APPLICATION NOTE 86

Using the Vector Impedance Meters

\

HEWLETT·

TABLE OF CONTENTS

[. INTRODUCTION . • . • . . . • . . . . . . . . . . . . . . . . . . " . . . . . . . . . . . . . . . . . .

n. EVALUATING Cm.lPONENlS

Direct Measurements . . . . . . . . . . Sorting Components . . . . . . . . . . . Characterizing Terminals and Fixtures. Characterizing Crystals . . . . . . . . .

nr. TIlAKSMISSIO:N LINE MEASUREMENTS

Characteristic Impedance . . . . . . . . Attenuation . . . . . . . . . . . . . . . .

Balanced Transmission Line Characteristics. Balanced Measurements Using a Broadband Dalun .

rv. TRAl\""SFORMER MEASURE7'ltENTS

Primary Inductance, Resistance, and Capacitance. Secondary Inductance and Resistance . TUrns Ratio . . . . l\'iub.mi Induclancc . . . . . . .

Leakage Induct.ance . . . . . .

Primary-to-secondary Capacitance.

V. MEASUREMENrs INVQINrNG nc BIAS

Varactor Junction Capacitance Transistol' Diasing . . . . Large Electrolytics . . . . . . Grounded Measuremenls . . . .

VI. MEASt.'RDI E�TS WrrH AC OR K'OISE SIGNALS PRESENT

Transducer Measurement. . . . . . . . . . . . .

Measurcmcnt oC Mixer Input aoo Output Impedance.

VTI. ACTIVE CmcUt'i' 1I1EASUREMENTS

vrn. X-Y PLOTTfNG

2

APPE!\DIX

General Immittancc Review . Nature of Q . . . . . . . . . •

Transmission Line Equations .

3

5 • 7 7

9 10 10 11

I. 1. [. I. 15 15

16 1(; I. 17

I' 20

21

23

25 27 29

SECTION I

INTROOUCTION

Like voltage and current, impedance i s an im­portant quantity to t h e design engineer. Unlike the other two parameters. however, impedance is more tedious for the engineer to measure. The basic dif­fi c u l t y with impedance measurement has been the lack o f suitable broadband, general purpose instru­mentation for the task. As a result, the design en­gineer is reduced to verifying his predicted results by using bridge techniques. Thus, the evaluation of wide ranges of impedance over wide frequency ranges has been a long, tedious process Involving point -by ­data collection and interpretation.

In recognition of the design engineer's require­ment for a general purpose Impedance measuring de­vice, Hewlett-packard engineers developed the 4800A

Ilnd 4815A Vector Impedance Meters to cover the fre­quency range between 5 Kz and 108 MHz. These in­struments are general purpose in the same sense as the broadband oscilloscope, the a c vacuum tube volt­meter, a n d the frequency count.cr. That is t o say, they allow the user the nexibiHty of rapidly making measurements over w i d e frequency a n d impedance l"anges directly in the parameter of interest to him. In essence, the instruments provide the design engi­neer with a fundamentally new class of instrumenta­tion.

IT one recalls the fundamental relationship in the series equivalent circuit, driving-poinllmpedance it; defined as the ratio of the voltage applied to the cur­rent entering one port. of the circuit. It may be rep­resented vectorially either as a point in the complex impedance plane having Cartesian coordinates R ± jx or polar coordinates IZI /.1.., where e i s t h e angle between the voltage and current vectors.

AUTOMAnc "'"

CONTRO L

r--

I •

r-TEST

�"

I ,

'"

t

Operation of the V e c t o r Impedance Meters i s based directly on this fundamental definition of driv­ing-point impedance, a n d t h e impedance vector is read out in the polar coordinates Z &. Figure 1 is a Simplified block diagram of the typical Vector Im­pedance r..-rcl.cr. In operation, a broadband oscillator applies a C W signal to an amplifier witll leveled out­put. Current from the amplifier then passes through the unknO'wn component, mounted across tenninals A and B, tllat is to be measured. Current passes (rom t h e B terminal t h r o u g h t h e ammeter t o ground. Thus, the current through the unknown is sensed by the ammeter a n d used to generate a leveled signal that, in turn, levels the output of the amplifier. The levellng then holds the current through tile unknown constant. Since Z = E/l and I Is now a constant, Z Is directly proportional t o t h e voltage across the un­known. T h e r e f o r e, a h i g h impedance, broadband voltmeter i s placed across terminals A and B, and the output is calibrated directly in impedance.

The magnitude of t h e driving-point impedance vector has been established, and it is only necessary to determine t h e phase angle existing between t h e voltage across and the current through the unknown. T o measure phase angle, nc outputs from botll the voltmel.cr and ammeter arc used to trigger a ;l;e1'0-crOSSing phase detector which is calibrated directly In phase angle.

TIlus, the Vector Impedance }.ieters represent a complete measuring system consisting of an excita­tion oscillator, broadband v o l t m e t e r, a m m e t e r, and phase meter. The entire system is calibrated in the parameter or interest, impedance, and the Inac­curacies of all elements in the system are taken into

VOLTAGE

T IMPEDANCE

CHANNEL M E TER

PHASE METfR

J C URRENT CHANNEL

1 �

Fig. I . Simplified Dlodl. ViaKr(ll/J _ VeClOr Impedance Meters.

3

account so that the driving - point impedance may be measured with known accuracy.

1\vo Vector Impedance Me t e r s a r e available , b o t h operating w i t h the same basic concept. The 4800A Vector Impedance l\Ieter covers the frequency range from 5 Hz to 500 kHz. It utilizes two measur­ing terminals. Both terminals are above ground and

Fig. J. VeCIOr impetiann' Melers.

The HP Motit'/4800A Vector Im{J�lhmce Meter euvers the 5 Hl tu 500 I:flz !rei{lIrncy rullge. Measllremrllfs are made aI/wo IlIIl!rcJllluled />anw/(I I'/tl8 t(trmina/.t.

The fir Modl'i 4815.4 HI-" Vector Impel/llncC' Meter covers ,/(p 500 kHz 10 108 MII� frequC'lIcy /w/!:e. Measllrillg terminals are ("Ol/lllillCd ilia probe whi"h 11111.1' be tlse.d t() make ill-circuit measure· mel/I.t re/erellced 10 Kr",md.

4

are connected directly to the amplifier circuits. The 4 8 1 5A R F Vector Impedance Meter operates from 500 kllz to 108 MHz and uses the synchronous sarn­pling technique to convert the R F signal to be mea-6ured to a narrow - band IF o f 5 kHz. It measures impedance a t t h e t i p 0 f a probe w i t h respect t o ground. Both instruments measure the unnormalized impedance, tltere(ore, they are not referenced to any characteristic system impedance. TIL e instruments are completely specified a s vector impedance sys­tems containing everything necessary t o m a k e di­rect' instantaneous driving - p o i n t impedance mea­ments.

Measurements 0 f passive circuits a n d devices are the most common measurements made with the Vector Impedance Meters. These measurements are made by simply connecting the "unknown" to the in­strument and setting the frequency as desired. The vector impedance readout glves the driving- point im­pedance of the "unknown" over the frequency range of interest. Perhaps the greatest single application of the Vector Impedance Meters is that of evaluating the actual impedance of real circuit elements such as re­sistors, inductors, and capacilors. The circuit ele­ments are never ideal, and calculating the exact im­pedance is di fficult. 1be Vector Impedance Meters not only give a fast measure of the actual impedances, but also. allow the engineer to evaluate the parasitics associated with real elements.

OULer areas of appl ication include measurements with d c biasing. Diodes, varactors, and electrolytic capacitors are some of the devices discussed in this category. The Vector Impedance Meters can even be used in some measurements where noise a n d other a c signals are present. Examples are transducers, m ixe rs, and antennas.

This Application Note describes a number of use­

ful measurements that can b e made with these de­vices and details some of the precautions to be taken and "techniques" that must be used.

SECTION 11 EVALUATING COMPONENTS

The Vector Imll·edance Meters aHer an extremely versatile technique for measuring the resistance, in­ductance, and capacitance 0 rei l' C II I t components. These devices may be connected diroctly to the mea­suring terminals and their complex Impedance exam­ined rapidly at specific frequencies of interest over a range of frequencies. The presence of phase infor­mation in this mCal:lUl'cment is quite valuable since it provides a good indicator of the extent to which the r e a I element approaches the ideal counterpart. At certain speCific frequencies, the value of series re­active elements In a y be measured with the readout directly in their primary units (microhenries or mi­Cl'ofarads). Wh e n measuring COnll)oncnts, t h e fol­lowing relationshil) should be kept in mind for series equivalent circuits.

R ", IZI Cos8

.. lZllsin8'forOO <."lSOO L 2; .. f

c

Fig. J. lJiscr"'t" ("(ml{Hm"IIIS ilia), b .. mt""JII"·,/1II high juqllcndu ... i,ll Iht" 48/5..1 b.l"cfippillK III .. m IOIII .. com,><m",11 ",,,,,millKw/"Pltr.

DIRECT MEASUREMENTS

T h e resistor mcasurement is, of course, the m o s t straightfol'ward. Si n c e rn.>quuncy d o e s not enter direclly into the formula, :tuv frequency where the phase angle is within ± 5° of 00 will yield the re­sistance a t that frequency. I t i s possible t o f i n d s o m e frequcncies where certain types of resistors will show a significant inductive 0 r even capacitive reactance. This is generally a result of lead induc­tance or inter-winding capacit:l.Ilce.

Figure 4 shows the type da t a that may be ob­tained f r o m measul'ements o f resistors. The fre­quency dependency of three 200-ohm, 1% precision resistors 0 r different constructions w a s measured. CUn'e 1 presents the data for a melal film resistor. It can be seen, that although the component is slightly capacitive (small negative phase angle), its \'alue is relatively constant over the range of intercst. Curve 2 presents data for an inductiveLy compensated wirc-

"00 ,

/ ! ,

/ , , ,, .

? �I /,

/, o· •

/ /"" V

.,.

L / -, -.

,

...

" , r .. 9 "

...

