Electronic integrator for Rogowski coil sensors

An integrator circuit for a current sensor such as a Rogowski coil. The integrator circuit includes an integrator having an input for receiving a signal from a current sensor and having an output providing a voltage signal. A high-pass filter has an input coupled to the output of the integrator and substantially removes the DC content from the voltage signal. A feedback loop has an input coupled to the output of the integrator and to the high-pass filter, and has an output providing the DC content of the voltage signal back to the input of the integrator. The integrator circuit can detect large current steps in the line conductor being monitored and can be used for line fault detection.

BACKGROUND

A Rogowski coil sensor12is shown inFIG. 1. The flexible coil12can be clamped around a current carrying conductor10without interrupting the conductor circuit, is linear over all practical current ranges, and forms therefore an attractive current sensor solution in the AC power industry. With the AC line centered in the coil12, the produced voltage V is:

Vi⁢⁢n=μ0⁢n⁢⁢A2⁢⁢π⁢⁢r⁢ⅆIⅆt⁢⁢A⁢<<r2
with μ0the permeability of free space, n the number of turns of the coil, A the coil cross section, r the radius as shown inFIG. 1, and I the line current. The voltage V is proportional to the differentiated line current, and signals V and I will thus show a 90 degree phase difference. A signal V in phase with current I will require integration. Integration will also lead to the correct current profile if higher harmonics are involved.

FIG. 1shows an example of an integrator14coupled to coil12at terminal16with basic harmonic transfer function:

VoutVi⁢⁢n=R1R3⁢11+j⁢⁢ω⁢⁢R1⁢C1
Integrator14with the shown component values is an inadequate solution when accurate phase information is required. The ωRC product at 60 Hz is only about 1.2 resulting in integrator14having a substantial undesirable phase shift at the line frequency. The desired phase shift of 90 degrees is obtained with removing R1so that the transfer function obtains the pure integrator form:

The practical problem with this transfer function is that the amplification at DC becomes infinite. As a result, the output can contain an undefined DC level that in essence represents the integration constant leaving the feedback capacitor C1DC charged. Scholastic indefinite integral calculus exercises ignore the integration constant, i.e. make it zero, and the challenge is now to extend this convenience to the present practical case. One remedy is to place a transistor (MOSFET) across the feedback capacitor C1to occasionally discharge it so that the OPAMP15DC output becomes redefined. This approach requires timing circuitry to discharge infrequently but still regularly at a preferred moment. When well implemented with near perfect MOSFETs this approach can provide for the remedy.

Accordingly, a need exists for an improved integrator for Rogowski coils or other current sensors.

SUMMARY

An integrator circuit for a current sensor, consistent with the present invention, includes an integrator having an input for receiving a signal from a current sensor and having an output providing a voltage signal. A high-pass filter has an input coupled to the output of the integrator and has an output, and the high-pass filter substantially removes a DC content from the voltage signal. A feedback loop has an input coupled to the output of the integrator and to the output of the high-pass filter, and has an output providing the DC content of the voltage signal to the input of the integrator.

DETAILED DESCRIPTION

An improved integrator for a Rogowski coil sensor, or other current sensors, is disclosed. Rogowski coil sensors are routinely used to monitor or measure 60 Hz line currents in the AC power grid. Developments in the smart grid infrastructure will require vast deployment of these sensors. Rogowski sensors are commercially available and require a preamplifier to bring the coil amplitude to an acceptable level. Generally, a Rogowski coil produces roughly 25 μV per ampere of line current, too weak to be directly useful for processing electronics that prefer an amplification of about 1 mV/A. Commercial preamplifiers or transducers are available, but they often produce a DC signal resulting in loss of the AC phase information. In order to obtain a phase accurate 60 Hz amplified replica of the line current, a new integrator design is disclosed. This integrator design is an improved solution to the remedy described above and involves adherence to the second transfer function above for the pure integrator form without requiring placement of any component across the feedback capacitor, using continuous monitoring of the DC level and forcing it to zero through stable feedback.

