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Inductanceand Capacitance
Measurements
Objectives
You *ill be able to:
l . SketchRC seriesand parallel equivalentcircuits for a capacitor, and write equations
rela:ingthe iwo circuits.
) Sketch Rl seriesand parallel equivalentcircuits for an inductor, and write equations
relatingthe two .;ircuits.
3. Explain the Q factor of an inductor and the D factor of a capacitor, and v.,ritethe equations for eachfactor.
4. Drau' circuit diagrams for the following ac bridges: simple capacitancebridge, seriesresistancecapacitancebridge, parallel-resistancocapacitancebridge, inductance comparisonbridge, Maxwell bridge, and Hay inductancebridge.
5. Erplain the operation of each of the bridges listed above, derive the equations for the
quantities to be measured,and discuss the advantagesand disadvantagesof each
brid-ee.
6. Sketch ac bridge circuit diagrams showing how a commercial rnultifunction impedance bridge uses a standard capacitor and three adjustable standard resistors to measurea wide rangeofcapacitancesand inductances.Explain.
7. Discussthe problemsinvolved in measuringsmall R, L, and C quantities,explain suitable measuring techniques,and calculate measuredquantities.
8. Sketch and explain the basic circuits for converting inductance and capacitance into
voltages for digital measurements.Discuss the specification and performance of a digital RIC meter.
9. Draw the circuit diagram for a Q meter, explain its operation and controls, and determine the Q of acoil from the Qmeter measurements.
189
Introduction
Inductancc, capacitai,ce,inductor Q factor, and capacitor D factor can all be measured precisely on ac bridges, which are adaptationsof the Wheatstonebridge. An ac
supply must be used, and the null detector musi be an ac instrument.A wide range
of ac bridge circuits are available for various specializedmeasurements.Some commercial ac bridges use only a standardcapacitor and three adjustablestandard resistors to construct several different types of inductanceand capacitancebridge circuits.
Special techniques must be employed for measuring very small inductance and capacitance quantities. For digital measurement,inductance,capacitance,and resistance
are first appiied to circuits that convert each quantity into a voltage. Capacitors and
inductors that are required to operate at high frequenciesare best measuredon a Q
meter.
3.1 RC AND RZ EQLIVALENT
CIRCUITS
Capacitor Equivalent Circuits
T h e e q u i v a l e n tc i rc u i to facapaci torconsi stsofapurecapaci tance
C pandaparal l el resistance Rp. as iliustrated in Figure 8-1(a). Cp representsthe actual capacitancevalue, and
ftp represents.the
resistanceof the dielectricor leakageresistancc.Capacitorsthat have a
high leakagecurrent flowing through the dielectric have a relatively low value of Rp in
their equivalent circuit. Vc,y iow teakagecurrents are representedby extremely large values of Rp. Examples of the tv,'c extremes are electrolytic capacitors that have high leakage currents (low parallel resistance), and plastic film capacitors which have very low
leakage (high parallel resistance).An electrolytic capacitor might easily have several microamperesof leakage crlrrent, while a capacitor with a plastic film dielectric could typically have a resistanceas high as 100 000 MO.
A parallel RC circuit has an equivalent seriesRC circuit [Figure 8-1(b)]. Either one
of the tu'o equivalentcircuits (seriesor parallel) may be usedto representa capacitorin a
circuit. It is found that capacitorswith a high-resistance
dielectric are best representedby
the seriesRC circuit, while those with a low-resistancedielectric should be represented
by the parallel equivalentcircuit. However, when the capacitoris measuredin terms of
the series C and R quantities, it is usually desirableto resolve them into the parallel
^,
1
'T
"l
(u) Parallel equivalent
circuit
190
(b) Series equivalent
circuit
Figure 8-1 A capacitor may be represented by either a parallel equivalent circuit or a
seriesequivalent circuit. The parallel equivalent circuit best represents capacitors that
have a low-resistancedielectric, while the
seriesequivalent circuit is most suitable for
capacitors with a high-resistancedielectric.
Inductanceand CapacitanceMeasurements
Chap. 8
*iln
equivalent circuit quantities. This is because
the (parallei) leakage resistancebest represents the quality of the capacitor dielectric.
Equations ihat rela; the series and parallel
equivalent circuits are derived belorv.
Refening to Figure g_l, the seriesimpedanceis
Zr= Rr- jX,
and the parallel admittance is
y-=a*; I
'n-
4 * t 4 = G e + iB e
where G is conductance and is susceptance.
The impedances of each circuit must be
^B
equal.
Thus,
giving
or
glvmg
Equating the real terms,
(8-1)
Equating the imaginary terms,
a,=
' R!^4
+ X !^
(8-2)
The equations aborre.can be shown to apply
also to equivarent series and parallel
RZ circuits, as well as RCcircuits
Sec. 8-l
RC and RZ Equivalent Circuits
tgt.
.-ffi
Inductor Equivalent Circuits
Inductor equivalentcircuits are illustrated in Figr"e 8-2. The :eries equivalentcircuit in
Figure 8-2(a) representsan inductor as 2 pur. inductanceL" in serieswith the resistance
oi its coil. Th;s seriesequivalentcircuit is normally the best way to representan inductor,
becausethe actualwinding resistanceis involved and this is an importantqua:rtity.Ideally, the winding resistanceshould be as small as possible,but this dependson the thicknessand length of the wire used to wind the coil. Physically small high-valueinductors
tend to have large resistancevalues,while large low-inductancecomponentsare likely to
have low resistances.
The parallel RL equivalentcircuit for an inductor [Figure 8-2(b)] can also be used.
As in the caseof the capacitorequivalentcircuits, it is sometimesmore convenientto use
a parallelRL equivalentcircuit rather than a seriescircuit. The equationsrelating the two
are derivedbelorv.
Referringto Figure 8-2, the seriescircuit irnpedanceis
Z,= R,+ jX.,
and the parallelcircuit admittanceis
.,
f
-P
=
I
-
-
RP
t-
I
" X,,
Yr= Go-iB,
Zr: Zp
R"+JX _1
GP - jBp
R.+x. =
R, +iX, =
glvlng
t
(Go+iBr\
Gp-iBp\Gr+ jBo )
Gp+ jBp
Grr+q
9
A
I
1I
I
(a) Series equivalent
circuit
192
il-1-t',
(b) Parallel equivalent
circuit
Figure 8-2 An inductor may be represented by either a parallel equivalent circuit or a
seriesequivalentcircuit. The seriesequivalent circuit is normally used, but it is sometimes convenient to employ the parallel
eouivalent circuit.
Inductance and CapacitanceMeasurements
Chap. 8
til
*--
Equrting the real terms,
ft,=
Gp
G,l+ r]
1/RP
rtR] + r/x]
I R;X; \
\ R:X: I
RoxS
(8-3)
xj +n]
Equatingthe imaginaryterms,
Y=
Bp
c| + n'z,
llxp
R;X; \
It"-"1
1/n] + ux] \ R;X; )
(8-4)
Like Equations8-1 and 8-2, Equations8-3 and 8-4 applv to both RC and RL circuits'
Q Factor of an Inductor
The quality of an inductor can be defined in terms of its power dissipation. An ideal inductor should have zero winding resistance,and therefore zero power dissipated in the
u,inding. A /oss,]'inductor has a relatively high winding resistance;consequentlyit does
dissipate some power. The quatity factor, ot Qfactor; of the inductor is the ratio of the inductir-ereactanceand resistanceat the operatingfrequency.
e=\='1"
R,
(8-5)
R"
where l, and R" refer to the componentsof an Rl seriesequivalent circuit [Figure 8-2(a)].
Ideally. ol. should be very much larger than R", so that a very large Q factor is obtainedas 1000 (depende faciors for typical inductors range from a low ofless than 5 to as high
on
frequency).
ing
As discussedearlier, an inductor may be representedby either a series equivalent
circuit or a parallel equivalent circuit. When the parallel equivalent circuit is employed,
the Q factor can be shown to be
Sec. 8-l
RC and RL Equivalent Circuits
193
FE-*-
.-ff|
Q=&-
Xp
R'
(8-0.1
^Ln
D Factor of a Capacitor
The quality of a capacitorcan be e;pressedin terms of its power dissipation.A very pure
capacitancehas a high dielectricresistance(low leakagecurrent) and virtually zero power
dissipation.A /ossy capacitor,which has a relatively low resistance(high leakagecurrent), dissipatessome power. The dissipationfactor D defines the quality of the capacitor.
