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Transient Analysis applying S-Parameters as
Operators
Heinrich Nuszkowski, Robert Wolf, Frank Ellinger, Senior Member, IEEE, Gerhard Fettweis, Fellow, IEEE
Abstract—We propose a calculation method for transient
analysis of linear time-invariant systems basing on the Mikusinski
operator. Since signal-flow charts and scattering parameters (Sparameters) are used to describe the analyzed networks, the
method is very vivid. The applied S-parameters form operators
describing transmission and reflection of power waves from one
node to an other of the network’s signal-flow chart. Although
the usage of operators exhibits some similarities to the Laplace
transform, the effect of the operators is interpreted in time
domain, and a transform of the signals is not required. Thus,
the presented calculation method is suitable to vividly analyze
dynamic processes on electrical lines.
Index Terms—Scattering parameters, S-parameters, transient
analysis, operators
I. I NTRODUCTION
T
HE characterization of radio frequency (RF) circuits
by scattering parameters (S-parameters) became state-ofthe-art since the publications of Penfield [1], Youla [2] and
Kurokawa [3]. In contrast to other n-port parameter sets like
Z- or Y-parameters, S-parameters can even be measured very
precisely for very high frequencies. Usually, S-parameters are
complex numbers describing the relationship of magnitude
and phase between incoming and outgoing power waves at
the ports in steady state. A transient analysis using power
waves is also possible [4]–[6]. In this case, the S-parameters
Sij are no complex number anymore but become impulse
responses sij (t) in time domain and transfer functions Sij (p)
dependent on the complex frequency p in frequency domain.
The outgoing power waves bi (t) can be derived from the
incoming ones aj (t) by convolution in time domain or by
multiplication of the transformed signals in frequency domain.
In this publication, we would like to propose a method for
transient analysis of dynamic processes using the S-parameters
as operators introduced by Mikusinski [7], [8]. Within the
book Lineare Netzwerke of Peter Vielhauer [9], the operators
are introduced to describe and calculate dynamic processes of
voltages and currents on transmission lines. Power waves and
S-parameters are not considered. This extension is what we
would like to present here.
Therefore, the basics of the usage of operators are presented
in section II. Section III introduces the description of electrical
networks by signal-flow charts and S-parameters forming
operators. The utilization of this method is demonstrated by
Manuscript received March 6, 2013. This work was partly funded by the
Federal Ministry of Education and Research (BMBF) in the excellence cluster
CoolSilicon, project CoolBroadcastRepeater.
H. Nuszkowski and G. Fettweis are with the Vodafone Chair Mobile
Communications Systems of the Technische Universität Dresden, Germany
R. Wolf and F. Ellinger are with the Chair for Circuit Design and Network
Theory of the Technische Universität Dresden, Germany
LTI-System
H
{x(t)}
{y(t)}
Fig. 1. Transmission model
basic examples in section IV followed by a summary in
section V.
II. M ATHEMATICAL BASICS
A. Transmission model
Fig. 1 shows the transmission model consisting of input
function {x(t)}, the output function {y(t)} and the transmission operator H. The curly brackets mean that the corresponding functions are causal and thereby equal to zero for negative
time t given by
(
0
t<0
{x(t)} =
.
(1)
x(t)
t≥0
The output function can be derived by {y(t)} = H{x(t)} ,
where H stands for the transmission operator in time domain.
In other words, the transmission operator specifies which
operations have to be performed on the input function in time
domain to obtain the output signal. Thus, it is not required to
transform the time functions into frequency domain or back.
The transmission operator of a linear time-invariant system
- can be derived out of the four basic operators: addition,
subtraction, multiplication and division,
- can be formed as a quotient of two causal and for t ≥ 0
continuous time functions, and
- can be converted into the complex transfer function by
substitution of the basic operators by the corresponding
transfer functions.
B. The four basic operators
The four basic operators are listed in Table I and are
explained in detail below like they are introduced in [9]. It
is important to consider that all quantities are normalized and
thereby dimensionless.
1) Weighting function: The product operator relating two
time functions {h(t)}{x(t)} has to be interpreted as convolution. The notation as product is eligible since the convolution
fulfills all axioms of the linear algebra for fields and rings
e.g. commutativity, associativity, distributivity. An important
special case reveals if a function is convoluted with the unity
step function {1} which leads to
Z t
Z t
{1}{x(t)} =
x(t − τ )dτ =
x(τ )dτ .
