Propagation constant and characteristic impedance: The propagation
constant and characteristic impedance of a transmission line are two
key physical parameters. They can be directly determined from its
measured S-parameters [1] as follows:
(Z C ) =
2
2
ZREF
(1 +
(1 −
2
S11 ) −S21
2
2
S11 ) −S21
0.30
3.5
unit, um
3.0
25
20
signal
2.5
50
40
0.25
solder
resist
0.20
FR4
ground
12
0.15
microstrip
2.0
0.10
1.5
0.05
0
10
20
30
frequency, GHz
40
Frequency-variant RLCG( f ) characteristics: The frequency-variant
capacitance and conductance can be determined by
C(f ) = Kg 1′r (f )10
(7)
G(f ) = G0 + 2pfKg 1′′r (f )10 = G0 + vC(f ) tan d(f )
(8)
The geometry-related constant (Kg) including the effective conductor
resistivity and metal roughness effect can be determined by combining
(1) and (2). The extracted parameters are shown in Fig. 2.
6
25
5
R
L
20
4
15
3
10
2
5
0
(1)
10
2.5
(2)
Frequency-variant complex permittivity: The frequency-variant
complex permittivity of a dielectric material can be represented with
the Debye model [2]
(3)
1(v) = 10 1′r (v) − j1′′r (v)
n = 10 11 +
D1′i / 1 + jvti − j s/v10
(4)
0.00
50
Fig. 1 Real and imaginary part of extracted permittivity
0
2
0.35
imaginary part of permittivity, e r
0.40
er
er
4.0
1
where γ, ZC and ZREF are the propagation constant, characteristic
impedance and measurement reference impedance, respectively.
Although the propagation constant can be stably extracted in the
broad frequency band, the characteristic impedance is sensitive to
resonance effects. To stably determine the transmission line parameters
in the broad frequency band, the frequency-variant complex permittivity
has to be determined.
i=1
4.5
C, pF/cm
e− g ℓ
⎡
⎤−1
2
2
2
2
2
2
S
−
S
+
1
−
(
2S
)
1
−
S
+
S
11
11
21
11
21
⎦
=⎣
+
2S21
(2S21 )2
where N is the number of total data points. β m and β e are the measured
and evaluated phase constants, respectively.
R, Ω/cm
Introduction: In very high-speed digital systems and microwave circuits, planar transmission lines are considered essential circuit components. In such high-frequency systems, signal loss due to metal
roughness effects, frequency-variant skin effects, dielectric polarisation
and electromagnetic radiation of the planar transmission lines have a
very significant effect on system performance. The field-solver-based
numerical analysis using nominal cross-sectional dimensions and
material parameters (i.e. resistivity, manufacturer-provided dielectric
constant and dissipation factor) may not be accurate. In practice, these
variations from nominal values may become more serious as the
metal pitch shrinks and the metal thickness becomes thinner.
Therefore, experimental characterisations of the transmission lines are
crucial for accurate circuit design.
In this Letter, for the experimental characterisations, we measured
S-parameters for the designed test patterns over a broad frequency
band (i.e. from 40 MHz to 50 GHz) by using a vector network analyser.
We determined the complex permittivity (dielectric constant and loss
tangent) of the inter-metal dielectric material, which is frequency-variant
rather than constant. Then, we determined the parameters of the
frequency-variant transmission line circuit model (i.e. the propagation
constant and characteristic impedance) in the measured frequency
band. As a result, the thin and fine transmission line loss mechanism
can be accurately characterised.
We propose that there are three critical frequencies: a frequency ( fskin)
where the skin effect becomes significant, a frequency ( fdie) where the
dielectric loss becomes significant, and a frequency ( frad) where the electromagnetic radiation loss becomes significant. Note that although the conductive loss follows the skin-effect model [1] below the critical
frequency frad, this is not really the case above the frequency frad.
Furthermore, it is noteworthy that the skin effect may not vary by the
square root of the frequency. Thus, a new rigorous skin-effect model is
necessary.
the complex permittivity can be determined as shown in Fig. 1 by using
a fitness function (Δ) [4, 5]
N
1
bm − be / maxbm 2
D= (6)
N i=1
20
G
30
40
0
50, GHz
6
C
5
2.0
4
1.5
3
1.0
2
0.5
1
0
0
G, mS/cm
New frequency-variant losses of planar thin film transmission lines are
experimentally investigated in a broad frequency range. The frequency-variant transmission line parameters are accurately determined
in the measured frequency band (i.e. from 40 MHz to 50 GHz). Then,
it is shown that there are three critical frequencies that characterise the
loss mechanism of thin film transmission lines. The conventional skineffect model is not accurate in thin and fine transmission lines.
real part of permittivity, e r
D. Kim, H. Kim and Y. Eo
where ε0 is the free space permittivity and εr is the relative permittivity.