Like the Q factor of a coil, D is simply the ratio of the component reactance(at a given
frequency)to the resistancemeasurableat its terminals. In the caseof the capacitor,the
resistanceinvolved in the D-factor calculationis that showu in the parallel equivalentcircuit. (This differs'from the inductor Q-factorcalculation,where the resistanceis that in
the seriesequivalentcircuit.) Using the pa rallel equivalentcircuit:
o=b
Rp
(8-7)
aCrR,,
Idealll', R, shouldbe very much larger than l/(.iiCo), giving a very small dissipation
factor. T1'picalll', D might range from 0.1 for electrolytic capacitors to less than 10< for
capacitorsu,ith a plastic film dielectric(againdependingon frequency).
\\rlren a seriesequivalentcircuit is used,the equation for dissipatiorrfactor can be
shown to be
D=
R " = coC J,
x"
(8-8)
ComparingEquation 8-7 to 8-6.andEquation 8-8 to 8-5, it is seenthat in each case
D is the inverse of Q.
Example8-1
An unknos'n circuit behavesas a 0.005 pF capacitor in series with a 8 kf,) resistor when
measuredat a frequency of I kHz. The terminal resistanceis measured by an ohmmeter
as 134 kQ. Determine the actual circuit componentsand the method of connection.
Solution
x"=
I
2nfC
2r.xll<*.I2x0.005 pF
: 3l .8 kC)
R"= 8 kO
194
Inductance and CapacitanceMeasurements
Chap. 8
Equation 8-I
R.r+X.r _ (8 k0)2 + (31.8kO)'?
o
' .P--& -
8 kc)
= 134kO
Equatiott8-2,
r = R : * ^,
^.
= 33.8kO
c,,=
'
1
2rJX,
= (8kq)2+ (3r.8k0)2
31.8ko
I
2rxl kH zx33.8kf)
_ 0.005 u.F
Since the measuredterminal resistanceis 134 kO, the circuit must consist of a
0.005 pF capacitorconnectedin parallel with a 134 kf) resistor.For a seriesconnectedcircuit, the terminal resistancewould be rnuchhigher than 134 k0.
8-2 AC BzuDGE THEORY
Circuit and Balance Equations
The basic circuit of an ac bridge is illustrated in Figure 8-3. This is exactly the same as
the Wtreatstonebridge circuit (Figure 7-3) except that impedancesare shown instead of
and an ac supply is used.The null detectormust be an ac instrumentsuch as
resistances.
an electronic galvanometer,headphones,or an oscilloscope.
\\hen the null detector indicates zero in the circuit of Figure 8-3, the alternating
volta-seacross points a and b is zero. This means (as in the Wheatstone brirfge) that the
voltage acrossZ, is exactly equal to that across 22, and the voltage across 23 equals the
r,oltagedrop acrossZa. Not only are the voltages equal in amplitude, they are also equal
ac supply
Figure 8-3 The basic ac bridge circuit is similar to the Wheatstone bridge except that
impedances are involved instead ofresistances. An ac supply must be employed, and the
null detector must be an ac instrument.
Sec. 8-2
AC Bridge Theory
195
nI*-
.-m
in phase.If the voltageswere equal in amplitude but not in phase,the ac null detector
would not indicatezero.
Vzr = Vzz
i1Z.= i2Z2
-*
and
V z t =V z q
or
iyZu= 1r7o
( l)
(2)
Dividing Equation I by Equation2,
irZr - iz4
itZt
izZ+
(8-e)
giving
As alreadystated,bridge balanceis obtainedonly when the voltagesat each terminal of the rrull detectorare equal in phaseas well as in magnitude.This results in Equation 8-9. u,hich involves complex quantities.In such an equation,the real parts of the
quantitieson eachside must be equal, and the imaginaryparts of the quantitiesmust also
be equal.Therefore,when deriving the balanceequationsfor a particularbridge, it is best
to expressthe impedancesin rectangular form rather than polar form. The real quantities
can then be equatedto obtain one balanceequation,and the imaginary (or7 quantities)
can be equatedto arrive at the other balanceequation.
The need for two balance equations arises from the fact that capacitancesand inductancesare never.pure;they must be definedas a combinationof R and C or R and I
(as discussedin Section8-1). One balanceequationpermitscalculationof L or C, and the
other is used for determining the R quantity.
Balance Procedure
As alreadyexplained,two componentadjustmentsare requiredto balancethe bridge (or
obtain a minimum indication on the null detector).Theseadjustmentsare ,?o/independent
of each other: one tends to affect the relative amplitudes of the voltages at each terminal
^f the null detector,and the other adjustmenthas a marked effect on the relative phase
differenceof thesevoltages.For example,Za inFigure 8-3 might consistof a variable capacitor in serieswith a variable resistor,as illustratedin Figure 8-4(a). Adjustment of Ca
ma1'make V7aequal in amplitude to V4 without bringing it into phase with V7j. The result is, of course, that the null detector voltage Vzt - Vz+is not zero [see Figure 8-4(b)].
Further adjustment of Co could alter the phase of VTabut will also alter its amplitude. If
Ra is now adjusted, Vn - Vzomight be further reducedby bringing the voltages closer together in phase. However, this cannot be achieved without altering the amplitude of V2a,
which is the voltage drop acrossRa and Ca [Figure 8-4(c)]. When the best null has been
obtained by adjustment of Ra, Ca is once again adjusted.This is likely to once more make
L96
Inductanceand CapacitanceMeasurements
Chap. 8
qI
/rt
_
\v z3
r/
\
v z4l
(a) Null detector voltage = (Vzz = Vzz)
vzt
Yz+
r:
-
t/
t/
r z3
(b)
I'zs and V.4
equal but not
in-phase
-rl
vzA
vzs - vzt
vz4
(rl
Vzg and V74
in-phase but
not equal in
amplitude
(d)
Vzs and, Vyn
equal and
in-phase
Figure 8-4 When an ac bridge is balanced, yzr must equal V^, and the two voltages
must he in phase.This requires altemately adjusting two quantities (Ra and Ca in this circ))it) unt'l the smallestpossiblenull detectorindication is achieved.
l/7. close toV..,in amplitude,but again has an unavoidableeffect on the phaserelationship. The procedureof alternately adjusting Ra and C4 to minimize the null detector voltage is continueduntil the smallestpossibleindication is obtained.Then, Vyais equal to
VTborh in magnitudeand phase[Figure 8-4(d)r.
AC Bridge Sensitivity
The same considerationsthat determined the sensitivity of a Wheatstone bridge apply t<i
ac bridge circuits. The.bridge sensitivity may be defined in terms of the smallest change
Sec. 8-2
AC Bridge Theory
t97
iR
in the measuredquantity that causesthe galvanometerto deflect from zero. Bridge sensitivity can be improved by using a more sensitive null detecior and/or by increasir-rgthe
level of supply voltage. The bridge sensitivity is analyzed by exactly the same method
used for the Wheatstone bridge, except that impedances are involved instead of resistances. Accuracy of measurementsis also determined in the sa;tle way as Wheatstone
bridge accuracy.
8-3 CAPACITANCE BRIDGES
iit
Simple Capacitance Bridge
The circuit of a simple capacitance bridge is illustrated in Figure 8-5(a). 21 is a standard
capacitorC1, and Q is the unknown capacitanceC,. 23 andZa arc known variable resistors. such as decaderesistanceboxes. When the bridge is balanced,21/23 = 7alZa (Equation 8-9) applies:
z,- = :11
toCr
Zz: R z
z":
and
-il
aC*
Z+ = R a
(a) Simple capacitancebridge
(b) Potential divider
substituted for R, and Ra
198
Figure 8-5 The simple capacitancebridge
measuresthe unknown capacitanceC, in
terms of standardcapacitor C1 and adjustable precision resistorsR3 and Ra. At balance,c,= cl3lR4. This circuit functions
only with capacitors that have very high resistancedielectrics.
Inductanceand CapacitanceMeasurements
Chap- 8
fr
-jllaCt
Therefore,
_
iJ!'cu
n3
R4
l=
I
C rR z
C'Rq
or
(8-r0)
glvrng
The actual resistancesof R3 and Ra zue not important if their ratio is knowno so a
potential-dividerresistancebox could be usedas shown in Figure 8-5(b).
Example 8-2
The standardcapacitancevalue in Figure 8-5 is Cy = 0.1 pF, and R3lRacan be set to any
ratio bet\\'een100:1 and 1:100.Calculatethe rangeof measurements
of unknown capacitance Q'.
Solution
Fnr r ation
R- l O
^
r
=-
CBt
R4
For R3/Ra= 100: I :
100
I
C. = 0.1 F.Fx
= l0 p.F
For R3/ Ra =l : 1 0 0 :
pFr
C,=0.1
#
= 0.001p.F
The foregoing analysis of the simple capacitancebridge assumesthat the capacitors
are absolutely pure, with effectively zero leakage current through the dielectric. If a resistance q,ere connected in series or in parallel with C. in Figure 8-5(a), and the rest ofthe
bridge components remain as shown, balance would be virtually impossible to achieve.
