What is the difference between phase velocity and group velocity in a dispersive transmission line?
Phase vs Group Velocity
For a non-dispersive transmission line (TEM-mode coax, stripline), the phase velocity is constant with frequency: vp = c/√εr. The group velocity equals the phase velocity, and all frequency components of a signal travel at the same speed. The signal waveform is preserved (no pulse spreading).
| 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
For a dispersive transmission line, the phase velocity varies with frequency: vp(f) = c/√εr(f). Different frequency components travel at different speeds, spreading the signal pulse as it propagates. The group velocity, which represents the speed of the pulse envelope, differs from the phase velocity. The group velocity is the velocity of energy and information transport.
Loss and Phase Stability
In waveguide, the dispersion arises from the cutoff phenomenon. Near the cutoff frequency, the phase velocity approaches infinity while the group velocity approaches zero. The product vp × vg = c² (in vacuum) or c²/εr (in dielectric-filled waveguide). This fundamental relationship means that the phase velocity exceeds c/√εr while the group velocity is less than c/√εr, with the product remaining constant.
- 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
Connector Interface
When evaluating the difference between phase velocity and group velocity in a dispersive transmission line?, engineers must account for the specific requirements of their target application. The optimal choice depends on the frequency range, power level, environmental conditions, and cost constraints of the overall system design.
Frequently Asked Questions
Does vp > c violate relativity?
No. Phase velocity greater than the speed of light does not violate special relativity because it does not represent the transfer of information or energy. A steady-state single-frequency wave carries no information. Information travels at the group velocity, which is always ≤ c.
When does dispersion matter practically?
Dispersion matters when the signal bandwidth is significant compared to the transmission line's cutoff bandwidth. Narrowband signals (<1% BW) experience negligible dispersion even in waveguide. Wideband signals (>10% BW) in waveguide can experience significant pulse spreading and phase distortion.
How do I measure group velocity?
Measure the phase of S21 across frequency. The group delay is tg = -dφ/dω. The group velocity is vg = length / tg. The VNA calculates group delay directly from the phase data. Non-constant group delay across frequency indicates dispersion.