(2)
0
0
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4) Shifting operator: The shifting operator delays a function. It also fulfills all axioms of the linear algebra. Particularly,
the power laws can be applied to describe the overall delay of
two cascaded delay transmission blocks,
TABLE I
BASIC OPERATORS
name
operator
effect
weighting
function
scaling operator
differentiation
operator
{h(t)}
R
{h(t)}{x(t)} = { 0t h(t − τ )x(τ )}
shifting operator
k
s
e −sT
e −sT1 e −sT2 = e −s(T1 +T2 ) .
k{x(t)} = {kx(t)}
s{x(t)} = {x′ (t)} + x(+0)
e −sT {x(t)} =
(
The rectangular function {rect(t/T − 1)}, which is an important test function, can be easily noted by unity step functions
and shifting operators
0
t<T
x(t − T ) t ≥ T
The output function is the integral of the input function.
Thus, the unity step function acts as the integration operator.
Since the integration operator is inverse to the differentiation
operator, which is discussed further below, and due to the
parallels to the Laplace transform, the integration operator can
be noted as
1
{1} = s−1 = .
(3)
s
2) Scaling operator: The scaling operator describes a proportionality between output and input function
{y(t)} = k{x(t)} = {kx(t)}.
(4)
Considering the definition {h(t)} = k the dirac distribution
δ(t) is expendable which is not the case if a proportional
transmission behavior is given in time domain without the
usage of operators
(x ∗ h)(t) = (x ∗ kδ)(t) = kx(t).
(12)
(5)
But it has to be carefully distinguished between the scaling
operator k and the step function {k} = k{1}.
3) Differentiation operator: The differentiation operator is
defined by
s{x(t)} = {x′ (t)} + x(+0).
(6)
Meaning that applying the differentiation operator to a function
{x(t)} results in its derivative and its right-hand limit at zero.
Due to this definition, some important conclusions can be
drawn. It is
1
(7)
s{1} = s = 1.
s
This relation can be derived either by employing (6) on the
unity step function {1} or by simplifying the differentiation
and the integration operator. Employing (6) on the function
e−αt leads to
1
.
(8)
s{e−αt } = −{αe−αt } + 1 ⇒ {e−αt } =
s+α
In a more general way, it can also be derived that
n−1
1
t
−αt
.
(9)
=
e
(s + α)n
(n − 1)!
By (??), exponentially damped sinusoidal and cosinusoidal
signals can be noted as operators in s by
o
−αt
1 n (−α+j ωt)
e
cos (ωt) =
e
+ e(−α−j ωt)
2
s+α
1
1
1
=
(10)
=
+
2 s + α − jω s + α + jω
(s + α)2 + ω 2
and
ω
.
(11)
{e−αt sin(ωt)} =
(s + α)2 + ω 2
{rect(t/T − 1)} = {1}(1 − e −sT ).
(13)
III. N ETWORK DESCRIPTION BY SIGNAL - FLOW CHARTS
The signal-flow chart is a very powerful and vivid method
to describe differential equation systems for instance of a
network. Especially, the combination of S-parameters and
signal-flow charts for the process of power transmission e.g.
in RF circuits is very helpful, since discontinuities, which
lead to power reflection, and feedback loops, which may
result in instability, can be found. The advantage of the
proposed calculation method is, that it is relatively easy to
formulate the solution approach for a transient analysis. All,
what has to be done, is replacing the classical S-parameters
of the signal-flow chart used for the steady state analysis
by the corresponding operator relations. Thus, the reflexion,
transmission and propagation processes between the notes of
the signal-flow chart can easily be understood.
By the operators for resistance, inductance, and capacitance,
the operators for impedances Z(s) or reflexion coefficients
Γ(s) can be derived. For a passive one-port, which is fully
characterized by the impedance operator Zl (s), the signal-flow
chart is shown in Fig. 2a. The power waves {a(t)} and {b(t)}
are going into and coming out of the one-port, respectively.
The operator Γl (s) stands for the reflexion coefficient of the
load impedance Zl (s) which can be calculated by
Γ(s) =
Z(s) − Z0
Z(s) − Z0
(14)
where Z0 is the reference impedance of the port which can be
chosen. The passive one-port is finally characterized by
{b(t)} = Γ(s){a(t)}.
(15)
To get an active one-port the signal-flow chart has to be
modified like shown in Fig. 2b. The description changes to
{b(t)} = {bg (t)} + Γg (s){a(t)},
(16)
where the operator Γg (s) stands for the the reflexion coefficient corresponding to the generator’s internal impedance
Zg (s). The wave {bg (t)} is the injected power wave which
the generator would deliver if it is terminated by the reference
impedance Z0 (s), and which can be calculated by
√
Z0
.
(17)
{bg (t)} = {ug (t)}
Zg (s) + Z0
Fig. 2c illustrates the signal-flow chart of a two-port,
described by the operator S-parameter set Sij (s), i, j = 1, 2
{b1 (t)}
S (s) S12 (s) {a1 (t)}
= 11
.
