Comprehensive theory of a power factor bridge

COMPREHENSIVE THEORY OF A POWER FACTOR BRIDGE. BY

J. C. BALSBAUGH

* and

ALFRED

HERZEN13ERG.t

The purpose of this paper is to give the complete theory of a bridge that may be used for high-precision power-factor The subject matter includes a general demeasurements. scription of the bridge, general bridge and shield equations, calculation of measured power-factor by the bridge, factors affecting the precision and accuracy of power-factor measurements, sensitivity of the bridge expressed in terms of the minimum detectable changes in power factor and capacitance, detecting circuit for balancing bridge and shield, method of obtaining a bridge balance, calculation of measured capacitance by bridge for obtaining dielectric constants, effect of shield unbalance on the power-factor and capacitance balance of the bridge, method of checking measured bridge power factor, calculation and measurement of the power factor of an oil sample, and the evaluation of the power factor of air condensers. A companion paper l deals specifically with the application of this theory to the measurement of the power factor of small oil samples and includes the details of construction of condensers and oil cells, general characteristics of the bridge, operation of the bridge for power-factor and capacitance measurements, characteristics of oil cells, and the procedure for measuring the power factor of small oil samples. I. GENERAL DESCRIPTION OF BRIDGE. A schematic diagram of the bridge with electrostatic shielding at bridge and ground potentials and operation switches is shown in Fig. I. The component parts of the

* Assistant

Professor, Elec. Eng. Dept., Mass. Inst. of Tech. t Research Assistant, Elec. Eng. Dept., Mass. Inst. of Tech. i “The M. I. T. Power-Factor Bridge and Associated Oil Cell, and the Power-Factor Measurement of Small Oil Samples,” by J. C. Balsbaugh, N. D. Kenney and A. Herzenberg, A. I. E. E., 1934. 49

50

J. C. BALSBAUGHAND A. HERZENBERG. [J. I;. I

bridge and shield circuits are indicated and explained in Fig.’I. This bridge is of the Schering type; but with symmetrical measuring arms, a variable condenser in one of the high-

+General

bridge

-

shield

-

ground

diagram.

voltage arms, and variable condensers for balancing the capacitance between the bridge arms and shield. The fixed and variable condensers (I, 2, and 3 in Fig. I) are shown in section in Figs. 2 and 3, respectively. These

High voltage section Bridge section Shield sections Ground sections Shield supportingshaft Insulationbetween bridge and shield section 10,ll. Spacers for 3,4 and 8,Q.

1. 2. 3.4. 5,6. 7. 8,Q.

12,13. Insulationbetweenhigh voltage section wd ground 14,15. Insulationbetween shield and ground 18. High voltage lead Bridge lead 17. Shield lead 18. 19.20. Ground leads

High tension aOOtiO,I BridSe lleotion Shield seotiom 6round seotiom

12.13. Insulationbetween shield dgr& 14.15. ShaPt and blook 16,17. dssembly bolts for 2.3.4.

1. 2. 3.4. 5.6.

Shield supportingshaft Inaulatim between bridge and shield 10.11. Inmlation betmsn highvoltage and ground

7.

18. 19.

6.9.

July, 1934.1

POWER FACTOR BRIDGE.

53

condensers are essentially the same in principle but differ in that the fixed condenser is of the cylindrical type and the variable condensers are of the conical type. Also the variable condensers are designed so that through a crank and threaded shaft the capacitance of these condensers may be varied. These condensers are designed with insulation between bridge and shield sections, between high-voltage and ground sections, and between shield and ground sections. The use of a ground between high-voltage and shield prevents variable surface leakage and conduction currents of the insulation for the high-voltage section from giving an unsteady shield balance. It is also important to have the insulation between bridge and shield effectively shielded from the high-voltage section to prevent any loss entering the bridge through this insulation. It is shown theoretically in Part VIII that the bridge measurements of power factor and capacitance may be made independent of the power factor of the insulation between bridge and shield. This fact may also be checked experimentally by taking bridge measurements with and without a resistance connected, between bridge and shield. Two types of oil cells which may be used for precision power-factor and capacitance measurements on small oil samples are shown in Figs. 4 and 5. The cell shown in Fig. 4 is constructed with a ground joint and with all of the electrical measuring surfaces mounted on a supporting tube so that the measuring surfaces may be removed from the assembled cell and effectively cleaned. For the cell in Fig. 4 the shield for the measuring section is placed outside the cell (may be made a part of the heating oven in which the cell is placed) so as to reduce the oil volume required for the cell and to reduce the number of surfaces to be cleaned. The cell shown in Fig. 5 is in principle the same as the cell in Fig. 4 except that the shield for the measuring section is placed within the cell and also that a copper tube is sealed into the glass tube through which the measuring section lead is brought out of the cell. The use of this copper tube connected to shield prevents surface leakage and conduction currents through the glass from the high-voltage lead and cylinders from entering the bridge section. The electrical measuring surfaces, supports, and mounting are made of nickel and the assembly

54

J. C. BALSBAUGH

AND A. HERZENBERG.

[J.F. I.

FIG.4.

1.

Pyrex

glass container

glass cap Pyrex glass supporting tube for electrical measur lng surfaces 4. High volts@ section 5,0. shield set tlons 7. Measuring section Fyrex glass rod for 8. supporting 5,6,7 Nickle band for 9. assembling 5,6,7,8. lO,l1,12,13,14. Nickle disks for supporting 4,5,6,7 on supporting tube 3 Support for 4 15. 16. Leaf to 4 Support for assembly 17. of 5,6,? 18. Shield for 19 Lead &I 7 19. 20 ?or 19 Seal 21 Lead to shield 22 seal for 21 23,24 Tubulations 25 Ground joint

2. 3.

-

Oil cell.

Pyrex

July, 1934.1

of these surfaces, spot-welding.

POWER FACTOR BRIDGE.

supports,

II.

GENEBAL

and leads is accomplished

BRIDGE

55 through

EQUATIONS.

The general circuit diagram of the bridge is shown in Fig. 6 in which bI and bz are the points of the bridge arms, s represents the shield, Z1, &, 27, and Zs the bridge arm impedances, Z3 the impedance between the high-voltage and shield, Zs the FIG.

5.

Oilcell.

shield balancing impedance, Z9 the galvanometer impedance or input impedance of an amplifier, Z4 and Zs the impedances between the bridge arms and. shield, and E the applied bridge voltage. Solving this general circuit diagram for 19 gives I9 = y

(1)

,

where N = Z~Z,Z3Z~Z~Z~

-

(

&++$+;+;+;++

6>

Z1Z3Z3Z4Z5Z6 >

(2)

56

J. C. BALSBAUCH AND A. HERZENBERG.

[J. F. I.

and D = 23242526

;

+ & + $ + ; 4

3

+

[Z1Zz(Z,

Z,Z8(& + z2 + z9>+ z9(&z8 + Z2Zdl

+z2z3z4&z8z9(.&

+z7>

+zlz3z5&&~9(~2

(g+;+;) 3

+z,>

(+

6

+A+;) 6

3 +

+ Zs + Z,)

6)

5

m-2.w*(~3

+

&?)(Z4

+

z5

+

(3)

z9>.

An inspection of (I), (2), and (3) will show that I9 will be equal to zero for any two of the following conditions: Z&3 = zzz,,

(4)

ZlZS =

23-G

(5)

z,z,

Z3Z8.

(6)

=

(4), (5), and (6) hold, then the remaining one Thus in Fig. 6 the current 19 will be equal to will also hold. zero for the condition of bl, bz, and s being at the same potential. It is furthermore interesting and important to note from (I) that even if (5) and (6) do not hold, that is, the shield is unbalanced, but if (4) and If any

two

of

z*

Zl

z=z hold, the current If the bridge the condition of

19 will still be zero. circuit is considered z4=z3=

then the current “’ = Z,Z,(Z,

(7)

through

by itself,

that

is, for

00,

(8)

Z9 becomes

E(ZzZ, - Z,ZsJ + ZS + Z,) + z7z8(& + +

* z2

+

z9>

ZdZ2Z7 + 21&j)

(9)

POWER FACTORBRIDGE.

July, 1934.1

57

While (I) and (9) give the exact value of 19 for the conditions assumed, it is sometimes desirable to use approximate Thus 19’in (9) may be approximated for 27 = 28 equations. bv I,

= 9

E

(22

-

.a

&&

27 227

+

z9

FIG.6.

Equivalent bridge and shield circuits.

and I9 from (I) may be approximated for Z7 z Zs by

(10)

58

J. C. BALSBAUGH

AND A.

[J. I?. I.

HERZENBERG.

