Transmission lines
in MMIC technology
Evangéline BENEVENT
Università Mediterranea di Reggio Calabria
DIMET
1
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
2
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
3
Transmission lines in MMIC technology
Propagation in transmission lines
Definition of propagation modes:
TE modes (Transverse Electric) have no electric field in the direction of
propagation.
TM modes (Transverse Magnetic) have no magnetic field in the direction of
propagation.
TEM modes (Transverse ElectroMagnetic) have no electric nor magnetic field
in the direction of propagation.
Hybrid modes have both electric and magnetic field components in the
direction of propagation.
Generally, transmission lines are used respecting certain
conditions in order to only have TEM modes of propagation.
4
Transmission lines in MMIC technology
Propagation in transmission lines
At low frequency:
At one time “t”, the points of the circuit linked by a conductor are at the same
potential:
(V A − VB )(t ) = (VC − VD )(t )
A
C
ZS
v (t)
EG
ZL
v (t)
∼
B
D
5
Transmission lines in MMIC technology
Propagation in transmission lines
At high frequency:
When the frequency “f” of the signal increases enough, the differences of
potential between two points linked by a conductor have not equal values any
more:
(VA − VB )(t ) ≠ (VC − VD )(t )
A
C
ZS
v (t)
EG
ZL
v‘ (t)
∼
B
D
ℓ
6
Transmission lines in MMIC technology
Propagation in transmission lines
At high frequency:
We can define the wavelength of the
signal which propagates at the
frequency “f” by the following
expression:
c
f
λ = (m )
where c is the speed of the light in
vacuum:
c = 3.10 8 (m / s )
Examples of wavelengths
at different frequencies:
Frequency f
Wavelength λ
50 Hz
6000 km
5 kHz
60 km
500 kHz
600 m
5 GHz
6 cm
50 GHz
6 mm
7
Transmission lines in MMIC technology
Propagation in transmission lines:
Each time the wavelength is in the same order of the physical dimensions of
the conductor linking to points, it is necessary to take into account the
propagation effects:
If ℓ ≈ λ ⇒ propagation effects !
Electrical length ≠ Physical length !
⇒ Theory of the transmission lines
8
Transmission lines in MMIC technology
Propagation in transmission lines:
Equivalent circuit of a transmission line with a TEM (or quasi-TEM)
propagation:
A transmission line is a structure with distributed elements (due to propagation
effects),
The elements of the equivalent circuit are given in units of length:
Resistance: R (Ω/m),
Inductance: L (H/m),
Associated to the conductors
Capacitance: C (F/m),
Conductance: G (S/m = Ω-1/m).
R ⇒ conductor losses !
Associated to the substrate
G ⇒ substrate losses !
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Transmission lines in MMIC technology
Propagation in transmission lines:
Two equivalent circuits for transmission lines:
Line without losses:
Ldx
Line with losses:
Rdx
Ldx
Cdx
x
x + dx
Z = jLω
ω
Gdx
Cdx
x
x + dx
Z = R + jLω
ω
Y = jCω
ω
Y = G + jCω
ω
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Transmission lines in MMIC technology
Propagation in transmission lines:
Equivalent circuit of a transmission line with a TEM (or quasi-TEM) propagation:
Coupled differential equations ⇒ Equation of telegraphers:
∂v
= −(R + jLω )i
∂x
By combining these two equations, we can obtain two independent equations:
∂ ²v
= γ ²v
∂x ²
∂ ²i
= γ ²i
∂x ²
Where γ is the propagation constant:
γ =
∂i
= −(G + jCω )v
∂x
(R + jLω )(G + jCω )
γ = α + jβ
γ = ZY
With α is the attenuation constant (Nepers/m) and β is the phase constant
(rad/m).
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Transmission lines in MMIC technology
Propagation in transmission lines:
Equivalent circuit of a transmission line with a TEM (or quasi-TEM) propagation:
The previous equations in voltage and current have as solutions a
combinations of waves propagating on the transmission line in opposite
directions:
V ( x ) = Vi e ( −γx ) + Vr e (γx )
I ( x ) = I i e ( −γx ) − I r e (γx ) =
1
Vi e ( −γx ) − Vr e ( γx )
ZC
(
)
Where Vi and Vr, and Ii and Ir are linked to the incident and reflected waves,
respectively.
ZC is the characteristic impedance:
ZC =
Vi
V
R + jLω
Z
=− r =
=
Ii
Ir
G + jCω
Y
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Transmission lines in MMIC technology
Propagation in transmission lines:
Incident and reflected waves:
From the previous expressions of voltage and current through the transmission
lines:
V ( x ) = Vi e ( −γx ) + Vr e (γx )
I ( x ) = I i e ( −γx ) − I r e (γx ) =
1
Vi e ( −γx ) − Vr e ( γx )
ZC
(
)
We can define the expressions in voltage and current of the incident and
reflected waves:
Vi ( x ) = Vi e ( −γx )
Ii (x) =
Vi ( −γx )
e
ZC
Vr ( x ) = Vr e ( γx )
Incident wave
Ir (x) =
Vr ( −γx )
e
ZC
Reflected wave
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Transmission lines in MMIC technology
Propagation in transmission lines:
Reflection coefficient and impedance at each point of the transmission line:
Two systems of axis are considered:
Towards the load,
Towards the generator.
I
V
0
ZL
x
L
Towards the load
L
Towards the
generator
s
0
14
Transmission lines in MMIC technology
Propagation in transmission lines:
Reflection coefficient and impedance at each point of the transmission line:
Reflection coefficient at each point of the transmission line:
Γ( x ) =
Vr ( x ) Vr ( 2γx )
= e
Vi ( x ) Vi
Impedance at each point of the transmission line:
( − γx )
Z( x ) =
Vi e
+vre
V (x)
=
1
I(x )
Vi e ( −γx ) − Vr e ( γx )
ZC
(
Vr ( 2γx )
e
Vi
1 + Γ( x )
= ZC
= ZC
V
1 − Γ( x )
1 − r e ( 2γx )
Vi
1+
( γx )
)
Reduced impedance:
z( x ) =
Z ( x ) 1 + Γ( x )
=
ZC
1 − Γ( x )
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Transmission lines in MMIC technology
Propagation in transmission lines:
Reflection coefficient and impedance at each point of the transmission line:
The reflection coefficient can be expressed from the impedance:
Γ( x ) =
z( x ) − 1 Z ( x ) − Z C
=
z( x ) + 1 Z ( x ) + Z C
At the abscissa x = L, the impedance is equal to the load impedance ZL:
Z( x ) = ZL
Then:
Γ(L ) =
Z L − ZC
= ΓL
Z L + ZC
If ZL = ZC then the reflection coefficient is null (Γ
Γ = 0) and the transmission
line is matched.
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Transmission lines in MMIC technology
Propagation in transmission lines:
Reflection coefficient and impedance at each point of the transmission line:
In the plane of the load (s=0), the reflection coefficient is:
ΓL = Γ(s = 0) =
We have seen that at each point of the transmission line:
Γ(s ) =
Vr ( 0 ) Vr
e =
Vi
Vi
Vr (s ) Vr ( −2γs )
= e
Vi (s ) Vi
So:
Γ(s ) = ΓL e ( −2γs )
From these expressions, one can obtain the expression of the reduced
impedance brought back at the entry:
z(s ) =
zL + th(γs )
1 + zL th(γs )
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Transmission lines in MMIC technology
Propagation in transmission lines:
Reflection coefficient and impedance at each point of the transmission line:
In the case of a transmission line without losses, the propagation constant is
the following:
γ = jβ
From the trigonometric properties, and replacing the propagation constant in the
expression of the impedance, one can obtained, for the reduced impedance
brought back at the entry:
z(s ) =
z L + jtg ( βs )
1 + jz L tg ( βs )
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Transmission lines in MMIC technology
Propagation in transmission lines:
Stationary waves ratio:
Generally, the reflection coefficient is not null, they exist stationary waves along
the transmission lines, and we can define the stationary waves ratio:
SWR =
Vmax
Vmin
=
1 + ΓL
1 − ΓL
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Transmission lines in MMIC technology
Propagation in transmission lines:
Transmission lines terminated by particular load:
Transmission lines terminated by a matched load: ZL = ZC
ΓL = 0 ⇒ no reflection,
SWR = 1 ⇒ progressive wave,
z (s) = 1 or Z (s) = ZC ⇒ the impedance at each point is equal to the
characteristic impedance of the transmission line,
The wave propagates along the transmission line at the phase speed:
vϕ =
ω
(m / s )
β
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Transmission lines in MMIC technology
Propagation in transmission lines:
Transmission lines terminated by particular load:
Transmission lines terminated by a short-circuit: ZL = 0
ΓL = -1 ⇒ total reflection,
SWR = ∞,
The impedance at each point of the transmission line is equal to:
z(s ) = jtg ( β s )
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Transmission lines in MMIC technology
Propagation in transmission lines:
Transmission lines terminated by particular load:
Transmission lines terminated by a open-circuit: ZL = ∞
ΓL = 1 ⇒ total reflection,
SWR = ∞,
The impedance at each point of the transmission line is equal to:
z(s ) =
1
jtg ( βs )
22
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
23
Transmission lines in MMIC technology
Topologies of transmission lines:
Choice of the topology of a transmission line:
Frequency of the signal,
Technology employed,
Power to transmit,
…Application…
Lines with several conductors:
Twisted pair line,
Coaxial cable,
Planar lines.
Transmission with only one conductor:
Waveguides.
24
Transmission lines in MMIC technology
Topologies of transmission lines:
Twisted pair line
Coaxial cable
Waveguides
25
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
26
Transmission lines in MMIC technology
Planar transmission lines:
Advantages of planar transmission lines:
technology of integrated circuit,
planar conductors,
little dimensions,
propagation with a TEM or quasi-TEM mode.
Structure of planar transmission lines:
Generally, one conductor supports the signal,
Another is the ground plane,
A (dielectric) substrate is used as mechanical support.
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Transmission lines in MMIC technology
Planar transmission lines:
Different topologies of planar transmission lines:
Microstrip line
Stripline
Coplanar line
Slotline
Etc…
Coupled microstrip lines
Coupled striplines
Coplanar strips
The microstrip line and the coplanar waveguide
are the most used transmission lines.
28
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
29
Transmission lines in MMIC technology
Microstrip line:
Structure:
W: width of the conductor strip,
t: thickness of the conductor strip,
h: thickness of the substrate.
σ
t
h
W
ε, tgδ
δ
Characteristics of materials:
σ: conductivity of the conductor,
ε: permittivity of the substrate,
tgδ: dielectric loss tangent of the substrate.
H
EM field:
Quasi-TEM propagation.
E
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Transmission lines in MMIC technology
Microstrip line:
Characteristic impedance:
Zc =
Z vaccum
2π ε eff
2
 h
2h  

lnC
+ 1+   
 W
W  


Effective permittivity:
ε eff =
ε r + 1 ε r − 1
2
σ
+
2
h 
1 + 10 
W

W
− ab
0.7528
 

h


C = 6 + (2π − 6) exp −  30.666 
 

W



 4  u 2 
u +   
  u 3 
1 
1
52  

a = 1+
ln
+
ln1 + 
 
49  u 4 + 0.432  18.7   18.1 




 ε − 0 .9 

b = 0.564 r
 εr + 3 
t
Zvacuum =
h
µ0
ε0
0.053
≈ 120π
ε, tgδ
δ
[1] E. Hammerstad, O. Jensen, « Accurate models for microstrip computer-aided design »,
1980MTT-S International Microwave Symposium Digest, Vol. 80, N°1, May 1980, pp. 407-409.
u=
W
h
µ 0 = 4π .10 −7 (H / m )
ε 0 = 8.842 .10 −12 (F / m )
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Transmission lines in MMIC technology
Microstrip line:
Example: εr = 10 (alumina), W = 5 µm to 1.54 mm.
Characteristic impedance of a microstrip line vs. W
Effective permittivity of a microstrip line vs. W
200
7.4
h = 1.54 mm
180
h = 1.54 mm
7.2
h = 635 µm
h = 635 µm
160
7.0
For Zc ≈ 50 Ω ⇒ W ≈ h
6.8
Effective permittivity
Zc (Ohms)
140
120
6.6
W = h = 635 µm
100
or
80
6.2
W = h = 1.54 mm
60
6.0
40
20
0.0000
6.4
5.8
0.0002
0.0004
0.0006
0.0008
W (m)
0.0010
0.0012
0.0014
0.0016
5.6
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.0016
W (m)
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Transmission lines in MMIC technology
Microstrip line:
Total attenuation constant: αt
αt = αd + αc
Attenuation due to the dielectric losses of the substrate: αd
αd =
ε r ε reff − 1 π
tan δ e
ε reff ε r − 1 λ0
Attenuation due to the losses in the conductors: αc
R
α c = s .K r .K i
Z cW
Rs = (δσ )
−1
2
δ=
ωµ 0 µ c σ
0.7

