connection through dielectric and backside via hole ground
* Taken from „Lumped elements for RF and microwave circuits”, Bahl [3]
Via holes
Parameters of a cilindrical via hole:
- inductance

 h + r 2 + h2
Lvia = 0.2 h − ln

r


 3
 + r − r 2 + h2
 2

(
)( pH )

where r and h are, expressed in microns, the radius and high of via hole, respectively,
- resistance
Rvia = RDC 1 +
f
fδ
fδ =
1
πµ0σt 2
where f is operating frequency, µ0 the free-space permeability, σ the conductivity of
metal, and t its thickness.
Microstrip Discontinuities
Step in width
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip Discontinuities
where:
C = 0.00137h
L1 =
+ 0.3  W1 / h + 0.264 
W  ε

1 − 2  reff 1
( pF )

 W / h + 0.8 
ε
W
−
0
.
258
2  reff 1


 1
ε reff 1 
Z 01
Lw1
L
Lw1 + Lw 2
L2 =
 Z
L = 0.000987h1 − 01
 Z 02

Lw 2
L
Lw1 + Lw 2
ε reff 1
ε reff 2
Lwi = Z 0i ε reffi / c
2

 (nH )


Note: all dimensions are
in micrometers.
Microstrip Discontinuities
Open ends
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip Discontinuities
where:
∆l =
ξ1 = 0.434907
ξ3 = 1 +
cZ 0C p
ε reff
∆l ξ1ξ 3ξ 5
=
h
ξ4
0.8544
0.81
ε reff
+ 0.26(W / h )
+ 0.236
0.8544
0.81
ε reff
− 0.189(W / h )
+ 0.87
[
0.5274 tan −1 0.084(W / h )
ε
[
1.9413 / ξ 2
0.9236
reff
0.371
(
W / h)
ξ2 = 1 +
2.35ε r + 1
]
]
ξ 4 = 1 + 0.037 tan −1 0.067(W / h )1.456 ⋅ {6 − 5 exp[0.036(1 − ε r )]}
ξ 5 = 1 − 0.218 exp(− 7.5W / h )
Microstrip Discontinuities
Gaps
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip Discontinuities
where:
C p = 0.5Ce
C g = 0.5C0 − 0.25Ce
0.8
m
0
C0
ε   s 
( pF / m) =  r    exp(k 0 )
W
 9. 6   W 
0.9
m
e
Ce
ε   s 
( pF / m) = 12 r    exp(k e )
W
 9. 6   W 
Microstrip Discontinuities
with:
W
[0.619 log(W / h ) − 0.3853]
k
for 0.1 < s/W < 1
k0 = 4.26 − 1.453 log(W / h )
m0 =
me = 0.8675
W 
ke = 2.043 
h
0.12
for 0.1 < s/W < 0.3
1.565
−1
(W / h )0.16
0.03
ke = 1.97 −
W /h
me =
for 0.3 < s/W < 1
Microstrip Discontinuities
Bends
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip Discontinuities
 (14ε r + 12.5)W / h − (1.83ε r − 2.25) 0.02ε r
+
C

( pF / m) = 
W /h
W /h
W
(9.5ε + 1.25)W / h + 5.2ε + 7
r
r

{
}
L
(nH / m) = 100 4 W / h − 4.21
h
for W/h < 1
for W/h > 1
Microstrip components – lumped
inductors
W
l
Its circuit representation
High-impedance line
Microstrip components – lumped
inductors
  l 
W +t
L(nH ) = 2 ⋅10 − 4 l ln
K g for l in µm
 + 1.193 + 0.2235
l 
 W + l 
W
Rs l 
 W 
5 < < 100
for
R=
1.4 + 0.217 ln 

t
2(W + t ) 
 5t 
t – conductor thickness
W
K g = 0.57 − 0.145 ln
h
h – substrate thickness
for
W
> 0.05
h
Rs – surface
resistance of a
conductor
Microstrip components – lumped
D
inductors
0
s
L(nH ) = 0.03937
a 2n 2
Kg
8a + 11c
W
D0 + D1
4
πanRs
R = 1. 5
W
a=
D1
c=
for a in µm
D0 − D1
2
For both inductors:
Circular spiral inductor
Q=
ωL
R
Microstrip components – lumped
capacitors
l
W
s
Interdigital capacitor
Its circuit representation
Microstrip components – lumped
capacitors
C ( pF ) = 3.937 ⋅10 −5 l (ε r + 1)[0.11(n − 3) + 0.252]
R=
4 Rs l
3 Wn
−1
 1 
 1 

Q = Q +Q = 
 + 
 ωCR 
 tg (δ ) 
−1
for l in µm
−1
C
−1
−1
d
n – number of fingers
tg(δ) – dielectric loss tangent
Microstrip components – lumped
capacitors
l
W
C=
d
Dielectric thin film
MIM capacitor
ε (W ⋅ l )
d
Rl
R= s
W
Microstrip components –
quasilumped elements
High impedance short-line element
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip components –
quasilumped elements
 2π
x = Z c sin 
λ
 g
B 1  π
=
tg
2 Z c  λ g

l



l


for
l<
β=
λg
8
2π
λg
 2π
x ≈ Zc 
λ
 g

l


B 1  π
≈
2 Z c  λ g

l


Microstrip components –
quasilumped elements
Low impedance short-line element
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip components –
quasilumped elements
B=
 2π
1
sin 
λ
Zc
 g
π
x
= Z c tg 
λ
2
 g

l



l


B≈
for
l<
λg
8
1
Zc
 2π

λ
 g
π
x
≈ Zc 
λ
2
 g

l



l


Microstrip components –
quasilumped elements
 2π
Yin = jYC tg 
λ
 g
Open-circuited stub
 2π
Yin ≈ jYC 
λ
 g
1
YC =
ZC

l



l


β=
for
l<
2π
λg
8
C = YC l / v p
λg
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip components –
quasilumped elements
 2π
Z in = jZ C tg 
λ
 g
 2π
Z in ≈ jZ C 
λ
 g
Short-circuited stub

l



l


for
l<
λg
8
L = ZCl / v p
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]
Microstrip components –
resonators
* Taken from „Microstrip Filters for RF/Microwave Applications”, Hong, Lancaster [2]