HP-AN1287-1_Understanding the Fundamental Principles of Vector Network Analysis

8/14/2019 HP-AN1287-1_Understanding the Fundamental Principles of Vector Network Analysis

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Unde rstanding theFun damen tal P r inc iplesof Vec tor Ne tw ork Ana lysis

Applica t ion Note 1287-1Table of Conten ts

Page

Introduction 2

Measurements in

Communications Systems 2

Im port an ce of Vector

Measurements 4

The Basis of Incident a nd

Reflected Power 5

The Smith Chart 5

Power Tran sfer Conditions 6

Network Analysis Terminology 9

Measur ing Group Delay 11

Network Characterization 12


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Measuremen ts inCommunicat ionsSys tems

2

Introduct ion Network ana lysis is the process by which designers an d man ufacturer smeas ure t he electr ical performan ce of the component s an d circuits u sed in

more complex systems. When t hese system s ar e conveying signals with

inform at ion cont ent, we ar e most concerned with gett ing the signal from

one point t o another with maximu m efficiency and minim um distortion.

Vector n etwork an alysis is a m ethod of accur at ely cha ra cterizing such

component s by measur ing their effect on th e amplitude a nd pha se of

swept-frequen cy and swept-power t est signals.

In t his application n ote, the fundam enta l principles of vector net work

ana lysis will be reviewed. The discussion includes the common par amet ers

tha t can be measur ed, including the concept of scatter ing param eters

(S-param eters). RF fundamenta ls such as tr ansmission lines and t he

Smith chart will also be reviewed.

Hewlett-Packar d Company offers a wide r ange of both scalar a nd vector

network a na lyzers for chara cterizing component s from DC t o 110 GHz.

These instr um ents ar e available with a wide ra nge of options t o simplifytestin g in both labora tory and production environment s.

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 versus

Nonl inear

Behavior

In a ny comm unications system, t he effect of signal distortion mu st be

considered. While we generally th ink of the dist ortion cau sed by nonlinear

effects (for example, when inter modulation products a re p roduced from

desired carrier s ignals), pur ely linear system s can also introduce signal

distortion. Linear systems can change th e time waveform of signals

passing through them by altering the am plitude or pha se relationships

of the spectral components tha t make u p th e signal.

Lets examine the difference between linear an d nonlinear behavior

mor e closely.

Linear devices impose magnitu de and pha se cha nges on input s ignals

(Figure 1). Any sinusoid appearing at the inpu t will also appear a t th e

output , and at the sa me frequency. No new signals ar e crea ted. Both a ctive

and passive nonlinear devices can shift an input signal in frequency or

add other frequen cy components , such as ha rm onic an d spur ious signals.

Large inpu t signals can drive norma lly linear devices into compr ession 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 tr an smission, th e amplitude response of the

device under t est (DUT) mu st be flat an d the pha se response must be

linear over the desired ban dwidth. As an exam ple, consider a s quar e-wave

signal rich in high-frequen cy components passing t hr ough a ban dpass filterthat passes selected frequencies with little at tenuat ion while at tenuat ing

frequencies outside of the pa ssband by var ying amounts.

Even if the filter h as linear pha se perform an ce, the out-of-band

component s of th e squar e wave will be att enua ted, leaving an outpu t

signal tha t, in this examp le, is more sinusoidal in na tur e (Figur e 2).

If the sam e square-wave input signal is passed t hrough a filter th at only

inverts the pha se of the third har monic, but leaves the har monic

am plitudes the sa me, the output will be more impulse-like in natu re

(Figure 3). While this is t ru e for t he examp le filter, in gener al, the out put

waveform will appear with ar bitra ry distortion, depending on th e

amplitude and phase n onlinearities.

Frequency

Magnitude

LinearNetwork

Frequency

Frequency

Time

0

360

180

Time

F(t) = sin wt + 1/3 sin 3wt + 1/5 sin 5wt

Figure 2.

