Inductance and Capacitance Measurements

"ff 

Inductance and Capacitance 

Measurements 

Objectives 

You *ill be able to: 

l. 

) 

Sketch RC series and parallel equivalent circuits for a 

rela:ing the iwo circuits. 

capacitor, and write equations 

Sketch Rl series and parallel equivalent circuits for an inductor, and write equations 

relating the two .;ircuits. 

3. Explain the Q factor of an inductor and the D factor of a capacitor, and v.,rite the equations 

for each factor. 

4. Drau' circuit diagrams for the following ac bridges: simple capacitance bridge, seriesresistance 

capacitance bridge, parallel-resistanco capacitance bridge, inductance comparison 

bridge, Maxwell bridge, and Hay inductance bridge. 

5. Erplain the operation of each of the bridges listed above, derive the equations for the 

quantities to be measured, and discuss the advantages and disadvantages of 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 measure 

a wide range ofcapacitances and inductances. Explain. 

7. Discuss the problems involved in measuring small R, L, and C quantities, explain suitable 

measuring techniques, and calculate measured quantities. 

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 adaptations of the Wheatstone bridge. 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 specialized measurements. Some commercial 

ac bridges use only a standard capacitor and three adjustable standard resistors 

to construct several different types of inductance and capacitance bridge 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 frequencies are best measured on a Q 

meter. 

3.1 RC AND RZ EQLIVALENT CIRCUITS 

Capacitor Equivalent Circuits 

Theequivalentcircuitofacapacitorconsistsofapurecapacitance Cpandaparallelresistance 

Rp. as iliustrated in Figure 8-1(a). Cp represents the actual capacitance value, and 

ftp represents.the resistance of the dielectric or leakage resistancc. Capacitors that have a 

high leakage current flowing through the dielectric have a relatively low value of Rp in 

their equivalent circuit. Vc,y iow teakage currents are represented by 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 microamperes 

of leakage crlrrent, while a capacitor with a plastic film dielectric could typically 

have a resistance as high as 100000 MO. 

A parallel RC circuit has an equivalent series RC circuit [Figure 8-1(b)]. Either one 

of the tu'o equivalent circuits (series or parallel) may be used to represent a capacitor in a 

circuit. It is found that capacitors with a high-resistance dielectric are best represented by 

the series RC circuit, while those with a low-resistance dielectric should be represented 

by the parallel equivalent circuit. However, when the capacitor is measured in terms of 

the series C and R quantities, it is usually desirable to resolve them into the parallel 

(u) Parallel equivalent 

circuit 

^, 1 

"l'T 

(b) Series equivalent 

circuit 

Figure 8-1 A capacitor may be represented 

by either a parallel equivalent circuit or a 

series equivalent circuit. The parallel equivalent 

circuit best represents capacitors that 

have a low-resistance dielectric, while the 

series equivalent circuit is most suitable for 

capacitors with a high-resistance dielectric. 

190 Inductance and Capacitance Measurements Chap. 8

*iln 

equivalent circuit quantities. This is because the (parallei) leakage resistance 

sents 

best 

the quality 

repre- 

of the capacitor dielectric. Equations ihat rela; the series 

equivalent 

and parallel 

circuits are derived belorv. 

Refening to Figure g_ l, the series impedance is 

and the parallel admittance is 

Thus, 

Zr= Rr- jX, 

y-=a*; I 

'n- 4*t4=Ge+iBe 

where G is conductance and ^B is susceptance. The impedances of each circuit must be 

equal. 

giving 

or 

glvmg 

Equating the real terms, 

(8-1) 

Equating the imaginary terms, 

a,= ^4 ^ ' R!+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 equivalent circuits are illustrated in Figr"e 8-2. The :eries equivalent circuit in 

Figure 8-2(a) represents an inductor as 2 pur. inductance L" in series with the resistance 

oi its coil. Th;s series equivalent circuit is normally the best way to represent an inductor, 

because the actual winding resistance is involved and this is an important qua:rtity. Ideally, 

the winding resistance should be as small as possible, but this depends on the thickness 

and length of the wire used to wind the coil. Physically small high-value inductors 

tend to have large resistance values, while large low-inductance components are likely to 

have low resistances. 

The parallel RL equivalent circuit for an inductor [Figure 8-2(b)] can also be used. 

As in the case of the capacitor equivalent circuits, it is sometimes more convenient to use 

a parallel RL equivalent circuit rather than a series circuit. The equations relating the two 

are derived belorv. 

Referring to Figure 8-2, the series circuit irnpedance is 

and the parallel circuit admittance is 

R" +JX 

Z,= R,+ jX., 

., I I 

f = - - t- 

-P 

RP " X,, 

Yr= Go-iB, 

Zr: Zp 

_1 

GP - jBp 

glvlng 

R.+x. = 

t (Go+iBr\ 

Gp-iBp\Gr+ jBo ) 

Gp+ jBp 

R, +iX, = Grr+ q 

9A 

I 

1I il t', 

I 

(a) Series equivalent 

circuit 

-1- 

(b) Parallel equivalent 

circuit 

Figure 8-2 An inductor may be represented 

by either a parallel equivalent circuit or a 

series equivalent circuit. The series equivalent 

circuit is normally used, but it is sometimes 

convenient to employ the parallel 

eouivalent circuit. 


til 

*-- 

Equrting the real terms, 

ft,= 

Gp 

G,l + r] 

1/RP I R;X; \ 

rtR] + r/x] \ R:X: I 

RoxS 

xj +n] 

(8-3) 

Equating the imaginary terms, 

Y= 

Bp 

c| + n'z, 

llxp 

1/n] + ux] 

I R;X; \ 

t"-"1 

\ R;X; ) 

(8-4) 

Like Equations 8-1 and 8-2, Equations 8-3 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; consequently it does 

dissipate some power. The quatity factor, ot Qfactor; of the inductor is the ratio of the inductir-e 

reactance and resistance at the operating frequency. 

e=\='1" 

R, R" 

(8-5) 

where l, and R" refer to the components of an Rl series equivalent circuit [Figure 8-2(a)]. 

Ideally. ol. should be very much larger than R", so that a very large Q factor is obtainede 

faciors for typical inductors range from a low ofless than 5 to as high as 1000 (depending 

on frequency). 

As discussed earlier, an inductor may be represented by 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' 

^Ln 

(8-0.1 

D Factor of a Capacitor 

The quality of a capacitor can be e;pressed in terms of its power dissipation. A very pure 

capacitance has a high dielectric resistance (low leakage current) and virtually zero power 

dissipation. A /ossy capacitor, which has a relatively low resistance (high leakage current), 

dissipates some 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 resistance measurable at its terminals. In the case of the capacitor, the 

resistance involved in the D-factor calculation is that showu in the parallel equivalent circuit. 