This is becausei1 and i2 could not be brought into phase, and consequently,i1R3 and i2R4
would not be in phase.As discussedin Section 8- 1, the equivalent circuit of a leaky capacitor is a pure capacitancein parallel with a pure resistance.Thus, the simple capacitance
bridge is suitable only for measurernentof capacitors with high-resistancedielectrics.
Sec. 8-3
CapacitanceBridges
r99
Series-ResistanceCapacitance Bridge
caDacirrnce
is r';prcsented
In the circuit shown in iigu;'e 3-6(a),the unkr:..'w:r
as a pure
capacitanceC5 in seri"s with a resirrance1.,. A standardadjustableresistanceR1 is connectedin serieswith standardcapacitorC1.The voltagedrop acrossR, balancesthe resisrrvc voltage ilrops in branch22when the bridge is balanced.The additionalresistorin series u,ith C increasesthe total resistive componentin 2., so that inconveniently small
values of l(1 are not required to achievebalance.Bridge balanceis most easily achieved
when each capacitivebranch has a substantialresistivecomponent.To obtain balance,R1
and either Rj or Ra are adjustedalternately.The .series-resistartce
c'apacitancebridge is
found to be most suitablefor capacitorswith a high-resistancedielectric (very low leakage currenl and low dissipation factor). When the bridge is balanced,Equation 8-9 apolies.
Zt -Zt
z3
giving
R r-j l /aC r
R3
z1
_ R ,-j l l aC ,
R.
(8 - r l )
Equatingthe real termsin Equation8-11,
R ,= R ,
R3
R"
(8-r 2)
glvln-s
Equatingthe ima-einary
termsin Equation8- 11,
1_1
roClR-1 oC,R+
(8- 13)
giving
The phasordiagram for the series-resistance
capacitancebridge at balanceis drawn
in Figure 8-6(bl. The voltage drops across23 andZ" are i1R3and i2Ro,respectively.These
two volta_ees
must be equal and in phasefor the bridge to be balanced.Thus, they are
drawn equal and in phase in the phasor diagram. Since R3 and Ra are resistive,i1 is in
phasewith ilRj and f2 is in phase with i2Ra.The impedanceof C1 is purely capacitive,
and current leads voltage by 90" in a pure capacitance.Therefore, the capacitor voltage
200
Inductanceand CapacitanceMeasurements
Chap. 8
-
r*f
(a) Circuit of series-resistancecapacitance bridge
irRs = i2Rn
.(b) Phasor diagram for balanced bridge
Figure 8-6 The series-resistance
capacitancebridge is similar to the simple capacitance
bridge. except that an adjustable series resistance (R1) is included to balance the resistive
component (R,) of 2". This bridge is most suitable for measuring capacitors with a highresis!ancedielectric.
drop i1X6r is drawn 90" lagging ir. Similarly, the voltage drop across c" is i2X6.5,and is
dran'n 90" lagging i2.The resistivevoltage drops l,R1 and i2Rsare in phasewith ir andi2,
respectively.
The total voltage drop across 21 is the phasor sum of i1R1 and i1X6.1,as illushated in Figure 8-6(b). Also, i2Z2 is the phasor sum of l2R" and i2Xs,. since i2z2
must be equal to and in phase with iiT, ifrt and i2R" are equal, as are i1X6 and.
izXcr.
Sec. 8-3
CapacitanceBridges
201
Example8-3
" .g
.i*er***
'.aaaF
=.aF*
A series-resistancecapacitancebndge [as in Figure 8-6(a)] has a 0.4 p,F standardcapacitor for C1, and R: = 10 k(r. Baiance is achieved with a l0OHz supply frequency when
Rr = 125 O and Ra = 14.7 kf,). Calculate the resistive and capacitive components of the
measuredcapacitur and its dissipation factor.
Solution
--##
-.#ipr..l|
,- aaa.#itaae
. --ffi
EquationS-13,
C,=
-#
0.1pF x 10kO
t4.7 k{l
+
= 0.068p.F
-....g
- *'#
..-w
Equarion 8-12,
o - R,Ro
-R3
125Ax M.7 kA
l 0 ko
= 183.8f)
Equation 8-8,
D = oC,R,
= 2n x 100Hz x 0.068pF x 183.8C)
:0.008
Parallel-Resistance Capacitance Bridge
The circuit of a parallel-resistance capacitance bridge is illustrated in Figure 8-7. In this
case,the unknown capacitanceis representedby its parallel equivalentcircuit; Crinparallel u'ith Ro.Z3 andZa are resistors,as before, either or both.of which may be adjustable.
Q is balancedby a standardcapacitor C1 in parallel with an adjustableresistor R1. Bridge
balanceis achievedby adjustmentor R1 and either R3 or Ra. The parallel-resistance
capacitancebridge is found to be most suitable for capacitors with a low resistancedielectric (relativell'high leakagecurrentand high dissipationfactor). At balance,Equation 8-9
onceagainapplies:
Z, -Zt
Z.
Z^
Also,
1l
-=Zt
Rr
_1
j(I/aC)
I
=- I + jaCl
Rr
Ll =
202
-
1
l l R t+ j aC t
InductanceandCapacitance
Measurements Chap.8
wsil'
dn*eho- 'rjffi
"
Figure 8-7 The parallel-resistancecapacitancebridge uses an adjustable resistance(Rr)
connected in parallel with C1 to balancethe resistive component (R) of Zz. This bridge is
most suitable for measuring capacitors with a low-resistancedielectric.
1=l*
and
4
I
Re jQlaCr)
T^
= *JaLp
Rp
or
4=l
l/Ro+ jaC,
.:
substltuttng
into Equation8-9,
I/(l/&+
jloCr) _ ll(llR,+ jaCo)
R3
Rn
1
1
R3(l/Rr +j<oCy)
R4(I/R.+ jaCo)
+i<,,cr)=n+
t.t*')
"(i
(8-14)
Equating the real terms in Equation 8-14,
R :-R o
Rr
Sec. 8-3
CapacitanceBridges
Re
203
(8-15)
giving
Equating the imaginary terms in Equation 8-14,
<oC1R3= aCrRa
(8-16)
glvlng
Note the similarity betweenEquations8-15 and 8-12, and betweenEquations 8-16 and
8 -1 3 .
Example 8-4
A parallel-resistancecapacitancebridge (as in Figure 8-7) has a standardcapacitance
value of Cr = 0. I pF and R: = l0 kO. Balanceis achievedat a supply frequency of 100
Hz when Rr = 3'75kO, R3= l0 kO, and Ra = 14.7kQ. Calculdtethe resistiveand capacitive componentsof the measuredcapacitor and its dissipation factor.
Solution
Equarions-16,
C,=+=t##4
= 0.068pF
Equarion 8-15,
R'&
^
375kC)x 14.7kQ
K^= 'R ^
r0ko
= 551.3kO
Equation 8-7,
o=
|
@C,R,
2r x I 00 Hz x 0.068pF x 551.3 kO
= 42.5x l0-3
815
-**t"
Calculate the parallel equivalent circuit for the C, and R" values determined in Example
8-3. Also determine the component values of R1 and Ra required to balance the calculated
Co and Ro values in a parallel-resistancecapacitance bridge. Assume that R3 remains
l0 ko.
204
Inductanceand CapacitanceMeasurements
Chap. 8
*ffi
ryEf.-'Et
rymt
' 'ry
..,*'"-*i'llf'P
Solution
x"=
1
2nfC"
2n x 100Hz x 0.068pF
=23.4kQ
Equation 8-1,
^
"n=
nl+ x? = (r83.8fi)'?+ e3.4kQ12
l83so
&
= 2.98MO
EquationS-2,
x,=
R?!x?
-
,4.r
(183.80)2 + (23.4kQ)2
23.4kQ
= 23.4k{l
C o=
'
I
2rrx 100H zx23.4kdl
|
2rJX,
:0.068 p.F
Front Equation 8-16,
o _ -CtRt _ 0.1 pFx l0kf)
Ce
0.068 p.F
rr4-
= 14.7kQ
From Equation 8-15,
, -
RtR,
R4
10k.0 x 2.98MO
4.7 kA
= 2.03MO
The capacitor, which was determined in Example 8-3 as having a series equivalent
circuit of 0.068 pF and 183.8O, was shown in Example 8-5 to have a parallel equivalent
circuit of 0.068 pF and298 MO. It was also shown that to measurethe capacitor on a
parallel-resistance
capacitancebridge,R1 (in Figure 8-7) would have to be 2.03 MO. This
is an inconvenientlylarge value for a precision adjustableresistor.So a capacitorwith a
high leaka-eeresistance(low D factor) is best measuredin terms of its series RC equivalent circuit.