, METAl. FILM RESISTOR

, W1REWOUND RESISTOR, INOUCTIYELY COMP£NSATEO

, WIREWOUNO RESISTOR, NO COMP£NSATION

, , , FREQUENCY(Mliz)

Fig. 4. Tn1iCII/ "mll "bllli""iJle 'rOIll mi'I1JU,i1!g '''.<i.<I<JTJ· ",ilh 1/" V"Cf()T /m{l(''''''I<''' M"'t"fs.

wound '·csistor. Note that its usefulness degenerates rapidly abovc 5111Hz due to the uncompensatcd induc­tance. Curvc 3 shows an uncompensated wlrewound rcsistor. n can be secn that it is practically useless above I MHz. In this illustration, the black curves show the absolute magnitude of impedance; the color curves show the value of the phase angle .

Reactive elements can be quickly characterized in tenns of Z, (J over the operating frequency range with the Vector Impedance Meters. In addition, both inductors a n d capacitors can b e measured in their primary units (,..1{ or ,..F) b y selecting a t"rcqullncy wherc thc phase angle is within 50 of ± 900 rcspec­tively. Also, the frequency should be rclated to 1/2" or approximately 0.159 by a power of ten.

f 0 _,_ x 10n 2 "

For case of operation, the 4800A frequency dial has "LC" inscribed at l'i.9 and the 4815A has an ar­row head at 1.59 MHz and 15.9 MHz. Thus, at the frequencies of 15.9 Hz, 159 Hlo, 1.59 kHz, 159 kl{z, 1.59 111Hz, and 15.9 MHz. the Impedance scales are calibrated directly i n microhcnries o r microfal'ads and components may be measured directly.

For Instance, the 4 8 0 0 A frequency dial may be

5

I

set to 1.59 kHz and the instrument used as a bridge would nonnally be used. However, in this applica­tion the 4800A will read an unknown component, to­gether w i t h all of its associated distributed para­meters, quickly and directly without the nul ling and balancing operation .required f o r bridge operation . The following table gives the scale factors that apply when the 4800A is operated in this mode.

Z Raoge Dir(X:1 Reading C Range Direct Reading L Raoge

" iO 1''' 11111 I'F 100 I'H ·1 mH

X," , "' III I'F , rnH iO mll

XIOIl .11'1' , "' 10 mH - lOO mU

X1K .011'1' .11'1' 100mH , H

XIOK 1000 pt' .011'1' , H '" H

X100K 100 pt' 10110 pI' '" H >0' H

X1M 10 pP 100 pI'

T ublt J. Direcl reuuing �'ullles wlren rlre 4/JOOA is opt'rutec/ ut /.59 "Ht.

The following measurements w e r e made o n a toroidal inductor w i t h a ferrite core. A s the fre­quency is v a r i ed, t h e inductor appears resistive , then inductive, then resistive at self-resonance, and finally capacitive.

Frequency Izl �I 159 Hz 1.10R 0°

15.9 MHz 1.91KR 90°

29.3 MHz 9G.OKQ 0°

43.5 MHz 3.10Kn _8So

L . Izl "lOnIOl'H at 15.9 MHz: n '" -2, L -19.1I'H

A capacitor can be evaluated In a similar man­ner. As the frequency is increased It appears capa­(.:tive, resistive a t resonance, a n d then inductiv e . The data below was obtained from a 1500 p F dipped mica capacitor. Note that the capacitor is self-reso­nant at 25.7 MHz.

Frequency IZI � 1.59 MHz 671"1 _880

25.7 101Hz 0.612 0°

60.0 101Hz 7.9n ,,0

c . I�I x IOn iOI-'F at 1.59: n· -1, C· 1500 pF

SORTING COMPONENTS

The straightforward operation of the 4800A Vec­tor Impedance Meter and high repeatability between measurements make the instrument valuable for go­no -go checkout. For example, resistors, capacl-

6

tors, and inductors can be sorted to assure that they fall within limits of some standard. To take advan­tage of the high repeatabiLity (on the order of 1 part in 103 under normal laboratory conditions) a digital voltmeter such as the -hp- 3440A may be connected to the magnitude analog output. An -hp- 5050A Re­corder can be tied in for recording purposes. Mea­surements can be made at approximately one reading every 5 seconds.

Table 2 shows the deviation of a group of Induc­tors w i t h rcspect t o n standard. T h e impedance magnitude of the standard at 120 kHz is 2.80 ohms and has an Inductance of 3.90 �H. Six of the Induc­tors checked on the 4800A Vector Impedance Meter fall within 1% o f t he standard; I. e. , 2.78 t o 2.83 ohms. Another six inductors arc less than the 2.80 ohms standard by at least 1%; and the last group of si" is greater than the standard by at least 1%.

No Go Go within 1% of standard No Go >- 1% Standard >+1%

2.67 2.78 2.80 2.81 2.84

2.73 2.78 2.82 2.84

2.7:1 2.79 2.83 2.85

2.75 2.87

2.77 2.88

2.77 3.03

L-3.90�H, f-120kHz, IZI in ohme

TI/blt: }. Go - no·go imlllctor check.

Capacitors call also be sorted using thc 4800A Vector Impedance Meter. As with inductors, the ca­pacitors can be chosen t o have a specified capaci­tance or a specified impedance at a given frequency. Tolerances c a n be set a s desired keeping In mind both the I"epeatability and accuracy (¥>%). The fol­lowing measurements were taken on a 1 pF :!:O.5 pF dipped mica capacitor at 159.2 kHz:

C (pF) Z (MIl)

C, 0.817 1.23

C2 0.847 1.19

C3 0.849 1.18

C, 0.853 1.17

C5 0.859 1.16

C, 0.992 1.01

C7 1.00 1.00 .5%

C, 1.03 0.98

C6 and C8 fall within :1:5% of C7. C7 is 1.00 pF to an accuracy of :i:7%. By USing the Vector Impedance 1I1eter, capacitors which vary by as much as :1:50% i n

this case can be quickly sorted to faH within 01,5% of a given value. Since t h e measuring terminals of the 4800A are balanced with respect to ground, extreme­ly small capacitors can be measurcd directly.

Similar measurements m a y b e made w i t h t h e 4815A; although, in general, the achievable repeat­ability is not as high as that with the 4800A.

CHARACTERIZING TERMINALS OR FIXTURES

Either the 4800A or the 4815A has the capability t o characterize the residual impedances associated w i t h i t s own terminals. Fixtures attached to t h e measuring terminals may be also characterized; and, if desired, their equivalent circuit developed. This allows mcasured data t o b e quickly and accurately corrected if necessary. For instllnce, the measur­ing terminal residual impedance of a typical 4800A may bc quickly determined by setting its frequency to 159 kHz and adjusting the impedance range swit.ch for an on-scale reading. In making this measurement on a standard production line instrument, an impedance reading o f 6 megohms, phase angle _900 w a s ob­tained, iodicatlng that the terminal residual placed in paraUel with the unknown component is .16 picofar­ads. If a fixture Is to be placed on these terminals it may be also characterized. In this I>articular case, It was desired to provide component mounting using quick conned spring clips for the component mount­ing adapter. The adapter was placed on the termi­nals a n d the new residual terminal Impedance was measured. The valuc with the component mounting adapter aUached to the terminals was 1.25 megohms , phase angle -900 indicating that the terminal capaci­tllncp- had been increased to .8 picofarads.

CHARACTERIZING CRYSTALS

Anotb.er component w h. i c h i s quickly and thor­oughly analyzed with the Vector Impedance Meters is the quartz crystal. Each resonant mode of a crystal may be represented by the equivalent circuit in Fig­ure 5.

Fig. 5. cqllivolelll circuit oj a cry.tWI rltJOtIOtrce.

This circuit exhibits a series and parallel reso­nancever)' close in frequency to the series resonance which occurs at the lower of the two frequencies. Rs i s observed directly by tuning t o series resonance and r e a cl i n g the impedance magnitude. Since t h e series resonant impedance is generally very small , t h e O.5-onm probe residual should b e subtracted when using the 4815A. If a counter such as the -hp-5245L, is connected to the R F output, the frequen­cies of series and parallel resonance may be deter­mined. One additional measurement will permit aU of the elements of the equivalent circuit to be com­puted.

If the Vector Impedance Meter is blOed to a fre­quency slightly below series resonance, the Imped­ance measured is the reactance of Gp and Cs In par­

allel. In othcr words, the total parallel capacitance, CT, can be measured. All of the values may now be computed from the following formulae.

L •

where fs Is the series resonance frequency and fp Is the parallel resonance frequency.

The Q of a crystal resonance is determined by the

bandwidth method. An external oscillator input per­mits a f r e q u e n c y synthesizer, such a s the -hp-5100A/5110A, to be used as a high-accuracy, hlgh­resolution signal source, if desired.

A typical crystal has a fundamenl.al resonance and a number of harmonically related resonances. It m a y also have a number of s p u r i o u s resonance modes. Ench of these takes 1ho form of the equIval­ent circuit, such as described above.

For many Cuodamental crystal rcsonunces. the impedance at parallel resonance will be greater than full s c a l e Qf t h e Vector Impedance Meter. The reading is brought on scale by placing a lOOK -ohm resistor in parallel with the crystal. The approxi­mate Impedance of the crystal can then be estimated. A metal film resistor is recommended for this pur­pose. Measure it separately nnd use this reading.

Tne following example s h o w s typical crystal measurements. Results of these measurements are shown graphically in Figure 6.

fa • 27.6658 MH7., Za '" 18.412 L!t!. fp = 27.6701 MHz:, Zp ,. 23.5KrI �

fs and fp are measured with nn HP 5245L Counter.

The O.S-olun probe residual should be subtracted from Zs.

R!j 18.4 - 0.5 • 17.91l

Rp 23.5KrI

CT Is fouod by setting the impedance meter to a frequency, just below series resonance, where the phase is approximately -900. The impedance at this point is 1820, /_82°.

7

_1_ .Z 1 (2.) (27 x 106) (182) • 35 pF

Any desired correction Cor stray capacitance should be subtracted from CT'

Using the above formulae:

Cp � eT '" 3:) pF

2 ''J, - r,l '" 10.5 IF Cs � eT �

L - 1 5.4 mR 4. 2 r 2 , c,

The Q of the crystal at series resonance is found by measuring the center frequency at series re80n­and the frequencies at both :1:45°.