III. CALCBLATIONOF MEASUREDPOWER FACTOR BY BRIDGE. A

solution

of

galvanometer

the

current

network will

in Fig.

in general

+

Assuming then

that

(12)

for

4

6 )I

5

shield

the

for

1- (12)

6

the shield

reduces

that

(;+;+A+;)

is balanced,

that

is, (5) and

(6)

to ZIZ,

Thus

3

z3z527

z3z4&z6

hold,

show to zero

$+$+$+$

+ ZaZ4&& +

ZZZ4Z6

6 will

be equal

and

= z,z,.

bridge

balanced,

(13) the

resistance

and

Z, and Z2 may be evaluated in terms of Z2 or Z1, respectively, 2, and Zs. This shows that it is not necessary to evaluate the impedances Z3 and Zs of capacitance

the shield mination

of

impedances

circuit of either

and the impedances Zq and Zr, for the deterTherefore one of the reasons for ZI or Z2.

the use of a shield

circuit

an impedance

as ZI

which The

such

can be accurately impedances

is to permit or Z,

in terms

the

measurement

of other

of

constants

determined.

ZI, Zz, ZI, and Zs are

&=X,--j--&,

(14)

Z2=Rz-j--$-,

(15)

z

7

z

= Rh I + =

2

- jwR2k~2C7 w2R,2k72C72’

R8k8 - jwR82k82G

8

I +

w2Rs2ks2Cg2’

(16) (17)

where Rc7

k, = R7

+

(If0 Rc7

’

Rc8

k8 = R8

+

Rcs

(19) -

July, 1934.1

Powm

Rc7 and Rc8 represent respectively, so that the measuring-capacitances stituting Z1, Z,, &, and respectively, in (13) and

FACTOR BRIDGE.

resistances in shunt with CT and C,, effect of losses in the power-factorSubmay be taken into account. Z8 from (14), (15)~ (IS), and (17)~ solving for Ii2 and C2 gives

RvW~ - R&Js

[

177

2

I

R&s

+

1

I +

+ RIG + ~2R&Risk,ksC~C,Cs , u2Rg2kg2Cs2

I +

u2Rs2kg2Cs2

c=R&G [

fc, =s

59

W2R,Ci(R&sC, - R,k,C,) + W2R~R&&,GG 1.

(20)

(21)

The power factor @f2 of impedance Z2 is given by p,2

=

R2 22

R2

=

wR2

Rz2 + -?WV,” >

l/2 =

(I +

C2

w~R~~CZ~)~‘ * ~

(22)

Let R2

wR&

= p2 =x,

2

(23)

where PZ is the tangent of the imperfection angle of Z2. Then the power factor fif2 of Z2 from (22) becomes

In general in this work, power factors will be expressed in terms of p as in (23), that is, the tangent of the angle equal to 909 minus the power-factor angle. Where the power factor is desired to a greater accuracy than as given by p, it may be obtained from (24) or from the sine of the angle whose tangent is given by p. For low values of power factor, wR2C2 or p2, is small in comparison with unity so that pf2 = cos 02

sp2.

The difference between P.B and pi for relatively power factors may be closely approximated by Pf2 VOL.

218,

NO.

1303-5

2

p2

-

3

p23

(25)

low (26)

60

[J. F. I.

J. C. BALSBAUGHAND A. HERZENBERG.

and the difference between P,z and PZ will be less than a given power factor p, for the condition of pZ equal to approximately Thus the difference between p, and $f2 as given by 3%. (23) and (22), respectively, will be less than IO+ for pZ of the order of one per cent. Substituting & from (20) and CZ from (21) in (23) gives

wR,k,C, - wRsksCs+ wRIC1 + w3R~R&k~k&C,C, ” =

I + w2RlCl(RsksCs - R,k,C,) $ w2R7Rsk,ksC&

*

(27)

Now if we assume that the power factors of the measuring capacitances CT and Cs are equal to zero, then k7 and kg are equal to unity. Also with CT and Cs relatively low so that OR& and wR8Cs are very small in comparison to unity, then (27) becomes ~2 = wR,C, - wRsCs + wR1G.

’

(28)

However, wRIC1 from (23) is the tangent of the imperfection angle of 21 and fir = wreck. (29) Then from (28) P2

=

w&C’,

-

wRsCs

-I-

PI.

(30)

Expressing R,, CT, and Rs, C8 similar to p2 and pl, from (23) gives

pz= ; - f + 8

Pl

.

(31)

Equation (30) shows that the power factor of impedance 2, is given in terms of resistances and capacitances of the low-tension bridge arms and also the power factor of the impedance 21. This means that it is necessary to investigate the power factor of the condenser forming Z1 in the bridge and determine if this power factor is significant in the range to which it is desired to express p2. It is obviously pertinent to investigate the difference between pZ as given by (27) and (30). Any difference between these two equations will be determined principally from the magnitudes of CT and C’Sand the power factors of the

July,

measuring capacitances.

p2=

Equation (27) may be written as

(m, - m3) + W%(*es - *c7) + *1&R*,, + I)(%*& (m7pC7

61

POWER FACTOR BRIDGE.

cm.1

+

I)(m*pc8

+

I>

+

+ I) +

mm31

mm%

+ PJ(m8 - m,) + m7m8(Pc7 -

9

(32)

Pdl

where *c7 and pc8 are the tangents of the imperfection angles of C7 and CS and are

-L

pc7 =

OR&

(33)

’

pea= -L

(34)

o&L’s ’

and m7 = wR&,

m8 = wRsCs.

(35)

In Table I there is given the difference between *Z as calculated from (32) and pz as calculated from (30) for different values of *I, *,7, PCs,and (m7 - m8). Thus with (m, - m8) TABLE pz

I.

from (3~) - p, from (30) (oalrrestimesro0).

0.5

PI

0 or 10-4 /‘ ,/ c/ ‘l t,

PC? = PC8 0 0 0 10-Z 10-Z 10-Z

m7

-

10-s 10-4 10-Z 10-S 10-4 10-g

ma

__-

I-

-1

-1

-1

-2

-1 -109

-5

-1 -129

-2 -20 -2032 -2 -21 -2096

= IO-~ or one per cent., ma = 0.1 (that is with a frequency of 60 cycles per second and Rg = 10~ ohms, CS from (35) equals26,525.8(10-12) farads), m7 = 0.11 (that is with R7 = 10~ ohms, CTfrom (35) equals 29, I 78.4( 10-12) farads) ; and pl = PC, PC8= o, the difference between p2 from (32) and p2 from (yo) is 109(10-6), with p2 from (30) being the larger value. The value of p2 from (32) is 9890(10-“) and that from (30) is IO,OOO(IO-6). The difference between p2 as calculated from (32) and (30) will d epend principally upon the magnitudes of

62

J. C. BALSBAUGH

AND A. HERZENBERG.

[J.F. I.

m7 and ma, that is C7 and CS, the magnitude of p,, and the magnitudes of and the differences between pe7 and p,,. For large values of m7 and ma the difference between p, from (32) and (30) may become quite substantial even though pl, pc7, and p, are equal to zero. It is therefore advisable in measuring either relatively large or small power factors that the capacitance of either CT or C’s be maintained relatively small so that the values of p may be obtained from (30). In determining the measured power factor of the bridge (pZ - pl) in (30) it is necessary to determine to the required accuracy the values of frequency, R7, Rs, CT, and Cg. It is obvious that any of the usual methods may be used in determining the frequency, R, and Rs,since a given percentage error in determining these values will give the same percentage error in (pZ - pl), for the condition of equal ratio arms, that is, R7 = Rs. When R7 f R8,the error in the calculated power factor will depend upon the errors in the determination of both R7,and Rsand also upon the magnitudes of C7 and G. Therefore for R7 = Rg,it is only in determining relatively large values of (pZ - PI) to a relatively high power-factor precision that it is necessary to determine quite accurately the values of frequency and Ii7 or Rs. R7may be adjusted to be very closely equal to Rg through reversal of the bridge arms as explained in the following. The determination of the values of C7 and C’Sin (28) is, however, more difficult. In general CT and C’Swill include capacitances connected in arms 7 and 8 of the bridge, respectively, and also shunt capacitances to ground of the resistances R7 and Ris and other shunt capacitances. As explained previously, one of the uses of a shield circuit is to permit a relatively simple evaluation of the constants of the impedances Z1 and & in the bridge arms. However, since it is impracticable to shield resistances, some other method must be utilized for an accurate determination of the difference between C7 and CS. It should be noted that arms 7 and 8 of the bridge circuit should be shielded at ground potential since this will tend to make the bridge balance independent of the shield balance as far as these shunt capacitances are concerned. Assume that the bridge and shield circuits are first balanced with the bridge arms connected direct in Fig. 6, that

July,

w34.1

POWER FACTOR BRIDGE.

63

is, arms 7 and 8 are connected to arms I and 2, respectively. Then the power factor measured by the bridge from.(30) is ~2

-

$1

=

u&C, - c&Ga,

(36)

where Csa is the capacitance to ground in arm 8. Now assume that the bridge arms are reversed, that is, arms 7 and 8 are connected to arms 2 and I, respectively. Then, assuming a difference between fiz and pr and the capacitance in arm 8 adjusted for balance, gives $2 - pr = CJR& - wR,C,.