The capacitor in Example 8-4 has a parallel RC equivalent circuit of 0.068 pF
and 551.3 kO. Conversion to the series equivalent circuit would demonstrate that this
capacitor is not conveniently measured as a series RC circuit. Thus, a capacitor with a
low leakage resistance (high D factor) is best measured as a parallel RC equivalent
circuit.
Capacitors with a very high leakageresistanceshould be neasured as seriesRC circuits. Capacitors with a low leakage resistance should be measured as parallel RC cir-
Sec. 8-3
CapacitanceBridges
205
:riTiffi.
I
:_*f*l*iiir
di-{*.ftiffidifrdt|L.
,ff
cuits. Capacitorsthat have neither a very high nor a
very low leakage resistance are best
rr,casurecas a parallerRC circuit, becausethis gives
u ii...t indicatioit of the capacitor
leakageresistance.
8.4 INDUCTANCE BRIDGES
Inductance Comparison Bridge
The circuit of the inductance comparison bridge
shown in Figure g_g is similar to the
series-resistancecapacitancebridge except
that inductors are in*volvedinstead of capacitors' The unknown inductance'represented
by its (seriesequivalentcircuit) inductance
z,
and R'' is measuredin terms of a precisestaniard
value inductor.zr is the standardinductor' R' is a variable standardresistor to balance
R,, R3 and Ra are standard resistors. Balanceof the bridge is achievedby alternatery
adjustingR1 and either R3 or Ra. At barance,
Equation8-9 once againapplies:
*[email protected]
.*klM
tt
Z,
=t t
24
R, + jaLl
R3
Rt
Rj
*
R,+ iaL"
R4
.aL,
= _& a;.L,
-R r
R 4' R o
( 8- 17)
Equatingthe real componentsin Equationg_17.
R r= R"
R3
R^
Figure 8-8 The inductance comparison
bridge uses a standard inductor Z, together
with adjustable precision resistors R1, R3
and Ro to measure an unknown inductor
in terms of its series equivalent circuit
Zand R-.
2M
lnductance and CapacitanceMeasurements
Chap. g
ij*..fl
glvlng
(8-13)
Equatingth" imaginarycomponentsin Equation8-17,
aLr
_ aL"
R3
R4
(8-1e)
gvlrrg
t"
"r-*"
An inductor that is marked as 500 mH is to be measuredon an inductancecomparison
brid-ee.The bridge usesa 100 mH standardinductor for L1, and a 5 kO standard resistor
for R.. If the coil resistanceof the 500 mH inductor is measuredas 270 f,). determine the
resistancesof R1 and R3(in Figure 8-8) at which balanceis likely to occur.
Solution
From Equation 8-19,
ro\3 -- R- o L r
L,
-- 5kOx100mH
500 mH
= 1kC )
Frotn Equation 8-18,
^
R"R"
270.f,x I kO
R4
5ko
= 54f)
Nlaxnell Bridge
Accurate pure standard capacitors are more easily constructed than standard inductors.
Consequently,it is desirableto be able to measureinductancein a bridge that usesa capacitance standardrather than an inductance standard."[he Manuell bridge (also known
as the Maneell-Wein bridge) is shown in Figure 8-9. In this circuit, the standardcapacitor
C3 is connectedin parallel with adjustable resistor R3. R1 is again an adjustable standard
resistor. and Ra may also be made adjustable. l,, and R, represent the inductor to be
measured.
The Maxwell bridge is found to be most suitable for measuring coils with a low Q
factor (i.e., where <oZ"is not much larger than &). To determine the expression for 7a
and 2,.
Sec.8-4
InductanceBridses
207
Figure 8-9 The N{axrvell bridge uses a standardcapacitor C3 and three adjustable preclsion resistors to measure an unknown inductor in terms of its series equivalent circuit, Z,
and R,. This bridge is most suitablefor measuringcoils with a low Q factor.
1=1_
23
1
=l
jllaQ
R3
+ jaC3
R3
I
'
7.-
llfu+ jaC3
Zz= R,+ joL"
and
Substitutingfor all componentsin Equation8-9,
Rr
R" +"1'rol,
tt(r/&+ jaQ)
R4
& * rrC. R, = &
R3
* j^ L ,
R4
(8-20)
R4
Equating the real componentsin Equation 8-20,
Rr
: : : : "-.
DD
r\?
or
-
R"
t\a
[-4&l* ' l
(8-21)
|
Equating the imaginary componentsin Equation 8-20,
208
Inductance and Capacitance Measurements
Chap. 8
qffi
..i-*ii#f*k-'*;sa$*r.rr.flfliiia.-I.i.iiiii!ktrii*ri*ri*'r'**idfl**ir*r*liaii-|.**r
lifu
tL'
uC 3R ,=
R4
(8-22)
glvrng
Example 8-7
A Maxq,ell inductancebridge usesa standardcapacitorof Cj = 0. I p,Fand operatesat a supp l y fre q u e n c y o fl 0 0H z.B al ancei sachi evedw henR=l1.26kA ,R 2= 410,f),andR o =JQ 6
f|. Calculate the inductance and resistanceof the measuredinductor, and determine its Q
factor.
Solutiott
Equatiort8-22,
L, = CzRtR+
= 0.1pF x 1.26kO x 500O
= 63mH
= -Eqlaf4q-q
Equariott8'21, R,=
- +&
R.
470Q
= 1.34kCl
Equation8-5,
Q=
tL'
2t x 1ooHz T 63 mH
R,
1.34kc).
0.03
Ha1-Inductance Bridge
The fla-r'bridge circuit in Figure 8-10 is similar to the Maxwell bridge,except that R3 and
C-r?r€ cont€cted in seriesinsteadof parallel,and the unknown inductanceis represented
as a parallel l,R circuit instead of a series circuit. The balance equations are found to be
exactl]' the same as those for the Maxwell bridge. It must be remembered,however, that
the measuredL, and R, are a parallel equivalent circuit. The equivalent series RI, circuit
can be determinedby substitutioninto Equations8-3 and 8-4.
\Vtrenthe bridge in Figure 8-10 is balanced,
Z, =4
Z^ Z^
Ra
glvlng
Rp
Sec. 8-4
InductanceBridses
.Ro
-
aLp
Ra
Rr
I
-t-
(8-23)
olC:Rr
209
.
'{l}rl
!6itl"trtffii:L
;*il
Figure 8-10 The Hay bridge uses a standardcapacitor C3 and three adjustable precision
resistors to measure an unknown inductor in terms of its parallel equivalent circuit, Lo
and Rr. This circuit is most suitable for inductors with a high Q factor.
Equatingthe real componentsin Equation8-23,
&= &
RP
Ri
(8-24)
Equatin-ethe imaginary componentsin Equation8-23,
R ^l
aLp - oC3R1
(8-2s)
giving
*
"n-t.
A Hay bridge operating at a supply frequency of 100 Hz is balanced when the components are Cr = 0.1 FF, Rr = I.26 kO, R3 = 75 O, and R4 = 500 f,). Calculate the inductance and resistance of the measured inductor. Also, determine the Q factor of the
coil.
zto
Inductance and CapacitanceMeasurements
Chap. 8
Yfi
st*Et@uattf:E*tf.*iflir;ffi
.-lI
Solution
Equation8-25,
Lp= C3Rfia
= 0.1p.Fx 1.26kO x 500O
= 63mH
Equations-24,
R"=
#
-
1'26k!l!5oo o
= 8.4kC)
Equation 8-6,
o= 3-P
tttLp
8.4kO
2r.xl 00H zx63mH
-
11)
Example 8-9
(a)
(b)
Calculate the series equivalent circuit for the Lp and Rp values determined in Example 8-8.
Determine the component values of R1 an.i.R3 required to balance the calculated L"
and R, values in the Maxwell bridge. Assume that R4 remains 500 O.
Solution
Xr=ZrJl-r=/11 xl 00H zx63mH
(a)
= 39.6O
Equation 8-3,
Rsc
_
p-L-
'
xj + R j
8.4kO x (39.6O)2
(39.6O)2+ (8.4kO)2
= 0.187O
Equation 8-4,
(8.4k0)2 x 39.6O
R:X,
,.
