Quantifying RF Mismatch Degradation from Substrate Stackup Variance
Substrate dielectric tolerance and etch variance shift trace impedance, raising return loss and degrading RF power amplifier efficiency in custom module designs.
Hammerstad Model
Meaning
A mathematical formula used to estimate the increase in conductor loss due to the roughness of the copper foil at high frequencies is essential for accurate electromagnetic simulations. This calculation, known as the Hammerstad model, applies a correction factor based on the root-mean-square roughness of the copper surface and the skin depth of the signal. It is widely used in EDA tools for PCB simulation up to several gigahertz.
Surface Equation
The model uses a simple formula that multiplies the smooth conductor losses by a scaling factor that approaches a value of two as the frequency increases. This scaling assumes that the micro-roughness profile consists of hemispherical structures that force the high-frequency current to travel a longer path. This longer path increases the effective series resistance of the transmission line.
Loss Prediction
As digital data rates exceed ten gigabits per second, the skin depth becomes smaller than the average copper roughness, causing the signal loss to increase rapidly. The model predicts this increase by adjusting the attenuation coefficient as a function of frequency. However, this prediction tends to underestimate losses at frequencies above fifteen gigahertz because it does not account for the complex fractal nature of real copper foil treatments.
Frequency Boundary
Engineers replace this model with more advanced multi-modal equations when simulating extremely high-speed channels such as PCIe Gen 6. This transition prevents under-designing the receiver equalization blocks.