(This differs'from the inductor Q-factor calculation, where the resistance is that in 

the series equivalent circuit.) Using the pa rallel equivalent circuit: 

o=b 

Rp 

aCrR,, 

(8-7) 

Idealll', R, should be 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 

capacitors u,ith a plastic film dielectric (again depending on frequency). 

\\rlren a series equivalent circuit is used, the equation for dissipatiorr factor can be 

shown to be 

D= R" =coCJ, 

x" 

(8-8) 

Comparing Equation 8-7 to 8-6.and Equation 8-8 to 8-5, it is seen that in each case 

D is the inverse of Q. 

Example 8-1 

An unknos'n circuit behaves as a 0.005 pF capacitor in series with a 8 kf,) resistor when 

measured at a frequency of I kHz. The terminal resistance is measured by an ohmmeter 

as 134 kQ. Determine the actual circuit components and the method of connection. 

Solution 

x"= 

I 

2nfC 

: 3l .8 kC) 

2r.xll

Equation 8-I 

Equatiott 8-2, 

o - R.r+ X.r _ 

'.P-&- 

= 134 kO 

(8 k0)2 + (31.8 kO)'? 

8 kc) 

r = R: *^, = (8 kq)2 + (3r.8 k0)2 

31.8 ko 

^. 

= 33.8 kO 

c,,= 1 

' 2rJX, 

2rxlkHzx33.8kf) 

I 

_ 0.005 u.F 

Since the measured terminal resistance is 134 kO, the circuit must consist of a 

0.005 pF capacitor connected in parallel with a 134 kf) resistor. For a seriesconnected 

circuit, the terminal resistance would be rnuch higher 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 Wtreatstone bridge circuit (Figure 7-3) except that impedances are shown instead of 

resistances. and an ac supply is used. The null detector must be an ac instrument such as 

an electronic galvanometer, headphones, or an oscilloscope. 

\\hen the null detector indicates zero in the circuit of Figure 8-3, the alternating 

volta-se across points a and b is zero. This means (as in the Wheatstone brirfge) that the 

voltage across Z, is exactly equal to that across 22, and the voltage across 23 equals the 

r,oltage drop across Za. 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 voltages were equal in amplitude but not in 

would not indicate zero. 

Vzr = Vzz 

phase, the ac null detector 

i1Z. = i2Z2 

(l) 

-* 

and 

or 

Vzt=Vzq 

iyZu= 1r7o 

(2) 

Dividing Equation I by Equation 2, 

irZr - 

itZt 

iz4 

izZ+ 

giving 

(8-e) 

As already stated, bridge balance is obtained only when the voltages at each terminal 

of the rrull detector are equal in phase as well as in magnitude. This results in Equation 

8-9. u,hich involves complex quantities. In such an equation, the real parts of the 

quantities on each side must be equal, and the imaginary parts of the quantities must also 

be equal. Therefore, when deriving the balance equations for a particular bridge, it is best 

to express the impedances in rectangular form rather than polar form. The real quantities 

can then be equated to obtain one balance equation, and the imaginary (or7 quantities) 

can be equated to arrive at the other balance equation. 

The need for two balance equations arises from the fact that capacitances and inductances 

are never.pure; they must be defined as a combination of R and C or R and I 

(as discussed in Section 8-1). One balance equation permits calculation of L or C, and the 

other is used for determining the R quantity. 

Balance Procedure 

As already explained, two component adjustments are required to balance the bridge (or 

obtain a minimum indication on the null detector). These adjustments are ,?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 adjustment has a marked effect on the relative phase 

difference of these voltages. For example, Za inFigure 8-3 might consist of a variable capacitor 

in series with a variable resistor, as illustrated in Figure 8-4(a). Adjustment of Ca 

ma1' make V7a equal 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 - Vzo might be further reduced by bringing the voltages closer together 

in phase. However, this cannot be achieved without altering the amplitude of V2a, 

which is the voltage drop across Ra 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 Inductance and Capacitance Measurements Chap. 8

qI 

/rt _ r/ \ 

\v z3 

v z4l 

(a) Null detector voltage = (Vzz = Vzz) 

Yz+ 

vzt 

vzA 

r: - t/ 

t/ 

rz3 

-rl 

vz4 

vzs - vzt 

(b) I'zs and V.4 

equal but not 

in-phase 

(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 smallest possible null detector indication is achieved. 

l/7. close toV..,in amplitude, but again has an unavoidable effect on the phase relationship. 

The procedure of alternately adjusting Ra and C4 to minimize the null detector voltage 

is continued until the smallest possible indication is obtained. Then, Vya is equal to 

VTborh in magnitude and phase [Figure 8-4(d)r. 

AC Bridge Sensitivity 

The same considerations that determined the sensitivity of a Wheatstone bridge apply t

iR 

in the measured quantity that causes the galvanometer to deflect from zero. Bridge sensitivity 

can be improved by using a more sensitive null detecior and/or by increasir-rg the 

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 measurements is also determined in the sa;tle way as Wheatstone 

bridge accuracy. 

8-3 CAPACITANCE BRIDGES 

Simple Capacitance Bridge 

i- 

it 

The circuit of a simple capacitance bridge is illustrated in Figure 8-5(a). 21 is a standard 

capacitor C1, and Q is the unknown capacitance C,. 23 and Za arc known variable resistors. 

such as decade resistance boxes. When the bridge is balanced, 21/23 = 7alZa (Equation 

8-9) applies: 

z, = :11 

- 

z": 

toCr 

- 

-il 

aC* 

Zz: Rz and Z+= Ra 

(a) Simple capacitance bridge 

(b) Potential divider 

substituted for R, and Ra 

Figure 8-5 The simple capacitance bridge 

measures the unknown capacitance C, in 

terms of standard capacitor C1 and adjustable 

precision resistors R3 and Ra. At balance, 

c,= cl3lR4. This circuit functions 

only with capacitors that have very high resistance 

dielectrics. 

198 Inductance and Capacitance Measurements Chap- 8

fr 

Therefore, 

-jllaCt 

n3 

_ 

iJ!'cu 

R4 

or 

l= 

CrRz 

I 

C'Rq 

glvrng 

(8-r0) 

The actual resistances of R3 and Ra zue not important if their ratio is knowno so a 

potential-divider resistance box could be used as shown in Figure 8-5(b). 

Example 8-2 

The standard capacitance value in Figure 8-5 is Cy = 0.1 pF, and R3lRa can be set to any 

ratio bet\\'een 100:1 and 1:100. Calculate the range of measurements of unknown capacitance 

Q'. 