Magnitude

Variation w ith

Frequency

Figure 3.

Pha se Variation

with Frequency


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4

Nonlinear Networks

Frequency Frequency

TimeTime

Saturation, crossover, intermodulation, and othernonlinear effects can cause signal distortion

Nonlinear devices a lso introduce distortion (Figure 4). For example, if an

am plifier is overdr iven, the outpu t signal clips becaus e th e am plifier is

sat ur at ed. The output signa l is no longer a pu re sinusoid, an d har monics

ar e present at mu ltiples of the input frequency. Passive devices ma y also

exhibit nonlinear behavior at high power levels, a good example of which is

an L-C filter th at u ses inductors with ma gnetic cores. Magnetic mat erials

often exhibit hyst eresis effects t hat ar e highly nonlinear.

Efficient t ra nsfer of power is an other funda ment al concern in

comm unications syst ems. In order t o efficiently convey, tr ans mit or

receive RF power, devices such as t ra nsm issions lines, ant enna s and

amplifiers must present the proper impedance match to the signal source.Impedance mismatches occur when th e real a nd imaginary par ts of input

and output impedan ces ar e not ideal between t wo connecting devices.

Measur ing both m agnitu de and ph ase of components is import ant for

several r easons. First, both measu rements are required to fully

characterize a linear network and ensure distortion-free transmission.

To design efficient mat ching networks, complex impedan ce must be

meas ured. Engineers developing m odels for comput er-aided-engineering

(CAE) circuit simulat ion program s require ma gnitude an d phase dat a for

accur ate m odels.

In addition, time-domain characterization requires magnitude and phaseinform at ion in order to perform an inverse-Fourier t ra nsform. Vector

err or correction, which improves mea sur ement accur acy by rem oving the

effects of inherent measurement-system errors, requires both magnitude

and ph ase dat a to build an effective error model. Pha se-measur ement

capability is very important even for scalar measurements such as return

loss, in ord er t o achieve a high level of accura cy (see Applying Error

Correction to Network Analyzer Measurements, Hewlett-Packard

Applicat ion Note 1287-3).

Figure 4.

Nonl inear

Induced

Distortion

Importance of Vector Measu reme nts


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5

The Basis of Incide ntand Reflected Powe r

In its fundamental form, network analysis involves the measurement of

incident, reflected, and transmitted waves that travel along transmission

lines. Using optical wavelengths as a n a na logy, when light st rikes a clear

lens (the incident energy), some of the light is reflected from t he lenssur face, but most of it contin ues th rough the lens (the tr ans mitt ed energy)

(Figure 5). If the lens ha s mir rored su rfaces, most of th e light will be

reflected an d little or n one will pass th rough it.

While the wavelengths ar e different for RF and microwave signals, the

principle is the sam e. Network a nalyzers accur at ely meas ur e the incident,

reflected, and t ra nsm itted ener gy, e.g., the ener gy that is lau nched onto

a tr an smission line, reflected back down the tr ans mission line toward the

source (due to impedence mismat ch), an d successfully tr ans mitt ed to the

terminat ing device (such a s an antenn a).

Incident

Reflected

Transmitted

Lightwave Analogy

Figure 5.

Lightwave

Analogy toHigh-Frequency

Device

Characterization

The Smith Chart The am ount of reflection tha t occur s when chara cterizing a device dependson the impeda nce tha t t he incident signal sees. Since any impedance

can be represent ed with real a nd imagina ry par ts (R+jX or G+jB), th ey

can be plotted on a rectilinear grid kn own a s th e complex impedance

plane. Unfortun at ely, an open circuit (a comm on RF impedence) appears

at infinity on the rea l axis, an d ther efore cannot be shown.

The polar plot is useful becau se th e entir e impedan ce plane is covered.

However, inst ead of plott ing imped an ce directly, the complex reflection

coefficient is displayed in vector form. The magnitude of the vector is the

distan ce from th e cent er of th e display, and ph ase is displayed as t he an gle

of vector r eferen ced to a flat line from t he center to the right-most edge.

