Current sensing circuit and current sensing assembly including the same
A current sensing circuit for use with a Rogowski coil arranged around a conductor having a primary current includes input terminals structured to receive an output voltage of the Rogowski coil, an analog to digital converter structured to convert a 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, and 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.
1. 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 structured to receive an output voltage of the Rogowski coil;
filtering elements 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, and
wherein the discrete-time transform implemented by the digital integrator is an impulse-invariant transform of the transfer function of the analog integrator, and
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
-
α
T
S
z
-
1
=
b
0
1
-
a
1
z
-
1
where T s is a sampling interval;
α
=
1
RC
;
a
1
=
e
-
α
T
S
=
e
-
T
S
RC
;
b
0
=
α
·
T
S
;
and z −1 denotes a one-sample delay.
2. The current sensing circuit of claim 1 , wherein the digital integrator has coefficients of a 1 and b 0 ; wherein a 1 is defined by the following equation:
a
1
=
cos
(
Δ
n
·
ω
e
)
cos
[
(
Δ
n
-
1
)
·
ω
e
]
wherein b 0 is defined by the following equation:
b 0 =√{square root over (1−2 a 1 cos ω e +a 1 2 )}
wherein Δn is a phase difference number of samples;
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.
3. The current sensing circuit of claim 1 , wherein the direct current blocker filter has a transfer function defined by the following equation:
H
(
z
)
=
y
(
z
)
x
(
z
)
=
b
0
+
b
1
z
-
1
1
-
a
1
z
-
1
where 0<a 1 <1, b 0 =1, and b 1 =−1.
4. The current sensing circuit of claim 1 , wherein at least one of the digital integrator and the direct current blocker filter are implemented with a digital biquadratic filter having a transfer function defined by the following equation:
H
(
z
)
=
y
(
z
)
x
(
z
)
=
b
0
+
b
1
z
-
1
+
b
2
z
-
2
1
-
a
1
z
-
1
-
a
2
z
-
2
where z −1 denotes a one sample delay, z −2 denotes a two sample delay, and a 1 , a 2 , b 0 , b 1 , and b 2 are coefficients.
5. The current sensing circuit of claim 4 , wherein the digital integrator is implemented with the digital biquadratic filter; wherein the discrete-time transform implemented by the digital integrator is an impulse-invariant transform of the transfer function of the analog integrator; and wherein a 1 ≠0, a 2 =0, b 0 ≠0, b 1 =0, and b 2 =0.
6. The current sensing circuit of claim 4 , wherein the digital integrator is implemented with the digital biquadratic filter; wherein the discrete-time transform implemented by the digital integrator is a bilinear transform of the transfer function of the analog integrator; and wherein a 1 ≠0, a 2 =0, b 0 ≠0, b 1 =b 0 , and b 2 =0.
7. The current sensing circuit of claim 4 , wherein the direct current blocker filter is implemented with the digital biquadratic filter; and wherein a 1 ≠0, a 2 =0, b 0 =1, b 1 =−1, and b 2 =0.
8. 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 structured to receive an output voltage of the Rogowski coil;
filtering elements 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, and
wherein the discrete-time transform implemented by the digital integrator is a bilinear transform of the transfer function of the analog integrator, and
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
2
+
α
T
S
+
α
T
S
2
+
α
T
S
z
-
1
1
-
2
-
α
T
S
2
+
α
T
S
Z
-
1
=
b
0
+
b
1
z
-
1
1
-
a
1
z
-
1
where T s is a sampling interval;
α
=
1
RC
;
a
1
=
2
-
α
T
s
2
+
α
T
s
,
b
0
=
b
1
=
α
T
s
2
+
α
T
s
,
and z −1 denotes a one-sample delay.
9. The current sensing circuit of claim 8 , wherein the digital integrator has coefficients of a 1 , b 0 , and b 1 ; wherein a 1 is defined by the following equation:
a
1
=
cos
[
(
Δ
n
+
1
2
)
·
ω
e
]
cos
[
(
Δ
n
-
1
2
)
·
ω
e
]
and wherein b0 and b1 are defined by the following equation:
b
0
=
b
1
=
1
-
2
a
1
cos
ω
e
+
a
1
2
2
·
(
1
+
cos
ω
e
)
wherein Δn is a phase difference number of samples;
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.
10. 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 structured to receive an output voltage of the Rogowski coil;
filtering elements 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, and
wherein the filtering elements include at least one ferrite bead.
11. A method of implementing a digital integrator, the method comprising:
providing a sampling frequency f s ;
providing a rated supply frequency f e ;
providing a phase difference number of samples Δn;
obtaining a power grid's normalized angular frequency at rated condition using the following equation:
ω
e
=
2
π
f
e
f
S
obtaining a first coefficient a 1 based on the following equation:
a
1
=
cos
(
Δ
n
·
ω
e
)
cos
[
(
Δ
n
-
1
)
·
ω
e
]
obtaining a second coefficient based on the following equation:
b 0 =√{square root over (1−2 a 1 cos ω e +a 1 2 )}
implementing the digital integrator as a digital filter using the first coefficient a 1 and the second coefficient b 0 .
12. The method of claim 11 , further comprising:
scaling the second coefficient b 0 to obtain a scaled second coefficient b′ 0 ; and
implementing the digital integrator using the scaled second coefficient b′ 0 .
13. The method of claim 11 , wherein the digital integrator is an impulse invariant transform of an analog integrator.
14. The method of claim 11 , wherein implementing the digital integrator further comprises implementing the digital integrator in a current sensing circuit for sensing a primary current in a conductor based on a voltage output of a Rogowski coil arranged around the conductor.
15. A method of implementing a digital integrator, the method comprising:
providing a sampling frequency f s ;
providing a rated supply frequency f e ;
providing a phase difference number of samples Δn;
obtaining a power grid's normalized angular frequency at rated condition using the following equation:
ω
e
=
2
π
f
e
f
S
obtaining a first coefficient a 1 based on the following equation:
a
1
=
cos
[
(
Δ
n
+
1
2
)
·
ω
e
]
cos
[
(
Δ
n
-
1
2
)
·
ω
e
]
obtaining a second coefficient b 0 and a third coefficient b 1 based on the following equation:
b
0
=
b
1
=
1
-
2
a
1
cos
ω
e
+
a
1
2
2
·
(
1
+
cos
ω
e
)
implementing the digital integrator as a digital filter using the first coefficient a 1 , the second coefficient b 0 , and the third coefficient b 1 .
16. The method of claim 15 , further comprising:
scaling the second coefficient b 0 to obtain a scaled second coefficient b′ 0 ;
scaling the third coefficient b 1 to obtain a scaled third coefficient b′ 1 ; and
implementing the digital integrator using the scaled second coefficient b′ 0 and the scaled third coefficient b′ 1 .
17. The method of claim 15 , wherein the digital integrator is a bilinear transform of an analog integrator.
18. The method of claim 15 , wherein implementing the digital integrator further comprises implementing the digital integrator in a current sensing circuit for sensing a primary current in a conductor based on a voltage output of a Rogowski coil arranged around the conductor.