(37)

Since in general R7 and Rs are not exactly equal, rebalance of the,bridge for (32) will require either an adjustment of the rebalance relative values of R,, Rs, or Cr and C2. Capacitance of the bridge may be obtained through a vernier variable condenser in one of the arms I or 2. The use of a vernier variable condenser in arms I or 2 will permit a closer capacitance balance to be obtained than is practicable through changes in R7 and R8. Since the required change in Cr or C2 will be of the order of IO-~ to IO+ of Cr or Cz, obviously the change in the absolute power factor of 2, or & (pl and p2, respectively) will be entirely negligible. Adding (36) and (37) gives for (pz - PI) p2

_

p,

=

wR

(csb- cau’ , 2

in which R = R, = R8. Equation (38) shows that the difference in the absolute power factors (Pz - PI) is now determined by the difference in the capacitances to ground of arm 8 in the two measurements. Since the shunt capacitance to ground of arm 8 is not changed in the two measurements, this method eliminates the necessity of determining this capacitance. The capacitance (C8b - Ca,) is then equal to the difference in capacitance of the condenser in arm 8 in the two balances. This difference may be obtained to a greater accuracy than the absolute value for any particular setting. This difference (CSb - C,,) may be determined directly by calibration with a standard

64

J. C. BALSBAUGHAND A. HERZENBERG.

[J. I;. I.

condenser or from the dimensions directly of a properly designed capacitance. For measuring power factors in the range of IO+ condensers C7 and Cs should be air condensers of a design such that humidity and temperature variations will not produce a measurable effect. With a condenser similar to the variable condenser in Fig. 3 used as a part of CTor Cs, obviously relatively small changes in capacitance may be obtained very easily from a properly designed condenser. Such a variable condenser as a part of C7 or Cs should be connected similarly as Item 2 in Fig. I except that the high-voltage section would now be connected to ground. This method requires only the evaluation of changes in capacitance to bridge. The power factor as measured from (38) may also be expressed in terms of larger differences in capacitance ( Cs6 - C&) in (38) by decreasing R, provided sufficient bridge sensitivity is available for these lower values of R. IV. SENSITIVITY

OF BRIDGE.

Assuming first the bridge circuit by itself, that is, neglecting the effect of the shield circuit, or that .Zqand Zs are infinite, then the voltage Eb across the bridge arms from (9) is

Eb= ls’zs =Z1Z*(Z7 +

(ZJ, - &Zs)ZsE

zs +

Zs) + Z,Z,(Z, + z@-,z,

+ zzs) + ZrZs)

(39)

and the voltage Eb will be equal to zero for the condition of zzz7 = zrzs. Now if Zr is changed from balance by AZ, then dEb AEb =aAZ1

(40)

and AZ1 = AR1 - jAXr.

(41)

Expressing AR, in terms of a change (A#,) in the tangent of the imperfection angle or power factor of Z1 from (23), with a positive Apl being an increase in the power factor of Zr, and AX, in terms of a change (ApI) equal to the ratio of the change AC, in capacitance of Zr to C,, with a positive Apr being an

July,1934.1

POWER FACTOR BRIDGE.

increase in the capacitance

X1=--$

65

of Z1, then (42)

1

AC, I AX1 = - = - Aq,Xl, CI WC1

(43)

and AZ1 in (41) becomes AZ1 = (API +&I)&.

(44)

The impedance & represents the impedance of the galvanometer when a galvanometer is used directly across the bridge arms or the input impedance of the amplifier when an amplifier is used. Then AEb from (40) becomes

A& =

‘s(Apl -I- j&)X1.

(45)

1

Equation (45) shows that a given change in the power factor of &, say Ap, = IO-~, gives the same magnitude of AEb or bridge unbalance as a corresponding capacitance From (39) change in Cl, that is, AC1 = IO-‘Cl. d&,

-= d&

d& (Zz.G - .&&)Z,Ezl

Za(- &&>E z,2

-

(46)

,

.G2

where Z, is the denominator of (39). Since the change is from bridge balance, that is, Z2Z7 = ZIZs, then

dEa -= dZ1

z1.q.G

- ZgZgE + ZB + Z,) + &ZdZl + Zz + Zd + ZdZ2-G + -c&)

*

(47)

-

(48)

This gives for AEt, in (45) -

AEb =

zlz2(z,+

(A$I + jAqJX1ZsZgE

-53 +

Z,)

+

Z,Zs(Z1

+

+

z2

uJ2.G

+

z9> +

Gz3>

With Z,, Zs, and Zs of the order of 10~ ohms and Z1 and Z2 equal to approximately 2.65(104) ohms (corresponding to

66

J. C. BALSBAUGHAND A. HERZENBERG.

[J. I?. I.

C1 and Cz equal to IOO(IO+) farads) then obviously 27, 28, or & may be neglected in comparison with Z1 or 22. Also assuming approximately equal ratio arms and the following: (49)

AEb from (48) may be written

A&

where

S

(51)

d’,(Ap~ + j&@&a,

Kb =

RZg 2R + Zg *

(52)

Now if we assume that a galvanometer or galvanometer with amplifier is used, which gives one mm. deflection for an input voltage or voltage across the bridge arms equal to E,, then from (51) the value of Ap, and Aq,,,for one mm. deflection of the galvanometer is equal to

Thus the change of Ap or Aq for a deflection of one mm. of the galvanometer decreases directly as E, decreases and decreases directly with an increase in frequency, capacitance C1 or Cz in high-tension arms of the bridge, the applied bridge voltage E and the value of Kb. The per-unit magnitude change in Rb for a per-unit change in R for Zs being a resistance equal to Rs is from (52)

$=$(2R;R,)

(Zg,=

R9)

(54)

and the per-unit magnitude change in Rb for a per-unit change in Ra is from (52) $$

= $(2R2;Rr)

(Z, = R,).

(55)

Thus the per-unit magnitude change in Rb for a given perunit change in either R or Re will be equal for Rs = 2R.

July, 1934.1

POWER FACTORBRIDGE.

67

Similarly for & = f jX, we get

(56) (57) and the per-unit magnitude change in Kb for a given per-unit change in either R or f jX, will be equal for Xg = 211. This shows that it is desirable to have Zs or RR,larger than R. A study of (52) also shows that Kb will be equal to R for & = 00 and that for & in range of practical values of R, Kb will be equal to some fraction of R. Obviously it is desirable to have large values of R (assuming that & may be varied with R) so that AEb for a given A$ or Ap will be increased or so that Ap, or Aq,will be decreased. However, as R is increased, the change in capacitance (C’ or Cs) for a given value of Ap decreases so that increased difficulty would be encountered in the power-factor measurements. Where the bridge is desired to measure the power factor of small oil samples the value of Cl is also limited by a convenient size of cell and the volume of oil required for testing. A study of AEb as in (48) obtained from (I) instead of (9) will give the effect of the shield circuit on the value of Apnz and Ap,,, for given bridge and detecting circuits. This will show that in general, for practical values of 23, 24, and Zg in Fig. 6, the effect of the shield will be to increase the value of AEb for a given Ap or A¶ and ‘for given bridge and detecting circuits. For very large values of the capacitances of Z3, Zq, and Zs in comparison with the capacitance of Zr the effect of the shield may decrease the value of AEb for a given Ap or Ap and for given bridge and detecting circuits. In general the effect of the shield on AEa will only be several per cent. (or smaller) of that as given by (51). V.

DETECTING

CIRCUIT

FOR

BALANCING

BRIDGE

AND

SHIELD.

The detector circuit for balancing both the bridge and shield is shown in Fig. 7. The detector circuit consists of a resistance-capacitance coupled amplifier feeding into either a

68

J. C. BALSBAUGHAND A. HERZENBERG.

[J. F. I.

vibration-type galvanometer or an alternating-current galvanometer with the field energized from a phase shifter. The complete amplifier is shown shielded at shield and ground potentials. The first stage of the amplifier is connected push-pull, so that the capacitances between shield and each arm of the bridge will be approximately equal. It is also possible to use a single tube for the first stage and then balance the capacitance between shield and either bridge arm by a condenser connected between one of the bridge arms and shield. The amount of voltage amplification required will depend upon the accuracy to which it is desired to balance, that is, FIG.7.

Amplifier and galvanometer.

the value of A$ or Aq which it is desired to be able to detect, upon the voltage AEb appearing across the bridge arms or input voltage to the amplifier for this particular value of Ap or A4 and the voltage sensitivity of the galvanometer. The magnitude of the voltage, AEb, appearing across the bridge arms for a given power factor or capacitance unbalance (Ap or Ap, respectively) of the bridge is given by (51). Then the required voltage amplification will be given by the ratio of AEb from (sI), for the desired value of Ap or Aq, to E, the voltage sensitivity of the galvanometer. Thus in (51) if we assume that El the bridge voltage is 1000 volts; Kb = R = 10~ ohms, corresponding to the condition of the input impedance of the amplifier, very large in comparison with R; w = 377, corresponding to a frequency

July, 1934.1

POWER FACTOR BRIDGE.