(18)
{b2 (t)}
S21 (s) S22 (s) {a2 (t)}
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Zg (s)
{a(t)}
{b(t)}
a)
{a(t)}
Zl (s) ⇒
{b(t)}
S21 (s)
1
{a(t)}
{bg (t)}
Γl (s) {ug (t)}
{a(t)}
⇒ Γg (s)
{b(t)}
{b(t)}
b)
{a1 (t)}
{b1 (t)}
{a1 (t)}
{a2 (t)}
⇒ S11 (s)
{b2 (t)}
{b1 (t)}
S
c)
{b2 (t)}
S22 (s)
S12 (s)
{a2 (t)}
Fig. 2. circuit and signal-flow charts for a) a passive one-port, b) an active one-port, and c) a two-port
Since the proposed calculation method is particularly suitable to analyze dynamic processes on transmission lines, the
S-parameter matrix of a transmission lines is an important
special case for a two-port. The matrix for a transmission line
of the length l is
1
Γ(1 − e −2γl ) (1 − Γ2 )e −γl
, (19)
S=
1 − Γ2 e −2γl (1 − Γ2 )e −γl Γ(1 − e −2γl )
where γ denotes the propagation constant and Γ the reflexion
coefficient. Both are operators and given by
p
(20)
γ(s) = (R′ + sL′ )(G′ + sC ′ ), and by
Zc (s) − Z0
Γ(s) =
,
(21)
Zc (s) + Z0
where Z0 is the chosen reference impedance at the corresponding port and Zc (s) is the characteristic impedance given by
r
R′ + sL′
.
(22)
Zc (s) =
G′ + sC ′
The parameters R′ , L′ , G′ , and C ′ are the characteristic
constants of the transmission line which are also used in
the Telegrapher’s Equations. For R′ = 0 and G′ = 0, the
transmission line is lossless, the characteristic impedance Zc
becomes a scaling operator. Additionally, the expression (19)
results in a shifting operator
e −γ(s)l = e −s
√
L′ C ′ l
= e −sτ0 .
(23)
If the reference impedance is additionally chosen equal to the
characteristic impedance Z0 = Zc then is Γ = 0 the S-parameter matrix can be simplified to
0
e −sτ0
.
(24)
S = −sτ0
e
0
Usually, just transverse electromagnetic (TEM) waves are
regarded so that the voltages and currents at the ports can be
calculated. Those can be determined using the power waves
by
p
{u(t)} = ({a(t)} + {b(t)}) Z0 , and
(25)
p
{i(t)} = ({a(t)} − {b(t)}) / Z0 .
(26)
IV. E XAMPLES
For the next examples, the transmission line is considered
lossless which reduces the effect of the transmission line to
an ideal delay without any distortion. By this constraint, the
handling of S-parameter operators for transient analysis can be
shown on simple examples. Lossy lines can also be examined
but the examples get more complicated and more computationally intensive. For more details we would like to refer
on the chapter Dynamische Vorgänge auf verlustbehafteten
Zg (s)
{bg (t)}
Z c , τ0
{ug (t)}
Zl (s)
{a1 (t)} e −sτ0 {b2 (t)}
Γl (s)
Γg (s)
{b1 (t)} e −sτ0 {a2 (t)}
Fig. 3. both-sided mismatched, lossless line; circuit and SFC
Leitungen in [9]. There is shown that the operator e −γ(s) ,
which can delay, damp, and distort a signal, of a lossy line
can be rearranged to
e −γ(s) = e −α0 l e −sτ0 (1 + {v(t)}).
(27)
−α0 l
The factor e
forms a scaling operator and describes a
linear damping, the factor e −sτ0 is a shifting operator and
works as the delay of the line, and {v(t)} is a weighting
function which describes the linear distortion of the lossy
transmission line.
Fig. 3 depicts the circuit and the signal-flow chart for a
lossless transmission line being terminated by impedances
Zg , Zl 6= Zc . The reference impedance is chosen equal to
the characteristic impedance Z0 = Zc . The evaluation of the
signal-flow chart e.g. for the power wave at the output reveals
{a2 (t)} = {bg (t)}Γl (s)e −sτ0 /D,
{b2 (t)} = {bg (t)}e
−sτ0
and
/D
(28)
(29)
involving the signal-flow chart determinant D given by
D = 1 − Γg (s)Γl (s)e −2sτ0 .
