MEASUREMENT OF SELF-HEATING AFFECTED DYNAMIC ERROR OF PRECISION WIRE-WOUND RESISTORS

This paper presents the thermal dynamic model of wire-wound resistors. In order to 
identify the thermal model an active bridge circuit is discussed. The power step response 
of the wire-wound resistors can be measured with the resolution of 1 ppm using the 
detailed circuit. Experimental results are also presented.


INTRODUCTION
Precision wire-wound resistors are widely used in the input circuits of electric instruments.These components form voltage dividers, current to voltage converters etc.The accuracy of the mentioned instruments is limited by the errors of these system elements.The transfer ratio of the input stages must be stable while the input signal varies in a relatively wide range (0..10 A, 0..300 V).Precision wire-wound resis- tors that form the input stages have the following error components: initial tolerance of resistance-can be corrected by calibration, long term drift-can be corrected by regular recalibration procedures, *e-mail: zszepes@mmt.bme.huparasitic impedances-can be trimmed at a given frequency by additional capacitors, temperature dependence is defined as the sensitivity to the ambient temperature variation.Since this error can be considered as a static error during the measurements because of the slow temperature vari- ation, this component can be reduced by stabilizing the environmental temperature or regular recalibration procedure of the transfer ratio.
There are other error components that are important only in high precision applications (e.g., metrology instruments, normal resistors, reference voltage dividers).
A significant error source is originated from the self-heating phenomenon.This error is referred as the thermal dynamic error (TDE).The thermal dynamic error is determined by the actual oper- ating current, that flows through the resistor.The current generates heat (Joule-heat), that increases the actual temperature of the resistor wire and indirectly changes the actual resistance.The resultant tem- perature change depends on the material structure of the resistor, the environmental heat sources, the ambient temperature and the geometry of the arrangement [1].The magnitude of the error depends strongly on the type of the resistor [2].A precision wire-wound resistor is built of a ceramic bobbin, the resistor wire wound on the bobbin and the system is encapsulated by a plastic case.The thermal dynamic behaviour may be described by a one-time-constant linear model, characterized by two parameters: the thermal resistance (R,, K/W) and the thermal time constant (T, s) [3].
The thermal dynamic error of a typical precision wire-wound resistor is in the range of 0-100 ppm if the applied power varies between zero and the nominal power of the component (0-1.5 W).The typical ther- mal time constants are in the range of 100-600 s.In the case of cali- bration instruments where the key task is the fast and high precision measurement (1 measurement per second, 10 ppm basic accuracy), the thermal dynamic error of the built-in precision wire-wound resis- tors must be corrected.The correction may be based on a special on- line error estimation.The high precision calibration instruments regularly contain an embedded computer that manages the measure- ment procedure.The identified thermal model based error estimation may be processed by the operating software of the embedded computer.
The model based self-correction procedure can increase the accu- racy of the final measurement results by one order of magnitude.The model identification procedure is a difficult measurement problem.The model parameters can be determined from the power step response of the investigated precision resistor.The relative resistance error must be measured with a resolution ppm.Since the dynamic error chara- cteristics can be examined in a long power step response measurement (usually longer than 15 minutes), the stability of the test set up is a significant requirement.In order to measure a significant thermal dynamic error on a precision resistor of 10 kf and a rated power of 1.5 W the test circuit must be capable to apply hundreds of volts on the component.This demand requires special circuitry (large common mode rejection ratio, proper shielding).
The measurement circuit is required to have a selective sensitivity to the investigated phenomenon, and needs to reject other error components (the TDE of other resistors, voltage dependence, parasitic impedances).
The active bridge circuit that meets the detailed requirements is introduced below.

THE ACTIVE BRIDGE CIRCUIT
Our investigation is focused on a commercially available precision wire-wound resistor.The resistor has the following parameters: Nominal resistance: 50 kf Initial tolerance: 0.01% Rated power: 1.5 W Temperature coefficient: Long term stability: 0.05%/2000 hours The thermal dynamic error measuring circuit can be seen in Figure 1.
Rx denotes the resistor under test, Rv is the reference resistor, RB1, RB2 are film type resistors.V stands for the input voltage source, A is an instrumentation amplifier, A2 is an operational amplifier.

FIGURE
Active bridge circuit to measure the thermal dynamic error of Rx.
The circuit operation is as follows: 1.The measured wire-wound resistor is Rx.The resistors denoted by Rx, RN, Rsl, Rs2 form the bridge.The relation between the nomi- nal resistances is as follows: Rx Rv, Rsl Rs2, Rsl Rx/50.The reference resistor Rv is composed of 16 resistors as can be seen in Figure 2.Each element has the same type as Rx.If the same voltage supplies Rx and RN, the dissipation of one element of Rv are 16 times smaller than that of Rx.Consequently the thermal dynamic error of Rv is negligible compared to that of Rx.The error of Rsl and Rs2 can also be disregarded because the voltage on these components is 50 times smaller than the voltage on Rx.
2. The bridge is supplied with V6 generator and the output of A2 operational amplifier.The VE voltage is amplified with A1 instru- mentation amplifier.If the bridge is balanced at low level V60 and FIGURE 2 Construction of Rv using 16 resistors of the same type as Rx.
the generator voltage is raised to Vol as a step sign, the peak value of the output of the circuit (VOUT) is proportional to the thermal dynamic error of Rx.
3. As a result of the feedback through A2 the common mode voltage of the instrumentation amplifier A is set to zero.The CMRR of the measurement circuit is increased CMRRToTAL CMRRA1, A2.
Vo is applied on Rx and Rv so the generator voltage can control the power dissipation of these elements.
VOUT can be expressed as follows: VOUT VG K A1.Ot, N OX aN + ax where K (2) In Eqs. ( 1)-(4) we can recognize the following error components that influences the output voltage: 1.The dissipation affected thermal dynamic error of Rx.
Sensitivity to the relative error of Rx neglecting the error of RN, RB1, RB2.
(7) SA1 OA1 Comparing (1) and ( 6), (7) equations, we can see that in order to minimize the effect of the instability of V and the instability of the gain of A the output voltage of the circuit VOUT must be set to zero using Cry, RrR components before measurement.
Substituting the actual values in (6) and (7) considering the actual Sa-, and the ppm resolution criteria, the following requirements are obtained: the relative error of V and A must be smaller than 3%.
Measuring the output voltage and using the calculated sensitivities the results shown in Figure 3 were obtained.Exponential curves were fitted to the results using least mean squares method.The thermal dynamic error of the investigated resistor due to VGo VGI step change in applied voltage can be expressed as follows: VG1 ll0W TDE(t) 4.20.(1 e (-t/400"6)) (ppm) VG1 180V TDE(t) 34.91.(1 e (-t/373"9)) (ppm) The average relative difference between the measured and the estimated values remains below 5%.

CONCLUSION
An active bridge circuit has been presented to determine the self- dissipation affected thermal dynamic error of precision wire-wound resistors.Using the described circuit measurement of resistance change with the resolution of ppm can be made.The achieved resolution makes it possible to measure the dynamic thermal characteristics of high precision wire-wound resistors.The configuration can be used for recording the static temperature characteristics, for identification of the thermal dynamic model, to estimate the long-term stability of precision resistors.
Using the identified thermal dynamic model the on-line error estimation can provide correction data for adequate measurement software.This on line self correcting procedure might reduce the influence of the self-heating affected instability of reference resistors.

FIGURE 3
FIGURE 3 The thermal dynamic error of the investigated resistor for different voltage steps.
circuit in Figure was built with the components detailed in the next table: