Circuit resistance: A not-so-simple property

IS THERE SOME KIND OF ELECTRICAL DEVICE that exhibits an efficiency of either 0% or 100%, depending upon how it's used? Yes, that's one of the complications of what we usually think of as a simple component-the humble resistor.

All electric circuits, a-c or d-c, include some combination of only three basic elements - resistance, inductance, and capacitance (see Figure 1). However, whereas the latter two of those elements may sometimes be negligible, resistance is always present. Whether in power grid or microchip, at high frequency or low, all electric circuit elements exhibit some resistance to current flow.

How much resistance is determined chiefly by an inherent material property we call resistivity. All materials will conduct electricity to some extent. The passage of current involves movement of electrons through the atomic structure of the material. The higher the resistivity, the greater the difficulty of that movement under the influence of an applied voltage.

If resistivity is relatively low, as in many common metals, the material is considered a conductor. What we instead call insulators are simply materials possessing extremely high resistivity, so that in normal usage the current that can flow is extremely low, even when applied voltage is high.

How much current can pass through any piece of material depends upon the dimensions involved. If a large crosssectional area is available for current flow, resistance is low. If the current path is long, resistance is higher. Thus, resistance R in ohms of any conducting body is expressed by this formula:

R = rho(L/A)

in which rho is defined as the bulk or volume resistivity, L is that length between points across which voltage is applied, and A is the cross-sectional area of that path.

This simple equation is subject to several important restrictions. First, the dimensional units must be consistent. For example, the value of rho is often stated in ohms per circular mil foot, or "ohms circular mils per foot." One ohm thus represents the resistance of a round wire one mil (0.001 inch) in diameter and one foot long. Using that in the formula for R requires that the conductor area A be given in circular mils, and the length L in feet. Although convenient for the standard wires used in ordinary power and lighting circuits, that's less well-suited to large conductors or special shapes. Some common resistivity units for copper, at 20°C:

10.37 ohms circular mils per foot

8.15 ohms square mils per foot

0.6788 microhms square inches per inch (or "microhm-inch")

1.724 microhm centimeters squared per centimeter (or "microhm-cm")

A second restriction on the resistance formula is that the current flow must be uniformly distributed across the area A throughout the length L. With alternating current, when the "skin effect" is present, more current will pass along the periphery of the area compared to the inner portion. That reduces the physical value of A to a smaller "effective" area, resulting in a higher resistance than the formula would indicate (see "Skin effect: what it is, what it does," in EA August 2006).

Similarly, the value of A must be the same throughout. If the length L is made up of several conductors having differing areas, the total resistance will not be exactly the same as the sum of the individual values calculated from each combination of length and area.

A simple example

Consider the simple situation of Figure 2. In the absence of any skin effect, current flowing through the larger conductor will be distributed uniformly across its entire area. Beyond the junction, the same current will become uniformly distributed across the smaller conductor area. But, as the dashed lines representing current flow indicate at (a), the flow immediately adjacent to the joint necessarily becomes distorted, so that resistance becomes variable.

Exact calculation of that portion of the overall resistance requires "field mapping" - see "How maps reveal field behavior" in EA April 2007. We need not go through that here. The overall resistance will lie somewhere between two extremes. Suppose we add to this hypothetical joint a strip of perfectly conducting material (that is, with no resistivity) as shown at (b). This will remove any distortion in current flow, to give us the lowest possible overall resistance through the joint, which is easily calculated. Next, we can imagine placing two strips of material having infinite resistivity as shown at (c). The resistance across the joint, again easily calculated, will now be the maximum possible. Somewhere between those upper and lower limits is the true value.

Returning to the basic resistance formula: the proper value of rho will depend upon the temperature of the material, which is often taken as a standard or reference temperature surrounding a conductor, such as 20°C, 30°C, or 40°C. That poses two problems.

First, actual ambient seldom matches such a convenient reference. Second, the temperature of any conducting path is influenced by the current flowing through it as well as by the surroundings. In a "pure" resistance, all the current involved is converted into heat, measured by the value of I^sup 2^R - current in amperes squared multiplied by resistance in ohms.

That heat will obviously raise the temperature of the conducting path. In typical metallic conductors, such as copper, higher temperature means higher resistance. As Figure 3 shows, copper undergoes a 4% rise in resistivity for every 10°C increase in temperature.

