Vector Calculus 13. The Fundamental Theorem for Line Integrals 13.3.
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Understanding theFundamental Principlesof Vector Network Analysis
Applica t ion Note 1287-1Table of Contents
Page
Introduction 2Measurements in
Communications Systems 2Im porta nce of Vector
Measurements 4The Ba sis of Incident a nd
Reflected Power 5The Smith C hart 5P ower Tra nsfer Conditions 6Netw ork Ana lysis Termin ology 9Measuring Group Delay 11Network Cha racterizat ion 12
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Measurements inCommunicationsSystems
2
Introduction Network an aly sis is the process by wh ich designers and ma nufa cturersmeasure t he electrical performa nce of th e components a nd circuits used in
more complex systems. When t hese systems a re conveying signa ls w ith
informat ion content, we a re most concerned wit h getting th e signal from
one point t o another wit h ma ximum efficiency a nd minimum distortion.
Vector netw ork ana lysis is a meth od of accurat ely cha ra cterizing such
components by measur ing their effect on th e amplitude a nd pha se of
swept-frequency an d sw ept-power test signa ls.
In t his application note, the fundam enta l principles of vector netw ork
an aly sis will be reviewed. The discussion includes the common para meters
tha t can be measur ed, including the concept of scat tering para meters
(S-parameters). RF fundamentals such as transmission lines and the
Smit h chart will also be reviewed.
Hew lett-P ackar d Company offers a wide range of both scala r an d vector
netw ork ana lyzers for chara cterizing components from DC to 110 GH z.
These instru ments a re ava ilable with a wide ra nge of options to simplifytesting in both labora tory a nd production environments.
Linear behavior:input and output frequencies
are the same (no additionalfrequencies created)
output frequency onlyundergoes magnitude andphase change
Time
A
to
Frequencyf1
Time
Sin 360 * f * t
Frequency
Aphase shift =to* 360 * f
1f
DUT
A * Sin 360 * f ( t t )
Input Output
Time
Frequency
Nonlinear behavior:output frequency may undergofrequency shift (e.g. with mixers)
additional frequencies created(harmonics, intermodulation)
f1
Figure 1.Linear versusNonlinearBehavior
In a ny communications system, t he effect of signa l distortion must be
considered. While we generally think of the distortion caused by nonlinear
effects (for example, when intermodulation products are produced from
desired carrier signa ls), purely linear systems can also introduce signa l
distortion. Linear sy stems can change the time wa veform of signa ls
passing thr ough them by altering the a mplitude or phase relationships
of the spectra l components tha t ma ke up the signa l.
Lets examine t he difference between linear an d nonlinear beha vior
more closely.
Linear devices impose magnit ude and pha se cha nges on input signa ls
(Figure 1). Any sinusoid appearing a t th e input will also appear a t th e
output, and a t th e same frequency. No new signa ls are creat ed. Both active
an d passive nonlinear devices can shift a n input signa l in frequency or
add oth er frequency components, such as h ar monic and spurious signals.
La rge input signa ls can dr ive normally linea r devices into compression or
saturation, causing nonlinear operation.
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Frequency FrequencyFrequency
Mag
nitude
Time
LinearNetwork
Time
F(t) = sin wt + 1/3 sin 3wt + 1/5 sin 5wt
For linear distortion-free tra nsmission, the a mplitude response of the
device under test (DU T) must be flat a nd t he phase response must be
linear over the desired ban dwidt h. As an example, consider a squa re-wa ve
signal rich in high-frequency components passing th rough a ba ndpass filtertha t passes selected frequencies w ith li t t le a t tenuat ion w hile a t tenuat ing
frequencies outside of the passba nd by var ying a mounts.
Even if t he filter ha s linear pha se performan ce, the out-of-band
components of the squar e wave will be at tenua ted, leaving a n output
signal t ha t, in th is example, is more sinusoidal in n at ure (Figure 2).