Solution 

Fnrration R- l O 

For R3/Ra = 100: I : 

^ 

r =- 

CBt 

R4 

C. = 0.1 F.F x 100 

I 

= l0 p.F 

ForR3/Ra=l:100: 

C,=0.1 pFr # 

= 0.001 p.F 

The foregoing analysis of the simple capacitance bridge assumes that 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 because i1 and i2 could not be brought into phase, and consequently, i1R3 and i2R4 

would not be in phase. As discussed in Section 8- 1, the equivalent circuit of a leaky capacitor 

is a pure capacitance in parallel with a pure resistance. Thus, the simple capacitance 

bridge is suitable only for measurernent of capacitors with high-resistance dielectrics. 

Sec. 8-3 Capacitance Bridges r99

Series-Resistance Capacitance Bridge 

In the circuit shown in iigu;'e 3-6(a), the unkr:..'w:r caDacirrnce is r';prcsented as a pure 

capacitance C5 in seri"s with a resirrance 1.,. A standard adjustable resistance R1 is connected 

in series with standard capacitor C1. The voltage drop across R, balances the resisrrvc 

voltage ilrops in branch 22when the bridge is balanced. The additional resistor in series 

u,ith C increases the total resistive component in 2., so that inconveniently small 

values of l(1 are not required to achieve balance. Bridge balance is most easily achieved 

when each capacitive branch has a substantial resistive component. To obtain balance, R1 

and either Rj or Ra are adjusted alternately. The .series-resistartce c'apacitance bridge is 

found to be most suitable for capacitors with a high-resistance dielectric (very low leakage 

currenl and low dissipation factor). When the bridge is balanced, Equation 8-9 apolies. 

Zt -Zt 

z1 z3 

giving 

Rr-jl/aCr _ R,-jllaC, 

R3 R. 

(8-rl) 

Equating the real terms in Equation 8-11, 

R,=R, 

R3 R" 

glvln-s (8- r 2) 

Equating the ima-einary terms in Equation 8- 11, 

1_1 

roClR-1 oC,R+ 

giving (8-13) 

The phasor diagram for the series-resistance capacitance bridge at balance is drawn 

in Figure 8-6(bl. The voltage drops across 23 and Z" are i1R3 and i2Ro, respectively. These 

two volta_ees must be equal and in phase for 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 

phase with ilRj and f2 is in phase with i2Ra. The impedance of C1 is purely capacitive, 

and current leads voltage by 90" in a pure capacitance. Therefore, the capacitor voltage 


- r*f 

(a) Circuit of series-resistance capacitance bridge 

irRs = i2Rn 

.(b) Phasor diagram for balanced bridge 

Figure 8-6 The series-resistance capacitance bridge 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!ance 

dielectric. 

drop i1X6r is drawn 90" lagging ir. Similarly, the voltage drop across c" is i2X6.5, and is 

dran'n 90" lagging i2.The resistive voltage drops l,R1 and i2Rs are in phase with ir and i2, 

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 

Capacitance Bridges 201

Example 8-3 

".g 

.i*er*** 

'.aaaF 

=.aF* 

--## 

-.#ipr..l| 

,- aaa.#itaae 

.--ffi 

-# 

-....g 

-*'# 

..-w 

A series-resistance capacitance bndge [as in Figure 8-6(a)] has a 0.4 p,F standard capacitor 

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 

measured capacitur and its dissipation factor. 

Solution 

EquationS-13, C,= + 

Equarion 8-12, 

= 0.068 p.F 

o - R,Ro 

-R3 

0.1 pF x 10 kO 

t4.7 k{l 

125 Ax M.7 kA 

l0 ko 

= 183.8 f) 

Equation 8-8, 

D = oC,R, 

= 2n x 100 Hz x 0.068 pF x 183.8 C) 

: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 capacitance is represented by its parallel equivalent circuit; Crinparallel 

u'ith Ro. Z3 and Za are resistors, as before, either or both.of which may be adjustable. 

Q is balanced by a standard capacitor C1 in parallel with an adjustable resistor R1. Bridge 

balance is achieved by adjustment or R1 and either R3 or Ra. The parallel-resistance capacitance 

bridge is found to be most suitable for capacitors with a low resistance dielectric 

(relativell'high leakage current and high dissipation factor). At balance, Equation 8-9 

once again applies: 

Z, -Zt 

Z. Z^ 

Also, 

1l 

-=- 

Zt 

Rr 

_1 

j(I/aC) 

I 

=- 

Rr 

+ jaCl 

1 

Ll= - 

llRt+ jaCt 

202 

Inductance and Capacitance Measurements Chap. 8

wsil' 

" dn*eho- 'rjffi 

Figure 8-7 The parallel-resistance capacitance bridge uses an adjustable resistance (Rr) 

connected in parallel with C1 to balance the resistive component (R) of Zz. This bridge is 

most suitable for measuring capacitors with a low-resistance dielectric. 


1=l* I 

4 Re jQlaCr) 

T^ 

= - *JaLp 

Rp 

or 

.: 

substltuttng into Equation 8-9, 

4=l 

l/Ro+ jaC, 

I/(l/&+ jloCr) _ ll(llR,+ jaCo) 

R3 

1 

Rn 

1 

R3(l/Rr +j

giving (8-1 5) 

Equating the imaginary terms in Equation 8-14, 

yEf.-'Et 

rymt 

*ffi 

..,*'"-*i'llf'P 

' 'ry 

Solution 

1 

x"= 

2nfC" 2n x 100 Hz x 0.068 pF 

=23.4kQ 

Equation 8-1, 

^ 

"n= 

nl+ x? 

& 

= 

(r83.8 fi)'? + e3.4kQ12 

l83so 

= 2.98 MO 

EquationS-2, x,= R?!x? - 

,4.r 

= 23.4 k{l 

(183.80)2 + (23.4kQ)2 

23.4kQ 

Co= 

' 

| 

2rJX, 

I 

2rr x 100 Hzx23.4kdl 

:0.068 p.F 

Front Equation 8-16, 

o _ CtRt _ 0.1 pFx l0kf) 

rr4- - 

Ce 0.068 p.F 

= 14.7 kQ 

From Equation 8-15, 

, - 

RtR, 

R4 

= 2.03 MO 

10 k.0 x 2.98MO 

4.7 kA 

The capacitor, which was determined in Example 8-3 as having a series equivalent 

circuit of 0.068 pF and 183.8 O, was shown in Example 8-5 to have a parallel equivalent 

circuit of 0.068 pF and298 MO. It was also shown that to measure the capacitor on a 

parallel-resistance capacitance bridge, R1 (in Figure 8-7) would have to be 2.03 MO. This 

is an inconveniently large value for a precision adjustable resistor. So a capacitor with a 

high leaka-ee resistance (low D factor) is best measured in 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 leakage resistance should be neasured as series RC circuits. 

Capacitors with a low leakage resistance should be measured as parallel RC cir- 

Sec. 8-3 Capacitance Bridges 205

:riTiffi. 

