HP Understanding the Fundamental Principles of Vector Network analysis.pdf

8/10/2019 HP Understanding the Fundamental Principles of Vector Network analysis.pdf

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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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3

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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4

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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7

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


8/14

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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9

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.


10/14

10

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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11

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.


12/14

12

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


13/14

13

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


14/14

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HP Understanding the Fundamental Principles of Vector Network analysis.pdf