Q

8

r, 6< 27 .66596 - 27.66572 27.66&84 1.15 x 105

,

,

, ,",

•

, •

\ ; I

r, 21.66S/I

J /

FREOUENcY (MHz)

\....

Fig. 6. 1mpt'donct! magnitude ami phasf' us a function of /requtll('Y in Ilrt! rf'gion of urits and porolld crystal rt.wnana.

SECTION III

TRANSMISSION LINE MEASUREMENTS

T h e Vector rmpedance Meters m a y be used to measure c h a r a c t e r i s t i c lmpedance, attenuation , length, and velocity of propagation o f transmission lines. The speed with which tile measurements can be accomplished on relatively short pieces of cable and the accuracy realized in using the simple, direct techniques described below, has made the instrument a valuable tool for the design engineer concerned with coaxial elements, as well as for the quality control engineer making spot checks during the manufacture of cable. These techniques are of primary interest at R F frequencies so that use of the R F Vector Im­pedance Meter Is Implied. Figure 7 shows the 481SA measuring coaxial cable parameters.

Fig. 7. 1"ht' 481JA RF Vector Impedunce MeUr may be /lstd /0 meaSllre transmission line parulllelerJ. Z. ". (mJ C are remlily detl!rmined.

CHARACTERISTIC IMPEDANCE

Several methods arc available for the measure­ment of characteristic impedance (Zo) with the RF Vector Impedance Meter. One of the most "atisfac­tory involves the familiar relationship, Zo =,,/Zi ZL' where Z i is the input impedance of a Quarter -wave­�ength line with a given termination, and Z L is the Impedance of the termination itself. For simplicity, this relation is used in the form Zo �R 1 R 2, where RI is a resistance measured directly on the R F Vec­tor Im pedance Meter terminals and R 2 is the input resistance of the Quarter-wave line terminated by RI. The actual procedure used for this measurement Is given below.

The R F Vector Impedance Meter Is s e t t o the desired measuring frequency. A piece of the Saml)le cable I s cut to a length corresponding t o approxi­mately o n e quarter-wavelength with both ends c u t back just enough to expose the center conductor and shield. If the cable dielectric Is solid coaxial poly­ethylene, this length m a y be taken d i r e c t l y from Figure S.

"' "'-\, " ...

-

'00

• ,

� , I": \,

'"

rllEQUENCY (MHz)

.

.

"' '" '00

Fig. 8. W(l\"elellglh '·S. fr<'qllt'lIcy i()r m(lxial cablt's hrl"illg poly· ('lhY/l'lIt'tlilllt'ctric.

To verify the correct cable length, one !:me! should be shorted and the other connected to Ihe 4S15A. The 4 S15 A should show a phase of zero degrees if the cablc is exactly �/4. IT it reads negative, the fre­qucncy should be lowered 0 r the c a b I e shortened. Likewise, a positive phase angle indicates too high a frequency or too short a cable. Since the character­istic impedance d o e s not c h a n ge significantly [or polyethylene at frequcncles above 20 MHz, it is usu­ally more convenient to adjust the frequency.

F o r the termination, s e l e c t a resistor whose resistance is roughly equal to the estimated charac­teristic impedance of the cable. If the latter cannot be estimated, 50 ohms will usually suffice. Remove the short circuit from the end of the quarter -wave­length I i n e and cOlUlect the reS istor i n i lS place, keeping the leads as short as possible. Now record the Z meter reading as R I ' The resistor should thcn be removed from the end of the cable and mea­sured directly across the probe to provide the value R2,

In a typical measurement, m a d e on a quarter­wavelength section o f RG 58/U cable, a r e s i s t o r which measured 47 ohms directly across the probe , was used to terminate the line. T h e line with this termination measured 64 otuns. Then,

Zo .. "'64 x 47 • 55f1

9

Since the line I s a quarter - w avelength section, the phase angle should be approximately zero.

An equally satisfacto ry method of determining Zo IS based on the relationship:

101,600 ne

where v is the velocity of l)rOpogation bctOl' In per­cenl and C is the cable capacitance in pl� per root .

The latter value Is determi ned by attach ing a very short length of the cable; i. e. , less than" 14, to the R F Veclo r Impedance Meter and measuring Cr di­rectly, T h e velocity o f I)J'Opaf,'Ution may be delcl'­mined as described in a later section,

A third method 0 f measuring Z 0 may be worth mentioning, although it is less satisfactory with re­spect to accuracy,

This method is impUed hy equation (3) on P-J.ge 29 in the Appendix, which indicatcs that the chal'l1otel'­istic impedance of a loss less I ine is equal to the ab­solute value of the reactance of a short-circuit sec­tion 1/8 wavelength long ,

To obtain the correct length, a quarter-wave set·­tion is rtrst established i n the manlier describl;'d ahove , a t a frequency twice t h e desit'ed meaSul'ing: frequency, This ft'Hquency is then hah'ed and the re­actance of the shol'l-eirculted sect i o n detel'm ine{1 from the phase reading of ±-15°,

ATIENUATIQN

A c o n ve n i c n t method or mc:\sul'ing attenua.tion, W:;lng short pieces of cable, is pl"o\'ideu by thceclua­lion,

60 x 8.69 uB o.L -

where a is dB per unit length, L is length, and S.U!) is the constant of proportionality. Here the valueof Z· is determined by measuring the panlllel resistance Of

l a piece of cable half-wavelength long open cireuited

at the rar end, If the frequency I s such that it half wuvercngth i!S less than approximately 4 feet, a onc or three -halves wavclcnbrth piece can be used, with no change in procedure, to minimi�c the effect or irreg­ularities in the cable. The attenuation in dR obtained for the length of cable tested can be readily adjusted to dB per 100 feet,

When the desired frequency has been selected, cut the cable to onc -haLf wavclength, and cut one cnd back only enough to make conta ct. Connect the cable with the far end open eircuited t.o the R F Vector lIn­pedance Meter. The meter should indicate a phase angle 0 f approximately zero. If the phase i s neg­ative, either lowel' the frequency or shorten the ca­ble slightly. On t h e other hand, a positive pba.se r equires a Slightly longer cable 01' higher frequency . With the phase equal to zero, the IZI met er reading is the propel" value for Z i to be used in the equation .

As a n example, a onc- half wavelength section o f RG-58/U cable at 77MHz (52" long) was found to have

10

a Zi of 2750 ohms. Applying this \'alue to the for­mula above, together with the known charactel"isLie impedance o[ 55 ohms, and adjusting for the length of Lhe section, the attenuation was found to be 3.!)8 dB per 100 feel.

BALANCED TRANSMISSION LINE

CHARACTERISTICS

Dual or balanced transmission lines, such as "twin lead", eannot be measured by direcL connection to the R F Vector Impedance �letel', but must be at­tached through a "balun". The funotion of the bulun is to supply an n F signal which is equal and opposite in phase to cach conductor of the halanced line, while providing a n equally hIgh re!Sistanee path t.o ground ror b oth elcments. The re al'e a numher of forms which can be used, depending on the !Situation, Sev­enll cOllllllet'eial types lire a\'a i l a b l e if dcsil'ed. Probably the simplest type of ba\un can be made by doubling a onc-half wavelenl,>1.h section of coaxial line and connecting the outer conductol'S lit the ends, 'nle outer conductol'!S are t h e n connected t o the probe ground and onl;' eenter conductor Is connected to the center pin, The balanced line may then be connected to both eenlcl' conductors . Fi gure 9 il lustrates such a balun connected at the end of :1 half wa\'eleng:th res­onant section. Since this t,}lIC of balun acts as a 2:1 voltage tl'llnsformel', or a -1:1 impedance transfOI'mer, the factor ·1 must he used in computing tile deSired charactCl'istic from the measurements m ade,

b� '------I I BALANCED

�",='�'�";'=� ,

- , ,

To make tranS!11iRsion I i n e measurelllents with lhis type balun, the following procedure is used,

CIW.raCleristic Impedance

1. !\leaS\lI"C the b a l ull, open-eil'cuited, on thc H. F VectOl' tm pedance !'Ilctel'. If t h e length is cor­rect, the phase should equal apl)l'oximalcly 00, If not, readjust the f n .. ,<!ueney slightly until the length is eOl"l"eet,

2. 'Measure and cut a section of the balanced line to b e measured slightly I o n g e r than one -quarter wavelength and attach it to the balun with the (ar end shorled, Shorten the line gradually by eULting off small pieces, until thl;' phaSe once again [s essentially 00. (Do not readjust frequency,)

3, Con!ICct a resistor, which is close t o t h e esti­mated line impedance and nonreactive, to the far end of the qual1.el'-Wave line und measure the im­pedance of the combination. Record th.is value as Z),

4. Measure the Z of the resistor connected directly to the probe. Call this value RZ.

5. Then, t h e c h a r a c t e r i s t i c impedance o f t h e quarter-wave cable is given by:

Attenuation

1. Mcasure the balun open circulted, making s u r c t h a t the phaSe is approximately equal t o zero. Record Z as RI.

2. Measure a half-wavelength section (or multiple thereof, to provide a minimum length of 4 feet), open circuited at the far end, and record the re­sult, IZlasR2'

3. Then the parallel l'esistance of the cable (R3 ), as seen through the balun, is:

4. Attenuation is:

dB _

100ft. -

Electrical Length

The measurement of electrical length of cables is based on quarter -wavclength resonances. Providing the cable is not too long, the cable will resonate at e a c h quarter-wa\'elength multiple. Therefore, the frequency difference between any two adjacent reso­nances will b e the frequency corresponding to one­quarter wavelength.

To make electrical length meaSurements, one end o f the cable should be shorted. This will aSsure a large standing wave to detect a resonance. Next, the impedance meter should be adjusted until a resonance frequency Is found. The impedance magnitude range should be switched to maintain an on -scale reading. Finally, the frequency should be adjustcd upward or downward until the next resonanCe is found. The fre­quency difference bctween these two adjaccnt reso­nances will b e the frequency of a quarter - wavelength cable. Check to be sure that these two frequencies are adjacent. One will show a large impedance, the other a small impedance. This necessarily follows since one is an even quarter -wavelength multiple and the other an odd multiple. All odd quarter-wavelength resonances will have a high impedance if the cable is shorted.