FIG. 2is a diagram of an integrator circuit20solution containing three OPAMPs (operational amplifiers)21,22, and23. The first OPAMP21is a pure integrator having an input receiving a signal from the Rogowski coil sensor at terminal16and having an output (25) providing the signal Va. The inverting input of OPAMP21may have a pull-down resistor to ground of, for example, 1 MΩ. OPAMP21is followed by a passive first order high-pass filter, formed by capacitor C2and resistor R2, having an input receiving the signal Vaand having an output delivering a DC free version of the integrated signal to the + input of the second OPAMP22serving as a voltage follower and having an output (26) providing the signal Vb. The high-pass filter can remove the DC content of the signal Vaor substantially remove the DC content by an acceptable amount for operation of circuit20. The voltage follower (OPAMP22) is used to minimize the load on the high impedance point formed by the high-pass filter output. The third OPAMP23is a difference amplifier (that in this case does not amplify) having an output (27) producing a copy of the DC content of the first OPAMP21output. Closing the DC feedback loop from the output Vcof OPAMP23to the + input V+of OPAMP21then guarantees that the integrator output of the first OPAMP21remains DC free.

Integrator circuit20receives a signal from a Rogowski coil sensor, or other current sensor, at terminal16and provides an output signal Voutat terminal24. Circuit20outputs a signal at terminal24related to the signal from the current sensor, for example a decaying ringing signal in response to a large current step in the line conductor monitored by the current sensor. Circuit20at output terminal24can be coupled to an analog-to-digital converter in order to provide a corresponding digital signal to a processor for use in smart grid infrastructure monitoring such as line fault analysis, for example. Circuit20can also be used to monitor and detect line faults in three-phase cables.

The performance of circuit20is shown inFIG. 3, comparing a line current measured by a current transformer (Hammond CT500A) and by integrator circuit20inFIG. 2. With the potentiometer R1(variable resistor) in circuit20adjusted to a 1 mV/A amplification, the traces inFIG. 3show almost perfect correspondence with a minute and generally acceptable phase difference.

The following provides harmonic analysis of the operation of circuit20. If the non-inverting input voltage V+ is at ground potential, then the transfer function of the first OPAMP21integrator is:

VaVin=-1j⁢⁢ω⁢⁢R1⁢C1=-1j⁢⁢ω⁢⁢τ1(1)
The function in equation (1) is that of an ideal integrator with a pole at DC or more realistically at a very low frequency with extremely high amplification. If the primary application forms an integrator for a 60 Hz signal, as in this exemplary case, then the integration of low frequency noise results in a slowly varying drift with large amplitude that becomes a nuisance. This nuisance can be avoided by occasionally resetting the integrator DC state or by limiting the amplification at low frequencies that are not of interest.

Accordingly, one remedy involves occasionally discharging the capacitor C1by shorting it through a MOSFET, effective but not the most elegant solution. Another common remedy to limit the amplification at low frequencies is to place a large feedback resistor Rfacross capacitor C1so that the amplification at DC is at most equal to the ratio Rf/R1of resistors Rfand R1. In practice, this ratio is still much larger than unity so that the slow drift nuisance is diminished but not removed. Further improvement can be obtained if the amplification tends to approach zero at DC. Such is the case with circuit20inFIG. 2as the following harmonic analysis demonstrates.

The voltages indicated inFIG. 2are all referenced to ground. With w the angular frequency, two basic transfer functions are provided by:

VbVa=-j⁢⁢ω⁢⁢R2⁢C21+j⁢⁢ω⁢⁢R2⁢C2=-j⁢⁢ω⁢⁢τ21+j⁢⁢ω⁢⁢τ2(2)
and with all resistors R's having equal values:
Vc=Vb−Va(3)
Considering an arbitrary V+input voltage, the derivation of the first OPAMP21integrator transfer function requires the following two equations:

-Vin+i⁢⁢R1+V+=0(4)-Vin+i⁡(R1+1j⁢⁢ω⁢⁢C1)+Va=0(5)
Removing current i from equations (4) and (5):

Va=-Vin⁢1j⁢⁢ω⁢⁢τ1+V+⁡(1+j⁢⁢ω⁢⁢τ1j⁢⁢ω⁢⁢τ1)(6)
Equating voltage V+to voltage Vcthrough direct feedback, as shown in circuit20, and combining equations (2), (3), and (6):

VaVin=-1j⁢⁢ω⁢⁢τ1+1+j⁢⁢ω⁢⁢τ11+j⁢⁢ω⁢⁢τ2(7)
Clearly, equation (7) approaches an ideal integrator if τ2>>τ1and a result reveals itself if the total integrator transfer function is derived:

VoutVin=VbVa⁢VaVin=j⁢⁢ω⁢⁢τ21+j⁢⁢ω⁢⁢τ2⁢-1j⁢⁢ω⁢⁢τ1+1+j⁢⁢ω⁢⁢τ11+j⁢⁢ω⁢⁢τ2=-1j⁢⁢ω⁢⁢τ1+2⁢⁢τ1⁢τ2+1j⁢⁢ω⁢⁢τ2(8)
The total transfer function in equation (8) becomes an excellent integrator when the following two conditions are both fulfilled:

1)⁢⁢τ2⪢T1⁢⁢2)⁢⁢ω⪢1τ2
The desirable benefit becomes apparent when w approaches DC because then the output is eliminated as a result of differentiation related to the term ωτ2. This is unlike a pure integrator in equation (1) where the amplification at DC tends to infinity.

The properties are further demonstrated by plotting the amplitude and phase responses of the transfer function in equation (8), as illustrated inFIGS. 4 and 5. The amplitude response of circuit20inFIG. 4clearly shows a resonance peak marking the absolute lower limit of the usable frequency range for integration purposes. A resonance frequency fresis obtained by setting the phase of equation (8) to zero with the amplitude the maximum τ2/(2τ1):

For a 60 Hz AC line current integrator, a design consideration can be to let fresnot exceed about 10 Hz. Circuit20requires a low noise environment, especially near fres, which for a monochromatic 60 Hz line signal can be comfortably assumed.

For further illustration purposes, the operation of circuit20can be compared against a first order filter circuit formed by placing a resistor Rfacross capacitor C1. Such comparison will demonstrate the benefits of circuit20with an extremely sharp transition from differentiation to integration extending over only a few hertz bandwidth.

The following provides transient analysis of the impulse response of circuit20. The Laplace transformed transfer function of circuit20is derived as:

vout⁡(s)vin⁡(s)=p⁡(s)=-1τ1⁢ss2+s⁢2τ2+1τ1⁢τ2=-1τ1⁢s(s+ω1)⁢(s+ω2)(10)
and as also shown in equation 10, by defining a root r:

r=τ2τ1-1(11)
The frequencies ω1and ω2are evaluated as complex conjugates:

ω1=1τ2⁢(1+j⁢⁢r)(12)ω2=1τ2⁢(1-j⁢⁢r)(13)
Based on Maxwell's equations, a Rogowski coil inductance terminated with a resistance is properly described by a first order differential equation. Impulse responses then have the form of exponential expressions of time. Analogously, an impulse response of exponential form can be obtained by differentiating a slow 100 mHz square wave from a wideband function generator over a 10 nF capacitor and a 50Ω resistor divider. Such an impulse is described by:

Vin⁡(t)=V0⁢ⅇ-tτ0=V0⁢ⅇ-ω0⁢t(14)
With ω0real and positive and with Laplace transform:

vin⁡(s)=V0s+ω0(15)
Substituting equation 15 in equation 10, the impulse response spectrum becomes:

vout⁡(s)=vin⁡(s)⁢p⁡(s)=-V0τ1⁢s(s+ω0)⁢(s+ω1)⁢(s+ω2)(16)
The inverse Laplace transformed impulse response of circuit20is provided in equation 17 with the input impulse content of V0τ0=M Iacwith M the Rogowski coil coefficient of mutual inductance and with Iacthe current step amplitude shown inFIG. 6.

Equation (17) is a very good approximation when |ω0|>>|ω1|, |ω2| and demonstrates the integrator circuit stability with always decaying impulse response solutions. With equation (17) it is possible to think of the integrator as an electronic tuning fork with resonance frequency well below the 60 Hz line frequency.

FIG. 6is a graph of the line current step response of circuit20, showing an actual line current step of 250 A and the resulting integrator circuit20ringing response.

The exemplary component values provided for the resistors and capacitors in circuit20are in ohms and farads. Other component values can be used depending upon, for example, a particular application of circuit20. When a component is described as coupled to another component, the components can be directly coupled or coupled through other components for electrical communication among them. The term input to a component can include a single input or multiple inputs. The term output from a component can include a single output or multiple outputs.