69

of 60 cycles per second of the applied voltage ; and a value of Ap or Ap equal to IO-~, the value of AEt, is equal to O.gq(IO-‘) volts. With a galvanometer having a voltage sensitivity of, volts per mm. deflection, the required amplificasay, S(IO-~) tion is approximately 8500. VI. BRIDGE

BALANCE.

The bridge circuit may be balanced by connecting the detecting circuit across the bridge arms and adjusting C7 and Cs and through adjusting either R7 and R8 or the variable condensers .ZzI and Zzz (Fig. 12). Of course the bridge and shield must be balanced successively, that is, first connecting the detecting circuit between the bridge arms and then adjusting for approximate bridge balance; then connecting the detecting circuit between either bridge arm and shield and adjusting for approximate shield balance and continuing this procedure until with the detecting circuit successively between the bridge arms and between bridge and shield there is no measurable deflection in the desired range of balance. The number of times it is necessary to continue this procedure of bridge and shield balancing will depend upon the effect of a shield unbalance on the bridge and the effect of a bridge unbalance on the shield. These effects will in general depend upon the coupling between bridge and shield, that is, the magnitude of Z4 and Z5 in comparison with Z1 and &, the degree of unbalance and the difference between Z4 and &. By adjusting either Z4 or Z5 until Z4 is approximately equal to Z5 the effect of a bridge unbalance on the shield or the effect of a shield unbalance on the bridge may be made quite small. A complete study of the effect of a shield unbalance on the bridge is given under Part VIII. If we assume that both bridge and shield circuits are balanced we have by (I 3)

Now if 2, is changed by AZ,, rebalance of the bridge may be obtained through a change in Zs equal to AZ8 and rebalancing of the shield. For small values of AZ, and neglecting second-

J. C. BALSBAUGHAND A. HERZENBERG.

70

[J. F. I.

order effects, from (13) ZrAZ8 = - ZsAZ,.

(58)

AZ1 may be expressed from (44) as (59)

AZ, = (API +&1)x1 and az, = $AR~

-+-~AC~

= (I ” u2R,1C’);Rg

-

2dRs3CgACs

(I + w~R&‘~~)~ -3

. awR,&ARg + uRs2(I - u~R~~C,~)AC~ * (I + ~~Ris~C’s~)~

(60)

Substituting (44), (59), and (60) in (58) and solving for AR* and AC8 gives AR* = &

[A~~(uR&

-

~1) + Aql (PNRECS +

1

ACs = &(r

;

01, (61)

p12) CAP& + plwRaCs) +

AdPI - (~&Cdl,

(62)

where p1 is given by (29). Equations (61) and (62) show that for the condition of pr and wR& small in comparison with unity, a capacitance change in either Z1 or Z2 is balanced principally by a change in R, or Rg and a power-factor change in either Zr or Zz is balanced principally by a change in either CT or Cs. These equations also show that if Cr and Cs are kept as low as possible for measuring the difference in power factor between p2 and pr, the changes in Ii8 or C8 for changes in Apl or Apr, respectively, may be made quite small. If the capacitance balance of the bridge is obtained through a change in the capacitance of Z2 (balancing through variable condensers Z2 in Fig. 6) instead of through RT or Rs then we may obtain equations giving Aq2 and AC* for change in API or Aq,, using a method similar to the one for the derivation

July, 1934.1

POWER FACTORBRIDGE.

of (61) and (62).

This will give

C R (I + u21G2G2) [Aad A42 = C:R; [(I

71

+ &.&Cs) + AA C&G

-

PI)]

+ dR,Re,C,C,) (I + &OR&) -

(w&G

’

- ~R,C,)(wRsCs - PI)]

(I + LO~RS~C~)[API(I + w~RTRJ,G) + A&&C’s - &C,)] AC8 = wR,[( I + w~R,R,C,C,) (I + P&&s) - ’ - (w&G - w&G) (&Cs - PI)] VII. CALCULATION

(63)

OF MEASURED

CAPACITANCE

(64)

BY BRIDGE.

For the calculation of the dielectric constant of an oil sample or other capacitance measurements it is necessary to determine capacitances from the bridge balance equations in terms of a known capacitance. This may be obtained from (21) giving C2 in terms of Cl. Equation (21) may be written in terms of ~27,ma, pc7, PCs,RT, Rg, and Cl, employing a method similar to the one used in obtaining p2 in (32). This gives [m*2 + (I + %P&)“l +m,p,,)(I +m*p,,> +pl(ms(I

c2 = %l[(I

-

m,(x

+

mspcs))

+wpc7> +

*

(@)

w&l

In Table II there are given values of C2afrom (69, divided by C2 from $ C1, for given values of m7, ma, PC?,PCs,and PI. 8 TABLE

ml-m.

103 10% 10-a lo* 10-r 103 10-b lo-” 10-p

10-4.

:

Yo6) l-1(10”)

10’.

:

I-lo(lo*)

1-l&~ 1-101(10-s) 1-110(10-g) : 1

: l-9(10*)

II.

10-p. l-1(:0-) l-100(10”) 1 1-2cio-s) 1-ZOo(l0-s)

I

1-l(i0-s) l-99(10*)

lo-‘. 1-1(10-q 1-10(1o-s) ~$JW)~~ 1-11(10-s) 1-1086(10-q 1-1(10-y 1-10(10-“) l-988(10*)

5x10-‘. l--4(10*) l-40(106) l-3984wJ? l-4(10? l-40(10*) 1-4031(10-s) 1-4(10-s) l-40(103 1-3983(10?

72

J. C. BALSBAUGHAND A. HERZENBERG.

[J. F. I.

Since Cz in (6.5) is determined from a known capacitance Cr, it is necessary to have the condenser representing C, either calibrated from a standard capacitance or have the design of Cr such that its capacitance may be determined to the desired accuracy from its dimensions. With a condenser of the design shown in Fig. 2 the relative dimensions may be made such that the capacitance may be calculated to sufficient accuracy from the dimensions. VIII.SHIELD BALANCE. In general it is desired to determine the effect of a shield unbalance on the power-factor and capacitance balance of the bridge. As shown in Part II the current through .Z9 or the potential across & (the detecting circuit) will be equal to zero, when & is connected across the bridge arms, for the condition of ZIZs = ZzZ7 and either ZIZs = 2327 or ZzZs = Z3Z8. These conditions will obviously be satisfied (within the limits of bridge sensitivity) if with the detecting circuit across successively the bridge arms and then between either bridge arm and shield there is no measurable unbalance. Under this condition the potentials of points br, bz, and s (Fig. 2) are equal and the bridge constants of power factor and capacitance may be determined directly from the relation 2l.Z~ = 222,. A study of (I) will also show that the current through & or the potential across &will be equal to zero for the condition If these two conditions of ZIZs = &.Z, and Z5/Z4 = Zz/Z,. hold then we may again obtain the bridge constants of power factor and capacitance directly from the relation ZIZs = Z,Z,, even though in this case ZiZs is not equal to Z3Zr and ZzZs is not equal to Z3Z8. The relation Z,/Z, = 2,/Z, may be obtained by having the detecting circuit Zg connected across the bridge arms and adjusting Zq, Z5, and Zs until a change in Zs gives no measurable unbalance of the bridge. The effect of a shield unbalance on the bridge readings of power factor and capacitance may be determined from (I). If it is assumed that the bridge and shield are balanced, that is, (4), (s), and (6) hold, then a change in Zg from balance to a value Zs + AZ6 will require a change in, say, Z1 from balance to a value Zr + AZ, to give a current through or potential

July, 1934.1

POWER FACTOR BRIDGE.

73

across the detecting circuit equal to zero with the detecting cirThis means that in general if the cuit across the bridge arms. shield is unbalanced the bridge circuit must also be unbalanced from the relation Z&Z8 = Z&‘~. Substituting in (I) for 2, and Zr, Zs + AZ, and Zr + AZ,, respectively, and solving for AZ,, for I9 equal to zero, gives A.ZG(Z,ZIZS - Z,Z,Zd . Az1 = Z,(Z,.Z, + 2425 + 2325 + Z,Z,) + Z3Z4Z5 + AZ,(Z,Z, + Z2-5 + 2324 + 2325 - 2125)

(66)

Now A& and AZ6 may be written AZ1 = AR1 - jAX,,

(67)

AZ, = ARs’ - jAX,‘,

(68)

where AR6’ and AX,’ are the changes in the equivalent series resistance and reactance of Zg. Solving (66), (67), and (68) for AR, and AX1 gives (AR&

+ AX,‘hz)(AR,‘h, + AX,% + hs) + (AXs’hl - AR,‘h,)(AX,‘h, - ARs’hs - hd)

AR1 = (ARs’h5+ AXs’& + h3)2+ (AX,‘h, and (AX& - AR,%)(AR,‘h, + AX,‘& - (ARs’hl + AX,‘hz)(AX,‘h5 Ax1 = (ARs’h5+ AXs’& + h3)2+ (AX6’h5 -

AR6’hs - h4)2

(69)

+ h3) ARs’hs - h4) ARC’& - h$ ’ (70)

where hl = h2

=

Xl3

{zh[k2k4($$5 -

I>1

I> -

k5@2~4 ~5) - W2

k&&5 Xl3 U&&$5 - I> - k5W4 - 41 - pLk&& + P5) - k& + $d]l, k&&5 -

[k2k4&1 +

_

+

kMP3P5 - I) + k&&&

- -&'[k,k5(@2 + P4) + +

k2k4&3 +

+ s5

~5) +

CPs(P3P4 -

(72)

- I)]

k2k5&3 + ~4)

k2k3&4 + I> -

P5)1)

(P3 + P4)1,

(73)

J. C. BALSBAUCHAND A. HERZENBERG.

74 h4=

-&+ 2

1-&‘[~3~6(P2P4 - I) + k2kdp3p4-

[J. I;. I.

