What is the effect of temperature on the electrical length of a transmission line?
Temperature and Electrical Length
Both the physical dimensions and the dielectric properties of a transmission line change with temperature. The physical length increases with temperature due to thermal expansion: ΔL/L = CTE × ΔT. The dielectric constant increases with temperature for most materials: Δεr/εr = TCε × ΔT. Both effects increase the electrical length, adding constructively to produce a phase shift that can be significant in precision systems.
| Parameter | Semi-Rigid | Conformable | Flexible |
|---|---|---|---|
| Loss (dB/m at 10 GHz) | 0.8-2.5 | 1.0-3.0 | 1.5-5.0 |
| Phase Stability | Excellent | Good | Fair |
| Bend Radius | Fixed after forming | Hand-formable | Continuous flex OK |
| Shielding (dB) | >120 | >90 | >60-90 |
| Cost (relative) | 2-5x | 1.5-3x | 1x |
Cable Selection Criteria
The total temperature coefficient of electrical length (TCEL) combines both effects: TCEL ≈ CTE + TCε/2. The factor of 1/2 on TCε arises because electrical length depends on √εeff, and d(√ε)/ε = 1/(2√ε) × dε/ε. For typical substrates: PTFE (CTE=17, TCε=280): TCEL ≈ 157 ppm/°C. FR4 (CTE=14, TCε=200): TCEL ≈ 114 ppm/°C. Low-TCε substrates are available for temperature-sensitive applications.
- Performance verification: confirm specifications against the application requirements before finalizing the design
- Environmental factors: temperature range, humidity, and vibration affect long-term reliability and parameter drift
- Cost vs. performance: evaluate whether the application demands premium components or standard commercial grades
Loss and Phase Stability
The phase change at frequency f for a line of length L over temperature change ΔT is: Δφ = 360° × f × L × √εeff × TCEL × ΔT / c. For a phased array with 10 cm feed lines at 10 GHz and 50°C temperature range: Δφ = 360 × 10⁹ × 0.1 × √3.3 × 157×10⁻⁶ × 50 / 3×10⁸ ≈ 17°. This is far too large for a phased array with ±5° tolerance and requires either temperature-compensated substrates or active phase calibration.
Frequently Asked Questions
What substrates have the lowest TCEL?
Ceramic substrates (Al2O3, AlN) have low CTE (6-7 ppm/°C) and low TCε. Rogers RO3006 (εr=6.15) has specified TCEL < 50 ppm/°C. Some vendors offer zero-TCEL substrates where CTE and TCε partially cancel, achieving TCEL < 10 ppm/°C. These specialty substrates are essential for phase-critical applications.
Does Invar solve the problem?
Invar has very low CTE (1.2 ppm/°C) and can be used as a substrate carrier to reduce physical expansion. However, the dielectric constant change still occurs, so Invar only addresses half the problem. Combining Invar carrier with a low-TCε dielectric minimizes total phase drift.
How do I compensate actively?
Measure the temperature at the substrate and apply a calculated phase correction through a voltage-controlled phase shifter or digital delay. The correction factor is calibrated at the factory by measuring phase vs temperature. This approach achieves <1° phase stability over the full temperature range.