Saturation of High-Frequency Current Transformers: Challenges and Solutions

When using high-frequency current transformers (HFCTs) to measure partial discharges (PDs) on power cables, core saturation caused by the 50-Hz operating current of the power cable is a major problem. Saturation leads to nonlinearity in the transfer function of the HFCT sensor and significantly reduces its sensitivity to the PD signals. To avoid magnetic saturation and ensure linear HFCT operation and thus maximum PD sensitivity, a split ferrite core can be used. However, finding the right air gap length between the two halves of the core is a complicated task, as the optimal length depends on several parameters and should be neither too small nor too large. This article deals with the air gap problem in detail and presents solutions for determining the optimal air gap length of HFCT cores. We present numerous measurements with exemplary split-core HFCTs from three different ferrite materials. Based on the obtained data, we then derive mathematical functions describing the optimal air gap length of different ferrite cores. Based on these optimal air gap functions, we then derive and validate an HFCT split-core model. Using the model, the optimal air gap function for any ferrite material can be calculated with a few simple equations, significantly speeding up the air gap design process. In this way, the model improves the computer-aided design of split-core HFCTs. With optimal air gap length, the HFCT does not saturate and its sensitivity for PD measurements is maximized. Our results show that for online monitoring of power cables, the optimal air gap length depends mainly on the amplitude of the 50-Hz current and thus changes with the load on the power cable. Therefore, the conventional HFCT design with constant air gap length is not practical. Instead, we want to develop an improved HFCT with active air gap control in the future.


I. INTRODUCTION
H IGH-FREQUENCY current transformers (HFCTs) are inductive sensors specially designed for measuring current signals with high-frequency (HF) content. They are therefore well-suited for measuring transient current pulses and are mainly used to detect partial discharges (PDs) on power cables, which is a common method for checking the condition of the dielectric insulation of these power cables. For this task, the HFCT sensors are installed at the ends of the power cables. If PD activity is detected, it indicates that the cable insulation is degenerating and cable failure may be imminent.
It is best to monitor the power cables online for PD pulses, i.e., during normal operation when the cable carries a high operating current at a frequency of 50 Hz (in some countries 60 Hz). During the PD measurement, the PD sensors are therefore exposed to strong magnetic fields caused by this operating current. Despite their high sensitivity, most HFCTs are less suitable for such online monitoring because the strong magnetic fields of the 50-Hz current drive them into magnetic saturation. Typical 50-Hz operating currents of power cables range from tens to hundreds of amperes at medium-or highvoltage (HV) levels. In comparison, the amplitude of the PD pulses superimposed on the operating current is usually in the mA-range and thus weak.
Saturation causes a nonlinear HFCT transfer function, so that the shape of the measured output voltage no longer matches the original input current. As a result, accurate PD measurements are no longer possible. Instead, the measured signal contains pulse-like voltage peaks due to the nonlinearity. These peaks falsify the measurement and mislead any PD detection algorithm. Therefore, magnetic saturation of the HFCT should be avoided as much as possible during PD measurements.
To avoid saturation during online PD monitoring and to ensure linear HFCT operation, air gaps are usually inserted into the ferrite core of the sensor. The longer the air gaps of such a split-core HFCT, the less prone it is to saturation [1]. This is common knowledge, but finding the optimal air gap length for a given HFCT design is a complicated task. It depends on both the material of the ferrite core (with nonlinear material properties) and the amplitude of the 50-Hz operating current of the power cable, which is not constant over time. If the air gaps are too short, the saturation is not reduced sufficiently; if the air gaps are too long, the sensitivity of the HFCT is unnecessarily reduced. There is no simple equation to calculate the optimal air gap length of a split-core HFCT and almost no high-quality literature on the subject (peer-reviewed journals). Only [1], [2], [3] are partly related to the topic and show some experimental data for HFCTs at different air gaps. However, all these measurements are only side results, and none of the publications focuses on air gap optimization. To avoid misunderstandings, it should be mentioned here that all the literature on HF voltage transformers for power electronics, so-called HFTs, is irrelevant to our topic. Our sensor is a current transformer (HFCT) for measuring PDs on power cables. Despite their similar name, the two devices have nothing in common. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Fig. 1. HFCT for measuring PDs on power cables. Secondary winding with three turns on a toroidal ferrite core made of Fair-Rite's No. 43 material. The best coupling between winding and core is achieved when the winding is evenly distributed over the core. This is especially important when measuring HF signals such as PDs.
This article tries to fill the identified research gap. Therefore, we perform measurements with three exemplary split-core HFCTs made of different ferrite materials to experimentally determine their optimal air gap length. The results are then fit to mathematical functions that depend on the 50-Hz operating current of the power cable. Based on the obtained data, we then develop an HFCT split-core model and validate it using the three measured optimal air gap functions. The model works well and can be used to easily calculate the optimal air gap length function for any HFCT core material, significantly speeding up the HFCT design process. In this way, the computer-aided HFCT design is simplified.
This article is an extension of a previously published conference paper of ours [4].
This article is structured as follows. In Section II, a comprehensive overview of the HFCT technology is provided. In Section III, the optimal air gap length of exemplary split-core HFCTs is measured experimentally. In Section IV, the analytical split-core model is derived and validated. Finally, a conclusion is drawn in Section V.