The foUov.1ng example will clarify the technique. First, the cable is shorted. The lowest frequency of resonance is 50.7 MHz and has an impedance of 3.62 KQ 00. Tuning upward in frequency, the next reso­nance is found at 99.9MHz and shows an impedance of 1.40Q &0. The difference between the two resonant frequencies is 49.2 MHz. rt follows that the frequency differences between theSe resonances is the frequency of a quarter - wavelength section. Thc e 1 e c t r I c a l

length, le' can now be calculated from the formula:

1 • • c

4ii"F

where c is 3 x I08m/sec and aF is the frequency dif­fercnce between adjacent quarter - wavelength reso­nances. For the example used le = 1.5 m.

The actual phySical length Ip of the cable may be detennined from t h e relationship Ip = le/�pro­viding the dielectric constant of the cable is known. Measurements of length with the 4815A are, however, limited to cable lengths o f about 150 meters of air line. Also, as a matter of technique, the first quar­ter-wave resonance should be found at the lowest fre­quency within t h e r a n g e of t h e instrwnent. TIlis technique a l s o allows cables t o be cut t o a known electrical length or two cables to be cut to thc same electrical length.

BALANCED MEASUREMENTS USING A

BROAD BAND BALUN

A broadband balW1 makes possible balanced mea­sUI'cments with the 4815A. The balun in effect puts a large reactance between the unknown and the probe ground. It also acts as a transmission line. In other words, the unknown appears as the termination of a transmission linc with length equivalcnt to that of the balun. Thus, to find the impedance of the unknown, the impedance transformation must be del.ermincd.

""'"'"'" Tlllt �o.o ...

Fi/:. 10. CO/IJ"/m'lion of {/ broodbond ba/"".

The diagram, Figure 1 0 , shows the construction of a broadband balun. It consists of a rigid coaxial cable approximately 7 Inchcs long and another con­ductor o f the s a m e diameter. Ferrite toroids are placed around both conductors to create a high imped­ance to ground. The inner conductor of the coax is connected to t h e other conductor and the two rigid cables are joined at one end as shown. A BNC con­nector is used to conncct to the 4815A probe.

The toroids around the o u t e r conductor o f the coax produce a high impedance between the balanced terminal output and ground. A s a result, b o t h the inner conductor and outer conductor are above ground. Thus, the two terminals see the same impedance to ground and are effectively balanced.

The impedance correction fOl" the electrical length of the balun is quickly accomplished with the help of a Z L.!J. chart. (The Z L.Jl Chart is a Smith Chart ex-

11

pressed in tenns of Z and e, rather than R and X. ) By measuring the impedance of the balun terminated in a short and locating this point on the Z L!i. chart , the wavelength at one frequency is detennined. All measurements w i t h an unknown terminal should be rotated the wavelength of the balun to account for the impedance characteristics of the balun. The example below describes the procedure in detail.

balun is found by measuring the open - circuit imped­ance. At 45 MH1., the balun described is 160Q /-850 . To determine the Significance of the shunt impedance o f the d e v i c c under measurement, the 160Q /-850 :mpedance must be rotated toward the load 0.06 }" . After rotating 160fl /-85° toward the load 0 . 0 6 }" , the Impedance shunting t h e balanced termi.nals i s 550fl

l..1.§0 as read on the Z L!2 Chart. Therefore, any mea-

Fig. 11. ZLfi Churt showing determillalion of .... aye/ength vf ba/,m ut 45 MHz.

Referring t o the Z IJl Chart i n Figure 11, t h e following steps are necessary to detennine the wave­length o f the balun at 45 MHz. First, the balun i s shorted. U n d e r shorted conditions , the impedance meter reads 20Q at an angle of 900. This point is then located on the Z L.§! Chart as point A. Next, since a short applied directly to the impedance meter probe ideally should read Of! a t an angle of 00, this pOint should be Located as point B. FinaUy , the wavelength of the ba\un is determined by reading the difference in wavelength between points A and B on the Z L!2. Chart . At 45 MHz, the balun described has a wavelength of 0.06 }., . Each impedance measurement made with the balun at45 MHz should be rotated 0.06 }., toward the load to correct for the impedance transformation due to balun wavelength. The correction must be deter­mined for each measurement frequency.

B e s i d e s correcting f o r the wavelength o f the b a l u n , a correction for the shunt Impedance effect may also be necessary. The shunt impedance of the

12

surement made using the balun is shunted by 550f! /700. Thcrefore measurements made with the balun require two corrections .

The procedure for making a measurement at 45 MH7. with these corrections is as foUows. First, measure the device under test with the batun, locate this point on the chart and rotate 0.06.\ back toward the load . Second, correct for the effect of the balun shunt impedance by converting to admittance and Cartesian coordinates and subtracting the admittance of the balun. Reconverting to impedance will yield the corrected balance measurement of the device under test. If the impedance of the device under test is appreciably less than the impedance of the balun, the shunt effect of the balun may be insignificant and therefore ignored for this measurement.

To measure the effectiveness 0 f the balun, one terminal should be shorted and the other opened. The

impedance i s t h e n measured. Next, t h e short i s switched to the open terminal and the impedance is again r e a d . A t the frequency of application, th.ese two impedance readings should agree. A t 45 MHz ,

the b a l u n measured 174Q 1_840 and 173�0 re­spectively. The frequency range is determined by the effectiveness of the balun. By adding more toroids , the low-frequency balance can be improved.

SECTION IV

TRANSFORMER MEASUREMENTS

One of th.e more sophisticated capabilities of the Vector Impedance Meters is the capability to quickly characterize transformer a n d s e r v o motors. The following paragraphs describe measurements with. a typical audio transfonner(Figurc 12) using the4S00A . Th.e vector impedance ch.aracteristics of this trans­former are sh.own in Figures 13 and 14.

Fig. 12. A number of mmstormer parameters ml1y be tie/ermined with both Vec/(lr Impedance Me/er.'" An iroll-core trallsformer, used (1/ /oll'er frequellcies. may be qui,"kly du",/Cleriud IIsing the 4800A

A plot of vector impedance, as a function of fre­quency, is probably the most complete data that can be given for a transformer under any given operating conditions. Referring to the plots in Figures 13 and 1 4 , several significant facts can be obtained. First , the bandwidth can be found from the 450 phase points which occur at 180 Hz and 40 kHz. Next, the imped­ance over the entire band can be checked for both p h a s e and m ag n i tu d e variations. T h e magnitude varies from 900 ohms to 2400 ohms about a mid-range value of 1300 ohms. Similarly, the phase varies some 35° about a mid-range phase of 10°. Phase variation over the bandwidth will cause phase distortion in th.e audio s i g n a l. Also, a high -impedance resonance occurs at IS kHz.

Referring to Figure 15 , measurements that can be made i nc I u de the primary inductancc, primary resistance , primary capacitance , secondary induc­tance , secondary resistance , secondary capacitance, leakage inductancc, mutual inductance , primary - t o ­secondary capacitance , and turns ratio.

Fig. 13. ['lot of impedance us a functioll of frequellcy for aI/ iron­core m/dio trallS/ormer terminated in 8 oil/liS.

FiX. f4. Plot 0/ {Jf1l4se M " tlmc/ioll of frequency for cm irOIl_C'ore audio transformer lermillllled ill 8 ohms.

r c ::���I C.

'. r� h\ '.

: L ,.f I -L C'T

C - C. -L T C'

i r i Ir ......

•

Fig. 15. The 4800A is ideal for measuring para­sitic elements such as are associated with all audio transJormer.

13

PRIMARY INDUCTANCE, RESISTANCE,

AND CAPACITANCE

To measure the primary inductance , resistance, a n d capacitance, t h e I>rimary i s cOIlllected to t h e 4800A terminals and Lhe secondary is left open. The frequency is adjusted so that the impedance Is Induc­tive, as indicated by a phase of approximately 900. Using a frequency that is a decade multiple of 15.9, as marked by the LC inscription on the dial, the induc­tance is read directly.

9 = 8So

f ::: IG.9 x l O H� IZI ::: 1 .02 x lKI1 L 1.02 H

The dynamic resistance in the primary is det.cr­mined at lhelowest measurement frequency where the phase angle is 600. Since it is usually impossible t.o find a low frequeocy within the frequency range of lhe 4800A, where the phase angle [s 00, it is best to use a convenient angle such as 600. At this phase angle, the resistance equals 1/2 the total impedance. Thus, t h e resistance is 1/2 the impedance read on the Z magnitude meter.

60' 8.3 kHl. 4.1 x lOKf! Iz1 /2 : 2.05 x [OK!! ::: 20.5Kf!

The primary capacitance is determined by selec­ting a frequcncy at w h i c h primary induc tance a n d capacitance resonate. For the purpose of these mea­surements , resonance is defined by a phase angle of 00. Since Lhe inductance and thc frequency are known. t h e transformer capacitance is calculated from Lhe formula;

1 C = .. 2 L

o r found o n the V e c t o r Impedance Calculator de­scribed in the Appendix.

• f L

0' 18.5 x 1 kn..: 1.02 H (as previously determined)

1

76 pF

If the resonance frequency is beyond the range of the 4800A, an external capacitor of knovm value can be connected in parallel w i t h t h e primary and the shunt subtracted from t h e above calculated capaci­tance.

14

SECONDARY INDUCTANCE AND

RESISTANCE

The inductrulce and resistance of the sccondary are measured following the same procedure used wilh the primary, except that the secondary is connected to thc 4800A terminals ruld the primary is left open. The rcsults arc as follows;

TURNS RATIO

L "" lG.3 mH ilp = G.G x lOOn

The turns ratio is quickly aplH'oximated by using the impedance t r a n s f e r properties assuming light coupling; i. e. , K :::::; l. A resistor, R , will read a n impedance of Z when connected across the secondary and measured from the primary. The ratio of R to z is related to the turns ratio:

where N : number secondarv turns number primary turns

For best results, R should be mueh Less than the pl'i­mary reactance. The phase shouLd be approximately zero to assure th:lt the reflected impedance has n o reactive component.

