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NOLTR 61-154
I—
COMMENTS ON THE TEMPERATURE COEFFICIENT OF RESISTANCE AS USED IN WIRE BRIDGE ELECTRO- EXPLOSIVE DEVICE ANALYSIS (U)
^
NOL NOVEMBER 1961
UNITED STATES NAVAL ORDNANCE LABORATORY, WHITE OAK, MARYLAND
in I
.-I
vO
a:
RSLUSKD TO ASTTA BY THE NAVAL ORDNAHOl LABORATORY ^Without ritstriotions □ For Release to Military and Ooverrroent
Acenoiu« Only, Q Approvnl bv .u'aps requiPfld for repeat«
to oontructora. □ Approval by BuWtp» raqulrad for ell
subsequent releas«.
NOLTR 6l-15^
COMMENTS ON THE TEMPERATORE COEFFICIENT OF RESISTANCE AS USED IN WIRE BRIDGE ELECTRO-EXPLOSIVE DEVICE ANALYSES
Prepared by:
J. N. AYRES
Approved by: Chief, ED Division
ABSTRACT: The resletance of an electro-explosive device (EED) at some temperature elevation, 0, above anblent can be expressed by R ■ R0 (l + ocö), where Rg is the resistance at ambient temperature, and (* is the temperature coefficient of resistance. Ihis equation is not consistent with the hand- book definitions of temperature resistance coefficient wherein the parameters R0 and o< are referenced to 0
oC conditions rather than ambient temperature conditions. Misuse of the numerical values can lead in some cases to signifi- cant computational errors. The source, magnitude, and correction of these errors is presented. Also the mathematical basis for an efficient test plan for the determination of «t , R0 , and t" is given.
PUBLISHED NOVEMBER 1961
Explosions Research Department U. S. Naval Ordnance Laboratory White Oak, Silver Spring, Maryland
NOLTR 61-15^ 1 Horeaber 1961
AB part of the Naval Ordnance Laboratory's effort on the HERO (Hazards of Electromagnetic Radiation to Ordnance) program an electro-thermal model has been postulated to explain the response of electro-explosive devices to various electrical environments. Mathematical Implementation of this model requires the use of an equation relating the resistance of the EED brldgevlre to Its temperature. Misunderstanding of the detail of this resistance- temperature relationship has been all too often encountered. It Is the purpose of the present report to explain the sources of confusion and thereby
eliminate the errors resulting therefrom.
Ihls work was carried out under Task NOL-W-B, Guided Missile Propulsion
Systems, Hazards of Electromagnetic Radiation to Ordnance (HERO). It should be of interest not only to the overall HERO program of the Navy but also to the field of electrical measurements in general.
W. D. COLEMAN
Captain, USN Coonander
C.Hl ARONSCN By alrection
11
NOLTR 6l-15^
CON'IKN'iy Page
INTRODUCTION - - 1
MATHEMATICAL EXPOSITION 2
ASSESSMENT OP ERRORS — - h Error In Resistance at Elevated Tenperature, T^pe 1 V Error In Resistance at Elevated Teiqperature, Qype 2 5 Error In Coefficient Due to Misinterpretation of Base Teaperature 6
DETERMINATION OF OL , t , AND R0 FOR AN EED BY THE THREE POINT METHOD 7 Measurements Required 7 Derivation 7
CONCUJSIONS - 8
ILLUSTRATIONS
Table 1 - Type 1. Errors in Estimate of Resistance at Elevated Temperatures (D»e to improper Use of Coefficient of Resistance) *
Table 2 - Type 2. Errors in Estimate of Resistance at Elevated Temperatures (Due to Improper Use of Coefficient of Resistance and Initial Resistance)
ill
NOLTR 61-154
COMMENTS ON THE TEMPERATURE COEFFICIENT OF RESISTANCE AS USED IN WIRB BRIDGE ELECTRO-EXPLOSIVE DEVICE ANALYSES
INTRODUCTION
1. Much work has been devoted to the application of a lunped-paraneter electro-thermal model to the study of the nature of wire bridge electro- explosive devices (EEDs). The work has led to a more fundamental under- standing of their transducing action (electrical signal In—explosive action out) and has aided in the solution of a number of rather diverse
problems.
2. Tbe experimental approach toward verification and use of this model has required measurement of the bridge-wire temperature elevation, 6, under various conditions of time, temperature, energy and power environment, and history. The verification was accomplished by utilizing the resistance- temperature property of the brldgewlre—I.e., the bridgewlre was used as its own resistance thermometer. TMs property is expressed by the simple linear relationship,
R - R0 (1 + 0C0), (1)
where R. is the initial resistance (at aofclent temperature), « is the temperature coefficient of resistance, and 9 Is the temperature elevation above ambient.
Equation (l) is used in the derivation of a large number of equations describing specific EED properties.