II. HFCT OVERVIEW
In this section, the operating principle of HFCT sensors is explained first in Section II-A. Second, an ideal HFCT model is presented in Section II-B. The HFCT prototype, which we use for the subsequent measurements, is presented in Section II-C. Finally, in Section II-D, the magnetic properties of ferrite materials are described.

A. Operating Principle
Inductive HFCT sensors are often used for PD measurements on power cables because they can be easily installed on existing cables. In addition, HFCTs are inexpensive, the coupling between the sensor and the device under test is galvanically isolated, and they are therefore suitable for online monitoring. An alternative inductive sensor is the coreless Rogowski coil, which is immune to saturation but much less sensitive in the HF range compared with HFCTs [5].
The construction of HFCTs is simple and is essentially based on a magnetic core and a secondary copper winding with n 2 turns wound around this core; see Fig. 1. The core is usually toroidal and made of a ferromagnetic material. The measurement bandwidth of an HFCT depends primarily on the selected core material. For good coupling in the HF range, ceramic ferrite materials are best. The operating principle is shown in Fig. 2. The toroid is placed around an electrical conductor to measure the current i 1 (t) flowing through it. This primary current flow causes a magnetic field H (t) around the conductor. Assuming that the primary conductor is straight and centered in the HFCT, the magnetic field H (t) can be calculated as follows [6]: where r c is the distance between the center of the primary conductor and the HFCT in m. For a toroidal HFCT core with outer radius r c,out and inner radius r c,in , the mean radius can be used, yielding The external magnetic field H (t) magnetizes the ferrite core. Provided that the core material has a constant permeability µ c , the resulting magnetic flux density B(t) can be calculated with the following equation: where µ 0 ≈ 1.25664 · 10 −6 (Vs/Am) is the vacuum permeability. The permeability of ferrites is high and approximately constant as long as the material is not saturated.
The magnetic flux density B(t) in the HFCT core induces a reverse current i 2 (t) in the secondary winding, which is proportional to the primary current i 1 (t) where n 1 = 1 is the number of turns of the primary winding, which usually consists of only one turn, and a is the turns ratio of the HFCT. Again, (4) is only valid at constant µ c , i.e., at unsaturated HFCT operation. The secondary current i 2 (t) then causes a measurable voltage u L (t) at the HFCT output, which is connected to an additional load resistor R L . The load resistance is usually 50 and is equal to the input resistance of the connected measuring device, e.g., the 50-input channel of an oscilloscope. The ratio between the measured HFCT output voltage u L and the input current i 1 of the primary conductor is the so-called transfer impedance Z T of the HFCT, which will be calculated in Section II-B.
To measure the currents flowing in a power cable, an HFCT can be installed at the end termination of a power cable, around either the inner or the outer conductor (shielding); see Fig. 3. Position 2 is the more optimal location because the PD signals are less disturbed. Installing the HFCT at position 1 is less suitable because the Earth path acts like a large antenna for noise and interference. Therefore, we prefer Fig. 3. For PD measurement, HFCTs are installed at the end of a power cable either around the cable shield (1), which is connected to earth/ground potential, or around the insulated part of the inner conductor (2), which is connected to HV potential. position 2, although the HFCT is then fully exposed to the power cable's operating current. For more insights into this discussion, see [7].

B. Ideal HFCT Model (Valid Only for Linear Operation)
The HFCT couples the primary current signal i 1 and transforms it to its secondary side u L . The quality of this transformation is defined by the frequency-dependent transfer impedance Z T ( f ) of the HFCT. For its calculation, all time-domain signals are first transformed into the frequency domain by Fourier transformation, e.g., u L (t) Fig. 4 shows the equivalent circuit of an ideal HFCT neglecting all the losses. The complex transfer impedance Z T ( f ) of the ideal HFCT (loss-free) can be determined by the impedance of the load resistor R L in parallel with the magnetizing inductance L m of the secondary winding, considering (4) [8] Z The coil inductance is calculated with [6] L where h c is the height or thickness of the HFCT ring core in m, and A c is its cross section in m 2 The transfer impedance corresponds to the sensitivity of the HFCT or its transfer function. It depends largely on the complex and frequency-dependent permeability µ c ( f ) of the selected core material [2], [9], [10], which is further discussed in Section II-D. For a given ferrite core, the complex permeability can usually be obtained from the manufacturer's website.
At low frequencies such as 50 Hz, the HFCT behaves like an inductance, i.e., jωL At high frequencies, the HFCT behaves more like a resistor, In this way, the load resistor limits the output voltage of the HFCT for high frequencies. If the HFCT were operated with a high impedance R L (open-circuit), the output voltage would otherwise theoretically rise to infinity (recall that this is an ideal model, i.e., parasitic effects are neglected; a more realistic model can be found in [8]). For a given low frequency such as 50 Hz and thus constant permeability µ c , (8) can also be given in the time domain [11] u where Φ is the magnetic flux through the area of one turn of n 2 . The area can be assumed to be the cross section of the core A c . With constant area A c over the entire toroid and assuming a homogeneous B field, the flux through one turn can be calculated as follows, and consequently (10) is valid: C. Initial HFCT Prototype for PD Detection In a previous publication, we showed that the bandwidth of most PD signals on power cables is <10 MHz [12]. A PD sensor should therefore have maximum sensitivity in this frequency band. Of course, some of the PDs on power cables reach higher bandwidths. However, measurement of the 0-10-MHz spectrum is also sufficient for their detection (neglecting higher frequency components only slightly reduces the measured PD amplitude, but detection is still possible).
Based on the bandwidth of 0-10 MHz, we optimized an HFCT [8], which can be seen in Fig. 1. Our optimized HFCT is built on a toroidal ferrite core with three windings on the secondary side. The core is made of a nickel-zinc (NiZn) ferrite from the manufacturer Fair-Rite (material No. 43) with a size of 63.5 × 102.6 × 15.9 mm (r c,in × r c,out × h c ). The output of the secondary winding can be connected to a measuring device via a BNC connector. Additional shielding of the sensor is not required, since uniformly wound toroidal coils are inherently immune to magnetic interference fields [13].
The measured transfer impedance Z T ( f ) of the optimized HFCT can be seen in Fig. 5. The frequency response is that of a high pass with a lower cutoff frequency of about 400 kHz. Up to 10 MHz, the transfer impedance is flat with a constant value of Z T ≈ 14 . Thus, when measuring HF signals, u L is proportional to i 1 (linear measurement), provided the HFCT core is free of saturation. A 50-Hz current generator and/or a PD generator are available to generate the input current for the HFCT. The HFCT output voltage u L is measured using an oscilloscope channel with an input impedance of R L = 50 .

D. Magnetic Properties of a Ferrite Core
The experimental setup for most of the measurements in this article can be seen in Fig. 6. An HFCT is installed around three conductors. Two of them are connected to a 50-Hz current generator (Omicron CMC-256-6), which can generate currents with amplitude up to I 1,50Hz = 75 A rms. The third conductor is connected to a PD calibrator (Haefely KAL 9511) which generates realistic PD pulses of 100 pC. These artificially generated PD pulses have an amplitude of u cal ≈ 0.64 V and a pulsewidth of t FWHM ≈ 8 ns (full width at half maximum). In the frequency domain, they have a flat frequency response up to about f −6 dB ≈ 63 MHz (pulse bandwidth). The resulting current pulse generated by the PD calibrator i cal can be seen in Fig. 12. The currents of both the current sources are simultaneously coupled by the HFCT, and the output voltage u L is measured with a wideband oscilloscope (Tektronix MDO 4024 C, R L = 50 ).
The transfer function of an HFCT depends largely on the permeability µ c of its core material. Permeability describes the relationship between the magnetic flux density B in a material in response to an external magnetizing field H . The plot of B versus H is called the magnetization curve of a magnetic material.
To measure the magnetization curve of our HFCT core, only the 50-Hz current source of Fig. 6 is turned on. The input current and output voltage of the HFCT are then recorded. Afterward, the H field is calculated using (1), and the B field is calculated using (10) The resulting B-H curve of the core material of our HFCT is shown in Fig. 7. The permeability corresponds to the slope of the magnetization curve. The figure shows the results for input currents with rms values I 1,50Hz of 2, 5, 20, and 50 A. It can be seen that the magnetization curve is linear at I 1,50Hz = 2 A with a permeability of µ c,lin ≈ 1000. From about I 1,50Hz = 2.5 A, the curve becomes increasingly nonlinear due to hysteresis. At currents I 1,50Hz > 15 A or H > H Sat , a second stage of nonlinearity due to saturation begins. At this point, the core material is saturated, and further increase in the Normally, the HFCT should be operated only in the linear range, i.e., at low input currents i 1 (compare I 1,50Hz = 2 A in Fig. 7). When measuring PDs or other HF signals offline, this is almost always fulfilled because their signal amplitudes are weak (mA range). At linear HFCT operation, the core permeability is equal to µ c,lin and constant. The model of (5) and (6) is only applicable for such linear operation.

III. HFCT OPERATION ON POWER CABLES
HFCTs should be operated in the linear operation mode only, but this is often not possible when monitoring power cables online. Thus, this section deals with nonlinear HFCT operation and how to avoid it. Therefore, in Section III-A, we show what happens when an HFCT is used in the nonlinear operation mode. Section III-B then presents an improved split-core HFCT. Subsequently, a method for quantifying the level of nonlinearity is introduced in Section III-C. Finally, Section III-D provides all the measurements used to experimentally determine the optimal air gap length of the split-core HFCT.

A. Nonlinear Operation
HFCTs are installed at power cables to detect PDs. However, the current in power cables is dominated by the 50-Hz operating current. These currents and their magnetic fields are usually strong enough to drive the HFCT core into saturation (typical operating currents for medium-voltage cables are in the range of several tens to hundreds of amperes). At saturation, the HFCT is operated at a nonlinear magnetization curve (compare I 1,50Hz = 50 A in Fig. 7), and thus the core permeability can no longer be assumed to be constant all the time, instead its nonlinearity must be considered. For an exemplary 50-Hz operating current, this process is shown in detail in Fig. 8. Since the amplitude of the sinusoidal input field H is too high for linear HFCT operation, the nonlinear part of the magnetization curve is also processed. As a result, the magnetic flux density B in the ferrite core is nonsinusoidal. The current induced in the secondary winding of the HFCT is proportional to the derivative of B with respect to timeḂ. Accordingly, the HFCT output signal contains voltage peaks (harmonic distortion [14]). Thus, due to nonlinearity, the shape of the measured HFCT output voltage u L and the original input current i 1 no longer matches (nonlinear measurement).
We performed further nonlinear measurements with our optimized HFCT using the experimental setup in Fig. 6. Therefore, the HFCT output voltage u L is measured at increasing input currents i 1 (50-Hz sine). Fig. 9 shows the results for input currents with rms values I 1,50Hz of 2, 10, 50, and 300 A.
At I 1,50Hz = 2 A, the measured output voltage is sinusoidal, which means that the input current is measured correctly (linear operation). As the current increases, the measured output voltage becomes more and more nonsinusoidal, i.e., its harmonic distortion increases. The starting point for nonlinear measurements depends on the selected core material. For ferrite material No. 43, the nonlinear operation starts at about I 1,50Hz > 2.5 A.

