A current sensing circuit for use with a Rogowski coil arranged around a conductor having a primary current, the current sensing circuit comprising: input terminals (102,104) structured to receive an output voltage of the Rogowski coil; filtering elements (106,108) structured to filter high frequency voltage from the output voltage and to output a filtered output voltage; an amplifier structured to receive the filtered output voltage and produce a differential voltage; an analog to digital converter structured to convert the differential voltage to a digital differential voltage signal; a digital integrator structured to receive the digital differential voltage signal, to implement a discrete-time transfer function that is a transform of a transfer function of an analog integrator, and to output a digital integrator output signal; a direct current blocker filter structured to remove a direct current bias from the digital integrator output signal and to output a digital current output signal that is proportional to the primary current in the conductor, wherein the discrete-time transform implemented by the digital integrator is an impulse-invariant transform of the transfer function of the analog integrator, wherein the analog integrator is an RC-filter having a resistance of R and a capacitance of C; and wherein the discrete-time transfer function of the digital integrator is defined by the following equation: H z = y z x z = α ⋅ T S 1 − e − αT S z − 1 = b 0 1 − a 1 z − 1 where T s is a sampling interval; α = 1 RC; a 1 = e − αT s = e − T s RC; b 0 = α · T s; and z -1 denotes a one-sample delay, wherein the digital integrator has coefficients of a 1 and b 0; wherein a 1 is defined by the following equation: a 1 = cos Δn ⋅ ω e cos Δn − 1 ⋅ ω e wherein b 0 is defined by the following equation: b 0 = 1 − 2 a 1 cos ω e + a 1 2 wherein Δn is the number of samples between the digital current output signal and the primary current in the conductor; ω e = 2 π f e f s, f e is a rated supply frequency, and f s is a sampling frequency of the analog to digital converter, wherein the rated supply frequency is related to the primary current by the following equation: i p t = A p ⋅ sin 2 πƒ e t where i p (t) is the primary current, and where A p is an amplitude of the primary current i p (t). input terminals (102,104) structured to receive an output voltage of the Rogowski coil; filtering elements (106,108) structured to filter high frequency voltage from the output voltage and to output a filtered output voltage; an amplifier structured to receive the filtered output voltage and produce a differential voltage; an analog to digital converter structured to convert the differential voltage to a digital differential voltage signal; a digital integrator structured to receive the digital differential voltage signal, to implement a discrete-time transfer function that is a transform of a transfer function of an analog integrator, and to output a digital integrator output signal; a direct current blocker filter structured to remove a direct current bias from the digital integrator output signal and to output a digital current output signal that is proportional to the primary current in the conductor, wherein the discrete-time transform implemented by the digital integrator is an impulse-invariant transform of the transfer function of the analog integrator, wherein the analog integrator is an RC-filter having a resistance of R and a capacitance of C; and wherein the discrete-time transfer function of the digital integrator is defined by the following equation: H z = y z x z = α ⋅ T S 1 − e − αT S z − 1 = b 0 1 − a 1 z − 1 where T s is a sampling interval; α = 1 RC; a 1 = e − αT s = e − T s RC; b 0 = α · T s; and z -1 denotes a one-sample delay, wherein the digital integrator has coefficients of a 1 and b 0; wherein a 1 is defined by the following equation: a 1 = cos Δn ⋅ ω e cos Δn − 1 ⋅ ω e wherein b 0 is defined by the following equation: b 0 = 1 − 2 a 1 cos ω e + a 1 2 wherein Δn is the number of samples between the digital current output signal and the primary current in the conductor; ω e = 2 π f e f s, f e is a rated supply frequency, and f s is a sampling frequency of the analog to digital converter, wherein the rated supply frequency is related to the primary current by the following equation: i p t = A p ⋅ sin 2 πƒ e t where i p (t) is the primary current, and where A p is an amplitude of the primary current i p (t).