The following r e s u l t s were obtained from the transformer in Figure 12.

, 0' IZI = 4.9 x lKf! R = loon

N =(�(2 1 =

Z 7

MUTUAL INDUCTANCE

�lutual inductance is determined by measuring the transformer inductance in series aiding a n d series opposing configurations , subtracting the smaller in­ductance from thc larger, and then dividing the re­mainder by four.

i\j

M ::: Laldlng - Lopposing

4

Opposing

86° 159.2 Hz 7.6 x l00Q 760 mB

1250 - 760 , 123 mll

• f

I z l '" L =

LEAKAGE INDUCTANCE

Aiding

" , 159.2 H� 1.25 x lKf! 1250 m H

Leakage inductance i s determined by shorting the secondary and measuring tile impedaoce of the pri­mary.

, [

I z l

81" 15.9 kH;o; 14R

r� - 140 �II

IT the leakage reactance ifS too small to be read di­rectly on the '1800A meter, a capacitor th:\! wl!l reso­nate w i t h the leakage inductance c a n be (;onnected across the l)rima!'y. lbe leakage inductance is then calculated from the formula:

L j

• ' C

or from the Vector Impedance Calculator.

With iron core transformers, a de component can be injected into the primary or secondary. Appropri-

ale de blocking capacitors are necessary w i t h the 4800A. (See Figure 16). Inductance as a function of de current can then be measured.

PRIMARY-TD-SECONDARY CAPACITANCE

Primary - to - secondary capacitance i s measured by connecting onc lead of the primary and onc lead of the secondary to the terminals of the Vector Imped­ance Meter. The other leads may be open 01' shorted . Phase angle should be approximately _900 .

, -880

[ = 15fl.2 Hz IZI 7.2 x 1 megR C = L:l7 p F

SECTION V

MEASUREMENTS INVOLVING OC BIAS

Il i s frequently necessary t o m a k e impedance measurements in the presence of d c bias. I\Ieasure­ments on both diodes 11 n d transistors a re common examples of this requirement. The 4815A can be used dil'eetly for these measurements providing the bias vollllr;e is II1HH than ;;0 volts. Above 50 volt8, block­ing techniques Illust be used to prevent the d e vultage from entering the probe,

Unlike the 48l5A, the 4S00A must have no dc bias sib'TIal applied directLy to the lenninals since they are connected directLy t o the amplifier channels. How­ever, b i a s measurements are possible if the d c is properly blocked . The circuit in Figure 16A shows one way to apply a bias. A blocking capacitor pre­vents d c from f1(lwing through the 4S00A. The im­pedancc of the capacitor must be small compared to that of the device under test. This can b e verified with the 4S00A. Since the d c bias suppLy appears in parallel with the unknown, the impedance of this por­tion of the circuit (dc supply plus its nssociated re­sistor) must have a high value with relation to the un­known. Ir th i s condition c a n n o t be aehievl;.>d, the

!lower s u p p l y impedance will reduce the observed value of the unknown to be measured. These readings c a n b e corrected, however, by making a separate measurement of the de supply impedance and manually eOl'l'ecting the data.

The d e supply m u s t b e lmgrounded unless the regulating resistor (R) is very large. A large resis­tance with grounded SUI)plies will is(llate the 4 8 O O A test signal from ground. See Figure 1GB.

The blocking technique described above c a n b e used to measure the dynam ic impedance of a diode . By biasing the diode from a kno\\'l'! de cur),ent source , the impedance can be recol'ded as a fWlctlon of cur­rent, IT the diode Is back biased, the junction capac­

itance, a s a function of voltage, can be measured .

r _1 _ _ I-.- UNKNOWN 00

SUPPL V

L ,

UNGROUNOEO SUPPLY

'"

R;>"'Zu '

r--I� oc 4800A UNKNOWN

SUPPLY

L� GROUNDEO SUPPLY

'"

r • A I�,ge R will elf..ctlvely i$(\late U>e te,ml ... l. flOm R'ourod.

fig. 16. Blocking circllil 10 keep de from 4800A amplifier circllit. 48/5A is oc <'o"pled ",ilir cl'pc,,:ilor ill probe".

VoHab'C-val'iablc capacitors and current - variable in­ductol's can be measured in a similar m annct·,

15

VARACTOR J UNCTION CAPACITANCE

Figures 1 7 and 18 illustrate the measurement of varactor junction callacitancc as a function of vollage, "1110 4800A is operated at 159,2 kHz so that capacitance may be read out directly in picofarads , A 419A Volt­mt:tcr is used to set the bias level aCl'OSS the varac­tor.

0 1 ' I MEOI/

'O--l H"""""'","", H .... l!IA

Fig, 17. Stlllp for mtasuring \'uroctor jum:tiQfl capac:i­IClIru as u fUIICI/OII 0/ voltage.

BIAS V{JLT .... OE (VOLTS)

Fig. 18. /un(lioll capacitallct of a lIaractor as a fUllc/ion of billS '·olta/{fl. J = /59.2 kHl..

TRANSISTOR BIASING

The biasing scheme shown in Figul'e 19A may be uscd to measure hlb of a transistor. By connectlng the circuit as shown In Figure 19B, the instrument will measure hic . If the same biasing current and

voltage are used for both the htb and hie measure­ments, beta can be approximated from the expression:

The abo\'e e}"l>rcsslon assumes that hre Is negligible, For measurements at the 4800A frequencies, the as­sumption is generally valid.

16

-­TERMiNAlS

-TERMINALS

" "

,.

'" Fig. /9. Using an appropriate biasing schtltlt (A), i/ is ponibk to ItItQS"'t h .. of U IrulUislor. By re­comr«lillg tlrt cjre:"it as ill (8), llit instrumtnt will !ne/mlfe "w. C. a",/ C. 11ft ,le blocking c:opaCilOfS. R, IIl1d R. art bilJ.t resiSlurs.

LARGE ElECTROlYTICS

A problcm that often arises when measuring large capacitors wilh a bias Is finding a larger capacitor for

blocking. The blocking capacitor need not be signifi­cantly larger if one Is willing to do some bricf calcu­lation. The following example demonstrates the tech­nique f o r measuring a (17 j.lF electrolytic capacitor biased by 45 volts.

"

O--------ll �--__, - + ....

T[llM1HALS

1..-_;;' ·1,1'1-'+ __ -'

Fig. ZOo LlIrg� tlee:trol)"lic CQpacito� may b� ",us­"re(/ by (onnectin/( rwo ((lpacitors if! series. thtl1 biasing.

The "unknown" 67 p F capacitor i s connected in series with a Similar capacitor of approximately equal value. Thc negative terminals are connected to the

4800A and the positive terminals joined. A 4 5 -volt battery is connected across the ''blocking capacitor" with regard to the correct polarity , Figure 20. Under these conditions. equal voltage appears across each capacitor. However, t h e net v o l t a g e between the 4800A terminals is zero.

The Z magnitude meter reads the series combina� lion of Cl and C2.

Izl = X ,.

C 2

1 w Izl

1 WCT

1 w lZI

If the blocking capacitor is selected to be at Icast ten limeS greater than the unknown, the term w IZI Cl , is l a r g e compared to one. T h e equation thcn be­comeS:

1 w Izl

When working with a biased capacitor, use ex­treme caution. A charged capacitor. placed directly across the terminals o f the 4 8 0 0 A, can discharge enough current to damab"e the diodes in the Instt'wnent.

Fig. ] I. St!wp for series meOS(lremt!n{ of varaClOrs.

" L.----��-----;l"�----�'�'�----�m BIAS VOLTAGE (VOLTS)

varactors can also be measured by placing them in series. Figures 21 and 22 illustrate t h e series measurement o f two MV863 varactors. These re­sults should be compared to t he above results ob­talned with one MV863.

Fig. ]2. JlIllc/ion capllciwlJce of /wo st!ries varacrors as a fullc/ioll of bios volrage. f = /59.2 kH�.

GROUNDED MEASUREMENTS

GroWlded measurements with t h e 4 S 0 0 A are a special case and are generally not recommended. A sl:mplified block diagram of the 4 S O O A circuit, Fig-

ure 23 will point out the reason for this. If a termin­al is grounded , either the oscillator or the current

channel is shorted. If the secondary of an Isolation transformer is connected across the tenninals, the grounded unknown may be placed across the primary. However, transfonners a r e generally restricted to narrow ranges of impedance and frequency.

MellSllr(lment condition 4800A 481SA

A suitable blockln, capacitor The probe contains a blocking Is necessary to pre"cnt DC capacitor suffiCient to block from entering the termInals. 50 volts de. Use external

DC bias Choose a capacitor that lIas blocking abovc 50 volts. negligible impcdaf\Ce com-pared to the unknown. Be SUfe It Is discharged before connecting it to the termInals.

AUllough the internal elr· The external sIgnal could beat eultTY is broadbarrl from S wIth harmonIcs of sampling H:t to 500 kH"'. each frcq· rale which varies from 0.5 10 ucncy range has bandpass 1 MH:t In 6p1lclng. The Inter-rntering. An external 81g- ference will cause the phase

External Signal nal. 2 decadeS from tile mea- and magnitude needles to (AC or Noise) 8urement frequency, Is swing wildly. By resetting

attenuated at least 14 dB. the frequency sllghtly, tile in-External interference will tcrferenee can often be effec-cause the pllase and magnl- lively filtered. tude needles to s"ing wildly.

Table 3. Meosllremell/s wirl! uc bios or e.r/ernal siglllll.

17

AUTOMATIC lEVEL

CONTR OL

-

r , VOlTAGE

T IMPEDANCE

- CHANNEL METER

TEST DEVrE

,

"'"' .... ,,' METER

� J CURRENT

CHANNEL

� Fill. 13, SimpUfied Block Diagram - VeCtor fmptd(mct Meters.