:_*f*l*iiir 

I di-{*.ftiffidifrdt|L. ,ff 

cuits. Capacitors that have neither a very high nor a very low leakage resistance 

rr,casurec 

are best 

as a paraller RC circuit, because this gives u ii...t indicatioit of the 

leakage 

capacitor 

resistance. 

8.4 INDUCTANCE BRIDGES 

Inductance Comparison Bridge 

*!@r...t 

.*klM 

The circuit of the inductance comparison bridge shown in Figure g_g 

series-resistance 

is similar to the 

capacitance bridge except that inductors are in*volved instead 

tors' The 

of 

unknown 

capaci- 

inductance' represented by its (series equivalent circuit) 


inductance 

R'' is measured 

z, 

in terms of a precise staniard value inductor. zr is the 

tor' 

standard 

R' is a 

induc- 

variable standard resistor to balance R,, R3 and Ra are standard 

ance 

resistors. 

of the bridge 

Bal- 

is achieved by alternatery adjusting R1 and either R3 

Equation 

or Ra. At barance, 

8-9 once again applies: 

tt 

=tt 

Z, 24 

R, + jaLl R,+ iaL" 

R3 

R4 

Rt .aL, * = _& a;.L, 

Rj 

-Rr 

R4'Ro 

(8-17) 

Equating the real components in Equation g_17. 

Rr= 

R3 

R" 

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 Z- 

and R-. 

2M 

lnductance and Capacitance Measurements 

Chap. g

ij*..fl 

glvlng (8-13) 

Equating th" imaginary components in Equation 8-17, 

aLr 

R3 

_ aL" 

R4 

gvlrrg 

(8-1e) 

"r-*" 

t" 

An inductor that is marked as 500 mH is to be measured on an inductance comparison 

brid-ee. The bridge uses a 100 mH standard inductor for L1, and a 5 kO standard resistor 

for R.. If the coil resistance of the 500 mH inductor is measured as 270 f,). determine the 

resistances of R1 and R3 (in Figure 8-8) at which balance is likely to occur. 

Solution 

From Equation 8-19, o -RoLr - 

r\3 - - - 

L, 

=1kC) 

5kOx100mH 

500 mH 

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 desirable to be able to measure inductance in a bridge that uses a capacitance 

standard rather than an inductance standard. "[he Manuell bridge (also known 

as the Maneell-Wein bridge) is shown in Figure 8-9. In this circuit, the standard capacitor 

C3 is connected in 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

Figure 8-9 The N{axrvell bridge uses a standard capacitor 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 suitable for measuring coils with a low Q factor. 


1=1_ 1 

=l 

23 R3 jllaQ R3 

I 

7.- ' 

llfu+ jaC3 

Zz= R,+ joL" 

Substituting for all components in Equation 8-9, 

Rr 

tt(r/&+ jaQ) 

+ jaC3 

R" +"1'rol, 

R4 

&*rrC.R,=& 

*j^L, 

R3 R4 R4 

(8-20) 

Equating the real components in Equation 8-20, 

::::"-. Rr - 

R" 

DD 

r\? 

t\a 

or 

[-4&l 

| *'l 

Equating the imaginary components in Equation 8-20, 

(8-21) 


qffi 

..i-*ii#f*k-'*;sa$*r.rr.flfliiia.-I.i.iiiii!ktrii*ri*ri*'r'**idfl**ir*r*liaii-|.**r lifu '{l}rl 

uC3R, = tL' 

R4 

glvrng (8-22) 

Example 8-7 

A Maxq,ell inductance bridge uses a standard capacitor of Cj = 0. I p,F and operates at a supplyfrequencyof 

l00Hz.BalanceisachievedwhenRl =1.26kA,R2=410,f),andRo=JQ6 

f|. Calculate the inductance and resistance of the measured inductor, and determine its Q 

factor. 

Solutiott 

Equatiort 8-22, L, = CzRtR+ 

= 0.1 pF x 1.26 kO x 500 O 

=63mH 

Equariott8'21, R,= +& = -Eqlaf4q-q 

- 

R. 470Q 

= 1.34 kCl 

Equation 8-5, 

Q = tL' - 2t x 1oo Hz T 63 mH 

R, 1.34 kc). 

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 series instead of parallel, and the unknown inductance is 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 measured L, and R, are a parallel equivalent circuit. The equivalent series RI, circuit 

can be determined by substitution into Equations 8-3 and 8-4. 

\Vtren the bridge in Figure 8-10 is balanced, 

Z, =4 

Z^ Z^ 

glvlng 

Ra .Ro Ra I 

t- 

(8-23) 

- 

Rp aLp 

- . 

Rr olC:Rr 

Sec. 8-4 Inductance Bridses 209

!6itl"trtffii:L 

;*il 

Figure 8-10 The Hay bridge uses a standard capacitor 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. 

Equating the real components in Equation 8-23, 

&=& 

RP Ri 

(8-24) 

Equatin-e the imaginary components in Equation 8-23, 

R^l 

aLp - oC3R1 

giving 

(8-2s) 

"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 Capacitance Measurements Chap. 8

Yfi 

st*Et@uattf:E*tf.*iflir;ffi 

.-lI 

Solution 

Equation 8-25, Lp= C3Rfia 

= 0.1 p.F x 1.26 kO x 500 O 

=63mH 

Equations-24, R"= - 1'26 k!l!5oo o 

# 

= 8.4 kC) 

Equation 8-6, 

o= 3-P 

tttLp 

8.4 kO 

2r.xl00Hzx63mH 

- 11) 

Example 8-9 

(a) Calculate the series equivalent circuit for the Lp and Rp values determined in Example 

8-8. 

(b) 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 

(a) 

Xr=ZrJl-r=/11xl00Hzx63mH 

Equation 8-3, 

p-L- 

_ 

' 

= 39.6 O 

Rsc 

xj+Rj 

= 0.187 O 

8.4 kO x (39.6 O)2 

(39.6 O)2 + (8.4 kO)2 

Equation 8-4, 

,. R:X, (8.4 k0)2 x 39.6 O 

"'- xj+ n/ (39.6 O)2 + (8.4 kO)2 

= 39.6 O 

, x, 39.6 O 

L'= 

W= rrtrooH, 

=63mH 

(b) From Equation 8-22, 

L, 63mH 

l{r ^ = - 

CtR+ 0.1 pFx500O 

= 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,. 

.,:f;.; 

ff*'*fri*t*'ii'd{*ir.in,i'+i4'i{iiF''i,$r*f,i*|iii*i.r|.tfft*?-''erffh'iffirrsg''Jill 

F:',Lnt Equatiott 8-2 l, 

RtRo 

o 

I\J - _ R, 

= 3.37 MO 

1.26 k{) x -ti)O sz 

0.r87 c) 

Example 8-9 demonstrates that the inductor parallel equivalent circuit determined 

in Example 8-8 actually represents a coil that has an inductance of 63 mH and a coil resistance 

of 0.187 O. The series equivalent circuit more correctly represents the measurable 

resistance and inductance of a coil. Conversely, the parallel CR equivalent circuit 

represents the measurable dielectric resistance and capacitance of a capacitor more correctly 

than a series CR equivalent circuit. 