The Debye model has several parameters: the real part of the relative
permittivity at high frequencies (ε∞), the variation in the real part of
relative permittivity (Δε′i), the relaxation time constant (τi) and the
dielectric conductivity (σ). Since the phase constant (β) of a dielectric
material can be represented as follows [3]:
1/2
2
m 10 mr 1′r (v)
1 + 1′′r (v)/1′r (v) + 1
b=v 0
(5)
2
L, nH/cm
Experimental characterisations of thin film
transmission line losses
Fig. 2 Frequency-variant RLCG
Therefore, the frequency-variant resistance and inductance can be
determined as
R(f ) = Re g2 / G(f ) + jvC(f )
(9)
L(f ) = Im g2 / G(f ) + jvC(f ) /v
(10)
The frequency-variant conductive loss can be modelled as
−1
R(f ) = l swd(f ) ,
∀t ≥ d(f )
(11)
where l, w, t and δ( f ) are the length, width, thickness and skin depth [3],
respectively. The classical skin-effect model that follows the square root
dependency with frequency overestimates the conductive resistance as
ELECTRONICS LETTERS 15th August 2013 Vol. 49 No. 17
shown in Fig. 3, which is given by
d(f )square
= 1/ pf ms
(12)
20
f ) uare
ffect, d( sq
R, W/cm
15
skin-e
10
fskin
ffect, d(
5
skin-e
frad
f ) eo
0
0
a, Np/cm
0.3
10
fdie
20
30
frad
a=
0.2
0.1
+
a C+a D
50, GHz
40
a rad
arad
aD
Conclusion: Thin film transmission lines are characterised with
S-parameter measurements from 40 MHz to 50 GHz. The frequencyvariant transmission line parameters are stably determined in the
measured frequency band. Then, it is shown that there are three critical
frequencies (i.e. fskin, fdie and frad) that characterise the transmission line
loss mechanism. Only the skin-effect model underestimates the total loss
in the frequency exceeding frad. As the frequency increases, the electromagnetic radiation loss can no longer be neglected. Furthermore, it is
shown that the skin effect may not follow the conventional square
root dependency with frequency.
aC
Acknowledgment: This work was supported by the Development of
Technologies
for
Next-generation
Electromagnetic
Wave
Measurement Standards, project of the Korea Research Institute of
Standards and Science under grant no. 12011016.
© The Institution of Engineering and Technology 2013
26 April 2013
doi: 10.1049/el.2013.1444
One or more of the Figures in this Letter are available in colour online.
0
Fig. 3 Transmission line losses with frequency-variations
Assuming a finite metal thickness, Eo [1] derived a more rigorous
modified skin-effect model, which is more accurate than (12)
(13)
d(f )eo = d(f )square 1 − exp −t/d(f )square (1 + t/w)
The critical frequency fskin where the skin effect becomes significant can
be defined as a frequency corresponding to d(f )eo ≃ t. The skin-effect
model of (13) is much more accurate than that of (12) as shown in
Fig. 3. In contrast, the critical frequency fdie at which the dielectric
loss becomes significant can be defined as a frequency where the loss
due to the dielectric (αD) is similar to the conductive loss (αC). Note
that the fdie ≃ 7 GHz. The skin-effect model using (13) is accurate up
to frad, but it underestimates the resistance above the critical frequency
frad. Above the critical frequency frad, it is clear that the radiation
effect may become an additional loss mechanism. The frad can be
defined as a frequency where the ηrad is not negligibly small, which is
defined as
−1
hrad ; R(f )measured − l swd(f )eo
(14)
D. Kim, H. Kim and Y. Eo (Department of Electronics and
Communication Engineering, Hanyang University, Ansan, Gyeonggido 426-791, Republic of Korea)
E-mail: eo@giga.hanyang.ac.kr
References
1 Eo, Y., and Eisenstadt, W.R.: ‘High-speed VLSI interconnect modeling
based on S-parameter measurements’, IEEE Trans. Compon. Hybrid.
Manuf. Technol., 1993, 16, (5), pp. 555–562
2 Djordjević, A.R., Biljić, R.M., Likar-Smiljanić, V.D., and Sakar, T.K.:
‘Wideband frequency-domain characterization of FR-4 and time-domain
causality’, IEEE Trans. Electromagn. Compat., 2001, 43, (4),
pp. 662–667
3 Hayt, W.H., and Buck, J.A.: ‘Engineering electromagnetics’
(McGraw-Hill, New York, 2001, 6th edn)
4 Davis, L.: ‘Handbook of genetic algorithms’ (Van Norstrand Reinhold,
New York, 1991)
5 Zhang, J., Koledintseva, M.Y., Antonini, G., Drewniak, J.L., Orlandi, A.,
and Rozanov, K.N.: ‘Planar transmission line method for characterization
of printed circuit board dielectrics’, Prog. Electromagn. Res., 102, 2010,
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ELECTRONICS LETTERS 15th August 2013 Vol. 49 No. 17