About via hole

Thermal resistance, resistance and inductance of a via hole

Fig.1 Via hole
Material Relative permitivity
μ_r Conductivity
σ[S/m] @295K Thermal conductivity
k [W/m/K] @293K

Silver 0.99998 6.21 × 10⁷ 418

Copper 0.999991 5.88 × 10⁷ 386

Gold 1.0 4.55 × 10⁷ 295

Aluminium 1.00002 3.65 × 10⁷ 204

Nickel 600 1.43 × 10⁷ 90

Via hole material:

μ_r : relative permeability

σ : conductance MS/m

k : thermal conductivity W/m/K

ε_r : relative permittivity

f : frequency MHz

h : via length mm

d : via diameter mm

t : plating thickness mm

θ : thermal resistance (plated via) K/W

R_dc : DC resistance (plated via) mΩ

δ : skin depth mm

R_ac : AC resistance (filled via) mΩ

L_via : inductance (filled via) nH

L_gnd : inductance (filled via to ground) nH

L_gnd : valid for heights less than 3% of wavelength.100μm ≤ height ≤ 631μm

Fig.2 Microstrip via hole to gound plane

Formulas used

$L_{via} = \frac{ \mu_0}{2 \pi } \biggl[ h \log\bigg( \frac{h + \sqrt{h^2 + r^2}}{r} \bigg) - \sqrt{h^2 + r^2} + \frac{h \mu_r}{4} + r \biggr]$

$L_{gnd}= \frac{ \mu_0}{2 \pi } \biggl[ h \log\bigg( \frac{h + \sqrt{h^2 + r^2}}{r} \bigg) + \frac{3}{2} \big(r - \sqrt{h^2 + r^2}\big) \biggr]$

where
h is length of the via. i.e.
r is radius of the via. i.e. d / 2
μ₀ is permitivity of free space.

Calculated inductance of via holes (example)

Fig.3 Inductance of via holes

Fig.4 Inductance of via holes to ground plane

REFERENCE

Brian C. Wadell, "Transmission line design handbook", Artech House, Boston, 1991.
F.W. Grover, "Inductance calculations", Van Nostrand, New York, 1946.
F.W. Grover, "Inductance Calculations", Dover Publications, 2004.
M. E. Goldfarb and R. A. Pucel, "Modeling Via Hole Grounds in Microstrip", IEEE Microwave and Guided Wave Letters, vol. 1, no. 6, pp. 135-137, June 1991.

EXTERNAL LINKS

Microstrip Via Hole Inductance

Fig.1 Via hole	Material	Relative permitivity μ_r	Conductivity σ[S/m] @295K	Thermal conductivity k [W/m/K] @293K
	Silver	0.99998	6.21 × 10⁷	418
	Copper	0.999991	5.88 × 10⁷	386
	Gold	1.0	4.55 × 10⁷	295
	Aluminium	1.00002	3.65 × 10⁷	204
	Nickel	600	1.43 × 10⁷	90

Via hole material:
μ_r : relative permeability
σ : conductance	MS/m
k : thermal conductivity	W/m/K
ε_r : relative permittivity
f : frequency	MHz
h : via length	mm
d : via diameter	mm
t : plating thickness	mm
θ : thermal resistance (plated via)	K/W
R_dc : DC resistance (plated via)	mΩ
δ : skin depth	mm
R_ac : AC resistance (filled via)	mΩ
L_via : inductance (filled via)	nH
L_gnd : inductance (filled via to ground)	nH

Formulas used

Calculated inductance of via holes (example)

REFERENCE

SEE ALSO

EXTERNAL LINKS