The (high) calculated value of Rj in Example 8-9 shows that the low-resistance 

(hi-eh-Ot coil cannot be conveniently measured on a Maxwell bridge. Thus, the Hay 

bridge is best for measurement of inductances with high Q. Similarly, it can be demonstrated 

that the lr4axwell bridge is best for measurement of low-B inductances, and that 

the Ha1'bridge is not suited to low-B inductance measurerneuts. 

Some inductors which have neither very low nor very high B factors may easily be 

measured on either type of bridge. In this case it is best to use the Maxwell circuit, because 

the inductor is then measured directly in terms of its (preferable) series equivalent 

circuit. 

8-5 MULTIFLT{CTTON IMPEDANCE BRIDGE 

All but one of the capacitance and inductance bridges discussed in the preceding sections 

can be constructed using a standard capacitor and three adjustable standard resistors. The 

sinele exception is the inductance comparison bridge (Figure 8-8). 

Figure 8-11 shows the circuits of five different bridges constructed from the four 

basic components. These are a Wheatstone bridge, a series-resistance capacitance bridge, 

a parallel-resistance capacitance bridge, a Maxwell bridge, and a Hay bridge. Al1 five circuits 

are normalll' provided in commercial impedance bridges. Such instruments contain 

the four basic components and appropriate switches to set the components into any one of 

the fir'e configurations. A null detector and internal ac and dc supplies are also usually included. 

8-6 ]\{EASURTNG StrtrA,LL C, & AND L QUANTITTES 

When measuring very small quantities of C L, or R, the strcty capacitance, inductance, 

and resistance of connecting leads can introduce considerable errors. This is minimized 

by connecting the unknown component directly to the bridge terminal or by means of 

very short connecting leads. Even when such precautions are observed, there are still 

srnall internal L, C, and R quantities in all instruments. These are termed residuals, and 

2r2 Inductance and Caoacitance Measurements Chao. 8

*reffi:' 

. f raiiiii**i5i| .*d 

(a) Wheatstone 

(b) Series capacitance 

(c) Parallel capacitance 

(d) I{a->iwell bridge 

(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-resistance capacitance bridge, 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€dance 

bridge has residuals of R = I x l0-3 ,f), C = 0.5 pf', and L = 0.2 pH. Obviously, 

these quantities can introduce serious errors if they are a substantial percentage of any 

measured quantify. 

The errors introduced by strays and residuals can be eliminated by a substitutiott 

technique (see Figure 8-12). In the case of a capacitance measurement, the bridge is first 

balanced with a larger caiacitor connected in place of the small capacitor to be measured. 

The small capacitor is then connected in parallel with the larger capacitor, and the bridge 

is readjusted for balance. The first measurement is the large capacitance C1 plus the stray 

and residual capacitance C". So the measured capacitance is C, + C". When the small capacitor 

C. is connected, the measured capacitance is C, + C" + C,. C, is found by subtracting 

the first measurement from the second. 

A similar approach is used for measurements of low value inductance and resistance, 

except that in this case the low value component must be connected in serie.s with 

the larger L or R quantity. The substitution technique can also be applied to other (nonbridge) 

measurement methods. 

Sec. 8-6 

Measuring Small C R, and, L Quantities 213

. llf, 

c, = f,llc. 

,nu, / 

crpaciturcc 

q llcn 

L.arge 

Stray capacilamc af l'cct+ 

nte3sUrcnrd aocurecy 

{b) $mall cag*horcrmccred 

in perallcl with large 

capacira fc mesrrrcmenl 

{c) It/t€err|rsnent grv*s 

c,ficoano 4ggo 

Hgurt &12 Smy clplcitu*e cln serirxtrly nffecr thc aocuracy of rnnaflrsnerr of e srmll c*pacitrx. 

For bcs accu::ry, tlrc unknown snnll capacior (C,) stxxH be

tlI 

frr- 

- 

J 

t/(f,,ll8p) - U.qp 

Frcm Example 84, 

so 

fo =553.1 kdl 

,t* 

r/54r.4lfl - r/5s3.r kG 

= 3O M{} 

S.7 DIGITAL I. C, AND f MNASUREMENTS 

Indurtance Mmsrernent 

lnductance and capacitance musl be first conveft€d into voltages befbrc an-y meas$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: 


when 

t=L- 

,tt, 

l'592v =r.592mA 

I kf,l 

V=l{Z^rlL)=l.S9?mAx?rr x I kHzx l0OmH 

= I V{rmsi 

L=200 mH. Vy. = ? Vi wben L = 300 rnH, Vs= 3 V; and so or. 

It is seen that the voltage developed across L is directly proportional to the inductive 

irnpeilance. A plwse-sensitive detectar [Figure 8-13(a)l is employd to resolve the 

inductor volhge into quadrature and in-phase voltages. These two componsnts represent 

the series equivalent circuit of the measured inductor The voltages are fed to digital measuring 

circuits to display the series equivalent circuit induclance 1., the dissipation fac&x 

(D = llQl, and/or the O factor. 

Capacitance Mereurrment 

Capacitive impedance is treated in a similar way to inductive impedance, except thaf the 

input voltage is developed across the capacitor and the output voltage is nreasured across 

the resistor [see Figure 8-13(b)]. 

In this case I = Vy'Xn and V6 =/rt. With V;= 1.592 Vrmg"f - I kHz, frr = I kd), and 

C= 0.1 FF: 

v. 

| = -r- =Vd2rfC) 

v 

/14 

= 1.592Y x2s x I kHzx0.l pF 

Scc. ll-7 l)igital 1., (', and ll Measuremenls 

213

;,iiw,}sd*il 

-"#F*]!# 

---.* 

4,ffi 

-."ffi 

''ffi 

s- 

ttl--a-lîtl 

*.*er*!d 

I/ 

(1.592 v 1 kIIz) 

Phase 

sensitive 

detector 

Quacirature 

component 

In-phase 

component 

(a) Linear conversion of inductive 

impedance into voltage 

v- 

(1.592 v 1 kl{z) 

Phase 

:ensitive 

detector 

"r 

In-phase 

component 

(b) Linear conversion of capacitive 

impedance into voltage 

Figure 8-13 Basic circuits for converting inuuctive and capacitive ^^,ipedances into voltage componenls 

rbr elecronic measurement. The loitages "re resolved into in-phase and quadrature compo_ 

nens tbr determination of the D and e factors. 

