SDAA566 Application note

SDAA566 October 2026

2.2 Mathematical Model and Sensitivity

For a Rogowski coil implemented on a PCB, the geometry is constrained by the substrate thickness and the layout area. The induced voltage for a rectangular or cylindrical PCB winding can be modeled by the following relationship:

The Rogowski coil (V_out) is as shown in Equation 1:

Equation 1.

B_{0} = \frac{μ_{0} I_{p} \sin (ω t)}{2 π R}

Where:

B₀ = magnetic field around a wire generated by the flow of current (Biot-Savart)
μ₀ = permeability of free space
ω = angular frequency = 2πf
I_p = primary current
R = distance from center of wire to a measuring point

The magnetic flux across the conductive trace is given in Equation 2 as:

Equation 2.

Φ = \int_{a}^{b} B_{0} \times d A

Where:

Φ = The magnetic flux across the conductive trace
dA = dr × h, which is the rectangular cross section area for element (dr) with PCB height (h)

Where:

dr = infinitesimal conductance at a distance R from the center of wire.
h = height of PCB

So, the total flux across the conductive trace is as shown in Equation 3:

Equation 3.

Φ = \frac{μ_{0} I_{p} \sin (ω t)}{2 π} \times h \times \ln (\frac{b}{a})

Where:

b = outer radius
a = inner radius

Proportional voltage to the rate of change of the magnetic flux (\Faraday's law of induction) is as shown in Equation 4. Assuming primary current (I_p) and angular frequency (w).

Equation 4.

V_{o u t} = N (\frac{d Φ}{d t})

Where:

N = number of turns

Substituting:

Equation 5.

V_{o u t} = \frac{N μ_{0} h}{2 π} \times \ln (\frac{b}{a}) \times \frac{d I_{p} \sin (ωt)}{d t}

Solving the derivative:

Equation 6.

V_{o u t} = \frac{N μ_{0} h}{2 π} \times {ωI}_{p} \times \cos (ωt) \times \ln (\frac{b}{a})

Note: (V_out: I_p) is referred to as sensitivity, typically measured in μV/A.

Note: Mutual inductance is part of V_out.

Equation 7.

M = \frac{N μ_{0} h}{2 π} \times \ln (\frac{b}{a})