I f the sa me square-wa ve input s ignal is passed through a f i lter tha t only
inverts the pha se of the third ha rmonic, but leaves the ha rmonic
am plitudes the sam e, the output w ill be more impulse-like in na ture
(Figure 3). While this is true for the example filter, in general, the output
waveform will appear with arbitrary distortion, depending on the
amplitude and phase nonlinearities.
Frequency
Magnitude
LinearNetwork
Frequency
Frequency
Time
0
360
180
Time
F(t) = sin wt + 1/3 sin 3wt + 1/5 sin 5wt
Figure 2.MagnitudeVariation withFrequency
Figure 3.Phase Variationwith Frequency
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Nonlinear Networks
Frequency Frequency
TimeTime
Saturation, crossover, intermodulation, and othernonlinear effects can cause signal distortion
Nonlinear devices also introduce distortion (Figure 4). For example, if an
am plifier is overdriven, th e output signa l clips because the a mplifier is
sat ura ted. The output signal is no longer a pure sinusoid, an d ha rmonics
ar e present a t mu ltiples of the input frequency. Pa ssive devices ma y a lso
exhibit n onlinear behavior at high power levels, a good exam ple of wh ich is
an L -C filter tha t uses inductors with m agn etic cores. Magn etic ma terials
often exhibit hyst eresis effects tha t a re highly nonlinear.
Efficient t ra nsfer of power is a nother funda menta l concern in
commun ications syst ems. In order t o efficiently convey, tra nsmit or
receive RF power, devices such as t ra nsmissions lines, antenna s a nd
am plifiers must present the proper impedance ma tch to t he signal source.Impedance mismatches occur when the real and imaginary parts of input
an d output impedan ces a re not ideal betw een t wo connecting devices.
Measuring both ma gnitude an d phase of components is importan t for
severa l reasons. First, both measurements a re required to fully
chara cterize a linea r netw ork and ensure distortion-free tra nsmission.
To design efficient ma tchin g netw orks, complex impedan ce must be
measured. Engineers developing models for computer-aided-engineering
(CAE) circuit simula tion progra ms require ma gnitude an d phase dat a for
accurat e models.
In addition, time-domain characterization requires magnitude and phaseinformat ion in order t o perform an inverse-Fourier tra nsform. Vector
error correction, w hich improves measur ement a ccura cy by removing the
effects of inherent measur ement-system errors, requires both ma gnitude
an d phase dat a t o build an effective error model. P ha se-measur ement
capability is very importa nt even for scalar m easurements such as retur n
loss, in order to a chieve a high level of accura cy (see Apply ing Err or
Correct ion to N etwork Anal yzer M easurements, H ewlet t-Pa ckard
Applicat ion N ote 1287-3).
Figure 4.NonlinearInducedDistortion
Importance ofVector Measurements
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Power TransferConditions
A perfectly m at ched condition mu st exist a t a connection betw een two
devices for ma ximum power tra nsfer into a load, given a source resista nce
of RS an d a load resista nce of RL. This condition occurs w hen R L = RS ,
an d is true w hether th e stimulus is a D C volta ge source or a source of
RF sine waves (Figure 7).
When t he source impedan ce is not purely resistive, ma ximum power
tra nsfer occurs wh en the load impedan ce is equa l to the complex conjugate
of the source impedance. This condition is met by reversing the sign of the
ima ginar y part of the impedance. For example, if RS = 0.6 + j 0.3, then
th e complex conjuga te is R S* = 0.6 j 0.3.
The need for efficient power t ra nsfer is one of the ma in rea sons for t he
use of tra nsmission lines at higher frequencies. At very low frequ encies
(wit h much lar ger wa velength s), a simple wire is adequa te for conducting
power. The resista nce of th e wire is r elatively low a nd ha s litt le effect on
low-frequency signals. The volta ge and current ar e the sam e no matt er
wh ere a mea surement is made on the wire.