=lmA 


Va=IR=lmAxlkO 

= I V (rms) 

vhen C:0.2 p,.4 Vn=2y) when C= 0.3 pF, Vn=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 

Inductance and Capacitance Measurements 

Chap. g

*.ngfifl3 

,#r 

quadrature and in-phase components, which in this case are proportional to the capacitor 

current. The displayed capacitance measurement is that of the parallel equivalent circuit 

(C).The dissipation factor (It) of the capacitor is also displayed. 

Capacitance Measurement on Digital Multimeters 

Some digital multimeters have a facility for measuring capacitance. 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 generator frequency 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 

measure of 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 

capacitance increase to 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*', 

*ct *irrV 

ri 

F+- Counting ->i 

tl 

Comparator 

output 

Clock pulses 

during tl can 

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

ffi 

3gilb..*I 

8-8 DIGITAL RCL METER 

The digital RCL meter shown in Figure 8-i5 can measure inductance, capacitance, resistance,conductance,anddissipationfactonThedesiredfunctionisselectedbypushbutton. 

The range switch is normally set to the automatic (AUTO) position for convenience. 

However, when a number of similar measurements are to be made, it is faster to use the 

appropriate range instead of the automatic range selection. The numerical value of the - 

measurement is indicated on the 3]-digit display, and the multiplier and measured quantity 

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-ring measurements (sec Section 7-6) is provided at the rear of the instrument. The 

ground terminal together u,ith the other four terminals is said to give the instrumentTiveterminal 

measurement capability. Bias terminals are also available at the rear of the instrument, 

so that a bias current can be passed through an inductor or a bias voltage applied 

to a capacitor during measurement. 

For R, L, C, and G, typical measurement accuracies ap +[0.257o + (1 + 0.002 R, L, 

C, or G) digitsl; for D, the measurement accuracy is +(ZVo + 0.010). 

Resistance measurements may be made directly on the digital LCR instrument in 

Figure 8-15 over a range of 2 Q to 2 MO. Conductance is measured directly over a range 

tlq q 

i[r[:: 

f--T-..]_ 

lL lc lelG lo 

ffi 

Figure 8-15 Digital impedance meter that can measure inductance, capacitance, resistance, 

conductance, anC Cissipation factor. (Courtesy of Electro Scientific Industries, Inc.) 


- *ttraFEEtnntFffi{ffi' 

ha'.l-iia*rf* srftnrF;ii'ir*rr.'ir*srir! -.lF 

of 2 pS to 20 S. Resistances between 2 MO and 1000 MO can be measured as conductances, 

and the resistance calculatcC: R = llG. For example, a resistance of 10 MO is 

measured as 0.100 pS. 

Inductance and capacitance measurements may 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 measured inductance is the series equivalent circuit quantity f. The Q factor 

of the inductor is calculated as the reciprocal of D: 

âL"1 = 

Q= =- 

R"D 

(see Section 8-1) 

Direct capacitance measurements give the parallel equivalent circuit quantity Co. ln 

this case D (for the parallel equivalent CR circi;it) is 

D = 

t 

aCoR o 

(see Section 8-l) 

\\/hen measuring lou, values of resistance or inductance, the connecting clips 

should first be shorted together and the residual R or L values noted (as indicated digitally). 

These values should then be subtracted from the measured value of the component. 

When measuring low capacitances, the connecting clips should first be placed as 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 

measured component cai:acitance. 

Return to Figure 8-13(a) and assume that a capacitor is connected in place ofthe inductor. 

The measured quantity is displayed as an inductance prefixed by a negative sign 

on the RLC meter in Figure 8-15. The capacitive impedance is equivalent to the impedance 

ofthe indicated inductance: 

tol" = 

I 

oCr 

or c,= -+a'L, 

For an indicated inductance of 100 mH, and a measuring frequency of I kHz, 

C,= (2rrxl kHz)'x l00mH 

= 0.25 p.F 

Similarly. inductance can be measured as capacitance when it is conveniento do so. 

The digital RCL meter shown in Figure 8-16 displays the measured quantity and 

the units of measurement. It also displays the equivalent circuit (parallel RC, series RL, 

etc.) of the measured quantity. 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 selected manually for measurement. 

Sec. 8-8 Digital RCI Meter 2r9

ffi*r* 

xia.'i8& 

,l 

* 

Figure 8-16 Digital RCZ merer that displays the equivalent circuit of the measured 

quantity, as well as the numerical value and the units. (@ 1991, John Fluke Mfg. co., Inc. 

All rights reserved. Reproduced with permission.) 

8-9 O METER 

Q-i\{eter Operation 

Inductors. capacitors, and resistors which have to operate at radio frequencies (RF) cannot 

be measured satisfactorily at lower frequencies. Instead, resonance methods are emplol'ed 

in which the unknown component may be tested at or near its normal operating 

frequency. The Q meter ts designed for measurin g the Q factor of a coil anci for measuring 

inductance, capacitance, and resistance at RF. 

The basic circuit of a Q meter shown in Figure 8-17 consists of a variable calibrated 

capacitor, a variable-frequency ac voltage source, and the coil to be investigated. Atl 

are connected in series. The capacitur voltage (V) and the source voltage (E) are monitored 

by voltmeters. The source is set to the desired:neasuring frequency, and its voltage 

is adjusted to a convenient level. Capacitor C is adjusted to obtain resonance, as indicated 

Coil 

terminals 

Signal generaror 

Capacitor 

terminals 

Figure 8'17 A basic B meter circuit consists of a stable ac supply, a variable capacitor, and a voltmetertomonitorthecapacitorvoltage.Whenthecircuitisinresonance, 

Vc=Vuand,e__Vs/E. 

220 

Inductance and Capacitance Measurements 

Chap. g

#riffi 

'* 

when the voltage across C is a maximum. If necessary, the source is readjusted to the desired 

outprri levei .,r'hen resonance is obtained. 

At resonance: 

Vc=Vt 


t=E 

R 

also o='L = 

1 

(8-26) 

R coCR 

and (8-27) 

Example 8-11 

\\'hen the circuit in Figure 8-17 is in resonance, E = 100 mV R = 5 O, and Xp = X, 

= 100 Q. 

h) Calculate the coil Q and the voltmeter indication. 

(b) Deterrnlne the Q factor and voltmeter indication for another coil that has R = l0 O 

and X1 = 100 C) at resonance. 