At constant voltage, however, higher resistance will decrease the current - thus tending to drive conductor temperature back down. An equilibrium point will be reached at which current, resistance, and heat dissipation from the conductor will all be in balance. But the value of R (or of the current) may not be what was originally calculated.

Let's look more closely now at the I^sup 2^R loss in a resistance. Measured in watts, it is converted entirely into heat. Whereas an inductive device such as a solenoid or a motor winding converts most of the energy supplied to it into some kind of useful mechanical work, and a capacitor stores energy that can be returned, the resistor "wastes" all of the power developed in it. That's why we can, in a sense, describe a resistor as "zero percent efficient."

On the other hand, consider the electric resistance heater. Since its sole purpose is to raise the temperature of its surroundings, we could describe it as "100 percent efficient." No other electrical device can make that claim.

Other purposes of resistors

Resistors serve other purposes as well, such as to reduce the voltage in one part of a circuit compared to another (the "dropping resistor"), or to control the flow of current under some operating condition (such as the "grounding resistor"). Nevertheless, each application involves the production of heat. Hence, two goals in resistor design are size reduction and ready dissipation of heat.

Size is important because the smaller the component, the greater will be its heat-dissipating surface area compared to its heat-producing volume. Whatever the basic dimensions, heat dissipation is improved by designing for free air circulation over the exposed surfaces.

Long-term stability and insensitivity to temperature variations are desirable in electronic circuit design. That has led to development of thin-film resistors, in which the resistive element is not a higher-resistivity material such as carbon, but a metallic alloy-normally a much better conductor. When the material is deposited on a substrate in an extremely thin layer, however, the value of A in the basic resistance formula becomes so small that R becomes quite high. Also, whereas copper or aluminum resistivities increase significantly with temperature, alloys used in such resistors are formulated to maintain constant resistivity over a wide temperature range. Influence on their performance of either ambient temperature swings or of self-heating is therefore minimized.

(In such thin layers of material, surface resistance becomes an important property. This is the resistance across the surface from one edge to another. It's expressed in "ohms per square." Per square what? That makes no difference. As Figure 4 shows, the ratio L/A remains constant for all sizes of square; doubling the square size increases both length and area by the same amount.)

At the other extreme, resistors required to dissipate large amounts of power (as in the secondary control of a large wound-rotor motor, or braking of a railway locomotive) will also use what are normally considered conductor materials for structural integrity (see "The high-power resistor: Not as simple as it seems," in EA December 1990).

Close control of resistance in circuit components themselves is thus an obvious concern of all circuit designers. Necessarily, however, all such components must be connected into the remainder of the circuitry. Joints, splices, and terminations of all kinds cannot be avoided. At all such locations, a resistance of zero ohms would be ideal. A great many operating problems and much catastrophic destruction of electrical apparatus have resulted directly from the impossibility of achieving that ideal condition. An understanding of what is-and is not-properly considered a "high resistance joint" is appropriate here.

When resistance develops heat within the relatively small volume of a circuit joint, it can cause local deterioration of the joint itself (for example, by expediting corrosion, which will further increase joint resistance through chemical change in the conductors, or by arcing-see Figure 5). That often leads technologists to be unduly concerned about the materials used in making permanent joints, particularly in large conductors carrying high currents. One example: the rotor bar connections to end rings in a squirrel-cage rotor.

Such joints are normally made by brazing or soldering. Though often used interchangeably, those two terms properly define two quite different processes. In brazing (sometimes appropriately called "silver soldering" or "hard soldering"), the filler material is a copper alloy such as a silicon or aluminum bronze, often with some silver or phosphorus content. Melting point of these materials ranges from 450°C to 900°C. In contrast, "soft soldering" uses a tin-lead alloy (or a newer lead-free alloy) that melts at 180°C to 230°C. Brazing yields a much stronger joint.

Whereas welding joins materials by melting each of them into the other, both forms of soldering melt only the filler alloy itself, fusing it to the components being joined as glue bonds pieces of wood. Ideally, soft soldering does not create a tight, low resistance joint, but only acts to maintain the contact already established. Figure 6 illustrates common types of wire splices in which soft soldering prevents any loosening in service.

In a brazed joint, the filler material becomes part of the conducting path, directly in series with the joined conductors (Figure 7). How important, then, is the resistance of the filler in that joint?

