Data Transmission Lines
and Their Characteristics
Overview
This application note discuss the general characteristics of
transmission lines and their derivations.Here,using a trans-
mission line model,the important parameters of character-
istics impedance and propagation delay are developed in
terms of their physical and electrical parameters.This appli-
cation note is a revid reprint of ction two of the Fairchild
Line Driver and Receiver Handbook.This application note,
the first of a three part ries(e AN-807and AN-808),
covers the following topics:
•Transmission Line Model
•Input Impedance of a Transmission Line
•Pha Shift and Propagation Velocity for the Transmis-
sion Line
•Summary—Characteristics Impedance and Propagation
Delay
Introduction
A data transmission line is compod of two or more con-
ductors transmitting electrical signals from one location to
another.A parallel transmission line is shown in Figure1.To
show how the signals(voltages and currents)on the line
relate to as yet undefined parameters,a transmission line
model is needed.
Transmission Line Model
Becau the wires A and B could not be ideal conductors,
they therefore must have some finite resistance.This
resistance/conductivity is determined by length and
cross-ctional area.Any line model,then,should posss
some ries resistance reprenting the finite conductivity of
the wires.It is convenient to establish this resistance as a
per-unit-length parameter.
Similarly,the insulating medium parating the two conduc-
tors could not be a perfect insulator becau some small
leakage current is always prent.The currents and di-
electric loss can be reprented as a shunt conductance
per unit length of line.To facilitate development of later
equations,conductance is the chon term instead of resis-
tance.
If the voltage between conductors A and B is not variable
with time,any voltage prent indicates a static electric field
between the conductors.From electrostatic theory it is
known that the voltage V produced by a static electric field E
is given by
(1)
This static electric field between the wires can only exist if
there are free charges of equal and opposite polarity on both
买东西英语wires as described by Coulomb’s law.
(2)
where E is the electric field in volts per meter,q is the charge
in Coulombs,e is the dielectric constant,and r is the distance
in meters.The free charges,accompanied by a voltage,
reprent a capacitance(C=q/V);so the line model must
include a shunt capacitive component.Since total capaci-
tance is dependent upon line length,it should be expresd
in a capacitance per-unit-length value.
It is known that a current flow in the conductors induces a
magnetic field or flux.This is determined by either Ampere’s
law
(3)
or the Biot-Savart law
(4)
where r=radius vector(meters)
,=length vector(meters)
I=current(amps)
B=magnetic flux density(Webers per meter)
H=magnetic field(amps per meter)
µ=permeability
National Semiconductor
Application Note806
Kenneth M.True
April1992trus
Data
Transmission
Lines
and
Their
Characteristics
AN-806©2002National Semiconductor Corporation
Transmission Line Model
(Continued)
If the magnetic flux (φ)linking the two wires is variable with time,then according to Faraday’s law
(5)
A small line ction can exhibit a voltage drop —in addition to a resistive drop —due to the changing magnetic flux (φ)within the ction loop.This voltage drop is the result of an inductance given as
(6)
Therefore,the line model should include a ries inductance per-unit-length term.In summary,it is determined that the model of a transmission line ction can be reprented by two ries terms of resistance and inductance and two shunt terms of capacitance and conductance.
From a circuit analysis point of view,the terms can be considered in any order,since an equivalent cir
cuit is being generated.Figure 2shows three possible arrangements of circuit elements.
For consistency,the circuit shown in Figure 2will be ud throughout the remainder of this application note.Figure 3shows how a transmission line model is constructed by ries connecting the short ctions into a ladder network.Before examining the pertinent properties of the model,some commen
ts are necessary on applicability and limita-tions.A real transmission line does not consist of an infinite number of small lumped ctions —rather,it is a distributed
network.For the lumped model to accurately reprent the transmission line (e Figure 3),the ction length must be quite small in comparison with the shortest wavelengths (highest frequencies)to be ud in analysis of the model.Within the limits,as differentials are taken,the ction length will approach zero and the model should exhibit the same (or at least very similar)characteristics as the actual distributed parameter transmission line.The model in Figure
01133601
I =CURRENT FLOW
,=LINE LENGTH
E =ELECTRIC FIELD H =MAGNETIC FIELD
FIGURE 1.Infinite Length Parallel Wire Transmission Line
01133602a 01133603b
01133604
c
FIGURE 2.Circuit Elements
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Transmission Line Model
(Continued)
3does not include cond order terms such as the increa in resistance due to skin effect or loss terms resulting from non-linear dielectrics.The terms and effects are discusd in the references rather than in this application note,since they tend to obscure the basic principles under consider-ation.For the prent,assume that the signals applied to the line have their minimum wavelengths a great deal longer than the ction length of the model and ignore the cond order terms.