6
90o
0o
180o+.2
.4.6
.8
1.0
90o
0
0 +R
+jX
jX
Smith chart mapsrectilinear impedanceplane onto polar plane
Rectilinear impedanceplane
Polar plane
Z = ZoL= 0
Constant X
Constant R
Z =L= 0O1
Smith chart
(open)
LZ = 0
= 180O1
(short)
Figure 6.Smith ChartReview
Sin ce there is a one-to-one correspondence betw een complex impeda nce
and reflection coefficient, the positive real half of the complex impedance
plane can be ma pped onto the polar display. The result is the Smit h char t.
All values of reactan ce and a ll positive values of resistance from 0 toinfinity fa ll with in th e outer circle of the Smith chart (Figure 6).
On th e Smith chart , loci of consta nt r esistance appear a s circles, wh ile loci
of consta nt reacta nce appear a s arcs . Impedances on the S mith chart are
alw ay s norma lized to the cha ra cteristic impedance of the component or
system of interest, usually 50 ohms for RF a nd microwa ve systems and
75 ohms for broadcast an d cable-television systems. A perfect termina tion
appears in th e center of the Smith cha rt.
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0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
LoadPower(normalized)
RL/ RS
RS
RL
Maximum power is transferredwhen RL= RS
For complex impedances,maximum power transfer occurswhen ZL= ZS*(conjugate match)
Zs= R + jX
ZL= Zs*= R jX
Figure 7.Power Transfer
At higher frequencies, wa velengths a re compar able to or sma ller than
the length of the conductors in a high-frequency circuit, a nd power
tra nsmission can be thought of in terms of tra veling waves. When the
tra nsmission line is termina ted in its cha ra cteristic impedance, ma ximumpower is tra nsferred to the load. When the termin at ion is not equa l to the
chara cteristic impedance, tha t par t of the signal th at is not absorbed by
the load is reflected back to the source.
If a tr an smission line is termina ted in its cha ra cteristic impedance, no
reflected signal occurs since all of the tr an smitt ed power is a bsorbed by th e
load (Figure 8). Looking a t t he envelope of th e RF signa l versus dista nce
along the tra nsmission line shows no standing w aves becau se without
reflections, energy flows in only one direction.
For reflection, a transmission line terminated in Zobehaves like an infinitely long transmission line
Zs= Zo
Zo
Vrefl= 0 (all the incidentpower is absorbed in the load)
Vinc
Zo= characteristic impedanceof transmission line
Figure 8.TransmissionLine Terminatedwith Zo
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When the tr an smission line is termina ted in a short circuit (wh ich can
susta in no voltage a nd t herefore dissipates zero power), a reflected w ave is
launched back along the line toward the source (Figure 9). The reflected
volta ge wave must be equal in ma gnitude to the incident volta ge wave an dbe 180 degrees out of phase with it at the plane of the load. The reflected
an d incident wa ves are equa l in magnit ude but tra veling in the opposite
directions.
If th e tra nsmission line is termina ted in a n open-circuit condition (which
can sust ain no current), the r eflected current w ave w ill be 180 degrees out
of phase w ith th e incident current w ave, while the reflected volta ge wave
will be in pha se with t he incident voltage wa ve at t he plane of the load.
This gua ra ntees tha t t he current a t t he open will be zero. The reflected and
incident current w aves are equa l in ma gnitude, but t ra veling in the
opposite directions. For both t he short a nd open cases, a st an ding wa ve
pat tern is set u p on the tra nsmission line. The voltage va lleys w ill be zero
an d th e volta ge peaks will be tw ice the incident voltage level.
If the tra nsmission line is termina ted wit h sa y a 25-ohm resistor, resulting
in a condition between full absorption an d full reflection, par t of the
incident power is absorbed an d pa rt is reflected. The a mplitude of the
reflected volta ge wa ve will be one-third t ha t of the incident w ave, a nd th e
tw o wa ves will be 180 degrees out of pha se at the plan e of the load. The
valleys of the sta nding-wa ve patt ern will no longer be zero, and the peaks
will be less tha n t hose of the short an d open cases. The ra tio of the peaks to
valleys will be 2:1.