 Z c  


K i = exp − 1.2

Z
vacuum

 

  ∆ 2 
K r = 1 + arctan1.4  
 δ  
π


2
Rs is the sheet resistance. δ is the skin depth. σ is the conductivity of the strip.
Ki is the current distribution factor.
Kr is a correction term due surface roughness. ∆ is the effective (rms) surface
roughness of the substrate.
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Transmission lines in MMIC technology
Microstrip line:
Attenuation constant of a microstrip line vs. frequency
0.045
Attenuation constant:
αt
0.040
Example:
εr = 10
tanδe = 0.001
σ = 45 MS/m
W = 50 µm
h = 635 µm
αd
αc
0.035
Attenuation constant (dB/mm)
0.030
0.025
0.020
0.015
0.010
0.005
⇒ The conductor losses are the preponderant
0.000
0
2lines.
4
6
factor of losses for microstrip
8
10
12
14
16
18
20
f (GHz)
34
Transmission lines in MMIC technology
Microstrip line:
Operating frequency:
In order to prevent higher-order transmission modes, the thickness of the
microstrip substrate should be limited to 10 % of the wavelength.
35
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
36
Transmission lines in MMIC technology
Coplanar waveguide:
Structure:
W: width of the conductor strip,
S: width of slots between central conductor and ground planes,
Wg: width of the ground planes,
σ
t: thickness of the conductor strip,
Wg
W
h: thickness of the substrate.
Characteristics of materials:
σ: conductivity of the conductor,
ε: permittivity of the substrate,
tgδ: dielectric loss tangent of the substrate.
h
S
t
ε, tgδ
δ
E
H
E
EM field:
Quasi-TEM propagation.
37
Transmission lines in MMIC technology
Coplanar waveguide:
Characteristic impedance:
Zc =
Z vacuum K ' (k 1 )
4 ε eff K (k 1 )
K: elliptic integral
k1 =
Effective permittivity:
ε eff = 1 +
ε r − 1 K (k 2 )K ' (k1 )
2
σ
Wg
W
ε, tgδ
δ
π
 1+ k ' 

ln 2

 1− k ' 
d = W + 2S
k ' = 1− k ²
For 0 ≤ k ≤ 0.707
S
t
h
W
d
K (k )
=
K ' (k )
K ' (k 2 )K (k1 )
 πW 
sinh

4d 

k2 =
 πd 
sinh 
 4h 
 1+ k 

ln 2

1
−
k
K (k )