Input Impedance of a Transmission Line
headmasterThe purpo of this ction is to determine the input imped-ance of a transmission ,what amount of input current
I IN is needed to produce a given voltage V IN across the line as a function of the LRCG parameters in the transmission line,(e Figure 4).
Combining the ries terms lR and lL together simplifies calculation of the ries impedance (Z s )as follows
Z s =,(R +j ωL)
(7)Likewi,combining lC and lG produces a parallel imped-ance Z p reprented by
(8)
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Since it is assumed that the line model in Figure 5is infinite in length,the impedance looking into any cross ction should be equal,that is Z 1=Z 2=Z 3,etc.So Figure 5can be simplified to the network in Figure 5where Z 0is the charac-teristic impedance of the line and Z in must equal this imped-ance (Z in =Z 0).From Figure 5,
(9)
01133605
FIGURE 3.A Transmission Line Model Compod of Short,Series Connected Sections
01133606
FIGURE 4.Series Connected Sections to Approximate a Distributed Transmission Line
01133607
a
01133608
b
FIGURE 5.Cascaded Network to Model Transmission Line
AN-806
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Input Impedance of a
Transmission Line(Continued)
Multiplying through both sides by(Z0+Z p)and collecting
terms yields
Z02−Z s Z0−Z s Z p=0(10)
which may be solved by using the quadratic formula to give
(11)
Substituting in the definition of Z s and Z p from Equation(7)
and Equation(8),Equation(11)now appears as
(12)
Now,as the ction length is reduced,all the parameters
(lR,lL,lG,and lC)decrea in the same proportion.This is
becau the per-unit-length line parameters R,L,G,and C
are constants for a given line.By sufficiently reducing,,the
terms in Equation(12)which contain l as multipliers will
become negligible when compared to the last term
which remains constant during the reduction process.Thus
Equation(12)can be rewritten asmoscow
wedding dress音译
(13)
particularly when the ction length,is taken to be very
small.Similarly,if a high enough frequency is assumed,
such that theωL andωC terms are much larger respectively
than the R and G terms,Z s=jω,L and Z p=1/jω,C can be
ud to arrive at a lossless line value of
(14)
In the lower frequency range,
the R and G terms dominate the impedance giving
(15)
A typical twisted pair would show an impedance versus
applied frequency curve similar to that shown in Figure6.
The Z0becomes constant above100kHz,since this is the
region where theωL andωC terms dominate and Equation
(13)reduces to Equation(14).This region above100kHz is
of primary interest,since the frequency spectrum of the fast
ri/fall time puls nt over the transmission line have a
fundamental frequency in the1-to-50MHz area with har-
monics extending upward in frequency.The expressions for
Z0in Equation(13),Equation(14)and Equation(15)do not
contain any reference to line length,so using Equation(14)
as the normal characteristic impedance expression,allows
the line to be replaced with a resistor of R0=Z0Ωneglecting
any small reactance.This is true when calculating the initial
voltage step produced on the line in respon to an input
current step,or an initial current step in respon to an input
voltage step.
Figure7shows a2V input step into a96Ωtransmission line
(top trace)and the input current required for line lengths of
150,300,450,1050,2100,and3750feet,respectively
(cond t of traces).The lower traces show the output
voltage waveform for the various line lengths.As can be
en,maximum input current is the same for all the different
line lengths,and depends only upon the input voltage and
the characteristic resistance of the line.Since R0=96Ωand
V IN=2V,then I IN=V IN/R0.20mA as shown by Figure7.