C _ + 'S Ö o P(t) p dt
where C is the thermal heat capacity of the bridge
© is the bridgewlre temperature elevation above ambient
t is the heat loss factor
P(t) is the power-time function.
HOLTR 6l-15^
3. Equation (l) Is of the sune form as Is used In the handbook definition of temperature-coefflclent-of-realstance.
R - R0 [l + «.(T - T.)] (2)
where
*> Is "the ratio of the change In resistance In a wire due to a change of teiqperature of 10C to Its resistance at 0oC".
Because T0, the base tenqperature, Is usually taken at 0oC, the above
equation Is often written
R - R0 (1 +\T)
which, though not the sane, is easily confused with Equation (l).
1*. Apparently, it is too easily forgotten that a specific value of * for a given material is meaningful only when consistent with the base (reference) tenqperature. Within the ordinary tenqperature limits Equation (l) describes the resistance-temperature relationship as a linear function. Occasionally the notion is encountered that ot is the slope of the resistance-ten?>erature
curve so that a specific numerical value of this slope will be independent of the temperature. The slope of the linear equation is, of course, the product OLR,. Since the product is independent of teii?)erature while the resistance, R0, will vary as a function of the tenperature at which it is determined, it can be seen that << will have a reciprocal relationship to R,.
5. Thusj it can be seen thet the values of «« and RQ in a specific system depend upon T0, the temperature upon which they are based. As a consequence, numerical solutions of the various electro-thermal equations will be in error unless the values of R0 and oc are handled properly. "Hie purpose of this report is to explain the differences and similarities between Equations (l) and (2), to show how the parameters should be corrected for differences in base temperatures, and to estimate the msgnltude of errors that might arise from failure to correct for the difference. •
MATHEMATICAL EXPOSITION
6. Ordinarily, handbook values of «* are given for a base tenqperature of 0oC. The general Equation (2) can be rewritten for this specific base tenqperature
NOLTR 61-15^
R - Rp [l + «T"J (3)
where R0 Is the resistance at 0 C, and
•* Is the ten^perature coefficient of resistance at O4^.
7« In order to reserve «• for the general equation (which visibly incorporates the base tenperature) and also to denote that Equation (l) uses the ambient temperature as the base temperature. Equation (l) is rewritten
R - 1^ [l + A©] - R,^ I 1 + A (T - Ta) ] . (U)
where B^ is the resistance at ambient temperature (rather than R )
Tft is the ambient temperature (in 0C),
A is the corresponding coefficient of resistance,
and T ■ 0 + Ta.
8. Since the individual values of « and R0 for a group of EEDs will ordinarily be given for a base temperature other than ambient, it will be necessary to compute Ra and A for use In the electro-thermal equations. From Equation (2) it can be seen that
1^ - R0 (1 +HTB -otT0). (5)
Equation (U) can then be rewritten and set equal to Equation (2):
R0 1^1 +«T - «<T0] « R0|l +ctTa - *T6"U 1 + AT - AT^l .
From this
A ' 1 + «* Ta - ot T0 ' ^
9. When the base temperature for particular * and Rö data is the usual value. I.e., when T0 - 0oC, then Equations (5) and (6) reduce to:
and (8)
Ra - Ro (1- ,-«Ta),
" 1 ^
A + * Ta
NOLTR 61-154
ASSESSMENT OF ERRORS
10. Error in Resistance at Elevated Tenqjerature, Type 1.
The usual situation is as follows:
R0 is interpreted to be the initial resistance as it is measured at anbient temperature (Ra) [consistent 1
«- is used as tabulated at some base teiqperattire and not corrected. [inconBistantT
9 is interpreted as T-Ta. [consistant]
Table 1
Type 1. Errors in Estimate of Resistance at Elevated Temperatures (Di>e to Improper Use of Coefficient of Resistance)
ohm ohm 0C
T, Elevated Ten?>erature (0c) 70 120 200 ?20 |
• 0.00075 o.o6£ 0.10^ 0.20^ o.kii |
0.001 0.10^ 0.1&f> 0.33^ 0.67^ |
0.002 0.36* 0.6T£ l.lki 2.C^
0.003 0.79^ 1.3^ 2.25i 3.60^ i
0.001+ 1.3316 2.29f, 3.56^ 5-33^ j
Ta - 20OC
T0 = 0oC
The actual equation used is
R - R,, [l +ot(T - Ta)]
NOLTR 61-154
The equation that should be used Is either
R - R0 ^1 +«-(T - T0)"|
or
R-Ra[l + A(T- Ta)] .
The error equation Is
P Actual-True "1 Actual E - 100 - 100 - 100 1 True True
^[l +öI(T - Ta)] - 100 ^ ; - 100
Ra j^l + A (T - Ta)]
which, upon substitution of Equation (6) for A, becomes
100 0<>2 (Ta-T0) (T-Ta)
1 + «■ (T . To)
This function has been evaluated (Table l) for various typical values of K. and T assuming Ta - 20oC and T0 « 0
oC.