B. Advanced HFCT Prototype With Split-Core
The measurements of Fig. 9 prove that the developed HFCT is not yet suitable for online monitoring of power cables. Linear operation must be ensured for all the operating currents of a power cable, not only for I 1,50Hz < 2.5 A. The most common method to increase the saturation capability of an HFCT is to insert air gaps into the ferrite core [1]. Thus, to improve our HFCT prototype, its core is split into half to create two air gaps. The modified split-core HFCT can be seen in Fig. 10. The length of each air gap d air can be changed as desired by inserting plastic pieces of different thicknesses (plastic has about the same permeability as air).
Inserting an air gap changes the magnetization curve of the HFCT ferrite core, as shown in Fig. 11. The figure shows B-H measurements of our split-core HFCT at increasing air gap length. The input current i 1 for the measurements is again a 50-Hz sine. It can be seen that the magnetization curves are stretched as the air gap length d air increases. Accordingly, saturation sets in at higher amplitudes of i 1 , i.e., the magnetic flux density B Sat at which saturation starts now and requires a higher input field H Sat . This means that the HFCT can be operated at higher 50-Hz currents without core saturation. It can also be seen that the longer the air gap, the more linear the magnetization curve (hysteresis becomes negligible). This ensures linear operation of the HFCT even at high 50-Hz input currents. Our split-core HFCT is therefore suitable for online monitoring of PDs on power cables carrying a high 50-Hz operating current.
On the other hand, it can be seen that the slope of the linear part of the magnetization curve, and thus the permeability µ c,lin , decreases due to the air gap. Thus, increasing the air gap length leads to less core saturation, but at the expense of PD sensitivity [2]. This is also evident from the PD measurements shown in Fig. 12. It can be seen that the amplitude of the HFCT output voltage u L decreases as the air gap length d air increases, while the original PD pulse i cal remains constant. The air gap should therefore be as long as necessary to ensure linear HFCT operation, but as short as possible. Fig. 9 indicates that calculating the total harmonic distortion (THD) of the measured HFCT output voltage u L is a good measure for determining the level of nonlinearity of its core. If the output voltage is sinusoidal, the THD approaches 0, which means the HFCT core is in the linear operation mode. To calculate the THD, the Fourier transform of the measured voltage must first be calculated, i.e., u L (t) F − → U L ( f ). Then the THD value can be calculated as follows [15]:

C. Calculate the Level of Nonlinearity
where U L,i is the ith harmonic (150 Hz, 250 Hz, . . .), and U L,1 is the fundamental component of the HFCT output voltage spectrum (50 Hz). The THD calculation is performed only up to the 25th harmonic ( f 25 = 2450 Hz) to suppress the influence of the omnipresent measurement noise on the result. Such HF noise can be seen, for example, in Fig. 9 at 2 A on the left. For a given split-core HFCT, the THD value depends on both the amplitude of the 50-Hz input current and the air gap length, THD = f (I 1,50Hz , d air ).
Applying (13) to the measured output voltages of Fig. 9, the following THD values are obtained: THD(2 A, 0 mm) = 0.0077 THD(10 A, 0 mm) = 0.3129 THD(50 A, 0 mm) = 2.3985 THD(300 A, 0 mm) = 11.0434. From the results, it can be concluded that a THD value less than 0.01 or 1% means that there is no core saturation. Recall that the measurements were recorded with an HFCT of ferrite material No. 43, which enters the nonlinear operation mode at input currents of about I 1,50Hz > 2.5 A. Ideally, the THD value would drop to 0, but this is unrealistic due to possible harmonics in the 50-Hz current of the power cable and the omnipresent measurement noise. According to our measurements, 1% is a good compromise for clearly detecting saturation while ensuring sufficient robustness against noise. Therefore, in this article, linear HFCT operation is defined by Fig. 9. Measurements with our initial HFCT prototype at increasing 50-Hz input currents i 1 (rms values I 1,50Hz of 2, 10, 50, and 300 A-from left to right). Nonlinearity can be seen in the measured HFCT output voltage u L . As the current increases, the degree of nonlinearity increases and the voltage peaks intensify. Fig. 10. Improved HFCT prototype. This is the same sensor as in Fig. 1, but with a split-core. The length of the air gaps can be adjusted as desired by inserting plastic pieces. The two halves of the core are held together with a rubber band.  THD ≤ 0.01, and values above 1% indicate nonlinear HFCT operation.