Consider Figure 8(a), in which a two-foot length of 1/4 by 1 inch copper bar is carrying a steady-state current of 300 amperes (either d-c, or a-c with negligible skin effect). "Standard" resistivity is 8.15 microhm square inches per foot at 20°C. Assuming actual bar temperature to be 75°C, the resistivity increases by 22%, the ratio of (234.5+75) divided by (234.5+20), to 9.89 microhms square inches per foot. Area is 0.25 square inches, length is 2 feet; actual bar resistance is therefore 79.1 microhm (millionths of an ohm), or 0.0000791 ohm.

What is the I^sup 2^R loss developed as heat in this bar? Multiplying current squared times bar resistance yields 7.12 watts. Now examine a small slice taken from the middle of the bar, at right angles to the direction of current flow. If it is 5 mils (0.05 inch) thick, its resistance will be 5/24,000 times 79.1 or 0.01468 microhm. Heat developed within the slice is 7.12 times 5/24,000 or 0.001483 watt-roughly 1/500 of the total loss in the entire bar.

Increasing resistivity

To illustrate the effect of a "high resistance joint" in this bar, we replace that same slice with one having a higher resistivity, such as a brazing alloy-see Figure 8(b). Assume for this example a material having three times the resistivity of copper.

Now, the slice resistance is three times 0.01468 or 0.04404 microhm. Total bar resistance becomes 0.03% higher than before, as does the total heat generated within the bar. Neither increase is of any significance to overall bar temperature or circuit behavior (voltage drop in particular). An overall conductor length of several feet would make the difference smaller yet.

The slice itself, of course, will be hotter-not, however, three times as hot. All the 5 mil slices of the solid, unjoined bar will give off heat through their sides to the surrounding air, as the illustration shows. Heat will not flow lengthwise along the bar, because the temperature of all slices is the same. But when a higher resistance is introduced at the joint, heat will flow axially to the lower resistance slices to left and right. That flow area is much larger than the area exposed to the sides and edges of the bar.

Furthermore, the axial heat flow path is directly into bar material of high thermal conductivity, rather than from bar surface to surrounding air. Hence, although the joint will exhibit a somewhat higher temperature than the rest of the bar, the difference will not be extreme-see Figure 9.

Calculating the exact variation is beyond our scope here.

Finally, of course, whatever the joint alloy used, actual current-carrying area will normally far exceed the crosssectional area of the conductor itself. Because a simple butt joint between flat bars isn't strong enough, an overlap joint is the usual choice, made large enough to reduce current density of 1,000 to 3,000 amperes per square inch in the bar to only 500 across the joint itself. Theoretically, then, joint brazing material could be at least twice the resistivity of the conductors themselves without adding any resistance.

The unavoidable stresses of vibration, electromagnetic forces (as in motor starting), or thermal expansion/contraction require a mechanically strong joint. However, strength is not the only reason for a large joint area. First, the brazing or soldering process may result in areas of relatively poor contact (see Figure 10). In most joints no thorough inspection is possible once the process is complete. Why isn't contact complete? During metal joining, air bubbles, pockets of flux, or contamination can cause the liquified solder or brazing alloy to fail to fully wet the joint surfaces, leaving some areas incompletely bonded (see Figure 11). The importance of proper cleaning beforehand cannot be overemphasized.

Evidently, the choice of a joint filler alloy should be based on strength, adhesion to the metals being joined, ease of use, and resistance to the joint environment, not on the material's resistivity. When a joint-brazed or even welded-does present unexpectedly high resistance, the usual cause is incompleteness of the bond, not the material involved.

Know your insulation!

If the conductors are insulated, even moderate local overheating could eventually cause insulation breakdown (Figure 12). That's normally avoided in two ways. For a simple splice between round wires, as in Figure 6, the solder adds considerably to the initial wire contact area, increasing the size of the joint to reduce its overall resistance. The same is often true for brazed joints.

Fuse clips, terminal screw connections, and bolted joints (of many types, some of which appear in Figures 13 and 14) are a different matter. Electrical integrity depends entirely upon surface contact between conductors. Surface condition and chemistry can change with fastener tightness, contamination, and thermal degradation (Figure 5). In any joint between "flat" surfaces, actual contact is never uniform. High and low areas of uncertain size are distributed throughout the joint. No single resistivity value can be assigned to such an interface. Most "high resistance joints" are of these types.

Resistance then, although probably the simplest basic property of any electrical circuit, can take some complex forms. They need to be fully understood by anyone involved in circuit design or system operation.

© 2008 Barks Publications Provided by ProQuest LLC. All Rights Reserved.