The dra wback of polar plots is tha t impedan ce values cannot be read

directly from the display.


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Pow er TransferCondit ions

A perfectly matched condition mu st exist a t a connection between two

devices for m aximum power t ra nsfer into a load, given a source resista nce

of RS an d a load resista nce of RL. This condit ion occurs when RL = RS,

and is t ru e whether the st imulus is a DC voltage source or a s ource of

RF sine wa ves (Figur e 7).

When th e source impedan ce is not pur ely resistive, maximum power

tr ans fer occurs when the load impeda nce is equal to th e complex conjugate

of th e source impedan ce. This condition is met by reversing t he sign of the

imaginar y part of th e impedance. For example, if RS = 0.6 + j 0.3, th en

the complex conjugate is RS* = 0.6 j 0.3.

The need for efficient power tr an sfer is one of the m ain r easons for t he

use of tra nsm ission lines at higher frequen cies. At very low frequencies

(with mu ch larger wavelengths), a simple wire is adequa te for conducting

power. The resistance of the wire is relatively low and has little effect on

low-frequency signals. The voltage and cur rent ar e the sa me no mat ter

where a measur ement 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 Chart

Review

Since there is a one-to-one correspondence between complex impedance

and reflection coefficient, the positive real half of the complex impedance

plane can be mapp ed ont o th e polar display. The result is th e Smith cha rt .

All values of reactan ce and a ll positive values of resistan ce from 0 t oinfinity fall within th e outer circle of the Sm ith char t (Figure 6).

On th e Smith chart , loci of const ant resista nce appear as circles, while loci

of constan t r eactance appear as a rcs. Impedances on t he Smith chart a re

always norm alized to the char acteristic impedance of the component or

system of inter est, usua lly 50 ohm s for RF a nd microwave system s and

75 ohms for broadcast and cable-television s ystems. A perfect term inat ion

appears in th e center of the Smith chart.


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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.

Powe r Transfer

At higher frequencies, wavelengths a re compa ra ble to or smaller th an

the length of the conductors in a high-frequen cy circuit, and power

tr ans mission can be th ought of in ter ms of tr aveling waves. When t he

transmission line is terminated in its characteristic impedance, maximumpower is tra nsferred t o th e load. When the ter mina tion is not equal to the

chara cteristic impedan ce, that p ar t of the signal th at is not absorbed by

the load is r eflected ba ck to the source.

If a tra nsm ission line is ter mina ted in its cha ra cteristic impedan ce, no

reflected signal occur s since all of the t ra nsm itted power is absorbed by th e

load (Figur e 8). Looking a t t he en velope of the RF signa l versus distan ce

along the tra nsm ission line shows no sta nding waves becau se without

reflections, ener gy flows in only one d irection.

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.

Transmission

Line Terminate d

wi th Zo


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When the tr ans mission line is termina ted in a short circuit (which can

sust ain n o volta ge and t herefore dissipates zero power), a r eflected wave is

laun ched back along th e line toward t he source (Figure 9). The reflected

voltage wave must be equal in ma gnitude t o the incident voltage wave andbe 180 degrees out of phas e with it at the plane of the load. The reflected

and incident waves are equal in ma gnitude but t ra veling in the opposite

directions.

If the tr an smission line is termina ted in a n open-circuit condition (which

can sust ain n o cur rent ), the r eflected curr ent wa ve will be 180 degrees out

of pha se with t he incident curr ent wave, while th e reflected volta ge wave

will be in phase with the incident volta ge wave at th e plane of th e load.

This guara ntees t hat the curr ent a t th e open will be zero. The reflected and

incident curr ent waves are equal in magnitude, but tra veling in t he

opposite directions. For both t he short an d open cases, a st an ding wave

pat tern is set up on t he tr an smission line. The voltage valleys will be zero

and the voltage peak s will be twice th e incident voltage level.

