How do I calculate the impedance of a differential pair on a PCB at multi-GHz data rates?

How do I calculate the impedance of a differential pair on a PCB at multi-GHz data rates? The differential impedance is determined by the trace geometry, dielectric properties, and coupling between the two traces, and must be precisely controlled for high-speed signal integrity: (1) Differential impedance definition: Z_diff = 2 × Z_single × (1 - k), where Z_single is the single-ended characteristic impedance and k is the coupling coefficient between the two traces (0 = no coupling, 1 = perfect coupling). Typical targets: 100 ohm differential (50 ohm single-ended per trace) for most standards (USB, PCIe, Ethernet). 85 ohm differential for some legacy protocols. 90 ohm differential for HDMI. (2) Key geometry parameters: stripline: traces sandwiched between two ground planes. More consistent impedance (shielded by ground planes on both sides). Typical for inner-layer differential pairs. Microstrip: traces on the outer layer with one ground plane below. Higher impedance for the same geometry (vs stripline, due to partial air dielectric). Z_diff depends on: trace width (W), trace spacing (S), dielectric height (H), and dielectric constant (Dk). For 100 ohm differential stripline: W ≈ 4 mil, S ≈ 5 mil, H ≈ 4 mil (each side), Dk = 3.5 (FR-4). These values scale with the substrate: low-loss materials (Dk 3.0-3.2) require wider traces for the same impedance. (3) Frequency dependence at multi-GHz: at DC-1 GHz: the impedance is well-predicted by 2D cross-section models (assuming TEM mode). At 10+ GHz: the dielectric constant (Dk) and loss tangent (Df) become frequency-dependent. Dk decreases by 3-8% from 1 GHz to 25 GHz for standard FR-4. This Dk variation changes the impedance by 2-4% over frequency. Roughness: copper surface roughness causes the effective conductor length to increase at high frequencies, adding impedance and loss. Standard FR-4: copper roughness (Rz) = 3-6 μm. Low-loss laminates: smooth copper (Rz < 1 μm) → better impedance consistency at > 10 GHz. (4) Calculation tools: 2D field solvers: Polar Instruments Si9000, Altium Impedance Calculator, HyperLynx. These solve the electromagnetic cross-section and compute Z_diff, Z_single, and propagation delay. 3D field solvers (for discontinuities): ANSYS HFSS, Dassault CST, Cadence Clarity. Manufacturer specifications: PCB fabricators (e.g., TTM, AT&S) provide impedance calculators based on their specific materials and processes.