In making high-precision resistors, it's important to control the effects of temperature on resistance. Resistor manufacturers have long known how to correct for changes in ambient temperature; yet, management of the “Joule effect” is less understood. Managing the Joule effect involves operating a system in which a resistor heats up and dissipates power as a result of the load it's handling. A new “Z-based” resistor foil technology greatly reduces the sensitivity of devices to changes in applied power, enabling a tenfold improvement for the effects of ambient temperature change and resistor self-heating (see sidebar on page 17). While maintaining the same form/fit function as conventional devices, new Z-based foil technology provides a breakthrough in accuracy for fixed-resistor applications, including feedback, gain setting, voltage division, and current sensing.
To understand the overall temperature effects on resistor precision, we must look at the details surrounding their temperature coefficients. The fundamental problem in selecting the best resistor for applications requiring high stability is that low-TCR resistors typically don't perform to their TCR specifications when self-heating raises their temperature. In high-precision resistors with TCR values of a few ppm/°C, this inaccuracy can lead to an unacceptable error in the estimate of the resistor's stability under load. For this reason, we need an additional figure of merit: a concept of “power coefficient of resistance” encompassing the effects of temperature changes as they relate to self-heating. For a better phonetic distinction from TCR, we prefer to call this wattage coefficient of resistance, or WCR.
Understanding TCR and WCR as distinct values enables designers to make the most appropriate resistor choice, depending on the power and temperature demands of a given application. If the current level is constant, but the ambient temperature is variable, then a low TCR is the most important specification to ensure device stability. If the ambient temperature is constant but the current level is variable, then we will need low WCR values to provide high resistor stability. If the current level and the ambient temperature are subject to variation, then both TCR and WCR need to be low to ensure stability.
With high precision, you can measure the temperature rise of the resistive layer caused by self-heating by building a thermally similar model and using a resistive material (e.g. Balco alloy) of high TCR (5,000 ppm/°C). First, measure the dc resistance value (R) at different temperatures (T) to establish the relationship between resistance change and temperature: R = f(T). Then, measure voltage (V) and current (I) at stabilized levels up to rated power. Power is the VI product, and resistance, V/I, may be translated into temperature using the established R = f(T) relationship.
A temperature sensor placed near the resistive layer provides similar results — but fails to indicate the resistive layer's average temperature. Usually this won't reflect actual temperature.
This testing usually shows a proportional relationship between load and temperature rise, which is often used by manufacturers to establish a value of thermal resistance, Rth in °C/W, for a given resistor. However, because the heat transfer mechanism of resistors allows for heat evacuation via conduction from the outside surface as well as through the leads or conduction through the p. c. board, the resulting Rth isn't exactly an inherent characteristic of a resistive device. In other words, the actual Rth depends on the mounting method, the substrate, the p. c. board material, and the size of the pads and traces.
You can also determine the manufacturers' Rth values with the assumption that the device's outer surface or p. c. board will remain stable (at ambient temperature). Therefore, when Rth doesn't refer to the environment, what's usually measured and published isn't strictly the true thermal resistance between two points of the resistor (resistive layer and body or heatsink) because part of that dissipated power is directed elsewhere.
If TCR isn't always the most critical figure, how can we determine WCR? To compare the strain, εa, under load in a foil chip with a rise in ambient temperature and self-heating:
εa = a×ΔTa
ΔTa = Change in ambient temperature
α = Coefficient of thermal expansion of the substrate material
This strain compensates for heated foil's increased resistance. However, if the foil's temperature rises by the same ΔT1 due to the heat generated by a load, P, the strain will not be the same. Part of the load's heat will be directed normally toward the substrate:
Ps = f1×P(f1<1)
Assuming a uniform distribution of power on the chip's surface, the thermal resistance of the foil/substrate interface, Rthi, will cause a temperature drop of
ΔTi = Ps×Rthi
This reduces the strain in the substrate's top surface by:
εI = α×ΔTI
Within the substrate, due to its thermal resistance, Rths, the temperature will drop further by:
ΔTs = Ps×Rths
In a plate that's free to expand, a linear temperature drop across its thickness, t, causes a difference in expansion between the top and the bottom surfaces — a strain difference between surfaces εbs = α×ΔTs — and the plate bends into a spherical shape of radius r = t/(α×ΔTs).
In a resistor chip, this effect may be constrained by external forces. When a chip is attached to a heatsink, the substrate isn't free to expand. Thermal stresses may be due to nonuniform distribution of the power on the chip's surface because the heat generating part of the foil doesn't cover the whole substrate, and some of the heat flows in a direction other than normal — possibly modifying the strain. As a result, the strain in the substrate's top surface will be further reduced by εts, smaller than the εbs of bottom surface, by a factor f2, with f2<1:
εts = f2×α×ΔTs
The total difference in the strain, for the same temperature in the substrate's top surface, due to change in ambient versus self-heating is:
εI+εts = α×ΔTi+f2×α×ΔTs = α× Ps(Rthi+f2×Rths)
When forcing the resistive foil to follow the strain in the substrate's top surface, the relationship between the relative resistance change and strain is the gage factor k, in this case k=2. Thus, a resistor of zero TCR will, under load P, show a reduction in value:
ΔR/R = 2α×f1P(Rthi+f2×Rths)
Its wattage coefficient of resistance is:
WCR = (ΔR/R)/P = 2α×f1(Rthi+f2×Rths)
The demand for precision resistors with high stability under changing load levels has led to the development of tailor-made solutions such as:
- Variations in ambient temperature,
- Load changes over time,
- Limits of allowed resistance change, and
- Time it takes to reach these limits after a given change in load.