18

SECTION VI

MEASUREMENTS WITH AC OR NOISE SIGNALS PRESENT

Generally, mensurements with e x t e r n a l a e 01' noise l>resent require the use of careful techniques . The impedance measuring circuits of both inslrwncnlS behave as high-galn voltmeters . Thus, it is I)Ossiblc \.0 ha\'e an cxtCI'Ik'\I unwanted signal 3.lnpUncd 10 the same order of magnitude as the internal test signal of the Vector Impedance �Ieter. This situation will , of course, result in a completely spurious impedance reading. Ibe condiUon is generally casy to recog­nize. however, since it almost always results in a complete inability to obtain a stable on-scale reading of either impedance or phase, 0 r both. Since' both Vector hnpcdance Meters hrL\'e an Internal oscillator that excites circuits under test through the measuring terminals, it is n o t necessary to h a v e another ac source present to make measurements. In measure­ments where interfering noise is likely, such as wilh antennas , electrical isolation of the measul'ing setup is often necessary. In addition , devices w h i c h have the property of cOfl\<erting some othtn' parameter to elecll"ical energy must be isolated from lhat type ex­citation. F o r instance , a I)iezoelectrlc transducer may have to be Isolated from mechanical vibl'alion to prevent the generation of unwanted ac signals,

E:'lch Vpctor Tmpedance "Mcter is susceptible to inwdering signals and noise in a dlHerent way. TIle 4S00A is a bl'oadband instrument over a continuous range of frequencies with an internal amplifier system that is Oat from 5 Hz to 5 0 0 kHz, To help elimin.ate interference, bandl)aSS flUe ring Is u s e d ovel' each d c c a d c rangc of frequency. The inlerference c a n sometimes be effectively filtered by choosing a freq­quency two decades away from interfenmce,

The '1815A, on the othcl' hand, uses a unique tech­nique of synchronous sampling to convert the R F to a 5 knz I F. Although the I F has a narrow bandwidth of 120 Hz, interference a t either t h e fundamental 0 1' harmonics of the Saml)ling rale Is com'cl'led to the IF fl'equcncy, Consequently , the 4815A c;ln b e consid­ered as a narrow -band Instrument at discrete rreq­uenclcs . Depending on lhefrequency ofmeasuremcnt , the sUlllpling harmonics will occur at Inter·vals spaccd up to 1 l\IHz apal't over t h e cnUre 0.5 to 108 l\'IHz range , Thus , the Instrument is susceptible to Inter­ference from broadband noise, The conditions under which measurement In the presence of noise can be made arc discussed in the following paragraphs . It should be I'cmembcred that a limitation of 50 volts de at the probe tip must not be exceeded.

At thc frequency of thc test signal, the pl'obe of the 481SA appears to be a constant - cUl'rent source: at aH other frequencies, it appears 10 be an essen­tially resistive iml)edance of approximately 25 ohms , Assuming t h a t the Intcrferlng Signal Is n o t a t the frequency of the test signal , there arc two ways I n which it may interferc with thc meaSU1'cment, First , it may o\'erloa<l the i np U t sampler and produce Inlel'modulatlon, This will occur if the unwanted sig-

nal amplitude Is g re a t e r than 0 . 5 volts nns I n t o 2 5 ohms,

Second, t h c interrcl'ing 81gnal may beat with a hal'monic of the sampling raLe and produce a spurious IF signal. If the unwanted Signal is an unmodulatcd carrier, the beats will be observed as the 4815A is swept In frequency, Thc measurement can be made by tuning the 481SA test signal frequency to a point between the beats, This tcchnique can be used whcn measuring the Impedance o f a mixer with the local oscillator excitation applied, If the signal is modu­lated (particularly frequency or phasc modulated), the chance of beating is greatly increased, and It may be impossible to find a point at which interference is not occurring. In this case, the measurement cannot be made accurately (or at all), dcl)(mding on the scverity of thc interference. This condition can be recognizcd b y exli'cmely n o i s }' meter indications and / or wild variations in impedance wHh minute changes in test signal fl'cquency, and/or complete lack of l'ange - to­range tracking 0 f the impedance magnitude scaies . Wide -band noise presents the same difficulties as a frequency or I)hase modulated signal, since the nW"Il ­ber o f interfcl'ing frequencies is i n f i n i t e . In this case, the measurement can be made only if the mag­nitude of the noise signal is small compared t o the lest signal level.

I� or the case where the interfering Signal present In the cIrcuit is at exactly the same frequcncy as the test signal, the m e a s u r e m e n t wi.l1 b e essentially meaningless since the signal will be convel'ted to IF as discussed under the second condition abo\'e. One exception to this rule Is t h e case of measurements within the feedlxl.ck loop of an osclllatol', If the source iml)edance of the 1001) is large compared to 25 ohms at the point of measurement , loading of the probe will stop oscillation. The impedance observed will have a nebrative real p a r t (negatlve reslstanec component) , since the oseillatol' is supplying powel" to the probe , The frequency at which the phase angle is 1800 is the frequency of oscillation, The magnitude of the nega­tive resistance varies inversely w i t h the amount of I)osllive feedback, This particular application Isdis­cussed in detail in the section on active circuit mca­surcments ,

TRANSDUCER MEASUREMENTS

Thcre Is an application for the Vector Impedance Meters in the measurement of transducerparametcrs . Sonictransduccrs are one example, Some precautions must be taken i n making these measurements , how­ever, since transducers are often broadband devices a n d may pick up n o i s e very effectively , This In­coming no i s e may saturale the measuring circuits m a k i n g it impossiblc to get on -scale indications . Mechanical and acoustical Isolalion of the device to be measured is often required 10 obtain useful results , Generally , w h e n noise is a pl"Oblem , Ule result is completc inability t o m a k e measurements , rather

19

" c-" " " " ,

i , , ,

, ,

, ,

"

. . .. ..

, ... ur ....

Fig. 24. Plot 0/ impedance and pllllu as " iWIClio" 0/ ,rl"llIrllcy Ior u liydrophol1t1.

�

�

�

'"

I w ,

, � -w

-'"

-�

-<>

� .... w

G{mmhos)

Fig. 25. Locus of j8 "'< {/ Ilme/ioll 01 G JO'Il hydroplu",e.

than erroneous rcndinb'S. The advantage of the Vec­tor Impedance M e t e r s i n th i s application I s , o f course, the capability to characterize Lhe h'lnsducer over its entire operating frequcncyranj{c, quickly and easily.

r-,lany transducer plots are expressed ill terms of R and jX ur G and JB, rather than Z and 8. IT H and X are the desired qua ntities , they may be m anually rep\oUed from a Z Lfl plot by resolving the rectan­gular coordinates . The general shape and discontin­uLies on the Z LJi curve will dclenninc the points that should be more closely exam ined on an R + jX 1)\01. Similarly, G and jB may be ma nually rcploUed using the reclpr'ocal of Z to obtain Y, taking the ncg­ative of the phasc a nglc reading, and ['csolving Y into the rcctanb'Uiar coordin�ltes , FigU!'llS 24 and 2:; show data taken with the Model 4800A from a hydrophone uscd [0" detecti ng undel'\�ater noises. The data was taken with Lhc hydrophone unloaded; L e " in air, al­though similar data can be obtained with thc unit im­mersed in wate " , In gcneral howevcr, this is a lab-

20

oratory mcasurement since there i s n o r m a l l y an excessive nmounl of interfering signal present in rIeld measurcments .

MEASUREMENT OF MIXER INPUT AND OUTPUT

It Is also possible to mensure the input and output impedance of a mixcr while It Is being excited with Lhe local osciltatol'. lIowevcr, spccial care must be takcn to sdect a mcasul'ing frequency such that the 4815:\ test sif,'nai will not m ix wilh thc local oscilla­tor signal, thus , producing an error signal. Suitable measul'ing freQuencies can be determined empIrically while making the mcaSUl'cmenl. Since an interfering error sIgnal Is extrem ely sensitive to the measure-'

m ent frequency, the measured impedancc will exhIbit large variations for very smal\ frequency deviations when a n intc rfering signal is causing a n error. De-

"'>--

.. ,

�

IOSI4A MIXER

CO"�"� '"

" ."" � ,

"'" "'

3 .. , •

"' V,M.

� ::--::..: ., • •

LO. 81AS LEVEL VOLTS

IS a -

10 � , ..., � -0 �

FiX_ 26. IlIpU/ impelJallCl' w " I0514A Mixer a., a fllllClirN! of local ".<eillwor bias.

tween a n y t wo successive frequencies producinglln error signal, thcrc should be a range of [rl'Quencles for which the measured impeda nce is essentiatly eon­stan\ . The impcdancc measul'ed w i t h i n this "con­stant" range Is the correct value.

Figul'e 26 Shows typical data obtained fI'om the -hp- Model lO;i14A i\Hxer, rhe Input impedance, as a function of local osc illator bias level, is plotted for a measurIng fl'equency of 30 i\lHz, It can be s e e n that the VSWR of the m ixet' improves from 1.6 to L02 as the bias level is varied, In addition , one may ex­amine the effect of changing thc loading, at the third port or the mixer, to v a l u e s other than 50 ohms , Thus , It Is possible to quickly examine the iml>edance at cach I>ort of the mixer and to determine the com­pensation n e c e s s a r y for establishing c o n j u g a t e matches ,

SECTION VII

ACTIVE CIRCUIT MEASUREMENTS

The accuracy and flexibility of the Vector Imped­ance Meters in making passive circuit measurements arc also realized in the case of active circuils. How­ever, two considerations must be taken into account . First, care must be Laken to exclude both ne and de Si gnals from the measurement. (See Sections V and V[). Second. the lest signal level of the Vector fJn­pedance Meters must be within the linear region of the circuit, since impedance Is defined only for lin­car circuits. Al s o, it should b e remembered that measurements on the 4815A are referenced to ground , while t h e 4800A makes ungroundcd me:u;uremenls . The In-circuit probe (Figu,rc 27) makes the 4815A es­pecially applicable to aelive clrcults.

Fig. ]7. The 4815A probe mllkes /1055ih!t' IUrtcr ill'circuit meUJ',.re· melllS of \'eClOr impedance.

