4.2 Resistivity, Temperature Coefficient, and the Wheatstone Bridge
Key Takeaways
- Resistivity (\u03c1) is an intrinsic material property measuring opposition to current, whereas resistance depends on physical dimensions.
- The resistance of a conductor is directly proportional to its length and resistivity, and inversely proportional to its cross-sectional area: R = \u03c1 * L / A.
- Pure metals have a Positive Temperature Coefficient (PTC) of resistance, while carbon, electrolytes, and semiconductors have a Negative Temperature Coefficient (NTC).
- NTC thermistors are commonly used in aviation engine temperature sensors (CHT/oil) and continuous-loop fire detection systems.
- A Wheatstone bridge is balanced when the voltage across the detector is zero, satisfying the relation: R1 / R2 = R3 / R4.
Resistivity and the Conductor Resistance Formula
For an aviation technician, calculating the resistance of aircraft wiring is essential to ensure that electrical runs do not introduce excessive voltage drops or thermal hazards. Resistance (R) is not only a property of an electrical component; it is a physical characteristic of any conductor that depends on three factors: the material's intrinsic resistivity, the conductor's length, and its cross-sectional area.
The relationship is governed by the resistivity formula: R = \u03c1 * L / A
Where:
- R: Resistance of the conductor in Ohms (\u03a9).
- \u03c1 (rho): Intrinsic resistivity of the material in Ohm-meters (\u03a9\u00b7m) at a reference temperature (usually 20\u00b0C).
- L: Length of the conductor in meters (m).
- A: Cross-sectional area of the conductor in square meters (m²). For a cylindrical wire, the area is calculated from the diameter (d) using A = \u03c0 * d² / 4.
Intrinsic Resistivity (\u03c1)
Resistivity is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. Materials with low resistivity are excellent conductors, while those with high resistivity are insulators.
| Material | Resistivity (\u03c1) at 20\u00b0C (\u03a9\u00b7m) | Temperature Coefficient (\u03b1) at 20\u00b0C (K⁻\u00b9) |
|---|---|---|
| Silver | 1.59 * 10⁻\u2078 | 0.0038 |
| Copper | 1.72 * 10⁻\u2078 | 0.00393 |
| Gold | 2.44 * 10⁻\u2078 | 0.0034 |
| Aluminum | 2.82 * 10⁻\u2078 | 0.0039 |
| Constantan | 4.90 * 10⁻\u2077 | 0.000008 |
| Carbon | 3.50 * 10⁻\u2075 | -0.0005 |
Aviation Conductor Materials: Copper vs. Aluminum
- Copper: With a resistivity of 1.72 * 10⁻\u2078 \u03a9\u00b7m, copper is the primary choice for aircraft electrical systems. It combines excellent conductivity, high ductility (allowing it to bend through structural runs without breaking), and high mechanical strength.
- Aluminum: Aluminum has a higher resistivity (2.82 * 10⁻\u2078 \u03a9\u00b7m) and is more prone to oxidation and vibration fatigue. However, it is significantly lighter than copper. In large commercial aircraft, aluminum is used for heavy, high-current power distribution cables (such as generator main feeders) where the weight savings outweigh the requirement for a larger conductor diameter.
- Circular Mils (AWG System): In American Wire Gauge (AWG) calculations, the area of a wire is often expressed in Circular Mils (cmil). A mil is equal to 0.001 inches. The circular mil area is calculated by squaring the diameter of the solid conductor in mils: Area (cmil) = (d in mils)\u00b2. In this system, the resistivity of copper is 10.4 \u03a9\u00b7cmil/ft, and the formula remains R = \u03c1 * L / A where L is in feet and A is in circular mils.
Temperature Coefficient of Resistance
Resistance is not constant; it changes as the temperature of the conductor changes. This variation is described by the temperature coefficient of resistance (\u03b1), which represents the fractional change in resistance per degree temperature change.
The formula to calculate resistance at a given temperature is: Rt = R0 * (1 + \u03b1 * (t - t0))
Where:
- Rt: Resistance at the final temperature t (in \u00b0C).
- R0: Resistance at the reference temperature t0 (usually 20\u00b0C).
- \u03b1: Temperature coefficient of resistance at reference temperature t0 (in K⁻\u00b9 or \u00b0C⁻\u00b9).
- t - t0: Temperature difference between final and reference temperatures.
Positive vs. Negative Temperature Coefficients (PTC and NTC)
- Positive Temperature Coefficient (PTC): In pure metals (such as copper and aluminum), resistance increases as temperature increases. When the metal is heated, the thermal vibration of the atoms in the crystal lattice increases. This increases electron scattering, making it more difficult for free electrons to flow through the material.
- Negative Temperature Coefficient (NTC): In carbon, semiconductors, and electrolytes, resistance decreases as temperature increases. Heating provides enough thermal energy to release valence electrons into the conduction band, increasing the density of charge carriers and reducing resistance.
Thermistors in Aircraft Systems
Thermistors are thermally sensitive resistors made from sintered metal oxides. They are classified into:
- NTC Thermistors: Their resistance drops sharply as temperature rises. They are widely used as temperature-sensing elements in Engine Cylinder Head Temperature (CHT) gauges, Oil Temperature sensors, and Continuous-loop Fire Detection systems (where heating causes the insulating core to conduct, triggering the fire alarm).
- PTC Thermistors: Their resistance rises sharply at a specific temperature threshold. They are used for self-regulating heater elements and resettable overcurrent protection devices (PolySwitches).
Variable Resistors: Potentiometers and Rheostats
Variable resistors allow technicians or systems to adjust resistance dynamically.
- Potentiometers: A three-terminal device used primarily as a voltage divider. The two outer terminals are connected across a voltage source, and the center terminal is connected to a sliding contact (wiper). As the wiper moves, the voltage at the center terminal varies from 0V to the supply voltage. Used for cockpit light dimmers, audio volume controls, and flap/rudder position sensors.
- Rheostats: A two-terminal device connected in series with the load to control current. It consists of a resistive track and a wiper, with one terminal connected to one end of the track and the other connected to the wiper. As resistance increases, circuit current decreases. Used in heavy-duty applications like motor speed controls and alternator field regulators.
The Wheatstone Bridge
The Wheatstone bridge is a highly sensitive and precise circuit configuration used to measure unknown resistances. It consists of four resistive arms arranged in a diamond shape, with a voltage source connected across two opposite nodes and a detector (such as a galvanometer or high-impedance voltmeter) connected across the other two nodes.
The Balanced Condition
The bridge is said to be balanced when the voltage drop across the detector is exactly zero, meaning Node B (V+) and Node C (V-) are at the same electrical potential. Under this balanced condition, no current flows through the detector.
By applying Ohm's Law and Kirchhoff's Laws to the balanced bridge: R1 / R2 = R3 / R4
Where R4 is typically the unknown resistor. Rearranging this equation allows us to find the unknown value: R4 = R3 * (R2 / R1)
Aviation Sensor Applications
- Resistive Temperature Detectors (RTD): In aircraft cabin climate control systems, one arm of a Wheatstone bridge is a platinum RTD sensor. As cabin temperature changes, the sensor resistance changes, unbalancing the bridge and generating an output voltage. The climate controller uses this voltage to adjust heating or cooling valves.
- Strain Gauges: Aircraft structural testing utilizes strain gauges bonded to wing spars. When the wing bends, the gauge stretches or compresses, changing its resistance. A Wheatstone bridge converts this minuscule resistance change into a measurable voltage change, allowing engineers to monitor structural load.
Worked Calculation Scenarios
Scenario 1: Wire Resistance Calculation
Calculate the resistance of a 60-meter aluminum feeder cable with a core diameter of 4 mm at 20\u00b0C.
- Identify Given Values:
- Length (L) = 60 m
- Diameter (d) = 4 mm = 0.004 m
- Resistivity (\u03c1) of Aluminum = 2.82 * 10\u207b\u2078 \u03a9\u00b7m
- Calculate Cross-Sectional Area (A):
- A = \u03c0 * d\u00b2 / 4 = \u03c0 * (0.004)\u00b2 / 4
- A = 3.14159 * 0.000016 / 4 = 1.2566 * 10\u207b\u2075 m\u00b2
- Apply the Resistivity Formula:
- R = \u03c1 * L / A
- R = (2.82 * 10\u207b\u2078 \u03a9\u00b7m) * (60 m) / (1.2566 * 10\u207b\u2075 m\u00b2)
- R = 1.692 * 10\u207b\u2076 / 1.2566 * 10\u207b\u2075 = 0.1346 \u03a9
- Answer: The wire resistance is approximately 0.135 \u03a9.
Scenario 2: Winding Resistance Under Heat
A copper motor winding has a resistance of 4.5 \u03a9 at 20\u00b0C. After a 2-hour flight, the winding temperature rises to 125\u00b0C. Calculate the winding resistance at this operating temperature. (\u03b1 for copper at 20\u00b0C is 0.00393 K\u207b\u00b9).
- Identify Given Values:
- R0 = 4.5 \u03a9
- t0 = 20\u00b0C, t = 125\u00b0C
- \u03b1 = 0.00393 K\u207b\u00b9
- Calculate Temperature Difference:
- \u0394t = 125 - 20 = 105\u00b0C (or 105 K)
- Apply the Temperature Coefficient Formula:
- Rt = R0 * (1 + \u03b1 * \u0394t)
- Rt = 4.5 * (1 + 0.00393 * 105)
- Rt = 4.5 * (1 + 0.41265)
- Rt = 4.5 * 1.41265 = 6.357 \u03a9
- Answer: The winding resistance increases to approximately 6.36 \u03a9.
Scenario 3: Wheatstone Bridge Balancing
A technician is calibrating a cabin temperature sensor bridge. R1 = 100 \u03a9, R2 = 500 \u03a9, and R3 is a standard reference resistor of 150 \u03a9. What resistance must the active sensor (R4) be to balance the bridge?
- Apply the Balance Formula:
- R1 / R2 = R3 / R4
- Substitute Known Values:
- 100 / 500 = 150 / R4
- 0.2 = 150 / R4
- Solve for R4:
- R4 = 150 / 0.2 = 750 \u03a9
- Answer: The sensor resistance must be 750 \u03a9 to balance the bridge.
How does doubling the length and halving the cross-sectional area of a wire affect its electrical resistance?
Which of the following electrical components exhibits a Negative Temperature Coefficient (NTC) of resistance?
In a balanced Wheatstone bridge, the four arms are R1, R2, R3, and R4. If R1 = 150 \u03a9, R2 = 300 \u03a9, and R3 = 75 \u03a9, what is the value of R4?
Why is aluminum used for large power distribution cables in some aircraft despite having higher resistivity than copper?