D. Finding the Optimal Air Gap Length
The effect of different air gap lengths on the HFCT output voltage u L can be seen in Fig. 13. For this figure, the 50-Hz input current is set to I 1,50Hz = 100 A. Then, the air gap Fig. 13. Increase in the air gap length d air at constant 50-Hz input current of I 1,50Hz = 100 A. As the air gap increases, the measured HFCT output voltage becomes more sinusoidal. At d air = 0.5 mm, the measurement is linear, i.e., u L is a sine. length of the split-core is increased. It can be clearly seen that the level of nonlinearity decreases as the air gap length d air increases, which can be seen from the fact that the measured HFCT output voltage u L becomes more and more sinusoidal. This can be confirmed by calculating the THD value of the measurements according to (13) THD(100 A, 0.1 mm) = 0.553 THD(100 A, 0.2 mm) = 0.162 THD(100 A, 0.5 mm) = 0.01. At a 50-Hz current of I 1,50Hz = 100 A, an air gap length of d air = 0.5 mm (on both sides) is required to ensure linear HFCT operation (based on the 1% criterion), i.e., until the output voltage is sinusoidal. In other words, at an operating current of 100 A, the optimal air gap is 0.5 mm on both the sides.
In the same way, the optimal air gap length d air,opt can be determined at various other 50-Hz currents. For this purpose, the input current of the split-core HFCT is varied between 2 and 600 A rms and the air gap length between 0 and 3 mm on each side. The THD of the HFCT output voltage u L is then calculated for all the measurements according to (13). The results are shown in Fig. 14.
It can be seen that the higher the 50-Hz input current, the higher the THD of the HFCT output voltage u L . It can also be seen that the air gap has a positive influence on the THD value. The longer the air gap, the later the HFCT enters the nonlinear operating mode (based on the 1% criterion). For example, an air gap of d air = 3 mm ensures linear operation up to a 50-Hz current of about 350 A, while with an air gap of 1 mm nonlinear operation already starts at about 170 A. Thus, as expected, with increasing air gap length, the HFCT can withstand higher 50-Hz currents without leaving the linear mode of operation.
The 1%-line marks the border between the two modes, i.e., the minimum air gap length to ensure linear HFCT operation.  . PD measurements with our split-core HFCT at various air gap lengths and 50-Hz currents. The amplitude of the measured HFCT output voltage u L (t) has been normalized and plotted on the z-axis. The blue line is the optimal air gap length function of the HFCT according to (14).
Since the air gap should only be as long as necessary, the minimum air gap length is equal to the optimal air gap length d air,opt . The optimal length depends on the 50-Hz input current and on the core material of the HFCT. For material No. 43 of our HFCT split-core, the 1%-line can be fit to the following mathematical function: d air,opt (I 1,50Hz ) = e I 1,50Hz 250 If the 50-Hz operating current of the power cable on which the HFCT is installed is known, this function can be used to optimally set its air gap. However, the operating current of a power cable is usually not constant, but varies between 0 and a maximum permissible value I 1,50Hz,max . Accordingly, d air,opt is not constant, but varies with the load of the power cable. Setting the air gap based on the maximum permissible current of the power cable is also not a good idea, as the air gap would be unnecessarily large at any lower current (sensitivity loss).
To further investigate the influence of the air gap on HFCT sensitivity, we performed PD measurements with the experimental setup of Fig. 6. This time, the split-core HFCT measures 100-pC PD pulses, while the 50-Hz input current is varied between 2 and 75 A rms and the air gap length between 0 and 3 mm on each side. The measurement results are shown in Fig. 15. The amplitude of the measured PD pulsê u L (t) is plotted on the z-axis, normalized to the measured PD amplitude at 0 mm and 0 A (offline measurement value; cf. Fig. 12). The optimum air gap length function of the split-core HFCT from (14) is plotted along with the measurements (blue line).
It can be seen that the maximum HFCT sensitivity is always reached near the blue line. If the air gap is shorter than the optimum, the sensitivity of the HFCT decreases sharply as it is operated under saturation. If the air gap is longer than the optimum, the sensitivity also decreases, but more slowly (unnecessarily large air gap). The air gap length should therefore always be close to the optimal length for maximum PD sensitivity.
IV. ANALYTICAL SPLIT-CORE HFCT MODEL FOR CALCULATING THE OPTIMAL AIR GAP LENGTH The experimental determination of the optimal air gap length of a given HFCT, as demonstrated in Fig. 14, is very time-consuming (more than 200 individual measurements are required). To speed up this process, in this section we develop an analytical split-core HFCT model to simulate the optimal air gap length function of HFCTs with cores of any ferrite material. The model is derived in the first Section IV-A and validated in the following Section IV-B.