The tra ditional w ay of determining RF impedance was t o measure VSWR
using a n RF probe/detector, a length of slotted t ra nsmission line, and a
VSWR meter. As th e probe wa s moved along the t ra nsmission line, the
relative position an d values of the peaks and valleys w ere noted on the
meter. From these measurements, impedance could be derived. Theprocedure wa s repeated a t different frequencies. Modern netw ork
an aly zers measure the incident a nd reflected wa ves directly during a
frequency sweep, an d impedance results can be display ed in any number of
forma ts (including VSWR).
8
Zs= Zo
Vrefl
Vinc
For reflection, a transmission line terminated ina short or open reflects all power back to source
In phase (0 ) for open
Out of phase (180 ) for shorto
o
Figure 9.TransmissionLine Terminatedwith Short, Open
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TRANSMISSION
Gain / Loss
S-ParametersS21,S12
GroupDelay
TransmissionCoefficient
InsertionPhase
REFLECTION
SWR
S-ParametersS11,S22 Reflection
Coefficient
Impedance,Admittance
R+jX,G+jB
ReturnLoss
, ,
Incident
Reflected
TransmittedR B
A
ReflectedIncident
AR
=Transmitted
Incident
BR
=
Network AnalysisTerminology
Now tha t w e understand the fundamentals of e lectromagnetic waves, we
must lear n the common terms used for measurin g them. Network ana lyzer
terminology generally denotes measurements of the incident wa ve with
the R or r eference cha nnel. The reflected wa ve is measured w ith t heA channel, and t he transmit t ed wave is measured with the B channel
(Figure 10). With t he am plitude and phase informa tion in t hese wa ves,
it is possible to qua ntify t he reflection and t ra nsmission cha ra cteristics
of a DU T. The reflection a nd t ra nsmission char acteristics can be expressed
as vector (magnit ude an d pha se), scala r (magnit ude only), or phase-only
qua ntit ies. For exam ple, return loss is a scalar m easurement of
reflection, w hile impedance is a vector reflection mea surement.
Ra tioed measur ements allow us to make reflection an d tra nsmission
measurement s tha t a re independent of both a bsolute power a nd
variations in source power versus frequency. Ratioed reflection is often
shown a s A/R a nd ra tioed tran smission as B /R, relat ing to th e
measurement channels in the instrument .
Figure 10.CommonTerms forHigh-FrequencyDeviceCharacterization
The most general term for ratioed reflection is the complex reflection
coefficient, or gamma (Figure 11). The magnitude portion of is called
or rho. The reflection coefficient is the ratio of the reflected signal voltage
level to the incident signa l voltage level. For example, a t ra nsmission line
termina ted in its cha ra cteristic impedance Zo, will ha ve all energy
tra nsferred to the load so Vrefl = 0 a n d = 0. When th e impedan ce of the
load, ZL is not equa l to the cha ra cteristic impedan ce, energy is reflected
and is greater t ha n zero. When the load impedan ce is equal t o a short oropen circuit, all energy is reflected a nd = 1. As a result, th e range of
possible values for is 0 to 1.
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VTransmittedVIncident
Transmission Coefficient = =VTransmitted
VIncident=
DUT
Gain (dB) = 20 LogV
Trans
VInc= 20 log
Insertion Loss (dB) = 20 LogV
Trans
VInc= 20 log
Figure 12.TransmissionParameters
Return loss is a wa y t o express the reflection coefficient in logarit hmic
terms (decibels). Return loss is the number of decibels that the reflected
signal is below t he incident signa l. Return loss is alwa ys expressed as a
positive number and var ies between infinity for a load a t the chara cteristic
impedance and 0 dB for an open or short circuit . Another common t erm
used to express reflection is voltage st an ding w ave r at io (VSWR), wh ich is
defined as t he ma ximum va lue of the RF envelope over the minimum value
of the RF envelope. It is relat ed to as (1 + )/(1 ). VSWR ra nges from
1 (no reflection) to infin ity (full r eflection).