= 
K ' (k )
π
For 0.707 ≤ k ≤ 1
[2] C.P. Wen, “Coplanar waveguide: a surface strip transmission line suitable for non-reciprocal
gyromagnetic device applications”, IEEE Trans. MTT, Vol. 17, p. 1087, 1969.
[3] K.C. Gupta, R. Garg, I.J. Bahl, “Microstrip lines and slotlines”, Artech House, Dedham, 1979.
38
Transmission lines in MMIC technology
Coplanar waveguide:
Example: h = 635 µm, εr = 10 (alumina), S = 5 to 100 µm, W = 5 µm to 200 µm.
Characteristic impedance of a coplanar waveguide vs. W
Effective permittivity of a colanar waveguide vs. W
140
5.50
5.49
S = 5 µm
120
S = 10 µm
5.48
S = 20 µm
100
Effective permittivity
Zc (Ohms)
S = 50 µm
S = 100 µm
80
5.47
5.46
5.45
60
50
5.44
40
5.43
20
5.42
0
20
40
60
80
100
120
140
160
180
W (µm)
Several values of (W,S) for Zc = 50 Ω
200
0
20
40
60
80
100
120
140
160
180
200
W (µm)
εeff ≈ (εr + 1)/2 = 5.5
39
Transmission lines in MMIC technology
Coplanar waveguide:
Propagation constant:
Different origins of losses:
Losses due to the dielectric substrate: αd
Losses due to the conductors: αc
Two additional phenomena at high frequencies:
Dispersion: εeff = f (f)
Losses by radiation: αr
⇒ Total attenuation αt = αd + αc + αr
40
Transmission lines in MMIC technology
Coplanar waveguide:
Phase constant of a coplanar waveguide vs. frequency
1000
Phase constant:
900
β=
λ0 =
2π
λg
=
2π
λ0
ε eff (rad / m )
c
1
3.10 8
=
=
(m)
f f ε 0 µ0
f
Example:
W = 20 µm
S = 10 µm
h = 635 µm
εr = 10
εeff = 5.5
Zc = 51 Ω
800
700
Phase constant (rad/m)
600
500
400
300
200
100
0
0
2
4
6
8
10
12
14
16
18
20
f (GHz)
41
Transmission lines in MMIC technology
Coplanar waveguide:
Dispersion of the effective permittivity:
β (f ) =
2π
λg
=
2π
λ0
ε eff (f )(rad / m )




ε r − ε eff 

ε eff (f ) =  ε eff +
−b 
 f  


1 + a

 fTE  

fTE =
c0
4h ε r − 1
2
W
ln(a ) = u. ln
S

+v

u = 0.54 − 0.64 x + 0.015x ²
v = 0.43 − 0.86 x + 0.540x ²
W 
x = ln 
 h 
b = 1.8
fTE is the cutoff frequency of the lowest TE mode.
εeff is the initial effective permittivity calculated from the previous expressions.
εr is the relative permittivity of the substrate.
h is the height of the substrate.
[4] G. Hasnain, A. Dienes, J.R.Whinnery, “Dispersion of picosecond pulses in coplanar transmission lines”, IEEE MTT, Vol.
MTT-34, N°6, June 1986, pp. 738-741.
42
Transmission lines in MMIC technology
Coplanar waveguide:
Example:
W = 70 µm
S = 40 µm
h = 635 µm
εr = 10
Dispersion of the effective permittivity:
Phase constant of a coplanar waveguide vs. frequency
Effective permittivity of a coplanar waveguide vs. frequency
60000
7.4
50000
β (f)
7.2
εeff (f)
β
7.0
εeff
Effective permittivity
Phase constant (rad/m)
6.8
40000
6.6
⇒ Very high frequency phenomena: f > 200 GHz
30000
6.4
6.2
20000
6.0
⇒ Not so high frequency phenomena if W ≈ h
5.8
10000
5.6
0
5.4
0
100
200
300
400
500
f (GHz)
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
f (GHz)
43
Transmission lines in MMIC technology
Coplanar waveguide:
Attenuation constant:
Losses due to the dielectric substrate: αd
αd =
q=
π εr
π εr
q tan δ e (Nepers / m ) = 8.686
q tan δ e (dB / m )
λ0 ε eff
λ0 ε eff
1 K (k 2 ) K ' (k1 )
2 K ' (k 2 ) K (k1 )
εr is the relative permittivity of the dielectric substrate.
tanδe is the dielectric loss tangent of the dielectric substrate.
εeff is the effective permittivity of the transmission line calculated from the
previous expressions.
The attenuation constant is expressed in Nepers/m. To be expressed in
dB, one can multiply the attenuation constant by 8.686 (1 Np = 8.686 dB).
[5] Rainee N. Simons“Coplanar waveguide circuits, components, and systems”, 2001 John Wiley and Sons, Inc.
44
Transmission lines in MMIC technology
Coplanar waveguide:
Attenuation constant:
Losses due to the conductors: αC
αc =
Rc + Rg
2ZC
2ZC
(dB / m )

 1 + k 1 
Rs
 4πW 

 − k 1 
π + ln
4W (1 − k 1 ²)K ²(k 1 ) 
 t 
 1 − k 1 
R linked to the ground planes:
Rg =
Rc + R g
R linked to the central conductor:
Rc =
(Nepers / m ) = 8.686

Rs
 4π (W + 2S )  1  1 + k 1 

 − ln
π + ln
4W (1 − k 1 ²)K ²(k 1 ) 
t

 k 1  1 − k1 
Sheet resistance:
Rs = (δσ )
−1
and skin effect:
δ=
2
ωµ 0 µ c σ
45
Transmission lines in MMIC technology
Coplanar waveguide:
Attenuation constant:
Losses due to the radiation: αr
  ε (f )  2 
 1 − eff
 
5
2
3/2


ε
π
 
r
  (W + 2S ) ε r
f3
α r =   .2. 
 3
ε eff (f )  c 0 K (k 0 ' )K (k 0 )
2



εr


k0 =
W
W + 2S
k 0 ' = 1− k 0 ²
c0 = 3.108 m/s (light velocity)
K is an elliptic function.
εeff (f) can be evaluated from the previous expressions of the dispersion.
εr is the relative permittivity of the substrate.
W and S are the width of the central strip and of the slot of the CPW,
respectively.
[6] M.Y. Frankel, S. Gupta, J.A. Valdmanis, G.A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar
transmission lines”, IEEE MTT, Vol. 39, N°6, June 1991, pp. 910-916
46
Transmission lines in MMIC technology
Coplanar waveguide:
Example:
W = 40 µm
S = 20 µm
h = 635 µm
εr = 10
tanδ
δe = 0.0001
t = 4 µm
σ = 45 MS/m
Total attenuation constant: αt
Attenuation constant of a coplanar waveguide vs. frequency
1.2
Attenuation constant of a coplanar waveguide vs. frequency
αt
0.8
αc
αd
0.0025
αr
0.0020
Attenuation constant (dB/mm)
Attenuation constant (dB/mm)
1.0
0.6
0.0015
0.0010
⇒ The losses due to the conductors are the main
factor of losses in planar transmission lines !
0.4
0.0005
0.2
0.0000
-60
-40
-20
0
20
40
60
80
100
120
140
f (GHz)
0.0
0
10
20
30
40
50
f (GHz)
60
70
80
90
100
47
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
48
Transmission lines in MMIC technology
Coaxial cable:
Features:
Widely used for high frequency measurements.
Ultra-large bandwidth.
The manufacturer guarantees that the coaxial cable correctly operates up to
a certain frequency.
Extremely low attenuation, depending on the dielectric material.
Various topologies:
choice depending on the frequency,
connections,
rigid or flexible cable…
49
Transmission lines in MMIC technology
Coaxial cable:
Structure:
TEM propagation:
[7] P-F. Combes, R. Crampagne, “Circuits passifs hyperfréquences, guides d’ondes métalliques”, Techniques de l’Ingénieur.
50
Transmission lines in MMIC technology
Coaxial cable:
a is the inner radius and b the outer radius.
Characteristic impedance:
Zc =
From the transmission line theory, one can obtain the total attenuation:
αt =
b
ln 
εr  a 
60
GZC
R
+
= αc + αd
2ZC
2
Attenuation due to the dielectric material:
α c (dB / m ) =
3.03.10 −9 ε r 1 + (b / a )
f
b
ln( b / a )
Attenuation due to the conductors:
α d (dB / m ) = 90.94.10 −9 ε r tan δ e .f
51
Transmission lines in MMIC technology
Coaxial cable:
RLCG elements:
L(H / m) =
µ0
ln(b / a)
2π
C (F / m ) =
2πε 0 ε r
ln( b / a )
R(Ω / m ) =
a+b
2πabδσ
G(S / m ) = ωC tan δ e
δ is the skin depth:
δ=
1
πfµ 0σ
52
Transmission lines in MMIC technology
Coaxial cable:
Cut-off frequency:
One can use a coaxial cable until the apparition of the first higher order
mode, the TE01 propagation mode.
The TE01 cut-off frequency is:
fc =
1
a+b
π
 µε
 2 
53
Transmission lines in MMIC technology
Propagation in transmission lines
Topologies of transmission lines
Planar transmission lines
Microstrip line
Coplanar waveguide
Coaxial cable
Waveguides
54
Transmission lines in MMIC technology
Waveguides:
Features:
Rectangular
waveguide
Propagation into hollow (empty) waveguide.
Rectangular or circular section.
TEmn or TMmn mode of propagation:
Circular
waveguide
⇒ Theory of transmission lines !
55
Transmission lines in MMIC technology
Waveguides:
Features:
Extremely high frequency.
High power applications.
Advantage of waveguides: not disturbed by the EM environment.
Even if a waveguide operates as a high-pass filter, the manufacturer
guarantees that the waveguide correctly operates for a certain bandwidth
because of disturbing higher frequency modes.
The bandwidth is directly linked to the size of the waveguide.
56
Transmission lines in MMIC technology
Waveguides:
Conventional rectangular waveguides:
Waveguide type
Frequency range (GHz)
Width a (mm)
Height b (mm)
WR650
1.12 – 1.70
165.10
82.55
WR510
1.45 – 2.20
129.54
64.77
WR430
1.70 – 2.60
109.22
54.61
WR340
2.20 – 3.30
86.36
43.18
…
…
…
…
WR18
40.0 – 60.0
4.775
2.388
WR14
50.0 – 75.0
3.759
1.880
WR12
60.0 – 90.0
3.0988
1.5494
WR10
75.0 – 110.0
2.5400
1.2700
57
Transmission lines in MMIC technology
Waveguides:
Propagation modes: TEmn or TMmn
m is the number of ½ wavelength variations of field in the “a” direction.
n is the number of ½ wavelength variations of field in the “b” direction.
Rectangular waveguide:
Circular waveguide:
b
a
TE10 mode
TE11 mode
58
Transmission lines in MMIC technology
Waveguides:
Rectangular waveguide: TE10 mode
The field expressions are normalized respect to the maximal value of
Ey which is obtained at the center of the waveguide at x = a/2:
E 0 = − jE
λc
λ
H0 = E0
ε
µ
The fields of the fundamental mode are:
E y = E 0 sin
H x = −H 0