(30)
Applying (17) and (25), the voltage at the output of the line
can be determined to be
{u2 (t)}
= {ug (t)}
=
Zc (1 + Γl (s))e −sτ0
1
Zg (s) + Zc
1 − Γg (s)Γl (s)e −2sτ0
{ug (t)}
(1 − Γg (s))(1 + Γl (s))e −sτ0
2
∞
X
(Γg (s)Γl (s))n e −2nsτ0 .
(31)
n=0
For further investigations, the voltage transmission operator
Hu (s) = {hu (t)} is used so that the equation results in
{u2 (t)} = Hu (s){ug (t)} = {hu (t)}{ug (t)}.
(32)
Similarly, the current transmission operator Hi (s) = {hi (t)}
can be defined by using (26).
A. Both-sided matching
In this case, the voltage transmission operator Hu (s) can be
reduced to Hu (s) = 12 e −sτ0 , since Γg = Γl = 0. Meaning
that the output voltage is half of the generator voltage and
delayed by the transmission delay τ0 .
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{u2 (t)}/U0
{u2 (t)}/U0
0.3
0.4
0.2
0
0
1
2
3
4
0.2
0.1
0
−0.1
5
t/τ0
0
1
2
3
4
t/τ0
5
6
7
8
Fig. 4. First example with Zg = Zl = 5Zc
Fig. 5. Second example Zg = 3Zc and Yl = 1/Zl = 1/Zc + sC
B. Termination by arbitrary resistances
In this case, the reflexion coefficients are arbitrary scaling
operators and (31) leads to
It can be seen that after a initial transition of approximately
t/τ0 = 6 the output voltage is almost equal to its final value
{u2 (t)}/U0 = 0.25.
Hu (s) =
∞
X
cn e −(2n+1)sτ0 ,
with
(33)
n=0
cn = 0.5(1 − Γg )(1 + Γl )(Γg Γl )n .
(34)
The input signal appears for the first time after the transmission
delay τ0 and afterwards periodically by 2τ0 , whereas the
echoes are exponentially damped expressed by
{u2 (t)} = c0 {ug (t − τ0 )} + c1 {ug (t − 3τ0 )} + · · · . (35)
Fig. 4 shows the first example with Zg = Zl = 5Zc . The
generator signal is a step function U0 {1}. After an initial
transient of approximately t/τ0 = 5 the normalized final value
of 0.5 is almost reached.
C. Termination by arbitrary impedances
If the lossless transmission line is terminated by an arbitrary
network of concentrated, linear, time invariant components
the corresponding impedance operator and the corresponding
reflexion coefficient is a rational function in s. The voltage
transmission operator is
Hu (s) =
∞
X
cn (s)e −(2n+1)sτ0 ,
(36)
n=0
where the factors cn (s)
cn (s) = 0.5(1 − Γg (s))(1 + Γl (s))(Γg (s)Γl (s))n
V. C ONCLUSION
The transmission operator H(s) introduced by Mikusinski
is an analogon to the complex transfer function in frequency
domain of the Laplace transform. The different approaches
complement each other and allow a vivid interpretation of
transmission processes in time and in frequency domain.
The pole–zero plot resulting from the Laplace transform is
a valuable and indispensable tool for network analysis and
synthesis. For considerations in time domain, the usage of
operators exhibits some advantages [9] which are for instance:
- The operator emerges directly within the mathematical of
the described system.
- A detour via a frequency domain is not required. Signals
and operations remain in time domain.
- Convergence investigations and resulting constrains can
be omitted.
- The problems revealing from the dirac distribution are
solved very easily and exactly.
Since the focus is on the transient analysis in time domain,
the presented method applying signal-flow charts and Sparameter operators is very convenient. The method proved its
applicability and its vividness for teaching dynamic processes
on transmission lines within the course Lineare Netzwerke and
the Technische Universität Dresden.
(37)
are also rational functions in s. An interpretation is possible by
partial fraction decomposition which requires that the degree
of the numerator is less than the degree of the denominator.
If this is not the case a polynomial long division has to be
executed to ensure this criteria. Thereby, (35) results in
{u2 (t)} = c0 (s){ug (t−τ0 )}+c1 (s){ug (t−3τ0 )}+· · · . (38)
After ascertainment of the operators cn (s), their effect on the
input signal can systematically be analyzed and visualized.
For the second example the generator’s internal impedance is
Zg = 3Zc , and the load admittance is Yl = 1/Zl = 1/Zc +sC.
For a stimulus by a step function, (37) and (38) lead to
1
1
{u2 (t)}
= {1 − e −αt }e −sτ0 − {αte −αt }e −3sτ0 −
U0
4
8
1
+ {(2 − αt)αte −αt }e −5sτ0 · · · , with (39)
32
2
1
=
.
(40)
α=
τ0
CZc
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