"'-
xj+ n/
(39.6O)2+ (8.4kO)2
= 39.6O
x,
,
L'=
W=
39.6O
rrtrooH,
=63mH
(b)
From Equation 8-22,
^
l{r
=
-
L,
63mH
C tR + 0.1pFx500O
= 1.26kQ
Sec. 8-4
Inductance Bridees
211
' cnFGF
?r?FBtrrFthjaEi€ifd*ri.+i$l+iitiiit*4i1"'fHl$$i$lg#${*.f*$T$ff&IffX}*Wlg!:lrg**igi
tt#lli#}!i;:i::r,.
ff*'*fri*t*'ii'd{*ir.in,i'+i4'i{iiF''i,$r*f,i*|iii*i.r|.tfft*?-''erffh'iffirrsg''Jill
F:',LntEquatiott 8-2 l,
o
I\J -_
R tR o
R,
1.26k{) x -ti)O
sz
0.r87c)
= 3.37MO
Example 8-9 demonstratesthat the inductor parallel equivalentcircuit determined
in Example 8-8 actually representsa coil that has an inductanceof 63 mH and a coil resistanceof 0.187 O. The seriesequivalentcircuit more correctly representsthe measurable resistanceand inductanceof a coil. Conversely,the parallel CR equivalent circuit
representsthe measurabledielectric resistanceand capacitanceof a capacitormore correctly than a seriesCR equivalentcircuit.
The (high) calculatedvalue of Rj in Example 8-9 shows that the low-resistance
(hi-eh-Otcoil cannot be convenientlymeasuredon a Maxwell bridge. Thus, the Hay
bridge is best for measurementof inductanceswith high Q. Similarly, it can be demonstratedthat the lr4axwellbridge is best for measurementof low-B inductances,and that
the Ha1'bridgeis not suitedto low-B inductancemeasurerneuts.
Some inductorswhich have neither very low nor very high B factorsmay easily be
measuredon either type of bridge. In this case it is best to use the Maxwell circuit, becausethe inductor is then measureddirectly in terms of its (preferable)seriesequivalent
circuit.
8-5 MULTIFLT{CTTON IMPEDANCE BRIDGE
All but one of the capacitanceand inductancebridgesdiscussedin the precedingsections
can be constructedusing a standardcapacitorand three adjustablestandardresistors.The
sineleexceptionis the inductancecomparisonbridge(Figure8-8).
Figure 8-11 shows the circuits of five different bridges constructedfrom the four
basic components.Theseare a Wheatstonebridge, a series-resistance
capacitancebridge,
a parallel-resistance
capacitancebridge, a Maxwell bridge, and a Hay bridge. Al1 five circuits are normalll'providedin commercialimpedancebridges.Suchinstrumentscontain
the four basiccomponentsand appropriateswitchesto set the componentsinto any one of
the fir'e configurations.A null detectorand internal ac and dc suppliesare also usually included.
8-6 ]\{EASURTNG StrtrA,LL C, & AND L QUANTITTES
When measuringvery small quantitiesof C L, or R, the strctycapacitance,inductance,
and resistanceof connectingleads can introduce considerableerrors. This is minimized
by connectingthe unknown component directly to the bridge terminal or by means of
very short connectingleads. Even when such precautionsare observed,there are still
srnall internal L, C, and R quantitiesin all instruments.These are termed residuals, and
2r2
Inductance and Caoacitance Measurements
Chao. 8
.,:f;.;
*reffi:'
raiiiii**i5i|
.f
(a) Wheatstone
(d) I{a->iwellbridge
(b) Seriescapacitance
(c) Parallel capacitance
(e) Hay bridge
Figure 8-11 The standard capacitor and tiree precision resistors typically contained in a commercial impedance bridge can be connected to function as a series-resistancecapacitancebridge, a parallel-resistance capacitance,a Wheatstone bridge, a Maxwell inductance bridge, or a Hay inductance
bridge.
instrument manufacturers normally list the residuals on the specification. A typical imp€dancebridge has residuals of R = I x l0-3 ,f), C = 0.5 pf', andL = 0.2 pH. Obviously,
these quantities can introduce serious errors if they are a substantial percentageof any
measuredquantify.
The errors introduced by strays and residuals can be eliminated by a substitutiott
technique(seeFigure 8-12). In the caseof a capacitancemeasurement,the bridge is first
balancedwith a larger caiacitor connectedin place of the small capacitorto be measured.
The small capacitor is then connected in parallel with the larger capacitor, and the bridge
is readjustedfor balance.The first measurementis the large capacitanceC1 plus the stray
and residualcapacitanceC". So the measuredcapacitanceis C, + C". When the small capacitor C. is connected, the measured capacitanceis C, + C" + C,. C, is found by subtracting the first measurementfrom the second.
A similar approach is used for measurementsof low value inductance and resistance, except that in this case the low value component must be connectedin serie.swith
the larger L or R quantity. The substitution technique can also be applied to other (nonbridge) measurementmethods.
Sec.8-6
MeasuringSmall C R, and,L Quantities
213
.*d
r. llf,
c,=f,llc.
, nu,
q llcn
/
crpaciturcc
L.arge
{b) $mall cag*horcrmccred
in perallcl with large
capacira fc mesrrrcmenl
Stray capacilamc af l'cct+
nte3sUrcnrd aocurecy
{c)
It/t€err|rsnentgrv*s
c,ficoano
4ggo
Hgurt &12 Smy clplcitu*e cln serirxtrly nffecrthc aocuracyof rnnaflrsnerr of e srmll c*pacitrx. For bcs accu::ry, tlrc unknownsnnll capacior (C,) stxxH be <marcoedin pralbl with a largcr
capacitrr. C. can lhen bcdrrcmincrl frofi thc rneuiurctlvalueof C,llCr.
&trl
"*m-tOn the bddge in Example84 u new balarce is obtainedwhen a small caprcitor (C,) is
connecte{tin parallelwith the measured
capacitor{,,. The new componentvaluesfor ba[alre as rtr = 369.3kO, fr3 = lO kfl, a1df' = 14.66kO. Detcrmirc tlrc vnltr of C. and
ils prallel resistiveco{nponentfir.
Matian
c^llc;=c:'+C'=
+&
fi.
O.l pt- x l0 kll
= rJ.682rl,F
t4.66k{}
pF - O.{}68pf'
C. = O.{I182pF - Cp= O.{1682
= 2fi) pF
R-llP-=
awl
Rtfr{ = 36tr.3rdl x 14.66kd)
ltr
t0 ktrA
= 541.4kO
rll
fr,' R, - &llR,,
- l _- _- - ::-
2t4
Indurtanu: arxl Capacitancc lVlcasurcnrcn$
Chlp. ll
J
tlI
frr- t/(f,,ll8p)- U.qp
Frcm Example 84,
fo =553.1kdl
so
,t*
r/54r.4lfl - r/5s3.rkG
= 3OM{}
S.7 DIGITAL
I. C, AND f MNASUREMENTS
Indurtance Mmsrernent
lnductance and capacitance musl be first conveft€d into voltages befbrc an-ymeas$r€ment
can be made by digital techniques.Figure 8-13 illustrates the bsrie mdhod.
In Figure 8-13(a) an ac voltage is applied to the noninyediug inprt terminal of an
operational amplifier. The input voltage is developed across resistor R1 to give a currcnt:
I = VlRr. This current also flows through the inductor giving a voltage drop: Vs = IX*, If
Vi = 1.592 Vrms,/- I kHz, fiq = I kf,l, and L = lt$ mH:
t=L,tt,
and
l'592v =r.592mA
I kf,l
V=l{Z^rlL)=l.S9?mAx?rr x I kHzx l0OmH
= I V{rmsi
when
L=200 mH. Vy.= ? Vi wbenL = 300 rnH, Vs= 3 V; andso or.
It is seenthat the voltagedevelopedacrossL is directly proportionalto the inducdetectar[Figure 8-13(a)l is employd to resolvethe
tive irnpeilance.A plwse-sensitive
inductor volhge into quadratureand in-phasevoltages.Thesetwo componsntsrepresent
the seriesequivalentcircuit of the measuredinductor The voltagesarefed to digital measuringcircuitsto displaythe seriesequivalentcircuit induclance1., the dissipationfac&x
(D = llQl, and/orthe O factor.
CapacitanceMereurrment
Capacitiveimpedanceis treatedin a similar way to inductiveimpedance,exceptthaf the
input voltageis developedacrossthe capacitorand the output voltage is nreasuredacross
theresistor[seeFigure8-13(b)].
In this caseI = Vy'Xnand V6=/rt. With V;= 1.592Vrmg"f - I kHz, frr = I kd), and
C= 0.1FF:
v.