I)

3 4 6

+

k&4(@3p,

+

%Ck3MPs

+

M4(p3

- &3

h6 =

he =

j$&

-

I> + +

+

k&3@4p,

P4)

$5)

+

CP6(P3+

Ck3kdPzP4

+

Udp3

k&3@,

P4) +

-

-

I)] +

+

P&II

(p3p4 -

I> +

~4)

kMP3P4

+

hk&P~ - I> + kzkdP4Ps - I>]

-

5

-

s6

(PIP6

-

Ck3Wz

(74)

I>],

-

I)

I),

(75)

+

P4)

+ hk4(p3 + Pd + k,W4

+

hW3

+

+

~4)

Pdl + 2 (PI+ pd, (76)

and, in general k,+-$,

pn =

12

(77)

1

wR,C,.

(78)

A study of (69) and (70) will show that AR1 and AX1 will both be zero for, first, the condition of both AR3’ and AX,’ zero and, second, the condition of

1

(P4P5+ I> + AP5 - P4) P4'+ 1 =

(PIP2 + I) + AP2 - Pd

k2(P12+ I>

1

(79)

for any values of either AR3’ or AX,‘. However, it can also be shown from (69) that for a given AX3’ and certain bridge and shield constants -AR1 may be made equal to zero by a certain value of AR3’. In this case, however, a large value of AXI will result. This means that even if the shield is not balanced as far as X3’ is concerned, the power-factor unbalance

July, 1934.1

POWER FACTORBRIDGE.

75

of the bridge may be made equal to zero through a compensating value of ARs’. This result may seem to indicate that it is not necessary in the shield balance to adjust X6’ for balance. However, it is necessary to have both AR,’ and AX,’ equal to zero, since (I) AR, in (69) cannot be made equal to zero by a definite value of AR,’ for any arbitrary value of AX,’ and certain bridge and shield constants, (2) the resulting large value of AX, due to adjustment of AR,’ to obtain the condition AR1 = o for a given AXs’ may introduce too great an error in the capacitance measurements, (3) in general a AX, will always remain; therefore when a vibration galvanometer is used the bridge unbalance will be so great as to prevent a satisfactory power-factor balance, and (4) even if an alternating-current galvanometer is used so that through the phase shifting of the galvanometer field it can be made insensitive to a given component of bridge unbalance, the large value of AX, will require an accuracy of phase-shifter setting which will be impracticable to obtain. The foregoing conclusions relative to an adjustment of ARC’to give AR1 = o for a given AXS’ will apply similarly to the case of adjusting AX,’ to give AXI’ = o, for a given AR,‘. The values of ARs’ and AX,’ may be determined from Zs (Fig. 6) and the changes in R6 and C, of Zs. Thus if Rs is changed to Rs + AR6 and Cc is changed to Cc + AC, where Cs and R6 refer to the balanced shield, then 2s + AZs = and

(Rs + ARs) - j4Rs + AW(Cs + AC,) 1 + w2(Rs + AR#(Cs + ACS)~ z

6

= RE - jwR& I + w2Rs2Cs2-

’

(go)

@I>

Now ARa’ and AX,’ may be determined from (68), (80) and 0-W. The accuracy with which the shield circuit must be balanced to prevent the bridge being unbalanced by a AR, or a AX1 greater than a given amount may be determined approximately from (69) and (70). In general the effect of pl, p2, p,, p4 and ps on the required accuracy of the shield balance will be only a small percentage of what can be obtained through 1’01..218, NO. 1303~-6

76

J. C. BALSBAUCHAND A. HERZENBERG.

IJ. F. I

Rs and Cs. Then from (69) and (70)~ with PI, Pz, P3, P4 and PS equal to zero, kz = I, and for points not far removed from shield balance (AR,‘X,’ - AXs’R,‘)[Xlka(k4 - kb) (ksks + ka + k4 + kdl + X12ARs’k3(k4 - k5) f AR1 = X,2 + zX1Xs’(k3k5 + ka + kq + k5) + ((XC’)~ + (R,‘)2)(k3ks + k3 + k4 + kd2 (AXs’Xs’ + AR,‘R,‘) [-&k,(k4 - k) (kaks + k3 + k4 + kb)J + X,2AX,‘k,(k, - ks) . Ax1 = Xl2 + zX,Xc’(kak5 + k3 + k4 + ks) + ((X,‘)2 + (Rc’)2)(k3k5 + k, + k4 + kJ2

(82)

(83)

In general, X6’ and R,’ are small in comparison with X1 and therefore AR, from (82) may be approximately written for a given value of ARs’, and AXS’ = 0, AR1 r AR,‘k,(k,

- KS).

(84)

Now, if it is desired to determine the change in AR,’ which &ill give a AR1 corresponding to a change in PI, or power factor of 21, of AP, = ARlwCl (85) we get from (84) and (85) AR,j’ is

API wCrk,(k, - k) ’

Thus for the condition API = IO+, Cr = IOO(IO+~) farads, kg = I, k4 = 5, kg = IO, and w = 377, AR6’ from (86) is approximately equal to 5.3 ohms. Obviously if C4 -and CS are made variable capacitances, they may be adjusted to be more nearly equal than as given by k4 = 5’ and kg = IO. Thus if we adjust k4 = 9 and kg = IO, the value of AR6’ from (86) (for the other constants remaining the same as for the foregoing example) becomes approximately 26.5 ohms. An inspection of (86) shows that the required accuracy of the shield balance in AR6’, for a given power-factor unbalance of the bridge and the shield balanced for AX,‘, becomes greater for an increase in frequency, increase in capacitance Cr, increase in capacitance from high-tension to shield, Ca,

July, 1934.1

POWER FACTOR BRIDGE.

77

and for an increase in the difference between the capacitances between bridge and shield, C4 and C6. Now if it is assumed that the shield is balanced for Rs, that is, ARs’ = o, but unbalanced by a Axe’, and again assuming that X6’ and Rs’ are small in comparison with X1 then from (83) AX, = AX~‘k~(k4 - k5). (87) Now for small changes in AX1 --=AC1 CI

AX, -= XI

-Aq,

(88)

and for small changes in Cs AX6’ =

wRc2AC6 I +

(89)

w2R,j2(C# ’

Then from (87), (88) and (89) r

AC

6

_

&(I

+ w2h2G2)

w~R~~&(&

-

k,)Cl

’

(90)

Thus with Aql = IO-+, that is, a capacitance unbalance of the bridge corresponding to IO%‘~, Rs = 10~ ohms, Cs = IO,OOO(IO-12)farads, KS = I, k4 = IO, k5 = 5, and w = 377, AC6 is equal to 140.9(IO-12) farads. This gives the approximate change in AC6 which will give a capacitance unbalance of the bridge equal to IO+C~. The required accuracy of adjusting Cs for shield balance, for a given capacitance unbalance of the bridge and the shield balanced for AR6’, becomes greater as the frequency and Ii6 are increased until w2Rs2Cs2becomes greater than unity, increases with an increase in Cl and CQ, and increases with an increase in the difference between C4 and C,. A study of (82) and (83) will show that for the condition of Rs and X6 small in comparison with X1, R, and the difference between kd and kg relatively small, a change in the shield from balance by AR6’ (with AX,’ = o) will give principally a power-factor unbalance of the bridge and similarly a change in the shield from balance by AX,’ (with ARs’ = o) will give principally a capacitance unbalance of the bridge. However, if X1 is not large in comparison with Rs and X6, and kl and

78

J. C. BALSBAUGH AND A. HERZENBERG.

[J. F. I.