The second consideraUon of possibLy driving the circuit into a nonlinear region i s unlikely to cause probLems since t h e test signal i !i deliberately kel)t small. The 4815A injects a constant current of 4. p.A , except on the 10 ohms Cull - s cale range where it is 12.6 p.A. Therefore, the voLtage across the unknown Is dependent on the impedance magnitude. The 4800A Injects a constant current of 270 p.Aon the XIO range , and 270 !lA on the XIOO range . On the XIK range , a constant voltage of 2 . 7 m V is appl icd, on the Xl OK it is 27 mY, on the XIOOK it I!; 207 mV, and on the X I M it is 2 . 7 V. Thus , the voltngc o r curn�nt respectively Is constant , depending on the impedan ce range .

Measuring the input impedance of an amplifier is one exampLe where i t i s first necessary t o decide whether the signal Icve\ will overdrive the circuit. Should the i np u t characteristics of t h e circuit b e known, Obscl'ving the output on a n OScilloscope may be helpful. In general, the problem adses only with amplifiers having a combination o[ high input imped­ance and high voltage gain. With the 4800A, high-gain circuits with l o w i n pu t impedance might be o\'cr­driven on the constant-voltage ranges.

A s a n example 0 f aclive circuit measurements that may be made with the 4815A, consider the tuned amplifier circuit in Figure 28.

"" , --

. 7,.H

2 1<42369

, ..

•

-OUT ,

Fig. 28. 5 Mllz tut!ed amplifier sl!ows t!"gmive u.<islat!C� Jmm 4 MHz (0 6 MHz.

Some of the measurements which can be made on the cir cuit are:

1. The effectiveness of the bypass capacitor. - It may be desirable to choose a capacitor which is series resonant at the operating frequency. This can be done quickly with the 4815A.

2 . The resonant frequency and Q of the tank circuit. ­Onc great advantage of the 4815A is that this mea­surement can be made "in-circuit" with all loads connected.

3. Output impedance.

4. Input impedance of t h e amplifier. - See Fig. 29.

The r e a l advantage of th.e V e c t o r Impedance Meters in this type application is that the various par­ameters may be measured and the effe cts of circuit changes or compensation may be directly observed as

they are accomplished. [n addition, variations as a function of frequency can be quickly verified.

For the particular circuit configuration shown in Figure 28 for instance, examlna.tlon of the input Im­pedance provides extremely v a l u a b l e information about cir cuit performance that Is not readily availabLe th.rough any othcr measurement technique. In this circuit, the I F amplificr Is required t o operate a t 5 MHz. However, when the input impedance is mea­sured over the range from 4 to 6 MHz, it is found to exhibit an impedance having a phase angle in the third quadrant of the complex impedance plane. Figure 29 graphically illustrates the input Impedance of the IF amplifier as a function of frequency. In this case, the

21

" ,

2.'

2.'

" . " S �" • .. � , N

, -.,. .. -120·

• • • ;. " FREQUENCY(MH�)

Fig, 29. Amplifier inp/U im{lt(/CIIICt _ /lined 5 Ml1z.

amplifitlr wtll be unstable and could oscillate with in­ductive sources. F'or example, a mixer might logi ­cally cause the c i r c u i t to oscillate. In o r d e r to achieve stable design, neutralization may be required . Effectiveness can be evaluated directly with the 4815A .

Anotherexample of lhetype circuit measurements possible with the 4815A is Illustrated by a low - h'cq­uency oscillator such as the one shown in Figure 30 . Since this circuit is regenerative , its natural oscil­lation must be damped :)0 thut nn Impedance mc:u;urc­ment is possible. Qtherw'[se, the 481SA reading will be meaningless. T h i s Is easily achieved w i t h the 481SA in lhis parttcular circuit since the input imj>ed­ance of thc probe is 25 ohms. The 25-ohm impedance loads the tank circuits, lowering the effective loop

gain and preventing oscillation. Yet, the constant­

current characteristics ofthc probe permit an accur­

ate measurement of the circuit impedance.

Measurement of this impedance Is useful in pre­dicting gain variation with frequency. The measure­ment is accomplished by tuning the OSCillator to var­ious frequencies a n d measuring the t a n k circuit . Referring to Figure 30, . the o s c i l l a t o r is tuned by adjusting the 8 to 200 pF capacitor. Impedance mea­surements are made by placing the probe between the transistor collector and ground. The 481SA should be set t o the frequency of oscillation. When the freq-

22

r-----------------r------. 1 >M..., I I

I I _ 0»>" I I � I I I I I : .- � �: I ....... I I I ..

I I : �. r : : l_ 1"" ------ 'iL-- L _ _ _ _ J - .

Fig. 30. Low-freqllellcy oscillator.

-

o 2 6 10 \4 18 22 26 30 34 3B 41 46 50 4 8 12 � � � � � � � « � �

FREQUENcY IN 101HZ

Fi/:. 3/. Circ"i/ im�dlmr� liS a fllllC/ioll of Jreqmmcy.

uency is properly set, the phase angle will be 1800 , indicating a pure resistance acting as a source Im ­pcd�mce. Variation of this resistance over the freq­uency range is a direct measurement of the variation In gain, and the t a n k c i r c u i t i m p e d a n c e may be adjusted 0 r optimized to keep i t within the desired limits. Figure 31 shows data taken on this oscillator circuit.

SECTION VIII

x-v PLOTTING

Both the 4815A and the 4800A can be used with an X-Y recorder to obtain continuous records of imped­ance and phase as a function of frequency. The analog outputs on the r e a r panel of the Veclor Impedance Meters produce a vollage proportional to the imped­ance meter deflection, phase meter deflection. and frequency dial position. These analogs can also be used to obtain impedance and phase as a fUnction of other p a r a m e t e r s , such as bias current. Besides being used for recording purposes , the an..'llog can be used in conjunction with an -hp- Model 3440A Digital V 0 1 t m c t e r to produce a digital readout for greater resolution. The 4800A combined with the -hp- !llodcl 3434A Comparator makes an ideal go - no - go imped­ance checkout system.

The Vector Impedance Meters can be frequency swept either m e c h a n i c a l l y or electronically. To sweep the instruments electrically, the r e a r panel oscillator loop should be disconnected and a suitable sweep oscillator should be connected to the external

,

,

•

,

•

,

•

,

,

•

."

FREQU[NCY(H�)

oscillator i n p u t. All range switches, including the rrequency range s w i t c h , must be set to the correct range.

The swept rat.e of the 4815A is l i m i t e d by the traCking rate of its phase lock loop to 1 MHz per sec­ond. When sweeping either electrically or mechani­cally , the sweep rate must not exceed this tracking rate. If it does, the "search" light on the 4815A front panel will light.

The sweep rate of the 4800A is limited by the time constant of the automatic level control feedback cir­cuit. Consequently, the maximum permissable sweep rate Is a function of the Impedance magnitude varia­tion. Each full - scale impedance magnitude variation may be swept no f a s t e r than 2 s e c o n d s . In other words, a resonance curve that rises and falls over a full-scale impcdance magnitude range , would requirc at least 4 seconds to sw·eep. See Figure 32.

a) t; • •

:: • +

� + •

• " ;;; w w

, " 0 w " e ,, � � " w

· ro " ro ..

• 2_5K 3K 3.5K 4K 4 . �K 51<

Fi/:. 31. Two-pen X-Y recording of IImed circuit.

23

APPENDIX

GENERAL IMMITTANCE REVIEW

IMPEDANCE

A review of impedance will aid in the understand­ing of the Vector Impedance Meter operation and ap­plications . The i m p e d a n c e o f any finite, linear , passive, two - terminal, ui � lateral network c a n b e represented by an equivalent parallel or Series n , L , C network. B r i d g e s lire based on this principle In b a I a n e i n g the equivalent resistance or reactance of the unknown.

Impedance conSists of two components; resistance a n d reactance. The resistive component dissipates e l e e t r i c a I energy whereas the reactive component stores eithcr c l e c t r i c a l or magnetic energy. The reactive c o m p o n e n t can again be broken into two parts, inductance lmd cllj:Jftcitance. Since i n d u c t i ve

reactance aod C31)3citive reactance cancel each other . only the predominant one will be observed. Thus . the eQuivalent circuit will be either RL or RC, depending on the frequency.

The following formulas apply to Lhc series eQui­valent circuit:

, - -.

I>--..,::-;.,..-..... VVI\r-. ;-i ---., Fig. JJ. SUit:5 t:qui�alent ("ir("II;r.

Resistance Rs

Reactance

Impedance

where

_ _ ,_ . where Col " 2 . w C s ' Rs *' iXs • Izl �

.j. Indicates inductive reactance - Indicates capacitive reactance

Impedance magnitude " Izl .. (Rs Z .j. x,/) 1/2

Impedance pha8e .. 8 .. tan - 1 (XS!R8)- cos - 1(RsI IZI)

I m p e d a n c e maybe plotted in either polar coor­dinates (Z, @) or rectangular c o o r d i n a t e s in the complex plane (R :I: jX).

The vector diagram for the series eQuivalent cir­cuit is as follows .

, "

,

.. '

Fig. J4. Vuwr diagram Jor serit:5 equivaJem dUllir.

The dIagram shows the v o l t ag e and current in phase for resistance. The voltage leads the current by 900 (+j) for Inductive reaclance and lags the cur­rent by 900 (-j) for capacitive r e a c t a n ce. Veclori­ally, the jXs adds with the -jXs to yield either a net inductive or capacitive reactance. The phase e is the angle between the voltage across and current through tile impedance tZI /!l.

A series RLe circuit exhibits an impedance drog at resonance. "fhe phase changes from -900 to +90 as frequency increases .

z

.. ,

( , •

) • .. , .

Fig. JoS.

AOMITIANCE

USing t h e admittance parameters, an analogous set of eQuations can be written for the parallel equl­valenL circuit.

Conductance • G ., llRp

Susceptanc:e .. B " _,_ .

'), w o" - 1

wf;,

2 5

Admittance .. Y .. liE = l/Z .. G � j8 " IYI I::i!.... where ... Indicates capacitive susceptance

- Indicates inductlve suseeptance

Admittance magnitude IYI .. (02 -t- 82) 1/2 Admittance magnitude - 8 = tan -1(8/G) = cos -1(G! IYI)

�" -,(RA)

T h e resuUing vector diagram using the admit­tance parameter In the parallel equivalent circuit is shown in Figure 36B.

