Feedback control in Hall-effect current sensors

Solving inverse problems by feedback iterations

A large class of problems can be easily carried in a forward direction, but applications often care about the inverse problem. In this article, the idea and significance of solving inverse problems through iterating forward processes will be demonstrated by an example of Hall-effect current sensors. First, the basics of the Hall effect sensor will be introduced, followed by a straightforward application of an open-loop type current sensing. Then, the closed-loop configuration will be given and discussed throughout including its equivalent sensor transfer function. Few commercial models utilizing each architecture will also be listed and compared in between. The key takeaway of this article is the demonstration of the utilization of the closed-loop system that effectively performs as an inverse problem solver.

The Hall-effect

Hall effect in a simple conductor that only has one type of charge carrier, “I_ref” is the reference current, “B_p” is the externally applied magnetic field, “t” is the conductor thickness, and “n,e” is the charge carrier density, and the charge correspondingly

Discovered by Edwin Hall [1], the classic Hall effect occurs in a current-carrying conductor and was a direct result of the Lorentz force [2]. Referring to the demonstrative figure shown above, a reference current “I_ref” was flowing in the x-direction while an externally uniform magnetic field “B_p” was applied perpendicularly in the z-direction. Driven by the Lorentz force, the moving charge carriers will be pushed in the y-direction, accumulated a net charge, and eventually null out the total force (please bare with the elementary particle model used here).

The voltage created can then be measured as the Hall voltage “V_hall”, proportion to the strength of the external field, the reference current, and also related to the charge carrier density “n” which is affected by temperature. This relationship enables a sensing mechanism of the magnetic field and opens a large categoryof applications including current sensing which will be discussed throughout this article.

Open-loop current sensors

Functional diagram of an open-loop type Hall-effect current sensor, the current to be sensed “I_p” was carried in the primary conductor that passes through the magnetic core colored in grey, the Hall-sensor was placed at the air gap and measures the generated magnetic field “B_p” as “V_hall”

Consists of a magnetic core that concentrated the field and a properly placed Hall-sensor, an open-loop current sensor directly measures the magnetic field “B_p” and inferred the current to be measured “I_p”. This signal process for this type of construction can be realized as the direct inversion of the sensor mechanism shown below.

Signal processing for open-loop sensors utilizes a direct inversion and suffers from a large set of errors

The accuracy of such architecture strongly relies on the linearity and the hysteresis of the magnetic core which is easily compromised under a higher current as depicted in the figure below. Furthermore, the gain of the Hall-sensor changes as the temperature drifts away as pointed out in the last section. These drawbacks limit both the dynamic and the steady-state performance of open-loop current sensors and lead to around 1-5% of full-scale sensing errors in practice.

Possible errors from the transfer of the measured current to the magnetic field

Closed-loop current sensors

Functional diagram of a closed-loop type Hall-effect current sensor, the error between the output of the Hall-sensor “V_hall” is fed to a regulator and then drives the current of the secondary coil “I_s” and forms a negative feedback mechanism

Alternatively, a closed-loop current sensor is an extension to the open-loop sensor with a secondary coil excited by a power amplifier (in the sense of a larger current rating). The polarity of the secondary coil is installed in a way that decreases the resulting magnetic field while the current “I_s” is increasing, which effectively forms a negative feedback loop with a regulator “K_reg” fed with the Hall-effect signal “V_hall” as depicted below.

Simplified block diagram showing the closed-loop controlled current of the secondary coil “I_s”

This construction provides a different perspective that converts an inversion problem into an iterative regulation problem by asking:

What is the “output” of the regulator that results in a zero error?

In this particular case (and in most sensor signal processing scenarios), the forward validating path was performed by the physics of the sensor configuration itself and the regulator should be designed such that the error asymptotically moves toward zero. Assuming an approximately linear sensor gain from the net current “I_tot” to the Hall voltage “V_hall” and denoted as “H_sense” and a proportional amplifier as the regulator “K_reg”, the transfer function from the primary current (to be sensed) to the secondary current can be derived as shown:

Derivation of the transfer function from the primary to the secondary current

The resulting transfer function is stable and indicates a near proportional between the primary and the secondary current with a large enough regulator gain “K_reg” within a certain bandwidth. The matched coefficient multiplied on the Laplace variable “s” also implies an identity at a higher frequency range. This is a result of the coupling nature between coils and extends the bandwidth even if the gain of the operational amplifier(s) is rolling off.

The most significant part of the closed-loop sensor is that the net magnetic field in the core is dynamically maintained at zero “I_tot, B_tot~0”, and avoids any gain error such as the temperature-dependent Hall-voltage, magnetic core saturation, nonlinearity, and ideally no hysteresis! As a result, most accurate current sensors (<0.1%) utilize closed-loop construction rather than an open-loop type.

Pros and cons with examples of off-shelf products

In order the quantified the difference between an open-loop and a closed-loop current sensor, two closely rated off-shelf sensors each representing one type of sensor both manufactured by LEM will be taken as a reference. If interested, you may also refer to this full spreadsheet.

Long story short, most closed-loop current sensors outperform and were not a surprise as discussed in the previous sections. Some of the few drawbacks might be a higher power consumption (to cancel out the primary current), higher unit price, and a larger physical size.

Further implication

The two types of sensor configuration each correspond to an approach to solving an inverse problem. Despite being potentially simpler, the open-loop sensor refers to a direct inversion that requires full knowledge (or modeling if you liked to) of the forward path and suffers from numerous types of errors. On the other hand, the closed-loop design suppresses the error due to the modeling of the forward path or simply does not require any, this is similar to for example Newton’s method and Backpropagation that regain inverse solution by iterating the forward process, cool.

References

[1] E. H. Hall, ‘On a New Action of the Magnet on Electric Currents’, American Journal of Mathematics, Vol. 2, №3, pp. 287–292, 1879.

[2] D. J. Griffiths, Introduction to Electrodynamics. Cambridge University Press, 2017.