the difference between kq and Kgare relatively large, then a change in the shield from balance of either ARC’ and AX,’ will give significant unbalances in the bridge of both power factor and capacitance. A study of (69) and (70) will also show that the effect of p1 and p5 will be to increase the powerfactor unbalance of the bridge due to AXs’ even though ARs’ is equal to zero. The effect of shield unbalance on the capacitance and power-factor balance of the bridge is shown in Figs. 8, 9! and IO. The capacitance and power-factor unbalance of the bridge is given in terms of Apl (eq. 85) and Aql (eq. 88). This means that if the bridge and shield are balanced and then the shield is changed to the values indicated, the tangent of the imperfection angle and capacitance of Z1 would have to be changed by Apl and Apl for the bridge to be brought back to balance. In Fig. 8 there is shown the effect of changing R6 (from balance), kg, fi4 and p5, for other given These curves show very clearly the desirabridge constants. bility of having k4 approximately equal to Kg. It is significant that Apl is greater than Apr for a given condition. In Figs. 9 and IO there is shown the effect of changing R6, p1 and Ps, first, for the condition of KS = IO and k, = kg = 2 and, second, for the condition of KS = 40 and k, = kg = 2. Thus in the case the shield is not balanced for phase, that is, independent of Rs’ there will be a value of Axe’. It may be noted from Fig. 9 that for p4 = fi5 = o and ko = 40 and k, = kg = 2, the value of Apl is equal to zero for R6 = 10,004.6 ohms. However, for this same condition from Fig. IO the value of Also from Fig. 9 for p4 = p5 = o and ks = IO Aq, is 97(10+). and k, = KS = IOO the value of Ap, is equal to zero for Rs = 9980.4 ohms. However, for this same condition from Fig. IO the value of Aq, is - 231 (IO-“). These curves show that if the shield is not balanced for phase, for the bridge and shield constants indicated, the value of Rs may be adjusted to give Apl = o, that is, the bridge is still balanced for power factor but there will remain a capacitance unbalance of the bridge. It is also shown that where the shield is not balanced for phase the effect of a power factor of 2, and 2, (p4 and fib) is to affect significantly the value of RE for API = 0, whereas the change in Aq is very small. Likewise it can be shown

July, I934.1

POWER FACTOR BRIDGE.

79

FIG.8. 2.0

1.6

16

.” 2 0.6

6

0.2

-0.4

-4

-0.6

-6

-8

-10

-12

-14 -1.6

-16

-16 -2.0 -20

0 9992

9996

Km0

10004

10006

80

J. C. BALSBAUGH AND A.

HERZENBERG.

[J. F. I.

FIG. 9.

for change in Rd tith valuss Of Apl

A 4 bridge shield

from and balance

July, 1934.1

POWER FACTORBRIDGE.

81

that if the shield is not balanced for Iis (that is, there is a value of A&‘) there will be a value of AX,’ for the bridge and shield constants indicated in Figs. 8, 9 and IO which will give Apl = o but there will remain a value of Ap,. FIG.

IO.

92

‘p 4 i

Y

90 08

-230.0 -230.1 -230.2 -230.3 -2S0.4 9990

9992 9992

loo02

9996 9996

1caOo

10006 loo04

10010 loo06

The shield circuit is balanced through the adjustment of Zs(&=’ and f jXs’) with the, detecting circuit connected between either bridge arm and shield. From (5) and (6) the value of Zs for balance is zz, &=~=~,

z&S 1

2

(91)

J. C. BALSBAUGH

82

AND A. HERZENBERG.

[J.F. 1

from which (R,, _ jX,‘>

= gIR7’(I

+ P12Q2;

.X3

--1x

R,‘(Pl

yP3

P3)

-

1

-

+

P?

Pl)

&‘(I +

-

PIP31

1

,

(92)

where R6’, R7’ and X6’, XT’ represent the equivalent series resistance and reactance of Z6 and 27, respectively. For relatively low values of PI and p3, X6’ and Rs’ are determined principally from the values of XT’ and R,‘, respectively, and the ratio of X3 to X1. There may be cases (due to the large inherent capacitance to ground of the shield circuit) in which a value of + jAX6’ will be required for the shield balance. This would mean that an inductance would be required in place of Cs. For the condition of the bridge and shield in balance the shield voltage V8 will be (93) from which E { [R,‘(x3P3

I/‘, = x,‘(p3’

+

R6’)

j[&‘(x3$3

+

-76’cx3

+

R6’)

+ 1) + 2X3(x6’

-

+

x6’)1

R6’(X3

+ p3R6’) +

+

(&‘)2

x6’)]) +

(x6’)2

’

(94)

An inspection of (94) shows that for the condition of Rs’ and X6’ small in comparison with X3, and p3 small, a change in X6’ gives principally a voltage in phase with E and a change in R6’ gives principally a voltage in quadrature with E. Therefore for adjusting X6’ we may use a component of the applied bridge voltage. The approximate value of voltage V,, in phase with the applied bridge voltage E which must be applied to the shield circuit for shield balancing, may be determined from the difference between the shield voltage V, and the voltage of either bridge arm. Thus V,l gzETz 1

-,“,“> 7

3

6

(95)

and the imaginary component of (95) must be balanced by Re.

July, ~34.1

POWER FACTORBRIDGE.

83

Equating Tr,lto the real components of (95) gives V*I

22

E(R,‘(XIpl + 117’)+ &‘@I + X7’>> X,Q12 + I) + 2X1(X7’ + MG’) + (R7’)2 + (x7’)2 E(RG’(X3$3 + R6’) + Xs’Wa + X6’)) . - X3Q32 + I> + 2X3(X6’ + W6’) + (.W2 + x6)2

(96)

The diagram of connections giving the method of obtaining a component of the applied voltage for the shield circuit is shown in Fig. I I.

7-

shield

‘6

Connections for introducing into shield circuit component of applied bridge voltage. IX.

METHOD

OF CHECFLING MEASURED

BRIDGE

POWER

FACTOR.

The power factor measured by the bridge may be conveniently and accurately checked by connecting a resistance Rc in series with the fixed impedance 2’1 as shown in Fig. I 2. This resistance -Rc should be connected in series with the bridge section of the impedance 21 and not in series with the high-tension section of Z1. Connecting Rc in the hightension side of Z1 requires an evaluation of the capacitance

84

J. C. BALSBAUGH

AND A. HERZENBERG.

[J.F. I.

from high-tension to shield and to bridge, in addition to greater shielding difficulties, whereas if Rc is connected in the bridge side of Z1 it is only necessary to know the capacitance from high-tension to bridge. FIG.12.

Circuit for checking measured bridge power factor.

When Rc is connected in series with the bridge section of 21 there will be a voltage drop across RC (with shields balanced) equal to Rc times the current flowing through the bridge section of Z1. Obviously therefore for the shields A and B in Fig. 12 on both sides of Rc to be balanced with respect

July,1934.1

POWER FACTOR BRIDGE.

85

to the bridge on corresponding sides of Rc requires an impedance 2 c’ connected across the shields A and B. The impedance drop across 2 c’ should be adjusted to be equal to the resistance drop across Rc. 1The resistance component of Zc’ will be approximately equal to the ratio of the capacitance of 2, to shield to the capacitance of Zr to bridge times the resistance Rc. Due to the probable difference in phase of the currents flowing from high-tension to bridge and to shield in Z1 and the capacitance of the shield to ground on the .Zr side of Rc an accurate adjustment of the potentials of the shield A and B may also require either an inductance or capacitance component in Zc’. The bridge circuit is balanced in the usual way by connecting the detecting circuit across the bridge arms br and bz in Fig. 6. The shield circuit on each side of Rc is balanced by connecting the detecting circuit between bridge and shield on corresponding sides of Rc and adjusting 26’ for shield balance on the impedance Z1 side of Rc and the usual shield balancing Z6 for the shield balance on the other side of Rc. With the resistance Rc shorted by means of switch S1 and Zc’ shorted by means of switch S2 balance of bridge and shield will give from (32) (R, = Rs = R and arms direct) $1 - Pz = wR(Cez~ -

C,,,).

(97)

Now if Rc is connected in the bridge circuit by opening S, and the bridge and shields on both sides of Rc are balanced, rebalance of the bridge will require an adjustment of Cs or CT and probably a very small change in capacitance of the vernier variable condenser Zz2. Assuming Cg is adjusted from Csdl to Csd2 for bridge balance (and the probable change in & will be entirely negligible) then the bridge measures where

PI + API -

P2

=

wR(Csr12-

C,,,),

(98)

Afi, = wR& and Cr is the capacitance from (98) gives

in farads of Z1.

A$, = wRcC, from which

differences

=

wR(Csd2

in capacitance

-

Subtracting

Cd,

(97) (99)

of C8 for different

86

J. C. BALSBAUGI-I AND A. HERZENBERG.

[J. F. I.

settings may be determined directly in terms of a given Apl or change in power factor. With the resistance Rc as shown in Fig. 12 perfect shielding is not provided by the shield circuit. It is therefore necessary to investigate the effect of this imperfect shielding on the accuracy of the bridge power-factor and capacitance balances. This may be done through (69) and (70) or (82) and (83). If the bridge and shield circuits on both sides of Rc are balanced then Z4 in (69) and (.70) or (82) and (83) will approximate the capacitance of the resistance Rc to shield and Z5 will be equal to infinity. Such a study will show that if the capacitance from R c to shield is kept relatively small in comparison with the capacitance of Zr this effect will be entirely negligible in the 1.0~~ range for values of Rc representing power factors in the range of IO-~. Furthermore the effect of the imperfect shielding of Rc will be greatly decreased by the location of Rc symmetrical with respect to shields A and B. It is of course possible to design Rc and the shield impedance in such a way that nearly perfect shielding of Rc may be obtained. This in general will not be required since it is unnecessary to check for values of Rc representing large power factors. Furthermore it would be simpler for checking large power factors to obtain differences in capacitance of a correctly designed condenser as shown in Part III. X. MEASUREMENT OF THE POWER FACTOR OF AN OIL SAMPLE.

Consider the bridge diagram of Fig. (13) in which 21 is a fixed impedance, Z,, and Zzz are variable impedances and 2, is the oil sample. If first the sample is connected to the shield, then balance of bridge and shield with arms connected direct (as shown in Fig. (13)) from (32) gives p, - ~1 = uR,C,dl -

uR&m

(100)

where p1 is the tangent of the imperfection angle of the impedance represented by 21, and PZ is the tangent of the imperfection angle of the combination represented by & Now if the oil sample 2, is connected into the bridge and &. circuit in place of z,, and Z1 is connected to shield, and as-

JOY, 1934.1

POWER FACTOR BRIDGE.

FIG. 13. E

Connections for oil sample power-factor measurements.

87

J. C. BALSBAUGH AND A. HERZENBERG.

88

[J. 1:. I.

suming C, = Cl, the capacitances of 2, and Z1, respectively, rebalance of the bridge may be obtained through adjusting either CT or Cs (assuming p,, the tangent of the imperfection angle of 2, is different than PI) and the variable capacitances C’s1or Czz, the capacitances of Zzl and Zzz, respectively. With the shield also balanced and assuming Cs adjusted from CBdl to C&Z for bridge balance, we get Pz - P, = &CTdr Subtracting

- w&Csd2.

(101)

from (100) gives for R, = R8 = R

(101)

p, - $‘I = wR(Cm -

c,dl>.

(102)

Thus the measured power factor of the bridge gives the difference between p, of the oil sample and pr of the impedance In general if Czl or Czz are which the sample replaced. adjusted between the two balances then PZ in (100) is not exactly equal to p2 in (101). However, the effect of this change will in general be negligible. It is shown from (102) that if it is desired to measure p, to a given accuracy, PI must either be negligible in this range or the value of pl must be known to the same accuracy. The value of $1 may be made negligible in the IO+ range by having the measuring section surfaces of Cr free from oxides and clean and under a relatively low gradient (see Part XI). The power factor of the oil sample p, may also be determined by first balancing the bridge with arms connected direct and reverse with 21 and 221 and Z2z in the respective high-tension arms. Thus for R7 = R8 = R. p2

-

PI

=

@R(C,dl

-

C,,,)

(arms direct),

(103)

~2

-

$1

=

wR(Csd2

-

C7d1)

(arms reverse),

(104)

in which it is assumed that CT is not changed in the two balances and CS is changed (assuming a value of p2 - pl) from Adding (103) and (104) gives Csdl to C8d2.

(p2 - pl) =

wR(C8d2 ; c6d1) = ~~~

Now if the oil sample, Z,, is connected in place of Z1 and the

July,1934.1

POWER FACTORBRIDGE.

89

bridge balanced with arms connected direct and reverse then

PZ - P, = wR(C~~~- C8d3) (arms direct), C~d3) (arms reverse),

p2 - fiS = wR(CW -

(106) (107)

in which again CT is not changed in the two balances and Cs is adjusted from &a to C&d (assuming a value of p2 - pS). Adding (106) and (107) gives p2

_

p,

cC8d4 ; C8d3) =

&

=

A,.

(108)

Subtracting (108) from (105) gives P,

-

PI

=

A,

-

(109)

A2,

from which again pS may be determined provided pr is known or is negligible in the desired accuracy range of p8. Of course if p2 of the variable capacitances is known or is negligible in the desired power-factor range then pS may be obtained directly from (108) or (101). From a construction and cost point of view it is less desirable to have a variable capacitance with a relatively low gradient than a fixed capacitance. Furthermore it is considerably easier to disassemble and clean a fixed capacitance than a variable capacitance. The power factor of the oil sample may also be determined by connecting the sample into the bridge circuit in parallel with ZzI and Z22. Thus if first the sample is connected to shield and Z1 is balanced against Z21 and Z2, with the arms connected direct and reverse, p2 - PI will be given by (105). Now if the sample is connected in parallel with ZzI and Z22, the bridge may again be balanced against Z1 with the arms connected direct and reverse and will give p2sl

-

$1

=

~R(Ctidc

-

C8dS)

=

A3,

(110)

where pzS’is the tangent of the imperfection angle of the combination Z,, Zzl’ and Z22’, Z2,’ and Zz2’ representing the impedances of Z21 and Z2, after balance in parallel with Z, against ZI. Balance of the bridge with the sample connected in parallel with Z21 and Z2, will require an adjustment of Z2, and C,, from an initial value of C2l

-I-

c22

2

Cl

(III)

[J. F. I.

J. C. BALSBAUGHAND A. HERZENBERG.

90 to

C2,’+ c22/ +

c.

?s Cl.

(112)

Subtracting (105) from (110) gives P28/ - pz = A3 - A,.

(113)

Thus in this case the power factor of .&(pI) is eliminated but now the values of pz for .Zzl and Z2, (for CzI + Czz 2 CI) in the first balances when 22, and 222 are balanced against &, and of p2’ for 221’and &z’ (for Czl’+ Czz’ z C, - C8) in the second balances when &‘, Zzz’ and Z, are balanced against ZI must be known. Also it is necessary to evaluate p, from p2*/. When two impedances Z, and’Zb are connected in parallel having capacitances C, and Cb, respectively, and tangents of imperfection angle p, and @b, respectively, the V&E of j&b of the combination is given by

Pab =

p&(pb2 c&b2

+ I> + PbCb(Pa2 + I) . + I> + cb(Pa2 + I>

(114)

Now if pzs’ is calculated from (I 13) knowing p2, A3 and Al, p8 may be obtained from (114) through fits’, p2’, CL’ + CZZ’ and C,. This gives

c,((p,‘)’ + I> f [cs”((pz’>”+ II2 - 4cdcz1’ + C22'>P28'((PZ'>" + IHPzs'(Pa - 4(&l + G2')(P28'- P2')211'2. cII5j P8 = 2(C2l’ + C22')(P28' - P”‘> p)ab

When p, and pb are very small in comparison with unity in (I 15) reduces to

Pab =Pya

ca

cb

+ Cb+PbCa +

(116)

cb)

and calculating p8 again from (I 16) instead of (I 14). gives p,

s p2sf

C21’

+

C22’

-I-

cs

C2l’

+

C22’ -

C-3

C8

(117)

The difference between p, as obtained from (I 17) and (I 15) will in general depend upon the relative values of pzs’, p2’, G’ -I- GE’ and C,. The magnitude of this difference

July, 1934.1 for

given

P~WEK ratios

FACTOR

CzI’ and

of

pzS’ and p2’ is given

BKII)GE.

Cz2’ +

in Table

91

C, to

C,, and

values

of

III.

TABLE

III.

ZZX

--

3

82’.

10-d

0 0 0 10-b

__-

10-Z 5(10-V 10-4 10-Z 5(10-*) 10-b 10-z 5(10-‘) 10-4 10-Z 5(10-?

3 3

:: 3 1I3 II3

I I3 1I3 * I3

II3

Table that

4(10-4) 4( IO-~)+48( IO-~) +6389( IO-~) 4(10P4)-3(10m4) 4(10-‘) -257(10-“) 2(10-1)+6059(~o-6)

4/3(10-4) 4/3(10-2)+1(10-0) 2/3(10-1)+74(10-“) 4/3(10-4) 4/3(1@) -34(10-“) 2/3(10-1)+41(10-6)

10-4 -

III

shows

that

ps as given

in general

by

for

there (I

the

may

17) and

be quite

(I 15).

power-factor

it is more

satisfactory

for a fixed

condenser

of approximately

by

placing

the

oil

The

the ease with

which

with

capacitances

different

change

main

in dielectric

\Zith in Fig.

the oil sample

of

with

substituted

a variable

latter may

method

is

be obtained

oil samples

and

the

etc.

for a fixed

condenser

(21

of 21 variable

changes

in capacitance

may be easily accomplished through the similar to that shown in Fig. 2 but with

the measuring

section

divided

sections

may

these

capacitance

with

to have the capacitance

each

of

show

of small

the oil sample

temperature,

fixed steps to take into account This of a condenser

the

balance

for different

constant

of the sample. design

advantage

differences

results

the same

in parallel

a capacitance

13) it is desirable

through

to substitute

sample

capacitance.

large

These

measurements

oil samples than

4(Io-4)

4( IO-~) e(10-1) 4(10-“) -3(10-4) 4(10-~) -3(10-‘) 2(10-‘)-3(10-4) 4/3(10-4) 4/3(10-‘) 2/3(10-I) 4/3(10-4)-I/3(IO-4) 4/3(10-‘)-1/3(10U4) 2/3(IO-2)-I/3(IO-4)

2(10-l)

10-1 100 0 0 0 10-4 10-4

-

between

pa from (117).