26

, • -, - ' . -

1 G , c. C

'

T

-' .

Pumlltl equ;valtlll circu;t

G' , •

' "

VtctQr diagram for parallel tquivaltm circuit Fig. 36.

, ••

•

'.

Fig. 37. Resolving the imptdanct triang1t of a equivalent urits circuit.

SERIES EQUIVALENT

The Vector Impedance Meters read out tile im­l)edance at the terminals of the unknown. The compo­nent values of the e q u i v a I e n t series circuit can be determined by resolving tile impedance triangle into the rectangular coordinates.

R. X, -

PARALLEL EQUIVALENT

Izl cos 8 IZI sin 8

Since admittance Is the reciprocal of impedance , It Is readily obtained by taking the reciprocal of the readings on the Izl meter and taking the negative of t h e phase reading. The components of the parallel equivalent circuit can be determined by resolving the admittance triangle.

•

•

-,. ,

: ' f G __

'.

Fig. JIJ. Rtsolving the udmittance triangle of u par· ulld equivalent circuit.

y

o B

m .ti -Iyl cos 8 IYJ sin 8

IY! f:.i

'/Rp >Ix"

CONVERSION FROM SERIES TO

PARALLEL EQUIVALENT

It can be seen that m e a s u r e m e n t s of Z, e are extremely c o n v e n i e n t since they allow relatively simple conversion between the e q u i v a l e n t parallel and series circuits . The c o m p o n e n t values of the parallel e q u i v a l e n t circuit can also be calculated directly from the series equivalent circuit as shown below .

,

R, 1 G . R '

• X 2 Rp , ,

-X, . _ _ ,_ B -R,' • X 2 ,

Xp

NATURE OF Q

Q has been used as a measure of impedance for many years. One bit of information contained in Q is bandwidth. High Q has always been a way to express narrow bandwidth, or selectivity. Likewise, low Q indicates broad bandwidth. Besides bandwidth , Q has provided a way to find a small resistive impedance in the presence of a large !'cactive impedance. Thus , real impedance behavior of circuit elements , such as induclors , is determined by Q.

The basic definltlon of Q is the ratio of the en­ergy dissipated to the energy stored per cycle in the system. The constant of proportionality is 2 " .

E stored Q '" 2 11 E dissipated per cycle

Since the energy dissipated per cycle is a function of frequency, Q becomes a fWlCtion of frequency. In the following sections, both parallel and series reso­nant eil'cuits wUI be examined in connection with Q ,

Resonance can bedcfined byone of three criteria; the f r e q u e n c y of zero phase angle, the frequency where the capacitive reactance equals the inductive react..'l.ncc, or the frequency of maximum Impedance (panlllel resonance) or minimum impedance (series resonance).

Resonance equations:

, 0' Xc XL Z Z maximum (parallel) Z - Z minimum (serics)

The abovc equatIons are not necessarily true at the same frequency, However, if Q is grealcr than 10 , the frequency will be approximateLy the same.

Derivation o f Q = wL/Rs' l/WCRs fo r RLC Series Resonant Circuits :

-,

"

,

Fig. 39. Energy re/mlcmsbips if! a" dememof)' lIC series circ/lil.

Working from the basic definition of Q ,

lotal encrgy s\.ored Q '" 2 " totaL cnCI'gy dissipated per cycle

The special serics RLC r e s o n a n t circuit formula (0) L/Rs will be derived in the following discussion .

The total energy stored in a series RLC r e s o n a n t circuit oscillates between the inductor and capacitor , AHnough oscillating, the total stored energy is con­stant at a given f r e q u e n cy. Assuming, at a given position of the cycle, all tne energy is stored in the inductor, tile expl'cssion of total stored energy be­comes:

The energydissipatcd by the series resistance is:

The energy dissipated per cycle is:

Edissipated/cycle " 1/2 Rs r2 (l/f)

The ratio of energy stored to energy dissipated per cycle equals Q/2 "" .

Since:

Q EstQred

- - oc---==-2 " Edissipated/cycle

Q 2 " !L

" � 1

� Q . _,_

wCR;;

1/2 L r2

Deriation o f Q = Rp/w Lp = CRp f o r RLC Parallel Resonant Circuits :

The parallel RLC resonance circuit Is shown in Fig­ure40A. An e q u i v a l e n t of this circuit is shown in Figure 40B. It is for this e q u i v a l e n t circuit that Q = Rp/(o) L appUes.

The impedance seen by the current , I, In circuit Figure 40A is:

(-J -, - ) (j w L -I- Rs) Z � w C

(-i � c) + (j Co) L + Rs)

Using the resonance equation:

1 X = l;C

27

The impedance becomes:

". X'

• - + (-jX) H,

The impedance magnitude is:

Izl - [(::r .

'12 1) � Q w L If Q > 10

T h e r e fo r e , IZI .. Q w L for large Q. From Ohm's Law, the current , I, through the circuit Is:

, • __ v_ Q w L Referring lo the equivalent parallel circuit, Figure 40B. t h e current flows entirely through Rp since a perfect parallel tank circuit has an Infinite impedance .

I'ZI = Q w L •

It follows that the current through the equivalent tank circuit is:

" vIR p

Equating thl' c:urrcnts :

28

• 0-

' 0-

......

or Q w L • �

w

R. � e f-o.

-

L

'" Fig. 40. (A) TQ.tk cifClli/ ccm/igllrQtiOIl.

(8) PQralle/ equi�Q{elt/ {JCTQSS lerminQ/s A B.

Rewriting:

wL

Swnmarlzing from the above discussion,

These relationships al'e true with large Q; I . e . • Q >10.

for resonance circuits

Q AS AN INDICATION OF SELECTIVITY

OR BANDWIDTH

By extending the analysis of power relationship In RLC circuits , a n expression describing t h e selec­tivity or response versus frequency in the vicinity of r e s o n a n c e can be derived. First, two reference points need be establisl\cd on the r e s 0 n a n e e curve . The points occur at frequencies at which the power in the circuit is o n e half the p o w e r at resonance. The reactance equals the resistance at these frequencies .

"/, .. ( , 2) '12 Rs ... X

If Rs · X

The impedance Z consiSts of equal resistive and re­acUve components. The re s i s t I v e component con­sumes power. A p p I y I n g the same voltage at the selected fn.rqucncies at resonance, will produce a current If = 0.70710. where To Is the current at reso­nance . The power dissipated Is then:

In other words , It is half Ihe powcr d i s s l p a l e d at resonance .

The next step is to examine th.e frequency rela­tionsh.lps Involved at the point where X = Ra. The net reactance changes due to two c o n t r ib u t i o n s in the same direction. One, there is a small change in in­ductive reactance resulting from a change i n freq­uency. Two , there is a s m a l l ch.ange in capacitive reactance. The net r e s u l t of these c h a n g e s is a change in reactance from zero reactance:

ax • 2 (2 tI" afL)

The change in (requency. a[. is the d i r r e r e n e e be­tween fo and e i t h e r hair p o w e r fl or f2' At h a l f power:

Adding the above equation:

2R .. 4 If ((2 - fl) L

Rewriting and multiplying by fo:

Q

Thus , the a b o v e expression for Q is a m e a s u r e of bandwidth.

Recalling the i n i t i a I assumption that X '" R , the phase at fl and f2 will be +45° and --45°, These freq­uency points afC now readily detennined on the Vec­tor Impedance Meters.

TRANSMISSION LINE EQUATIONS

The general formula for a l o s s I e s s line of unit length, having a characteristic impedance of Zo and terminated in an impedance ZL is:

(11

where the phase constant f3 = 2 11 / A , and A " wave­length. Now if the l i n e is 1/2 w a v e l e n g t h l o n g , 1 " ,\ /4 and:

J3 1 = 2 . T

, ,

Substituting in (1) above:

• 2 radians, or 900

and

(21

If the line is lis wa\'elength l o n g and is s h o r t cir­cuited, then 1 '" A /S, fj 1 = 'IT /4 ra:dian or 45° , and ZL = O. Substituting in (1):

fjZa x (:1)

'The c a I c ul a t o r may also be used to determine scale factor when component measurements are being made with the Vector Impedance Meters. That is to say, the calculator serves as a scale factor "nano­graph" when the Vector Impedance Meters are set to a frequencb at which r e a c t a n c e is direct reading (1.592 x 10 as discussed on Page 5). For instance , if the 4800A f r e q u e n c y is set to 1.59 kHz, the Xl

In a similar manner it can be srrown that the in­put i m p e d a n c e of a 1/8 wavehmgth Hne that is open circuited is:

(4)

For the purpose of deriving a means of measurhig the attenuation of a transmission line, the g e n e r a l expression for a line with loss is given below. The impedance, Z i1 , looking into a line with loss having a characteristic impedance of Zo and terminated in an impedance ZR can be expressed as:

CR + z" tanh 1'1 ) ZH lanh 1'1 (5)

where, 1 = a 1 + j j3 1 , and j3 = 2 " / >" .

In the case 0 f a half - wavelength line:

1 }. /2 " 2 . ; , , }. /2 - • and n o 1 + , . Also,

Lunh n tanh ( a 1 j " ) - tanh 0 1 . IT . , is small, then tanh . 1

lanh 1'1 - a l

Substituting i n ( 5 ) above:

(, , 2 .1 '" 2 "H

, 0 '. +

a I and

Dividing numerator and demoninator ofthe fraction on the right by ZR, we obtain:

{G)

If the half -wavelength cable is open - circuited; L e . , ZR = 00 , (5) will reduce to:

, .1 ,

. ,

. , Zo/Zi

l·nepers. Then,

Z o /Zil x 8.G�1 dB, where Zi1 is

resistive and is measured directly.

range is direct reading in microhenries from 100 j.lH to 1 , OO O j.lH. This may be determined from the cal­culator by setting the frequency 1.59 kHz opposite the alTOW. The direct reading inductance range is then read opposite the appropriate impedance. In the ex­ample above, 100 j.lH is read opposite 111 and 1 ,000 j.lH is read opposite 1011.

29

HEWLETT PACKARD

1 MAR &8 HINTED IN U.S.A.