A. Derivation of the Split-Core HFCT Model
In Section II-D, we showed that the magnetization curve of an HFCT becomes nonlinear due to hysteresis and saturation. For our split-core HFCT prototype made of ferrite material No. 43, hysteresis starts at 50-Hz currents of I 1,50Hz ≥ 2.5 A and saturation at I 1,50Hz ≥ 15 A, as shown in Fig. 7 (measured at an air gap of 0 mm). As the air gap increases, the magnetization curve becomes more linear and saturation becomes the main source of nonlinearity (see Fig. 11). For the following model, we therefore assume that the influence of hysteresis as a source of nonlinearity is negligible.
Under this assumption, the saturation point is equal to the point where nonlinear operation starts. For input fields between 0 and H Sat , linear HFCT operation with constant permeability can be assumed accordingly. This linear model permeability µ c,mod is determined only by the saturation point of the core material. Using the measurements from Fig. 7 and (3), the following value is calculated for the ferrite material No. 43 of our HFCT: The magnetization curve used to determine µ c,mod should be measured at a frequency of 50 Hz, since saturation can only be expected at this frequency. The further HFCT model is based on the ideas of the magnetic circuit theory (applicable when the core permeability is constant, as assumed for µ c,mod ). According to this theory, each material has a certain reluctance R M that determines the magnetic flux Φ in that material. If there are different materials in a magnetic circuit, their reluctance can be combined. The total reluctance then determines the magnetic flux in the circuit.
A split-core HFCT forms a magnetic circuit consisting of two different materials in series, the split ferrite core, and the two air gaps (the influence of the copper of the secondary winding on the magnetic circuit is negligible). The reluctance of the ferrite core can be calculated as follows [1]: where µ c,mod is the linear model permeability of (15). A c is the cross section of the toroid, and l c is its magnetic path length. This length corresponds to the mean core length l c = (r c,Out + r c,In ) · π.
(17) The reluctance of the air gaps is calculated in a similar way [1] where 2 d air accounts for the total air gap length of both the air gaps of the split-core. A air is the cross section of the air gaps, i.e., the area where the magnetic flux is located in the air gap. When calculating the air gap reluctance, the fringing flux phenomenon must be considered. Fringing means that the magnetic flux spreads not only across the cross section of the core but also in the adjacent volume outside the core. The effective cross section of the air gap A air is therefore not equal to the core cross section A c but increases due to the fringing flux. Accordingly, the reluctance of the air gap decreases compared with the fringing-flux-free state, i.e., the fringing flux "shortens" the air gap. To account for this phenomenon, a fringing flux factor can be calculated [1] This factor depends only on the length of the air gap. Without air gap, k FF is equal to 1 and increases with the air gap length. The total reluctance of the HFCT is the sum of the split-core and air gap reluctance R M,tot (d air ) = R M,c + R M,air (d air ).
(20) The ferrite material has a high permeability and therefore a low reluctance. In comparison, the air gap reluctance is very high even for small air gap lengths, R M,air ≫ R M,c . Thus, controlling the air gap is equivalent to controlling the overall HFCT reluctance.
Using the following approach, an effective permeability µ c,eff can be defined for the magnetic circuit of the split-core HFCT: Solving (21) for the effective permeability .
The effective permeability describes the field enhancement in the HFCT core according to (3) at different air gap lengths. The larger the air gap, the later B Sat is reached, compare Fig. 11. Since the magnetic flux density for core saturation B Sat is constant and known for a given material, the corresponding maximum input field H Sat at which linear operation is still guaranteed can now be calculated using the effective permeability .
(23) Fig. 16. Validation of the split-core HFCT model using the optimal air gap length functions of three split-core HFCTs. The comparison of measurement and simulation shows only minor deviations, especially for the blue and violet materials.
With (1), this gives a relationship between I 1,50Hz and d air The result corresponds to the maximum input current at which linear HFCT operation is ensured as a function of the air gap length. The inverse function d air (I 1,50Hz ) is equal to the optimal air gap length function of the HFCT.