=Vd2rfC)
| = -rv
/14
= 1.592Yx2s x I kHzx0.l pF
Scc. ll-7
l)igital 1.,(', and ll Measuremenls
213
;,iiw,}sd*il
Quacirature
component
I/
Phase
sensitive
detector
(1.592v 1 kIIz)
4,ffi
-." f f i
''ff i
s-
In-phase
component
t t l- - a - l ^ i t l
*.*er*! d
- " #F*] ! #
---.*
(a)
Linear conversion of inductive
impedance into voltage
v-
Phase
:ensitive
detector
(1.592v 1 kl{z)
In-phase
component
"r
(b) Linear conversion of capacitive
impedance into voltage
Figure 8-13 Basic circuits for converting inuuctive and
capacitive ^^,ipedancesinto voltage componenls rbr elecronic measurement. The loitages
"re resolved into in-phase and quadrature compo_
nens tbr determination of the D and factors.
e
= l mA
and
V a= IR = l mA xl kO
= I V (rms)
v h e n C :0 .2 p ,.4Vn= 2y) w hen C = 0.3 pF, V n= 3y;and
so on.
The voltage developed across R is directry proportional
to the capacitive imped_
ance' The phase sensitive detector
[Figure 8-13(b)] resolves the resistor voltase into
216
Inductanceand CapacitanceMeasurements
Chap. g
,#r
*.ngfifl3
quadrature and in-phase components, which in this case are proportional to the capacitor
current. The displayedcapacitancemeasurementis that of the parallel equivalent circuit
(C).The dissipationfactor (It) of the capacitoris also displayed.
Capacitance Measurement on Digital Multimeters
Some digital multimeters have a facility for measuringcapacitance.This normallv involves charging the capacitor at a constant rate, and monitoring the time taken to arrive at a given terminal voltage. In the ramp generator digital voltmeter system in
Figure 6-1, the ramp is produced by using a constant current to charge a capacitor.
Figure 8-14 shows the basic method. Transistor Q1, together with resistors R,, Rr,
and R3. produce the constant charging current to capacitor Cr when Q2 is off. C1 is
discharged when Q2 switches on. (A similar circuit is treated in more detail in
Section 9-4.)
As alreaciy explained for the digital voltmeter, a ramp time (rr) of I s and a
clock generatorfrequency of 1 kHz result in a count of 1000 clock pulses, which is
then read as a voltage. If Vi remains fixed at 1 V the display could be read as a
measureof the capacitor in the ramp generator.A I pF capacitor might produce the
I s counting time, so that the display is read as 1.000 pF. A change of capacitance
to 0.5 pF would give a 0.5 s counting time and a display of 0.500 p.F. Similarly, a
capacitanceincreaseto 1.5 pF'would produce a 1.5 s counting time and a 1.500 pF
display. In this way, the digital voltmeter is readily converted into a digital capacitance meter.
(Fixed
quantity) -+- V1
[-L]
i*',
Comparator
output
*ct *irrV
ri
F+- Counting ->i
tl
Cl oc k pul s es
d uri ng tl c an
be a measure
of capacitance
(a) Ramp generator circuit
(b) Waveforms
Figure 8-14 Basic ramp generator circuit and waveforms for a digital voltmeter. If V; is a fixed
quantity, time Ir is directly proportional to capacitor C1, and the digital output can be read as a measure of the capacitance.
Sec. 8-7
Digital t C and R Instruments
217
3gilb..*I
ffi
8-8 DIGITAL RCL METER
The digital RCL meter shown in Figure 8-i5 can measureinductance, capacitance, resistance,conductance,anddissipationfactonThedesiredfunctionisselectedbypushbutton. The range switch is normally set to the automatic (AUTO) position for convenience.
However, when a number of similar measurementsare to be made, it is faster to use the
appropriate range instead of the automatic range selection. The numerical value of the
measurementis indicated on the 3]-digit display, and the multiplier and measuredquantity are identified by LED indicating lamps.
Four (cunent and potential) terminals are provided for connection of the component to be measured.(See Section'7-4 for four-terminal resistors.) For general use each
pair of current and voltage terminals are joined together at two spring clips (known as
Kelvin clips) which facilitate quick connection of components. A ground terminal for
guard-ringmeasurements(sec Section7-6) is provided at the rear of the instrument.The
ground terminal together u,ith the other four terminals is said to give the instrumentTiveterminal measurementcapability. Bias terminals are also available at the rear of the instrument, so that a bias current can be passedthrough an inductor or a bias voltage applied to a capacitor during measurement.
For R, L, C, and G, typical measurementaccuraciesap +[0.257o + (1 + 0.002 R, L,
C, or G) digitsl; for D, the measurementaccuracyis +(ZVo+ 0.010).
Resistancemeasurementsmay be made directly on the digital LCR instrument in
Figure 8-15 over a range of 2 Q to 2 MO. Conductanceis measureddirectly over a range
t lq q
i[r[::
f--T-..]_
lL lc
l el G
lo
ffi
Figure 8-15 Digital impedance meter that can measure inductance, capacitance, resistance, conductance,anC Cissipationfactor. (Courtesy of Electro Scientific Industries, Inc.)
218
Inductance and CapacitanceMeasurements
Chap. 8
-
- *ttraFEEtnntFffi{ffi'
ha'.l-iia*rf*
srftnrF;ii'ir*rr.'ir*srir!
-.lF
of 2 pS to 20 S. Resistancesbetween2 MO and 1000 MO can be measuredas conductances,and the resistancecalculatcC:R = llG. For example,a resistanceof 10 MO is
measuredas 0.100 pS.
Inductance and capacitance measurementsmay be made directly over a range of
200 pH to 200 H, and 200 pF to 2000 pF, respectively.The dissipation factor D is determined by pressing and holding in the D button while the L or C button is still selected.
The directly measuredinductance is the seriesequivalent circuit quantity f. The Q factor
of the inductor is calculated as the reciprocal of D:
^ a L " 1 = =Q=
R"D
(seeSection8-1)
Direct capacitancemeasurementsgive the parallelequivalentcircuit quantity Co. ln
this caseD (for the parallel equivalentCR circi;it) is
D=
t
aCoRo
(seeSection8-l)
\\/hen measuring lou, values of resistanceor inductance,the connecting clips
should first be shortedtogetherand the residualR or L valuesnoted (as indicateddigitally). Thesevalues should then be subtractedfrom the measuredvalue of the component.
When measuringlow capacitances,
the connectingclips shouldfirst be placedas close together as the terminals of the component to be measured(i.e., without connecting the
component). The indicated residual capacitance is noted and then subtracted from the
measuredcomponentcai:acitance.
Return to Figure 8-13(a) and assumethat a capacitoris connectedin place ofthe inductor. The measuredquantity is displayedas an inductanceprefixed by a negative sign
on the RLC meter in Figure 8-15. The capacitiveimpedanceis equivalentto the impedanceofthe indicatedinductance:
tol" =
I
oCr
c,=-+-
or
a'L,
For an indicatedinductanceof 100 mH, and a measuringfrequencyof I kHz,
C ,=
(2rrxl kH z)' x l 00mH
= 0.25p.F
Similarly. inductancecan be measuredas capacitancewhen it is convenientto do so.
The digital RCL meter shown in Figure 8-16 displays the measuredquantity and
the units of measurement.It also displays the equivalentcircuit (parallel RC, seriesRL,
etc.) of the measuredquantity. ln RCL AUTO mode of operation,the dominating component is measured,and its equivalent circuir is displayed. Any one of several parameters
(Q, D, R* R,, etc.) may be selectedmanually for measurement.
Sec.8-8
Digital RCI Meter
2r9
xia.'i8&
ffi*r*
,l
*
Figure 8-16 Digital RCZ merer that displays the equivalent circuit of the measured
quantity,as well as the numericalvalue and the units. (@ 1991,John Fluke Mfg. co., Inc.
All rights reserved.Reproducedwith permission.)
8-9 O METER
Q-i\{eter Operation
Inductors.capacitors,and resistorswhich have to operateat radio frequencies(RF) cannot be measuredsatisfactorilyat lower frequencies.Instead,resonancemethods are emplol'ed in which the unknown componentmay be testedat or near its normal
operating
frequency.The Q meter ts designedfor measuring the factor of a coil ancifor measurQ
ing inductance,capacitance,and resistanceat RF.
The basic circuit of a Q meter shown in Figure 8-17 consistsof a variable calibrated capacitor,a variable-frequencyac voltage source,and the coil to be investigated.Atl
are connectedin series.The capaciturvoltage (V) and the source voltage (E) are monitored by voltmeters.The sourceis set to the desired:neasuringfrequency,and its voltage
is adjustedto a convenientlevel. CapacitorC is adjustedto obtain resonance,as indicated
C oi l
terminals
Signal generaror
Capacitor
terminals
Figure 8'17 A basic B meter circuit consistsof a stableac supply,a variable
capacitor,and a voltm e te r to m o n ito rthecapaci torvol tage.Whentheci rcui ti si nresonance,
V c=V uand,e__V s/E .
220
Inductanceand CapacitanceMeasurements
Chap. g
'*
r#riffi
when the voltage across C is a maximum. If necessary,the source is readjusted to the desired outprrilevei .,r'henresonanceis obtained.
