01 Inputs
02 Solved Parameters
Characteristic impedance
— Ω
W/h = —
εeff @ DC / @ f
—
at f: —
Trace width
— mm
solved from Z₀
Guided wavelength λg
— mm
free-space λ₀: —
Phase velocity vp
— ×10⁸ m/s
— c
Group velocity vg
— ×10⁸ m/s
— c, dispersive
Conductor loss αc
— dB/m
skin depth: —
Dielectric loss αd
— dB/m
tan δ: —
Total loss αtotal
— dB/m
Conductor
Dielectric
03 Physical length for electrical length
At f = — GHz using dispersive εeff(f)
| El. length | Physical | Use case |
|---|
—
04 Method & equations
Quasi-static synthesis/analysis: Hammerstad–Jensen closed-form. Frequency dispersion of εeff: Kirschning–Jansen (1982). Conductor loss: Hammerstad–Wheeler / Pozar; dielectric loss: standard tan δ relation. Valid to a few percent for W/h ∈ [0.1, 10] and εr ≤ 16. Always verify with full-wave EM (HFSS/ADS) before tape-out.
Effective dielectric constant & impedance (Hammerstad–Jensen)
W/h ≤ 1: ε_eff = (εr+1)/2 + (εr-1)/2 · [1/√(1+12h/W) + 0.04(1-W/h)²]
W/h ≥ 1: ε_eff = (εr+1)/2 + (εr-1)/2 · 1/√(1+12h/W)
W/h ≤ 1: Z0 = (60/√ε_eff) · ln( 8h/W + W/(4h) )
W/h ≥ 1: Z0 = (120π/√ε_eff) / [ W/h + 1.393 + 0.667·ln(W/h + 1.444) ]
Synthesis — given Z₀, solve W/h (Hammerstad)
A = Z0/60·√((εr+1)/2) + (εr-1)/(εr+1)·(0.23 + 0.11/εr)
W/h = 8e^A / (e^(2A) - 2) [ used when result < 2 ]
B = 377π / (2·Z0·√εr)
W/h = (2/π)·[ B-1-ln(2B-1) + (εr-1)/(2εr)·(ln(B-1)+0.39-0.61/εr) ] [ W/h ≥ 2 ]
Frequency dispersion — Kirschning & Jansen (1982)
ε_eff(f) = εr - (εr - ε_eff(0)) / (1 + P(f)), f in GHz, h in cm
P1 = 0.27488 + (W/h)[0.6315 + 0.525/(1+0.0157fh)^20] - 0.065683·e^(-8.7513·W/h)
P2 = 0.33622·[1 - e^(-0.03442εr)]
P3 = 0.0363·e^(-4.6·W/h)·{1 - e^[-(fh/3.87)^4.97]}
P4 = 1 + 2.751·[1 - e^(-(εr/15.916)^8)]
P = P1·P2·[(0.1844+P3·P4)·f·h]^1.5763
v_g derived numerically: v_g = v_p / (1 + (f/ε_eff)·dε_eff/df)
Loss — conductor (Hammerstad–Wheeler/Pozar) & dielectric
Rs = √(π·f·μ0/σ) (surface resistance)
δ = √(2/(ω·μ0·σ)) (skin depth)
W/h ≤ 1/(2π): α_c = 8.686·(Rs/(2π Z0 h))·[1-(W_eff/4h)²]·(32-(W_eff/h)²)/(32+(W_eff/h)²)
W/h ≥ 1/(2π): α_c = 8.686·(Rs/(Z0 h))·[1-(W_eff/4h)²]·{ W_eff/h + (0.667 W_eff/h)/(W_eff/h+1.444) }
α_d = 8.686·π·εr·tanδ / (λ0·√ε_eff) · (ε_eff - 1)/(εr - 1) [dB/m]
05 Substrate library
| Material | εr | tan δ | Typical application |
|---|
Click any row to load that substrate. Values are nominal at 10 GHz — confirm against current datasheet before finalizing layout.