Download Validity of non-physical RLGC models for simulating lossy

Validity of Non-Physical RLGC models for
Simulating Lossy Transmission Lines
Woopoung Kim* and Madhavan Swaminathan
School of ECE, Georgia Institute of Technology, Atlanta, GA 30332, USA
ABSTRACT ; A non-physical RLGC model for transmission lines is developed through the propagating wave behavior
analysis. The non-physical RLGC model is different from the conventional RLGC model with physical RLGC values in
the sense that it uses non-physical parameters. The parameters of the non-physical RLGC model are made up of
frequency-dependent characteristic impedances and propagation constants. A lossy transmission line is simulated using
the non-physical RLGC model. This has been compared with the transmission line equation (quasi-TEM assumption) and
TDR measurements, which show good agreement.
I. INTRODUCTION
The complexity and operating frequency of the state-of-the-art systems and circuits require the accurate simdation of
transmission limes. For the simulation of the transmission lines, several kinds of models have been developed in SPICE,
such as T-models, U-models, and W-models[l]. T-models yield the exact response of lossless transmission lines by using
delay equations. For lossy transmission lines, U-models and W-models were developed based on RLGC models with
physical parameters, which we categorize as ‘physical RLGC models’ in this paper. The U-models use frequency.
independent RLGC values and the W-models can incorporate the skin effect of the conductors, frequency-dependent
dielectric loss, and the causality condition accounting for wave propagation on transmission lines. While the physical
RLGC models assume that the RLGC values must be physical, a new model introduced in this paper was derived from
the propagating waves and consequently satisfies the wave behavior in the transmission line. This model is quite similar
to the conventional RLGC model except that the new model does not use physical values. Hence these RLGC models are
called ‘Non-Physical RLGC models’. The advantage of using non-physical RLGC models is that it IS easier to derive
these parameters from measurements without any consideration on the transmission line structure.
11. NON-PHYSICAL RLGC MODEL
The starting point to derive the Non-Physical RLGC model is the transmission line equation with quasi-TEM assumption
as shown in Fig. 1. Since the input impedance in Fig .I is equivalent to the result from plane wave analysis, it represents
the behavior of propagating waves on the transmission line[2]. It is important to note that the input impedance is the
result in the frequency domain. The input impedance equation in Fig. 1 is the starting point for modeling the transmission
lines. Good transmission line models should yield the same result as the input impedance for a lossy transmission line.
Since the input impedance equation is in the steady state, the actual behavior of propagating waves can be translated into
the time domain as in Fig. 2. Since the transient representation in Fig. 2 has been derived from the steady state
representation in Fig.1, Fig. 2 is an accurate representation of the transmission line in the time domain.
Since the RLGC models represent the steady-state behavior of the transmission lines, we derived the steady-state
behavior from the transient signals in Fig.2 to construct the RLGC model. Summing all the signals in the positive z
direction at a point z in Fig. 2 yields the steady-state voltage(V7 propagating in the +z direction . Similarly, summing all
the signals in the negative z direction at z yields the steady-state voltage(V) propagating in the -z direction. The v‘ and
V can be expressed as :
V + ( z )= V;e-P
(1)
v-(z) = V;e” = V,’r’e-’”eP
(2)
where r’ is the reflection coefficient at the load, I is the length of the transmission line, y is the complex propagation
constant, and z is the position on the transmission line.
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Fig. 1. Transmission line equation of the input impedance
under quasi-TEM assumption [2].
Fig. 2. Transient analysis of the transmission line in Fig.1
and its causality conditions.
From (1) and(2), the voltage and the current at z in the steady state are expressed as:
v ( z ) = v + ( r ) t ~ ~ ( r ) = t&+re-’”ep
~,’e~~
(3)
where 20 is the characteristic impedance of the transmission line. Equations (3) and (4) show the steady-state wave
behavior of the propagating signals on the transmission line.
For a small length Az of the transmission line, let the voltage and current at z be V1 and I1 respectively,
and the voltage and current at ztAz be V2 and I2 respectively. Then from equations (3) and (4), equations (9,(6), (7),
and (8) can be derived.
Using the voltage and current values derived in (5)-(8), the
equivalent circuit for a length Az can be constructed as shown in
Fig. 3. Based on circuit theory, the voltages and currents in Fig.