Solution 

(a) 

1=E= 

R 

loomv =2omA 

sf,) 

Vt= l/r= I Y, 

=20mAxl00O 

=2Y 

o= v, = 2v 

=zo 

- E l00mV 

(b) 

For the second coil 

E 

t=A= 

100 mV 

=l0mA 

100 

V1= Vr= | Nt 

=l0mAxl00O 

=lV 

V, lV 

o- - = - =lo 

E 100 mV 

Sec. 8-9 QMeter 221

{ei, 

'*{ffi6ffi;#*iit!;,}{"#G$k'iia':}Esl;&'6'ii'{sT*i;ltrF 

*!'-*F 

rr 

...'+i,.r$i 

4Erffi 

Q Meter Controls 

Example 8-ll shows that wiren Q = 20 the capacitor voltmeter indicates 2 Y and when 

Q = l0 the voltmeter indicates I V. Clearly, the voltmeter can be calibrated to indicate the 

coil B directly [see Figure 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 prob- 

Iem 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 sometimes be 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 measured on the capacitor 

voltmeter are correct. If E is set to the 2 position, measured Q values must be 

multiplied by 2. Instruments that have a signal generator with a stabilized output do not 

use a meter for monitoring the source voltage (i.e., there is no multiply-Q-by meter). In 

this case. the voltage level of the supply is selected by means of a switch, and this switch 

becomes a Q-meter range control. 

If the adjustable capacitor in the Q meter circuit is calibrated and its capacitance indicated 

on a dial, it can be used to measure the coil inductance. From Equation 8-26, 

'ffi 

I 2 

t a t Capacitor voltmeter calibrated 

to monitor O 

(b) Supply voltmeter calibrated as a 

multiply-Q-by meter 

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 muldply- 

Q- by meter, and the capacitance dial can indicate coil inductance as well as capacitance. 


lrT fil tttt1t" tt 

*!*F"t{*$i*{&rfiffiS 

-:fffitr**. 

*i*{i#*r$i*r*riasi**ii*ar{,4ril!*iiidi*Ldftrd' 

''ffi' 

'flJlt 

I 

Q"ffc 

Suppose that f = l.592MHz, and resonance is obtained with C = 100 pF. 

L- 

I 

(2:nxl.592MHz)" x l00pF 

_ 100 p.H 

I 

a'C 

When resonance is obtained at the same frequency with C = 200 pF, L - 

50 pH. Also, if 

C = 50 pF at l.592MHz, L is calculated as 200 pH. It is seen that the capacitance dial 

can be calibrated to indicate the coil inductance directly (in addition to capacitance) [Figure 

8-18(c)1. 

If the capacitor dial is calibrated to indicate inductance when/= l.592MHz, any 

change in/changes the inductance scale. For/= 15.92MHz and C = 100 pF, 

L- 

(2n x 15.92MHz)2 x 100 pF 

=1p.H 

With C : 200 pF and 50 pF, I becomes 0.5 pH and 2 p"H, respectively. Therefore, if the 

frequencf is changed in multiples of 10, the inductance scale can still be used with an appropriate 

multiplying factor. 

As an alternative to using a fixed frequency and adjusting the capacitor, it is sometrmes 

convenient to leave C fixed and adjust/to obtain resonance. 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 resistance and 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 

source resistance R6, the circuit current at resonance is 

E 

,E 

I_ 

instead of 

RE+ R 

R 

Also, the indicated Q factor of the coil is 

instead of the actual coll Q, which is 

u= 

Q= 

aL 

Rr+R 

aL 

R 

Obviously, R6 must be much smaller than the resistance of any coil to be investigated. 

Similarlv. residual inductance must be held to a minimum to avoid measurement errors. 

Sec.8-9 QMeter 223

ffir 

In a practical Qmeter, the output resistance of the signal generator is around 0.02 O, and 

the residual inductance n'"y typically b:0.015 pH 

Commercial QMeter 

-# 

.**&.,@ 

--r#rff .#*.ia 

..geF 

.# 

The Q meter shown in Figure 8- l9 has a meter for indicating circuit Q and a Q LIMIT 

(rangel switch. A frequency dial with a window is included, and controls are provided for 

frequency range selection and for continuous adjustment of frequency. The L/C dial indicates 

the circuit Z and C and is adjusted by the series capacitor control identified as UC. 

The -\C control (alongside the L/C control) provides fine adjustment of the series capacitor. 

Its dial indicates the capacitance as a plus (+) or minus (-) quantity. The total resonating 

capacitance is the sum or difference of that indicated on the two capacitance dials. 

-\Q ZERO COARSE and FINE controls are situated to the right of the Q indicating meter. 

These are used to measure the difference in O between two or more coils that have close- 

11, equal Q factors. 

\leasuring Procedures 

\Iediunr-range inductance measurement (direct connection). Coils with inductances 

of up to about 100 mH can be connected directly to the inductance terminals. 

as explained earlier. The signal generator is set to the desired frequency, and its 

output level is adjusted to -qive 

a convenient Q-foctor range. With the AC capacitor 

dial set to zero, the B capacitor control is adjusted to 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 generator is set to a specified 

frequencr'. When some oih"r frequency is employed, the inductance can be calculated 

from.f and C (Equation 8-25). With the coll Q and L known, its resistance can also be 

calcuiated. 

Figure 8-19 HP4342A O meter has a deflection meter for indicaring Q, a frequency 

dial. and an UC dial. (Courtesv of Hewlett-Packard.) 

Inductance and Capacitance Measurements Chap. 8

i "'fEFJt 

Tffi.- 

:Hrlr@F.fi.r".Jr.t," 

'{flJ} 

Example 8-12 

With the sigi,-l generator frequency of a Q meter set to 1.25 MHz, the Q of a coil is measured 

as 98 when C = 147 pF. Determine the coil inductance and resistance. 

Solution 

From Equation 8-26, 

L= 

tt 

^ 

a'C 

=-^ 

(2nfi"C 

(2r x 1.25 MHz)2 x 147 pF 

=ll0pH 


âL 

u=- 

K 

- 

2r.fL 

o98 

= 8.8 f) 

2r x 1.25 MHz x I l0 pH 

High-impedance measurements (parallel connection). Inductan:es greater 

than 100 mH, capacitancesmallcr than 400 pF, and high-value resistances are best measured 

bv connecting them in parallel with the capacitor terminals. 

For measurement of parallei-connected inductance (lp), the circuit is first resonated 

using a reference inductor (or work coil).The values of C and Q are recorded as Cl and 

Qr Lp is nou' connected, and the circuit is readjusted for resonance to obtain C2 and Q2. 

The parameters of the unknown inductance are now determined from the following equations: 

,- | 

' 

a-(C2 - C1) 

(8-28) 

O: 

QtQz(Cz- C) 

C{Qz- Q) 

(8-2e) 

To measure a parallel-connected capacitance (Cp), the circuit is first resonated 

using a reference inductor, as before. The values of C1 and Q1 are noted. Then the capacitor 

is connected. Resonance is again found by adjusting the resonating capacitor to give a 

value C2. Normally, the circuit Q is not affected. The unknown capacitance is 

Sec.8-9 QMeter 225

B*o"Y,:t 

ffiis 

i*Fe*I'.i-nFi+rj'."{ii{,*.+ 

.ry 

r*.r 

i**i5 

(8-30) 

Large-value resistors (Rp) connected in parallel with the, iesouatlng capacitor alter 

the circuit p, but no capacitance adjrlstment is necessary (unless Rp also has capacitance 

or inductance). Once again, the circuit is first resonated using a reference inductor. Then 

Rp is connected, and the change in Q factor (AQ) is measured. The unknown resistance is 

calculated from 

.-*d 

.l# 

..*d 

/R-? I I 

'..'.ff 

*

ytfltr*. :rfiii*{idti*lt *it6i*fiiriis! tIt^t 

REVTEiV 

QUBSTTONS 

8-1 Sketch RC series and parallel equivalent circuits for a capacitor. Discuss ile capacitor 

types best represented by each cil'^,-rit. 