If the tra nsm ission line is term inat ed with sa y a 25-ohm resistor, resu lting

in a condition between full absorption an d full reflection, part of the

incident power is absorbed and par t is r eflected. The am plitude of th e

reflected volta ge wave will be one-third tha t of the incident wave, and the

two waves will be 180 degrees out of phas e at the plane of the load. The

valleys of the st an ding-wave patt ern will no longer be zero, and th e peaks

will be less th an those of the sh ort an d open cases. The r at io of th e peaks t o

valleys will be 2:1.

The tr aditional way of deter mining RF impedan ce was to measu re VSWR

using an RF pr obe/detector, a length of slotted tr an smission line, and a

VSWR meter. As th e probe was moved along the tr an smission line, the

relat ive position and values of the peaks and valleys were noted on th e

meter. Fr om these mea sur ement s, impedance could be derived. Theprocedure wa s repea ted at different frequencies. Modern network

ana lyzers mea sur e the incident a nd reflected waves directly during a

frequency sweep, and impedance result s can be displayed in any nu mber of

formats (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.

Transmission

Line Terminate d

with Short , Open


9/14


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.

Transmission

Parameters

Retur n loss is a way to express t he r eflection coefficient in logarith mic

term s (decibels). Retur n loss is th e num ber of decibels th at the reflected

signal is below the incident signal. Return loss is a lways expressed as a

positive number a nd var ies between infinity for a load a t th e cha ra cteristic

impedance an d 0 dB for a n open or short circuit. Another comm on term

used to express reflection is voltage standing wave ratio (VSWR), which is

defined as t he ma ximum value of the RF en velope over th e minimu m value

of th e RF envelope. It is relat ed to as (1 + )/(1 ). VSWR ra nges fr om

1 (no r eflection) to in finity (full r eflection).

The tr an smission coefficient is defined as t he t ra nsm itted volta ge divided

by the incident voltage (Figure 12). If the absolute value of the tr an smitt edvoltage is greater th an the a bsolute valu e of the incident volta ge, a DUT or

system is said to have gain. If the absolute value of the tra nsm itted volta ge

is less tha n t he absolute value of the incident voltage, the DUT or system

is said to have att enua tion or inser tion loss. The phase portion of the

tr an smission coefficient is called insertion ph ase.

=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.

Reflection

Parameters


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11

Direct exam inat ion of insertion pha se usu ally does not pr ovide useful

inform at ion. This is becau se the inser tion phase ha s a lar ge (negative)

slope with r espect to frequency due to th e electr ical length of the DUT. The

slope is proportional to the length of the DUT. Since it is only deviationfrom linear ph ase th at caus es distortion in commu nications systems, it is

desirable to remove the linear portion of the phas e response to ana lyze

the r emain ing nonlinear portion. This can be done by using th e electr ical

delay featu re of a net work a nalyzer to mat hema tically cancel the a verage

electr ical length of the DUT. The r esult is a high-resolution display of

phas e distortion or deviation from linear pha se (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.

Deviat ion from

Linear Phase

Figure 14.

What Is Group

Delay?

MeasuringGroup De lay

Anoth er u seful measu re of phase distort ion is group delay (Figure 14).

This parameter is a measur e of the tran sit time of a signal thr ough a DUT

versus frequency. Group delay can be calculat ed by different iating t he

DUTs pha se response versu s frequency. It r educes the linear portion of thephas e response to a const an t value, and tr ans form s the deviations from

linear pha se into deviations from const an t gr oup delay, (which cau ses

phas e distortion in comm unications systems). The avera ge delay

represents th e average signal tr ansit t ime through a DUT.