At resonance:
and
V c= V t
1
=
o='L
also
R
t=E
R
coCR
(8-26)
(8-27)
and
Example8-11
\\'hen the circuit in Figure 8-17 is in resonance,E = 100 mV R = 5 O, and Xp = X,
= 1 0 0Q.
h) Calculatethe coil Q and the voltmeterindication.
(b) Deterrnlne the Q factor and voltmeter indication for another coil that has R = l0 O
and X1 = 100 C) at resonance.
Solution
1=E=
R
(a)
loomv =2omA
sf,)
Vt= l/r= I Y,
= 20mA xl 00O
= 2Y
o
- =
(b)
v, =
E
2v
l 00mV
=zo
For the second coil
E
t= A =
100mV
= l 0mA
100
V1= Vr= | Nt
= l 0mA xl 00O
=lV
o-
Sec. 8-9
QMeter
V,
lV
=lo
- = E
100mV
221
{ei,
rr
*!'-*F
'*{ffi6ffi;#*iit!;,}{"#G$k'iia':}Esl;&'6'ii'{sT*i;ltrF
. . . ' + i , . r$ i
4Erffi
Q Meter Controls
Example 8-ll shows that wiren Q = 20 the capacitorvoltmeter indicates2 Y and when
Q = l0 the voltmeter indicates I V. Clearly, the voltmeter can be calibrated to indicate the
coil B directly [seeFigure 8-18(a)].
Ii the ac supply voltage in Example 8-11 is halved, the circuit current is also
halved. This results in V6'and V1 becoming half of the values calculated. Thus, instead of
indicating 2 Y for a Q of 20, the capacitor voltmeter would indicate only 1 V. The probIem of supply voltage stability can be avoided by always setting the signal generator voltage to the correct level or by having the signal generator output voltage precisely stabilized. However, it can sometimesbe convenient to adjust the supply to other voltage
levels.If the 100 mV position on the supply voltmeter is marked as 1, and the 50 mV position is marked as 2, and so on, the supply voltmeter becomes a multiply-Q-by meter
[Figure 8-18(b)]. When E is set to give a I indication, all B values measuredon the capacitor voltmeter are correct. If E is set to the 2 position, measuredQ values must be
multiplied by 2. Instrumentsthat have a signal generatorwith a stabilizedoutput do not
use a meter for monitoring the sourcevoltage (i.e., there is no multiply-Q-by meter). In
this case.the voltagelevel of the supply is selectedby meansof a switch, and this switch
becomesa Q-meterrange control.
If the adjustablecapacitor in the Q meter circuit is calibrated and its capacitanceindicatedon a dial, it can be usedto measurethe coil inductance.From Equation 8-26,
'ffi
I
2
(b) Supply voltmeter calibrated as a
multiply-Q-by meter
t a t Capacitorvoltmetercalibrated
to monitor O
100
(c) Capacitance dial calibrated to
indicate coil inductance
Figure 8-18 With the Q-meter supply voltage (E) set to a convenient level, the capacitor voltmeter can directly indicate Q, the supply voltmeter can function as a muldplyQ- by meter, and the capacitancedial can indicate coil inductance as well as capacitance.
222
Inductance and CapacitanceMeasurements
Chap. 8
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Supposethatf = l.592MHz, and resonanceis obtainedwith C = 100 pF.
L-
I
(2:nxl.592MHz)"x l00pF
_ 100 p.H
pH. Also, if
When resonanceis obtained at the same frequency with C = 200 pF, L
- 50
C = 50 pF at l.592MHz, L is calculatedas 200 pH. It is seenthat the capacitancedial
can be calibratedto indicatethe coil inductancedirectly (in addition to capacitance)[Figu re 8 -1 8 (c )1 .
If the capacitordial is calibratedto indicateinductancewhen/= l.592MHz, any
changein/changes the inductancescale.For/= 15.92MHz and C = 100 pF,
L-
(2n x 15.92MHz)2x 100 pF
= 1p.H
With C : 200 pF and 50 pF, I becomes0.5 pH and 2 p"H, respectively.Therefore, if the
frequencf is changedin multiples of 10, the inductancescalecan still be used with an appropriatemultiplying factor.
As an alternative to using a fixed frequency and adjusting the capacitor, it is sometrmesconvenientto leave C fixed and adjust/to obtainresonance.In this case,the inductance scaie on ihe capacitor dial is no longer usable.However, Equation 8-26 still applies,
so Z can be calculated from the C and fvalues.
Residuals
Residual resistanceand inductance in the Q meter circuit can be an important source of
error when the signal generator voltage is not metered. If the signal generator has a
sourceresistanceR6, the circuit currentat resonanceis
I_
E
insteadof
RE+ R
,E
R
Also, the indicated Q factor of the coil is
u=
aL
R r+ R
insteadof the actual coll Q, which is
Q=
aL
R
Obviously, R6 must be much smaller than the resistanceof any coil to be investigated.
Similarlv. residual inductance must be held to a minimum to avoid measurementerrors.
Sec.8-9
QMeter
223
ffir
In a practicalQmeter, the outputresistance
of the signalgeneratoris around0.02 O, and
the residualinductancen'"y typicallyb:0.015 pH
Commercial QMeter
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The Q meter shown in Figure 8- l9 has a meter for indicating circuit Q and a Q LIMIT
(rangel switch. A frequencydial with a window is included,and controls are provided for
frequencyrangeselectionand for continuousadjustmentof frequency.The L/C dial indicatesthe circuit Z and C and is adjustedby the seriescapacitorcontrol identified as UC.
The -\C control (alongsidethe L/C control) providesfine adjustmentof the seriescapacitor. Its dial indicatesthe capacitanceas a plus (+) or minus (-) quantity.The total resonating capacitanceis the sum or differenceof that indicatedon the two capacitancedials.
-\Q ZERO COARSE and FINE controlsare situatedto the right of the Q indicating meter.
Theseare usedto measurethe differencein O betweentwo or more coils that have close11,equalQ factors.
\leasuring Procedures
\Iediunr-range inductance measurement(direct connection). Coils with inductancesof up to about 100 mH can be connecteddirectly to the inductanceterminals. as explained earlier. The signal generatoris set to the desired frequency,and its
output level is adjusted to
a convenient Q-foctor range. With the AC capacitor
-qive
dial set to zero, the B capacitorcontrol is adjustedto give maximum deflection on the
Q metei. Thc Q factor of the coil is now read directly fiom the meter. The coil inductance mav also b: read from the C/L dial if the signal generatoris set to a specified
frequencr'.When some oih"r frequency is employed, the inductancecan be calculated
from.f and C (Equation 8-25). With the coll Q and L known, its resistancecan also be
calcuiated.
Figure 8-19 HP4342A O meter has a deflection meter for indicaring Q, a frequency
dial. and an UC dial. (Courtesvof Hewlett-Packard.)
Inductanceand CapacitanceMeasurements
Chap. 8
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Example8-12
With the sigi,-l generatorfrequency of a Q meter set to 1.25 MHz, the Q of a coil is measuredas 98 when C = 147 pF. Determinethe coil inductanceand resistance.
Solution
From Equation 8-26,
tt
=-^
L=
^
(2nfi"C
a'C
(2r x 1.25MHz)2x 147pF
= l l 0pH
and
u^aL= -
K
2r x 1.25MHz x I l0 pH
2r.fL
-
o98
= 8.8f)
High-impedance measurements (parallel connection). Inductan:es greater
than 100 mH, capacitancessmallcr than 400 pF, and high-valueresistancesare best measuredbv connectingthem in parallelwith the capacitorterminals.
For measurementof parallei-connectedinductance (lp), the circuit is first resonated
using a referenceinductor (or work coil).The values of C and Q are recorded as Cl and
Qr Lp is nou' connected,and the circuit is readjustedfor resonanceto obtain C2 and Q2.
The parametersof the unknown inductance are now determined from the following equations:
,'
O:
|
a-(C2 - C1)
QtQz(C z- C )
C { Qz- Q)
(8-28)
(8-2e)
To measure a parallel-connected capacitance (Cp), the circuit is first resonated
using a referenceinductor, as before. The values of C1 and Q1 are noted. Then the capacitor is connected.Resonanceis again found by adjustingthe resonatingcapacitorto give a
value C2. Normally, the circuit Q is not affected. The unknown capacitanceis
Sec.8-9
QMeter
225
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ffiis
B*o"Y,:t
r*.r
.ry
(8-30)
Large-valueresistors(Rp) connectedin parallel with the,iesouatlng capacitoralter
the circuit p, but no capacitanceadjrlstmentis necessary(unlessRp also has capacitance
or inductance). Once again, the circuit is first resonated using a reference inductor. Then
Rp is connected,and the changein Q factor (AQ) is measured.The unknown resistanceis
calculated from
.-* d
/R -? I I
.l #
..* d
'..'.ff
*<*
Small values of resisLow-impedance measurements (series connection).
tance, small inductors,and large capacitorscan be measuredby placing them in series
with the referenceinductor. The componentto be measuredis connectedbetween the
LO terminal of the Q meter and the low potential terminal of the reference inductor.