8-2 Derive equations for converting a series RC circuit i:rto its equivalent parallel circuit. 

8-3 Sketch RL series and parallel equivalent circuits for an inductor. Explain which of 

the two equivalent circuits best represents an inductor. 

8-4 Derive equations for converting a parallel Rl circuit into its equivalent series circuit. 

8-5 Define the O factor of an inductor. Write the equations for inductor Q factor with 

Rl series and parallel equivalent circuits. 

8-6 Define the D factor of a capacitor. Write the equations for capacitor D factor with 

RC series and parallel equivalent circuits. 

8-7 Sketch the basic circuit for an ac bridge and explain its operation- Discuss the adjustment 

procedure for obtaining bridge balance, and derive the balance equations. 

8-8 Drar'.' the circuit diagrarn of a simple capacitance bridge. Derive the balance equation. 

and discuss the limitations of the bridge. 

8-9 Sketch the circuit diagram of a series-resistance capacitance bridge. Derive the 

equations for the measured capacitance and its resistive component. 

8-10 Dra*'the phasor diagram for a series-resistance capacitance bridge at balance. Explain. 

8-11 Sketch thc circuit diagram of a parallel-resistance capacitance bridge. Derive the 

equations for the measured capacitance and its resistive component. Discuss the 

different applications of series RC and parallel RC bridges. 

8-12 Sketch the circuit diagram of an inductance comparisorr bridge. Derive the equations 

fcr the resistive and inductive components of the measured inductor. 

8-13 Sketch the circuit diagram of a Maxwell bridge. Derive the equations for the resistile 

and inductive components of the measured inductor. 

8-14 Sketch the circuit dia-eram of a Hay inductance bridge. Derive the equations for the 

resistii'e and inductive components of the measured inductor. Discuss the various 

applications of the Maxwell and Hay bridges. 

8-15 Sketch ac bridge circuit diagrams showing how a standard capacitor and three adjustable 

standard resistors may be used to measurc capacitance as a series RC circuit. 

capacitance as a parallel RC circuit, inductance as a series RL circuit, and inductance 

as a parallel RL circuit. 

8-16 Discuss the problems involved in measuring small C R, and L quantities, and explain 

suitable measuring techniques. 

8-17 Sketch the basic circuits for converting inductance and capacitance into voltages 

for digital measurements. Explain the operation of each circuit. 

8-18 Draw a circuit and waveforms to show how capacitance can be measured on a digital 

multimeter. Exolain. 

Revieu' Questions 227

;li 

]Tie*rorl@*ili:$.lf€f{llEffi-,ffffffq:.43-rii""drr€!{!frE€Edf,lFFtfF!st?n+';"i4*i{tt+:1 

r;r:;}':;c1""':::i" 

,cnInl 

;ar !.r 

"iit, 

8-19 Draw the basic circuit diagram for 2 Q mcter, explain its 

equation for Q factor. 

8-20 Draw a practical Q-meter circuit, and discuss the various 

meter neasurements. 

operation, and write the 

controls irrvolved in Q- 

8-21 Discuss the various methods of connecting compcnents to a Q meter for measurement. 

Explain briefly. 

PROBLEMS 

8-1 A circuit behaves as a 0.01 p"F capacitor in series with a 15 kf,) resistance when 

measured at a frequency of I kHz. If the terminal resistance is measured as 3l .l 

kQ. determine the circuit components and the connection method. 

8-2 When measured at a frequency of 100 kHz, an unknown circuit behaves as a 1000 

pF capacitor anii a 1.8 kf) resistor connected in series. The terminal resistance is 

measured as greater than 10 Mf,). Determine the actual circuit components and the 

connection method. 

8-3 A sirnple capacitance bridge, as in Figure 8-5, uses a 0. I pF standard capacitor and 

nvo standard resistors each of which is adjustable from I k0 to 200 k0. Determine 

the minimum and maximum capacitance values that can be measured on the bridge. 

8-4 A series-resistance capacitance bridge, as in Figure 8-6, has a I kHz supply frequency. 

The bridge components at balance are C, = 0. 1 pF, Rr = 109.5 O, R. = 1 

kQ. and R+ = 2.I k0. Calculate the resistive and capacitive components of the measured 

capacitor, and determine the capacitor dissipation factor. 

8-5 A parallel-resistance c;rpacitance bridge (Figure 8-7) uses a 0.1 pF capacitor for C', 

and the supply frequency is I kI{2. At balance, Rt = 541 f,), R, = I kC), and R, = 

666 Q. Determine the parallel RC components of the measured capacitor, and calculate 

the capacitor dissipation factor. 

8-6 Calculate the parallel equivalent circuit components (C, and Ro) for the measured 

capacitor in Problem 8-4. Also, determine the values of R1 and Ra required to balance 

C, and Ro when the bridge is operated as a parallel-resistance capacitor bridge. 

Assume that R3 remains I kO. 

8-7 An inductance comparison bridge (Figure 8-8) has Zr = 100 pH and R+ = 10 k,f). 

\\/hen measuring an unknown inductance, null is detected with R' = 3-7.1 O and 

R: = 2l .93 k0. The supply frequency is 1 MHz. Calculate the measured inductance 

and its resistive component. Also, determine the 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 measured 

on a Maxwell bridge (Figure 8-9). The bridge uses a 0.1 pF standard capacitor 

and a I kO standard resistor for R1. Calculate the resistance values of Rj and Ra 

at which balance is likely to be achieved. 

8-9 A Maxu'ell bridge with a l0 kHz supply frequency has a 0. I pF standard capacitor 

and a 100 O standard resistor for R1. Resistors R3 and Ra can each be adjusted from 

1000 to I kO. Calculate the range of inductances and Qfactors that can be measured 

on the bridge. 


**emffir . ..# 

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 measured inductor. 

8-11 Calculate the series equivalent circuit components L" and R" for the Lo and Ro quantities 

determined in Problem 8-10. Also, determine the resistances of 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 resonance when 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 indicates 5 V when a coil is in 

resonance. If the coil has R = 3.3 f,) and X, = 66 O at resonance, calculate the coil 

Q and rhe supply voltage. 

Problems 229

Inductance and Capacitance Measurements

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