12/14

12

Depending on t he device, both deviation from linear phas e an d group delay

ma y be meas ur ed, since both can be importa nt. Specifying a maximum

peak-to-peak pha se ripple in a device may not be sufficient to completely

chara cterize it, since the slope of the pha se ripple depends on the n um berof ripples th at occur per unit of frequency. Group delay tak es th is into

account because it is th e differentiat ed phas e response. Group delay is

often a m ore easily interpr eted 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 u nkn own linear two-port device, we

must make m easurements under various conditions a nd compute a set of

par am eters. These par am eters can be used to completely describe th e

electr ical beha vior of our device (or net work), even u nder source an d loadconditions other tha n when we made our mea sur ement s. Low-frequency

device or net work char acterizat ion is usua lly based on measu remen t of

H, Y, and Z para meter s. To do this, th e total voltage and cur rent at t he

input or out put port s of a device or nodes of a network m ust be measu red.

Fur therm ore, measurement s must be made with open-circuit and

short-circuit conditions.

Since it is difficult t o measur e total curr ent or volta ge at h igher

frequencies, S-parameters are generally measured instead (Figure 16).

These parameter s relate t o familiar measur ements such as gain, loss,

and reflection coefficient. They a re r elatively simple to mea sur e, and do

not requ ire connection of und esirable loads t o the DUT. The mea sur ed

S-para meter s of multiple devices can be cascaded to predict overall system

performance. S-parameters are readily used in both linear and nonlinearCAE circuit simula tion tools, and H , Y, and Z para meter s can be der ived

from S-parameters when necessary.

The nu mber of S-par am eters for a given device is equal to the squ ar e of

the n um ber of ports. For example, a two-port device has four S-para meter s.

The num bering convention for S-para meters is t hat the first n umber

following the S is t he port at which ener gy emerges, and th e second n umber

is the port a t which energy enters. So S21 is a meas ur e of power emerging

from P ort 2 as a r esult of applying an RF st imulus t o Port 1. When th e

numbers a re th e same (e.g. S11), a r eflection measu rem ent is indicated.

Figure 15.

Why Meas ure

Group Delay?

NetworkCharacterizat ion


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, an d Z

Parameters

(Why UseS-parameters?)

Figure 17.

Measuring

S-Parameters

Forward S-para meters ar e determined by measuring the magnitude and

phas e of th e incident, reflected, an d tra nsm itted signals when th e outpu t

is termin ated in a load tha t is precisely equal to the cha ra cteristic

impedance of the t est system . In th e case of a simple two-port net work,

S11 is equivalent to the input complex reflection coefficient or impedance of

th e DUT, while S21 is the forward complex transmission coefficient. By

placing the source at the output port of the DUT and t erminating th e input

port in a perfect load, it is possible to meas ur e the other two (reverse)

S-parameters. Parameter S22 is equivalent to th e outpu t complex reflection

coefficient or output impedance of the DUT while S12 is the reverse

complex transmission coefficient (Figure 17).

Exploring the Architectures of Network Analyzers , Hewlett-Packard

Applicat ion Note 1287-2.

Applying Error Correction to Network Analyzer Measurements,

Hewlett-Packar d Application Note 1287-3.

Network A nalyzer Measurem ents: Filter and Am plifier Exam ples,

Hewlett-Packar d Application Note 1287-4.

Sugges ted Reading


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of f i ce l i s t ing , v i s i t our web s i t e ,http://www.hp.com/go/tmdir. Youcan also contact one of the followingc e n t e r s a n d a s k f o r a t e s t a n dmeasurement sales representative .

United States:Hewlett-Packard CompanyTest and Measu rement Call CenterP.O. Box 4026En glewood, CO 80155-40261 800 452 4844

Canada:Hewlett-Packard Canada Ltd.5150 Spectru m WayMississauga, OntarioL4W 5G1(905) 206 4725

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Japan:Hewlett-Packard J apan Ltd.Measurement Assistance Center9-1, Takakura-Cho, Hachioji-Shi,Tokyo 192, J apa nTel: (81-426) 56-7832Fax: (81-426) 56-7840

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HP-AN1287-1_Understanding the Fundamental Principles of Vector Network Analysis