The other end of the reference inductor is connected to the HI terminal of the B meter.
Initially. a low-resistanceshorting strap is connectedto short-out the unknown component. The circuit is now tuned for resonance(using an internal coil), and the values of
Q1 and C1 are noted. The shorting strap is removed, and the circuit is retuned for resonance.
When a pure resistanceis involved, circuit resonanceshould not be affected by removal of the shortingstrap.However,the circuit Q should be reduced.The changeto Q2
is measuredas AQ. The series-connected
resistanceis now calculatedas
p"= --49-
(8-32)
<ttCrQrQz
A small series-connectedinductance (1") affects both the Q factor and the circuit
resonance.The circuit is initially resonatedwith L" shorted,and the capacitorvalue (C1)
is noted.The shortingstrap is removed,and the capacitoris readjustedfor resonanceand
its neu' r'alue(C) is recorded.The inductanceis now calculatedas
(8-33)
With a large series-connectedcapacitor (Cs), the circuit is first resonated with a
shorting strap acrossthe capacitor terminals. The strap is removed, and the circuit capacitor is readjustedfor resonance.In this case, the Q of the circuit should be largely unaffected. The series-connectedcapacitanceis
(8-34)
226
Inductanceand CapacitanceMeasurements
Chap. 8
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QU B S TTON S
8-1 SketchRC seriesand parallelequivalentcircuits for a capacitor.Discussile capacitor typesbestrepresentedby eachcil'^,-rit.
8-2 Derive equationsfor converting a seriesRC circuit i:rto its equivalentparallel circuit.
8-3 SketchRL seriesand parallel equivalentcircuits for an inductor. Explain which of
the two equivalentcircuits bestrepresentsan inductor.
8-4 Derive equationsfor converting a parallel Rl circuit into its equivalentseriescircuit.
8-5 Define the O factor of an inductor. Write the equationsfor inductor Q factor with
Rl seriesand parallelequivalentcircuits.
8-6 Define the D factor of a capacitor.Write the equationsfor capacitorD factor with
RC seriesand parallelequivalentcircuits.
8-7 Sketch the basic circuit for an ac bridge and explain its operation-Discussthe adjustment procedurefor obtainingbridge balance,and derive the balanceequations.
8-8 Drar'.'the circuit diagrarnof a simple capacitancebridge. Derive the balanceequation. and discussthe limitations of the bridge.
8-9 Sketch the circuit diagram of a series-resistance
capacitancebridge. Derive the
equationsfor the measuredcapacitanceand its resistivecomponent.
8-10 Dra*'the phasordiagram for a series-resistance
capacitancebridge at balance.Explain.
8-11 Sketch thc circuit diagram of a parallel-resistance
capacitancebridge. Derive the
equationsfor the measuredcapacitanceand its resistive component.Discuss the
different applicationsof seriesRC and parallelRC bridges.
8-12 Sketch the circuit diagram of an inductancecomparisorrbridge. Derive the equations fcr the resistiveand inductivecomponentsof the measuredinductor.
8-13 Sketchthe circuit diagram of a Maxwell bridge. Derive the equationsfor the resistile and inductivecomponentsof the measuredinductor.
8-14 Sketchthe circuit dia-eramof a Hay inductancebridge.Derive the equationsfor the
resistii'eand inductive componentsof the measuredinductor. Discuss the various
applicationsof the Maxwell and Hay bridges.
8-15 Sketchac bridge circuit diagramsshowing how a standardcapacitorand three adjustable standardresistorsmay be used to measurccapacitanceas a seriesRC circuit. capacitanceas a parallel RC circuit, inductanceas a seriesRL circuit, and inductanceas a parallelRL circuit.
8-16 Discussthe problemsinvolved in measuringsmall C R, and L quantities,and explain suitablemeasuringtechniques.
8-17 Sketch the basic circuits for converting inductanceand capacitanceinto voltages
for digital measurements.
Explain the operationof eachcircuit.
8-18 Draw a circuit and waveformsto show how capacitancecan be measuredon a digital multimeter.Exolain.
Revieu'Questions
227
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8-19 Draw the basic circuit diagram for 2 Q mcter, explain its operation,and write the
equation for Q factor.
8-20 Draw a practical Q-meter circuit, and discussthe various control si rrvolvedin Q meterneasurements.
8-21 Discussthe various methodsof connectingcompcnentsto a Q meter for measurement. Explain briefly.
P R OB LE MS
8-1 A circuit behavesas a 0.01 p"F capacitorin serieswith a 15 kf,) resistancewhen
measuredat a frequencyof I kHz. If the terminal resistanceis measuredas 3l .l
kQ. determinethe circuit componentsand the connectionmethod.
8-2 When measuredat a frequencyof 100 kHz, an unknown circuit behavesas a 1000
pF capacitor anii a 1.8 kf) resistorconnectedin series.The terminal resistanceis
measuredas greaterthan 10 Mf,). Determinethe actualcircuit componentsand the
connectionmethod.
8-3 A sirnple capacitancebridge, as in Figure 8-5, usesa 0. I pF standardcapacitorand
nvo standardresistorseachof which is adjustablefrom I k0 to 200 k0. Determine
the minimum and maximum capacitancevaluesthat can be measuredon the bridge.
8-4 A series-resistance
capacitancebridge, as in Figure 8-6, has a I kHz supply frequency.The bridge componentsat balance are C, = 0. 1 pF, Rr = 109.5 O, R. = 1
kQ. and R+ = 2.I k0. Calculatethe resistiveand capacitivecomponentsof the measuredcapacitor,and determinethe capacitordissipationfactor.
8-5 A parallel-resistance
c;rpacitance
bridge (Figure 8-7) usesa 0.1 pF capacitorfor C',
and the supply frequencyis I kI{2. At balance,Rt = 541 f,), R, = I kC), and R, =
666 Q. Determinethe parallel RC componentsof the measuredcapacitor,and calculatethe capacitordissipationfactor.
8-6 Calculate the parallel equivalentcircuit components(C, and Ro) for the measured
capacitorin Problem 8-4. Also, determinethe values of R1 and Ra required to balanceC, and Ro when the bridge is operatedas a parallel-resistance
capacitorbridge.
Assumethat R3remains I kO.
8-7 An inductancecomparisonbridge (Figure 8-8) has Zr = 100 pH and R+ = 10 k,f).
\\/hen measuringan unknown inductance,null is detectedwith R' = 3-7.1O and
R: = 2l .93 k0. The supply frequencyis 1 MHz. Calculatethe measuredinductance
and its resistivecomponent.Also, determinethe Q factor of the inductor.
8-8 An inductor with a marked value of 100 mH and a Q of 2l at I kHz is to be measuredon a Maxwell bridge (Figure 8-9). The bridge usesa 0.1 pF standardcapacitor and a I kO standardresistorfor R1. Calculatethe resistancevaluesof Rj and Ra
at which balanceis likely to be achieved.
8-9 A Maxu'ell bridge with a l0 kHz supply frequencyhas a 0. I pF standardcapacitor
and a 100 O standardresistorfor R1. ResistorsR3 and Ra can eachbe adjustedfrom
100 0 to I kO. Calculatethe range of inductancesand Qfactors that can be measuredon the bridge.
228
Inductance and Capacitance Measurements
Chap. 8
" iit,
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8-10 A Hay bridge (Figure 8-10) with a 500 Hz supply frequency has C3 = 0.5 F.F and
R+ = 900 O. If balance is achieved when R1 = 466 O and R3 = 46.1 A, calculate the
inductance,resistance, and Q factor of the measuredinductor.
8-11 Calculate the seriesequivalent circuit componentsL" and R" for the Lo and Ro quantities determined in Problem 8-10. Also, determine the resistancesof R1 and R3 required to balance Z, and R, when the circuit components are connected as a
Maxwell bridge. Assume that R4 and C3 remain 900 O and 0.5 pB respectively.
8-12 The Q-meter circuit in Figure 8-17 is in resonancewhen E = 200 mV R = 3 ,C),and
Xt= Xc = 95 C).Calculate the coil B and the voltmeter indication.
8-13 The voltmeter in the Q-meter circuit in Figure 8-17 indicates5 V when a coil is in
resonance.If the coil has R = 3.3 f,) and X, = 66 O at resonance,calculatethe coil
Q and rhe supply voltage.
Problems
229