The tra nsmission coefficient is defined as the t ra nsmitt ed volta ge divided
by th e incident volta ge (Figure 12). If th e absolute value of the tra nsmitt edvoltage is greater t ha n th e absolute value of the incident volta ge, a DU T or
system is said to have gain. If the absolute value of the tra nsmitt ed volta ge
is less than the a bsolute va lue of the incident volta ge, the DU T or system
is said t o have a ttenua tion or insertion loss. The phase portion of the
tra nsmission coefficient is called insertion pha se.
=Z
L ZO
ZL + OZ
ReflectionCoefficient
=Vreflected
Vincident=
=
dB
No reflection(ZL= Zo)
RL
VSWR
0 1
Full reflection(ZL= open, short)
0 dB
1
Return loss =20 log(),
VSWR =EmaxEmin
=1 + 1
Voltage Standing Wave RatioEmaxEmin
Figure 11.ReflectionParameters
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Direct examina tion of insertion phase usua lly does not provide useful
informat ion. This is becau se the insertion phase ha s a large (negative)
slope w ith r espect t o frequency due to the electrical lengt h of the DU T. The
slope is proport iona l to the lengt h of the DU T. Since it is only deviat ionfrom linear phase tha t causes distortion in communicat ions systems, it is
desirable to remove the linear portion of the pha se response to an alyze
the rema ining nonlinear portion. This can be done by using the electrical
delay feat ure of a network ana lyzer to ma thematically cancel the a verage
electr ical lengt h of the DU T. The result is a high -resolution displa y of
phase dist ortion or deviation from linear phase (Figure 13).
Deviation from constant groupdelay indicates distortion
Average delay indicates transit time
GroupDelay
Frequency
Group Delay
Average Delay
to
tg
Group Delay (t )g
=1
360o
=
d d df
in radians
in radians/sec
in degrees
in Hzf
2=( )f
Phase
*
Frequency
Use electrical delay to removelinear portion of phase response
Linear electricallength added
+ yields
Frequency
(Electrical delay function)
Frequency
RF filter responseDeviation fromlinear phase
Phase1
/Div
o
Phase45/D
iv
o
Frequency
Low resolution High resolution
Figure 13.Deviation fromLinear Phase
Figure 14.What Is GroupDelay?
MeasuringGroup Delay
Another useful measure of phase distortion is group delay (Figure 14).
This param eter is a measure of the tra nsit t ime of a s ignal thr ough a DU T
versus frequency. Gr oup delay can be calculated by differentiat ing the
DUTs phase response versus frequency. It reduces the linear portion of thephase response to a consta nt va lue, and tra nsforms the deviations from
linear pha se into deviat ions from constan t group delay, (wh ich causes
phase distortion in commun ications systems). The a verage delay
represents th e average signal tra nsit t ime thr ough a DU T.
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Depending on the device, both deviation from linear phase a nd group delay
ma y be measured, since both can be importa nt. Specifying a ma ximum
peak-to-peak pha se ripple in a device ma y not be sufficient to completely
chara cterize it , since the slope of th e phase ripple depends on the n umberof ripples tha t occur per unit of frequency. G roup delay ta kes this into
account because it is the differentiat ed phase response. Group delay is
often a more easily int erpreted indication of pha se distortion (Figure 15).
Same peak-to-peak phase ripple can result in different group delay
Phase
Phase
Group
Delay
Group
Delay
dd
f
f
f
f
dd
In order to completely chara cterize an unkn own linear tw o-port device, we
must m ake measur ements under var ious conditions and compute a set of
para meters. These para meters can be used to completely describe the
electr ical beha vior of our device (or netw ork), even under source an d loadconditions other tha n w hen we ma de our mea surements. Low-frequency
device or netw ork cha ra cterization is usua lly based on measurement of
H, Y, and Z para meters. To do this, th e total volta ge and current at the
input or output ports of a device or nodes of a netw ork must be mea sured.