2πz 
πx
exp − j

a
λg 


2πz 
λ
πx
sin exp − j

λg
a
λg 


2πz 
λ
πx
H z = j H0
cos exp − j

λc
a
λg 

λc = 2a
fc =
c0
λc
=
c0
2a
  λ ² 
λg = λ 1 − 

  4a ² 
−1/ 2
59
Transmission lines in MMIC technology
Waveguides:
Rectangular waveguide:
Attenuation in rectangular waveguide:
α t (Nepers / m ) =
1 Pw
1
= 3/2
2 Pd a
2πεcρ −
 fc 
 
f 
3/2
 a  f
+ 
 2b  fc
f 
1−  c 
f 
2
c=
1
εµ
Pw is the power lost in the walls of the waveguide.
Pd is the power transmitted by the dielectric material.
ε is the permittivity of the material filling the waveguide.
c is the speed in the material filling the waveguide.
fc is the cutoff frequency of the waveguide.
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Transmission lines in MMIC technology
Waveguides:
Circular waveguide: TE modes (fundamental = TE11 mode)
The fields of the TE modes are:
µ λc
E ρ = −H
ε λ
µ λc
Eθ = j H
ε λ
Hρ = − jH
Hθ = −H
n
J n (u )
u
J n ' (u )
λc
J ' (u )
λg n
λc J n (u )
n
λg
u
u = kc ρ
kc =
2π
λc
λc =
2πa
u ' mn
The value of H is fixed by measuring the
power transmitted by the waveguide.
a is the radius of the circular waveguide.
Jn is the Bessel’s function of first kind
and at the order n.
Roots and their derivatives of the Bessel’s functions
umn
u’mn
U0,1 = 2.405
U’1,1 = 1.841
U1,1 = 3.832
U’2,1 = 3.054
U2,1 = 5.136
U’0,1 = 3.831
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Transmission lines in MMIC technology
Waveguides:
Circular waveguide:
Attenuation in a circular waveguide for the TE11 mode:
(fundamental mode)
α t (Nepers / m ) =
5.5.10 −5
a3 / 2
f

 fc



−1 / 2
1 f

+
2.38  fc
f

 fc



3/2
2

 − 1

fc is the cutoff frequency of the circular waveguide.
a is the radius of the circular waveguide.
62
Transmission lines
in MMIC technology
Evangéline BENEVENT
Università Mediterranea di Reggio Calabria
DIMET
Thank you for your attention !
63