For example, our 0.3W standard through-hole resistor shows a slope of -0.8ppm/°C TCR over a 25°C to 50°C range. With an Rth of 100°C/W and a WCR of 280ppm/W, the same foil's temperature rise due to self-heating results in a slope of 2.8 ppm/°C (Fig. 1, on page 20). Fig. 2, on page 20, shows a current sensing resistor rated at 3W in air and 10W on a heatsink (at 25°C ambient), engineered for low TCR at room temperature.
Fig. 3 shows the same style of resistor engineered for the best combination of TCR and WCR. Fig. 4 is the response of a VCS332 heatsink-mounted current sensor to a step pulse of 10W. Its thermal time constant improved by a factor of 3.
The Z-foil resistor has opened up a new area of performance, providing a TCR unseen before in a single chip device — and its technology is useful in many ancillary products (photo, on page 18). It allows use of the same foil inside power resistors and current-sensing resistors with a combination of low-temperature coefficient and improved thermal resistance. This takes current measurement to a new level, because it significantly reduces the variation of resistance with current change. In addition, we can use the new Z-foil in surface-mount chip resistors, again improving performance to a level never achievable before.
The method of mounting the resistor affects WCR — especially for surface- and heatsink-mounted devices. Therefore, you can't precisely assess WCR from data supplied by a resistor's manufacturer. If a stability of a few parts per million is specified, the stability under changing load should be checked after circuit assembly under conditions that correspond to the actual application.
For applications requiring high stability under changing load conditions, specify resistors of low WCR. For steady loads and changing ambient temperatures, low-TCR resistors will provide the best stability. When ambient and power dissipation are changing, the design requires low TCR and WCR to ensure high stability.
Foil Resistor Technology
Bulk Metal® foil technology resistors employ a chip resistor element consisting of a homogeneous metal, photo-fabricated to a specific pattern on a ceramic substrate. Various chip sizes and configurations are employed to provide the variations in power, size, and other characteristics. The combination of materials and construction provides its unique resistor characteristics. Two predictable and opposing phenomena within the composite structure of the resistance alloy and its substrate are the key to the low TCR of Bulk Metal foil resistors.
- Resistivity of the resistive alloy changes directly with temperature. (Resistance of the foil increases when temperature increases.)
- The coefficient of thermal expansion of the alloy and substrate are different, resulting in a compressive stress on the resistive alloy when temperature increases. (Resistance of the foil decreases due to compression caused by the temperature increases.)
The temperature coefficient of these resistors is the result of matching the variation in resistivity of the alloy with temperature and variation of the alloy's resistance under stress. These two effects occur simultaneously with changes in temperature. The result is a low and predictable TCR. These foil resistors accomplish this TCR automatically, without selection, and regardless of the resistance value or the date of manufacture — even if years apart. The shape of the deposited resistive pattern plays an important part in its performance. This can be seen by examining its equivalent circuit consisting of a resistor in series with an inductance and in parallel with a capacitance. In spiraled and wirewound resistors, these reactances are created by the loops and spaces formed by the spirals or turns of wire. In planar resistors, the geometry of the lines of the resistor patterns is intentionally designed to counteract these reactances. The opposing directions of current prevent the build-up of mutual inductance and reduce the capacitive effects by placing the inter-conductor capacitances in series.
In pulse applications, these reactive distortions result in a poor replication of the input. For example, a pulse width of one nanosecond would be missed by the wirewound resistor, whereas the foil resistor achieves full replication in the time allotted.
In high-frequency applications, reactive distortions also cause changes in apparent resistance with changes in frequency. Foil resistors provide good response in the 100Ω range out to 100 MHz — and all values have good response out to 1 MHz.
Resistors can be a source of noise, independent of the presence of a fundamental signal. Resistors made of conductive particles in a non-conductive binder are most likely to generate noise. In carbon composition and thick film resistors, conduction takes place at points of contact between the conductive particles within the binder matrix. These contacts constitute a high resistance conduction site that is a noise source. These sites are sensitive to distortion resulting from expansion mismatch, moisture swelling, mechanical strain, and voltage input levels.
Bulk Metal foil resistors have the lowest noise of any resistor technology. Their conduction is across the inter-granular boundaries of the alloy. The inter-granular current path from one or more metal crystals to another involves multiple and long current paths through the boundaries, reducing chance for noise generation. In addition, the photo lithography and fabrication techniques employed in the manufacture of foil resistors results in more uniform current paths than found in other resistors.
Low-ohm, Z-foil resistors employ four terminal leads in current sensing applications, improving performance to a level never before achievable. The extra pair of leads provides a Kelvin connection that eliminates the voltage drop error in lead resistance, a major factor in the total resistance. As you can see in this sidebar figure, Kelvin leads connect to the resistor — without its leads — and carry a much lower current than the current sense resistor itself.
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