11. Error In Resistance at Elevated Temperature, Type 2.
For the situation where R0 and (X- veCLues, measiired at base ten^erature T0, are both used as If the base temperature had been Ta, the actual equation used Is:
when
R - R0 [ 1 + oc(T - Ta)]
R - R0 ^1 + t(T - T0)"]
should have been used.
NOLTR 61-154
Table 2
Type 2. Errors In Estimate of Resistance at Elevated Tenperatures (Due to Improper Use of Coefficient of Resistance and Initial Resistance)
1 *- 1 ohm I ohm "C
T, Elevated Temperature (0C) j
0 20 80 iko 300 1
j 0.0005 -1.01^ -1.00^ -0.98)t -0.9^ -0.88361
\ 0.001 -2.0«$ -2.00^ -1.92^ -1.79)6 -I.5Ö56
\ 0.002 -k.ni -4.00^ -3.70)6 -3.2356 -2.56)6
0.004 -e.'rvf, -8.00^ -6.9056 -5A056 -3.77%
Ta - 20oC
T0 - 0oC
The error equation in this case is
R0 [l + ^(T - Ta)] - R0 [ 1 + *(T - T,)] E - 100
R0 L 1 +^(T - T0 )^
100 QCCT, - Ta)
1 +oc(T - T0)
- 100
The nagnltudes of the errors have been evaluated (Table 2) assuming Ta - 20oC and T0 - 0oC.
12. Error in Coefficient Due to Misinterpretation of Base Temperature. ( BC used when A should have been used):
The error equation is
T Actual * True E - 100
I True ]- - 1^1
NOLTR 61-15U
vfalch upon substitution of Equation (6) for A becomes
E - 100 ot[Ta - T0] .
TbaiB It can be seen that, for the case where Ta - 20oC, and T0 - 0
oC, the error In the value of the coefficient will be + 1.0^, + 2.0^, + h.Cffo, and + 8.0^ for values of *. correspondingly of 0.0005, 0.001, 0.002, and O.OOl* ohms/öhm^C.
EKERMINATION OF OC , i , AHD R 0 FOR AN BED BY THE THREE POINT METHOD
13. MeasurementB Required. Specific Instrumentation for making the following determinations Is not spelled out. numerous equivalent methods are available. Riere are, of course, a number of Instruments, peculiarly suited for this work which have been reported In references (l), (2), and (3).
a. Determine the resistance at »mblent temperature with negligible current through the bridge*. Ihis is the resistance R_.
b. For a current I through the bridgewlre, measure the •stabilized resistance Rg that the bridgewlre attains at equilibrium.
c. Find a temperature Te for which the bridgewlre reslatance reaches the same value Re obtained in step b.
Ik. Derivation. From the above measurements, the values for Ra, Ta, Rg, and Te, «ore available. Substituting these values Into
R# - R^ [ 1 + A (Te - Ta)]
gives sufficient information to compute A, R0 , « , R0, and o< ,
A ■ **'**
Ra (Te-Ta)
R = ^ (Te - Tp)- Rg (TR - T0 ) 0 mm
••■e ■'•a
*Bj^ negligible is meant a current low enough so that the elevation of the bridgewlre from heating by this current flow is small compared to the elevation to Te (see step c).
NOLTR 61-154
X Re " Ra
Ra (Te - T0) - Re (Ta - T0)
« _
Ra^e " ReTa Te " ^
Re " Ra RaTe -ReTa
PHirthemore,, since the heating current, I, was determined In step h, the power to raise the hridgewlre to Te can be computed as I Rg. Under steady state (constant power) conditions the equation for the thermal model (see
footnote to paragraph (l)) can be solved:
lO - P(t).
Therefore "i can be computed from experimental data by:
i2*. I2R.
CWWCIUSIOIIS
Ik. It can be seen that in many practical situations the improper choice of base temperature for either or both parameters will Introduce negligible errors. Other situations (for instance with Tungsten whose <* is about 0.003 ohms/ohm/0C) the error may be significant. In any case it is simple to make the proper choice of parameter and to correct available data consistent with
the choice.
and R0 has been used to 15. Wie three-point method for determining ot , 1 make many hundreds of sets of determinations. It is an efficient method and capable of accuracy In the order of l^t or better, depending, of course, upon
the quality of the instrumentation.
8
NOLTR 61-15^
REFERENCES
(1) L. A. Roeenthal, "Electrothermal Equations for EEDs", NavOrd Report 668k, 15 Aug 1959.
(2) L. A. Rosenthal, "The Harmonic Generation Technique for the Determination of Thermal Characteristics of Wire-Bridge EEDs", NavOrd Report 6691, 9 Sep 1959.
(3) L. A. Roeenthal, "Cooling Curve Generator and Its Application to EEDs", NavWcps Report 7313, 15 Dec i960.
•
9
NOLTR 6l-15^
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