A popular method for estimating the input current into a line
in respon to an input voltage is the formula
C(dv/dt)=i
where C is the total capacitance of the line(C=C per foot x
length of line)and dv/dt is the slew rate of the input signal.If
the3750-foot line,with a characteristic capacitance per unit
length of16pF/ft is ud,the formula C total=(C x,)would
yield a total lumped capacitance of0.06µF.Using this C(dv/
dt)=i formula with(dv/dt=2V/10ns)as in the scope photo
would yield
This is clearly not the ca!Actually,since the line imped-
ance is approximately100Ω,20mA are required to produce
2V across the line.If a signal with a ri time long enough to
encompass the time delay of the line is ud(t r@τ),then
the C(dv/dt)=i formula will yield a resonable estimate of the
peak input current required.In the example,if the dv/dt is
2V/20µs(t r=20µs>τ=6µs),then i=2V/20µs x0.06µF
=6mA,which is verified by Figure8.
Figure8shows that C(dv/dt)=i only when the ri time is
greater than the time delay of the line(t r@τ).The maximum
input current requirement will be with a fast ri time step,
01133609
FIGURE6.Characteristics Impedance
versus Frequency
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Input Impedance of a
Transmission Line (Continued)
but the line is esntially resistive,so V IN /I IN =R 0=Z 0will give the actual drive current needed.The effects will be discusd later in Application Note 807.
Pha Shift and Propagation
Velocity for the Transmission Line
There will probably be some pha shift and loss of signal v 2with respect to v 1becau of the reactive and resistive parts
of Z s and Z p in the model (Figure 5).Each small ction of the line (,)will contribute to the total pha shift and ampli-tude reduction if a number of ctions are cascaded as in Figure 5.So,it is important to determine the pha shift and signal amplitude loss contributed by each ction.
01133610
01133611
,=150,300,450,1050,2100,3750ft.
24AWG TWISTED PAIR R 0.96Ω
FIGURE 7.Input Current Into a 96ΩTransmission Line for a 2V Input Step for Various LIne Lengths
01133612
01133613
R 0=96Ω,δ=1.6ns/ft.
FIGURE 8.Input Current Into Line with Controlled Ri Time t r >2π
AN-806
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Pha Shift and Propagation
Velocity for the Transmission Line
(Continued)
Using Figure5,v2can be expresd as
(16)
or
(17)
全国一卷答案and further simplification yields
(18)
Remember that a per-unit-length constant,normally calledγ
is needed.This shows the reduction in amplitude and the
change in the pha per unit length of the ctions.
γ,=α,=jβ,(19)
Since
v2=ν1−γ,=ν1−α,+ν,−jβ,(20)
where v1α,is a signal attenuation and v1−jβ,is the change
in pha from v1to v2,
(21)
Thus,taking the natural log of both sides of Equation(18)
(22)
Substituting Equation(13)for Z0and Y p for,/Z p
(23)
Now when allowing the ction length,to become small,
Y p=,(G+jωC)
will be very small compared to the constant
since the expression for Z0does not contain a reference to
the ction length,.So Equation(23)can be rewritten as
(24)
By using the ries expansion for the natural log:
(25)
nootbook
and keeping in mind the
value will be much less than one becau the ction length
is allowed to become very small,the higher order expansion
terms can be neglected,thereby reducing Equation(24)to
(26)
If Equation(26)is divided by the ction length,
(27)
the propagation constant per unit length is obtained.If the
resistive components R and G are further neglected by
assuming the line is reasonably short,Equation(26)can be
reduced to read
(28)
Equation(28)shows that the lossless transmission line has
one very important property:signals introduced on the line
have a constant pha shift per unit length with no change in
amplitude.This progressive pha shift along the line actu-
ally reprents a wave traveling down the line with a velocity
equal to the inver of the pha shift per ction.This
velocity is
(29)
for lossless lines.Becau the LRCG parameters of the line
are independent of frequency except for tho upper fre-
quency constraints previously discusd,the signal velocity
given by Equation(29)is also independent of signal fre-
quency.In the practical world with long lines,there is in fact
a frequency dependence of the signal velocity.This caus
sharp edged puls to become rounded and distorted.More
on the long line effects will be discusd in Application
Note807.
Summary—Characteristic
Impedance and Propagation Delay
Every transmission line has a characteristic impedance Z0,
and both voltage and current at any point on the line arediligence
related by the formula
In terms of the per-unit-length parameters LRCG,
Since R!jωL and G!jωC for most lines at frequencies
above100kHz,the characteristic impedance is best ap-
proximated by the lossless line expression
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