Ps frOIFl(11.j).

PZS’. __-

into

sections.

be taken

out

The

lead

the

condenser

of

from

through the shield supporting shaft (Fig. 2) and into a shielded switch box. This will permit any section of the condenser (except either

of

course

to bridge

capacitance condenser.

the

end

or to shield

equal

to that

VOL.218, NO. 1303-7

shield

sections)

and thereby of the

to

give

oil cell

be connected

approximately

which

replaces

a the

92

J. C. BALSBAUGH

AND

A. HERZENHERG.

[J. I;. I.

XI.EVALUATION OF POWER FACTOR OF AIR CONDENSERS.

Consider the diagrammatic plan of the bridge circuit in Fig. 14, In this diagram &,, 212, &a, and 2r4 represent fixed condensers of identical type of construction (Fig. 2) with the measuring and high-tension sections made from the same kind of metal, and of approximately equal capacitances but with different spacings. The measuring surfaces of each of these condensers are initially similarly treated to remove the surface oxides (when these may be removed) and then cleaned with carbon tetrachloride. The object is to obtain similar surfaces in each of the capacitances. The connections as shown in Fig. 14 are such that any of the fixed capacitances may be connected either to bridge or shield and the low-tension measuring arms may be connected direct or reverse. With Zn, 212, 213 and 214 successively balanced against 22, and Z&, obviously we can obtain the following differences in power factors, expressed as Br, &, B, and B4.

Pll -

p2 = Bl,

(118)

plz - pz = &,

(119)

p13 - p, = B3,

(120)

PI4 -

(121)

Pz = B4,

where p, represents the tangent of the imperfection angle of impedances ZzI and Zzz and pr2, fi13, $14 and p14 represent the tangents of the imperfection angles of Zn, 212, 213 and 214, respectively. From (IIS), (119)~ (120) and (121) can be obtained the difference in power factors of any of the fixed capacitances. The value of p11 of the fixed impedance Z,I from (23) is p11 = w&G,

(122)

where Cn is the capacitance of Zn in farads, IL is the equivalent series resistance and w equals 27r times the frequency in cycles per second. The equivalent series resistance &I may be expressed as

fill =

Wll co2E2(c,,y

+

wl(pll)2 I

dE2( c,,y

(123)

July, 1934.1

POWER FACTOR BRIDGE.

FIG. 14. E

__..__~--_; Circuit for evaluation of power-factor of condensers.

93

91

J. C. BALSBAUGH AKD ‘4. HERZENBERG.

[J. F. I.

In general pn2 will be quite small in comparison with unity and therefore from (122) and (123) pll

=

_-!L oE2Cn

(124)

’

where Wn is the loss in watts in Zn and E is the applied voltage to Zn in volts. In (123) if Rn from the first term represents a pn corresponding to IO-* then the second term represents a value of RI1 corresponding to a prr of IO-~ which will be negligible. The loss Wn of Zn may be assumed to be proportional to the surface area and to some power of the voltage gradient. Then the loss IVn may be expressed as wn = K~Lr(rnaEna~

+ rnb&?),

(125)

in which KA is a constant, rlla and rn b are the inner and outer radii, respectively, of 2 11, ElIa and &a are the voltage gradients on the surface areas corresponding to rlr a and rll b, respectively, and Ln is the length of the measuring section of .&. The voltage gradients Ena and El1 b may be expressed as

(126)

(127)

Also the capacitance

Cn of Zn may be expressed as

Ll farads, Cn = K;-----inrl’b

(128)

r11a

where Ks is a constant. (128) and (124) gives *11

-

KA-(rlla IF2 Ks

w

Substituting

+

rllb)'-'

(125), (126), (127) and

( h~)l-z-

(129)

By (129) and analogous equations for PE, PM and fi14 and the differences between any values of pn, p12, PI3 and p14

POWER FACTOR BRIDGE.

Juk 1934.1

9.5

from (IIS), (IIS), (120) and (I~I), the values of $11, #IS p13 and p14 may be evaluated and also the value of any p as a function of applied voltage. Figure 15 shows the results obtained from a series of tests made on a group of four condensers, the high-voltage, shield and measuring surfaces of which were made from standard brass tubing. These condensers each had a capacitance of approximately 86.8 X IO- E farads and the spacings of Zu, &, Z13 and .Z14were approximately 1/16”, 3/16”, 7/16” and 15/I@‘, respectively, and the external diameter of the bridge The tests cylinder was approximately 2” in each condenser. were made with an applied bridge voltage of 800 volts at a frequency of 5.5 cycles per second. The values plotted are the differences between PII, P12, PI4 and Pu. The measuring surfaces of these condensers were initially polished with a fine emery cloth to remove surface oxides as completely as possible Following this and then cleaned with carbon tetrachloride. the condensers were assembled without touching the measuring surfaces and differences between absolute power factors An evaluation of the obtained from bridge measurements. absolute power factors as described previously was made and found to be Absolute Power Factor.

211..

z12 ZIS.. Z,a..

..

1

3.1(10-6, I.O( 10-G)

0.4(10--6)

...

0.2(10-9

Approximate Average Gradient.

13,000

volts

4,300 volts 1,800 volts 900 volts

per per per per

inch inch inch inch

It was also found that the loss in the condensers was proportional to the square of the applied voltage and thus the absolute power factor of a given condenser would be independent of the applied voltage. This conclusion was also checked through measurements made at different applied voltages. At time A in Fig. 15 the measuring surfaces of 214 were dampened by means of a wet cloth. It is interesting to note that immediately following dampening of the surfaces, the value of (PI4 - &) decreased by approximately the value of P 14. Following this the absolute power factor of .Z14increased at first quite slowly and finally quite rapidly and then ap-

??

[J. I;. I.

J. C. BALSBAUGHAND A. HERZENBERG.

96

proached an approximately constant value of 7( IO-~). This condenser was then disassembled and it was noted that the measuring surfaces had become slightly oxidized. Further. more after this condenser was cleaned (time B Fig. 15) similarly to the initial cleaning the absolute power factor of the condenser returned to approximately its initial value. This series of tests was performed a second time with a FIG.

15.

1.

7.

I

I

6.

5.

4.

s. 1.

1.

I

I

I

,

I

D.,.

different set of brass tubing and the results were substantially the same. While these power factors are relatively low, it is evident that even with the brass surfaces cleaned as effectively as possible the power factor of a brass condenser will be significant in the IO-~ range with a gradient of the order of 5000 volts per inch. A series of tests was also made on brass tubing as received, which in general has a substantial amount of surface oxides, and it was found that the power factor for a gradient of 10,000 volts per inch may be substantially greater than IO-~. In this case cleaning the surfaces with carbon

July,1934.1

POWER

FACTOR

97

BRIDGE.

tetrachloride had only a small effect on the power factor. It was also found that the power factor of the condenser varied in the range of the square of the voltage gradient. However, due to the difficulty in obtaining surfaces with the same amount of surface oxides it was not possible to obtain this value to a high degree of accuracy. A number of tests were also made with the measuring surfaces of the condensers made of aluminum and also of chromium plated brass. It was found in these tests, that in general, with the surfaces cleaned as effectively as possible, the power factor was greater than with brass surfaces and similar voltage gradients. BIBLIOGRAPHY. I. H.

SCHERING, “Tgitigkeitsbericht

der

Instrzmerzf, 40, 1920, p. 124. 2. GIEBE AND ZICKXEK, “Verlustmessungen II, 1922, p. 109. 3. H. SCHERING, “ Die Isolierstoffe der 1924, P. 369. 4. G. HAUFFE, “Zur 5. 6. 7. 8.

9. IO.

Theorie

Phys.-T’echn.

Reichsanstalt,”

Archiv f. El.,

an Kondensatoren,” Elektrotechnik,”

der Scheringschen

Briicke,”

Zs. f.

Berlin Archivf.

(Springer), El., 17, 1926,

p. 422. L. TSCHIASSNB, “Die Messgenauigkeit der Scheringbriicke,” Archiv f. El., 18, 1927, p. 248. H. L. CURTIS, “Shielding and Guarding Electrical Apparatus,” A. I. E. E. Trans., 48, 1929, p. 1263. KWJWENHO~EN AND BAFJOS, “High Sensitivity Power Factor Bridge,” A. I. E. E. Ttans., 51, 1932, p. 202. J. H. KING AND H. D. RANDALL, JR., M.I.T. S.M. Thesis, 1932, “Calculation of the Effect of Shielding on the Sensitivity and Precision of PowerFactor Measurements with a Modified Schering Bridge.” L. J. BERHERICH, “ Measurement of the Phase Angles of Shielded Resistors,” Physics, Vol. 3, No. 6, Dec. 1932, pp. 296-313. J. C. BALS~AUGH AND P. H. MOON, “A Bridge for Precision Power-Factor Measurements of Small Oil Samples,” A. I. E. E. Trans., 52, 1933, p. 528.