Furt hermore, measur ements must be ma de with open-circuit a nd
short-circuit conditions.
Since it is difficult to measure tota l current or volta ge at higher
frequencies, S-par am eters ar e generally mea sured instea d (Figure 16).
These par am eters relate to familia r measur ements such as ga in, loss,
an d reflection coefficient. They a re relat ively simple to measur e, an d do
not req uire connection of und esira ble loa ds to th e DU T. The mea sured
S-para meters of mult iple devices can be cascaded to predict overall syst em
performa nce. S-para meters ar e readily used in both linear an d nonlinearCAE circuit simula tion tools, and H, Y, and Z para meters can be derived
from S-para meters w hen necessar y.
The number of S-para meters for a given device is equa l to the sq uar e of
the num ber of ports. For example, a t wo-port device has four S-para meters.
The numbering convention for S-para meters is tha t t he first number
following the S is the port a t w hich energy emerges, and t he second number
is the port a t w hich energy enters. So S21 is a measur e of power emerging
from Port 2 as a result of applying a n RF st imulus to Port 1. When the
numbers a re the sa me (e.g. S11), a reflection m easurement is indicated.
Figure 15.Why MeasureGroup Delay?
NetworkCharacterization
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H,Y, and Z parametersHard to measure total voltage and currentat device ports at high frequencies
Active devices may oscillate or self-destruct with shorts or opens
S-parametersRelate to familiar measurements (gain, loss, reflection coefficient, etc.)
Relatively easy to measure
Can cascade S-parameters of multipledevices to predict system performance
Analytically convenient
CAD programs
Flow-graph analysis
Can compute H, Y, or Z parameters from S-parameters if desired
Incident TransmittedS21
S11Reflected S22
Reflected
Transmitted Incident
b1
a1 b2
a2S12
DUT
b1 = S11a1+ S12a2b2 = S21 a1+ S22a2
Port 1 Port 2
S11 =Reflected
Incident=
b1a1
a2=0
S21 =Transmitted
Incident=
b2a1
a2=0
21
1
2
1
Incident TransmittedS21
S11Reflectedb1
a1
b2
Z0Load
a2=0DUT
Forward
IncidentTransmitted S12
S 22Reflected
b2
a2
1b
a1=0
DUTZ0Load
Reverse
S22=Reflected
Incident=
ba
a =0
S12=Transmitted
Incident=
b
2aa =0
Figure 16.Limitations of H,
Y, and ZParameters
(Why UseS-parameters?)
Figure 17.MeasuringS-Parameters
Forward S-param eters a re determined by measuring the ma gnitude and
phase of the incident, reflected, an d tra nsmitt ed signals wh en the output
is termina ted in a load tha t is precisely equa l to the cha ra cteristic
impedance of the t est system. I n t he case of a simple tw o-port netw ork,
S 11 is equivalent to the input complex reflection coefficient or impedance of
th e DU T, wh ile S21 is th e forwar d complex tra nsmission coefficient. B y
placing the source at th e output port of the DU T an d termina ting th e input
port in a perfect load, it is possible to measure the other two (reverse)
S-param eters . Pa rameter S 22 is equivalent to the output complex reflection
coefficient or output impedance of the DUT while S 12 is the r everse
complex tr a nsm ission coefficient (Figu re 17).
Explori ng th e Ar chitectur es of N etwork Anal yzers, Hewlet t-Pa ckard
Application Note 1287-2.
Appl ying Er ror Corr ect i on to Network Anal yzer M easur ements,
Hewlett-Packard Application Note 1287-3.
Network A nal yzer M easur ements: Fil ter and Ampli f ier Exampl es,
Hewlett-Packard Application Note 1287-4.
Suggested Reading
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