3(a) can be expressed as :
v,= v, -x*II
I, =I, - Y * v,= (1 t x * Y )* I , - Y *v,
Fig. 3. RLGC model of the transmission line
(9)
Replacing the currents and voltages with the equations (9,(6), (7), and (8)
X
v , = ~ , - x * =I ,V,’(I--)ex . *tvof(lt-)r’e~’”e*
-
y -y‘ e-,At e-?J
zo
+ &+be+e*c
ZO
(IO)
If yAz is small,
e-* zI-yAz,e* =It@
ify~zzO
(11)
The approximation in (1 1) results in:
x = Zo*
(12)
Similarly, Y has the relationship
From (12) and (13), the parameters of the Non-Physical RLGC model in Fig. 3 (b) are
R eq = Re(X)
Geq = Re(Y)
Le9 = h ( X ) I w
(14)
Ceq = Im(Y) I w
Although these parameters are not physical values, the Non-Physical RLGC parameters satisfy the wave behavior on the
transmission line.
111. COMPAUSON WITH PHYSICAL RLGC MODELS
where Zois the characteristic impedance, V, is the phase velocity, and LphFa..~ and CphFlul are the physical inductance and
capacitance of the transmission lie respectively. These assumptions come from the physical considerahon of the
transmission line. Then, the parameters in Eq.(14) become
Req=Z,a&
Gq=%
Leq = LPb,,&
Ces = C-&
zo
(15)
These parameters have similar values to those of physical U G C models. For a lossless transmission line, that is, a+,
these parameters are exactly the same as those of the physical RLGC models. However, as the loss(a) increases, the
difference between physical RLCC models and Non-Physical RLGC models become wider. Interestingly, Re9 and Geq
are not independent in the Non-Physical RLGC models unlike physical RLGC models. It is important to note that the
characteristic impedance can be real in the non-physical RLGC model unlike the physical RLGC model where the
characteristic impedance is always complex. This is illustrated in Eq.(16).
The non-physical RLGC model was derived from arbifmy characteristic impedance and propagation constant, unlike the
physical RLGC model of which the characteristic iTedance and propagation constant are defined by the physical RLGC
parameters. Hence, the non-physical RLGC model can simulate any characteristic impedance and propagation constant.
From the measurement of characteristic impedances and propagation constants, transmission lines can be easily simulated
using the non-physical RLGC models.
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IV. Simulation with Non-physical RLGC models
To simulate the transmission line in Fig. 1 in the time domain using the non-physical RLGC model, the following
.parameters were used:
Source impedance = 50 a,
Rise time = 36 ps, Amplitude of the step pulse = 250 mV,
ZO=25.7natDC (TDRmeasurement),a= 1.41e-9*f+ 1.63 Np/m, p=2*pi*f/c*sqrt(3.1611)
length = 5 cm, termination : short
/m
where f is the frequency, c is the speed of light in air, and 3.161 1 is the effective dielectric constant. Both a and p were
extracted from the measurement of a transmission line on FR4. Fig. 4 shows the input impedances derived from the
equation in Fig. 1 and the non-physical RLGC model. Az is recommended less than O.Ol*(wavelengthof the maximum
frequency) in the non-physical RLGC model simulation, where the error at the maximum frequency is around 5%. They
match well in the frequency domain as shown in Fig. 4. Finally, the TDR(Time Domain Reflectomeby) signal of the
lossy transmission line was simulated using the transient analysis in Fig. 2 and plotted in Fig. 5 with a measured signal
by TDR. The non-physical RLGC with the causality conditions in Fig. 5 was derived by applying the causality condition
in Fig. 2. The non-physical RLGC without the causality conditions was simulated by the RLGC equivalent circuit. The
transient analysis of physical RLGC model was derived by W-element simulation in Hspice based on the physical crosssection of the transmission line. The non-physical RLGC model with the causality conditions shows good agreement with
the measurement. Also, the non-physical RLGC model without the causality conditions yields an acceptable result. The
non-physical RLGC models show better correlation with measurements since the non-physical RLGC model satisfies the
actual propagating wave behavior on the transmission line.
0
-
;
4.05
2
9.1
9.15
9.2
9.25
4.3
Fig. 4. Real (a) and imaginary (b) of the input
impedances of the lossy transmission line
Fig. 5. TDR measurement and simulations using the
transient analysis in Fig. 2 of the lossy line
V. CONCLUSION
A Non-Physical RLGC model for transmission lines was developed through the wave behavior analysis on the
transmission lines. The Non-Physical RLGC model satisfies the behavior of the propagating waves on transmission lines
and shows good agreement in the frequency domain. In the time domain, the non-physical RLGC models show good
agreement with measurements with the causality condition enforced as in Fig. 2.
REFERENCES
[I] Stur-HspiceMunuulRelease 2001, Avant!, pp. 17-1 I, 2001.
[Z] M. F. Iskander, ELECTROMAGNETICFIELDS AND WAVES, New Jersey: Prentice-Hall. Inc, pp. 385-389,1992
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