B. Model Validation
To validate the split-core HFCT model, the optimal air gap length function of three different HFCTs is calculated according to (24) and compared with measurements. The design of all the three HFCTs is the same as in Fig. 10, but with cores made of different ferrite materials. The shape and size of all the three ferrite cores is identical. The first HFCT is that of ferrite material No. 43, whose optimal air gap length function is given in (14). The second HFCT consists of a ferrite material suitable for the HF range, but of unknown origin. Its measured optimal air gap length function is given in (25). The core of the third HFCT is made of material No. 78 from the manufacturer Fair-Rite, which is a manganese-zinc (MnZn) ferrite and is more suitable for lower frequencies < 1.5 MHz. Its measured optimal air gap length function is given in (26). All the data of the additional measurements can be found in the Appendix. The validation plot of Fig. 16 compares the measurements with the simulation results from our split-core HFCT model.
Looking at the blue curve shows that the split-core model is good at predicting the optimal air gap length of the HFCT made of material No. 43. The deviation between measurement and simulation is small. For the unknown material, the deviation between measurement and model is also negligible. The deviation is greater for the HFCT made of material No. 78. Here, the saturation capability of the ferrite material is overestimated by the model by about 10%, i.e., the simulated air gap length is too short. An air gap that is too short is unacceptable.
Thus, it can be concluded that the split-core model works accurately for ferrites used in the HF range of 3-30 MHz (mostly NiZn ferrites). If the recommended frequency range of the ferrite is lower than this (mostly MnZn ferrites), the model becomes less accurate and should not be used. However, since most HFCTs are built on NiZn ferrites to achieve measurement bandwidths in the HF range, the split-core model is a useful tool for the computer-aided HFCT design.
To use the split-core model with HFCTs made of other ferrite materials, only their magnetization curve at d air = 0 mm Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
needs to be known. Therefore, the measurements from Fig. 7 need to be made to determine B Sat and H Sat of the ferrite core. Then, µ c,mod can be calculated according to (15). After this, the optimal air gap length function can be easily calculated. We programmed our split-core model in MATLAB-the code is attached to this article.
It should be noted that the simulation result depends on the graphical construction of B Sat and H Sat . This step should therefore be carried out carefully.
When the split-core HFCT is operated with an optimal air gap length, the model of Section II-B provides a good approximation of its transfer function Z T ( f ).

V. CONCLUSION
This article deals with PD measurements on power cables. We have investigated how best to avoid saturation of HFCT sensors when monitoring power cables online. Our approach is based on air gaps in the HFCT core and we have analyzed how long the air gaps need to be. We show how to determine the optimal air gap length that maximizes the sensitivity of the HFCT to PD pulses while avoiding magnetic saturation. Based on the results, split-core HFCTs optimized for online power cable monitoring can be built.
In summary, we have shown two ways to determine the optimal air gap length of split-core HFCTs, once experimentally and once by simulation. For the experiments, we have used exemplary split-core HFCTs made of three different ferrite materials. The measurement procedure has been explained in detail, and the evaluation shows that the optimal air gap length of an HFCT depends on the amplitude of the 50-Hz operating current of the power cable. Afterward, a simulation model for split-core HFCTs has been derived step by step and validated against the measurements. The model can be used to calculate the optimal air gap length, which is much faster than measuring it. The model can be used for HFCTs made of different ferrite core materials. The simulation and measurement results agree with sufficient accuracy. The simulation results are most accurate with ferrites, which are suitable for the HF range of 3-30 MHz. Thus, the model is well-suited for most HFCT core materials and allows for easy and computer-aided development of split-core HFCTs.
Setting the optimal air gap ensures that the core material is always free from saturation and the HFCT achieves the highest possible sensitivity for measuring PDs. Successful online monitoring of power cables is only possible in this way. Since the current in the power cable changes constantly depending on the load, the optimal air gap length also changes over time. Therefore, we plan to develop a split-core HFCT with active air gap control in the future.
If the HFCT design has a constant air gap, the air gap length should be chosen too large rather than too small. Avoiding saturation should be the top priority.
To distinguish between linear and nonlinear HFCT operation and thus determine the optimal air gap length function, a THD threshold of 1% has been used in this article, but in other environments, harmonics and noise may result in a higher background noise level. Then the THD threshold should be adjusted accordingly. In general, a dynamic threshold that adapts to the actual background noise level would be best and should be researched in the future.

MEASUREMENTS OF THE OTHER TWO SPLIT-CORE HFCTS
To validate the split-core model, we repeated the measurements to experimentally determine the optimal air gap length function for the two additional HFCTs.
The core of the first HFCT is made of an unknown ferrite material suitable for the HF range. The magnetization curve of this material is shown in Fig. 17 for an air gap of 0 mm. The function of the optimal air gap length function is obtained from the measurements in Fig. 18 and is as follows: (25) The core of the second HFCT is made of the ferrite material No. 78 from the manufacturer Fair-Rite. The recommended frequency range for this material is <1.5 MHz. The magnetization curve of this material is shown in Fig. 19 for an air gap of 0 mm. The function of the optimal air gap length function is obtained from the measurements in Fig. 20 and is as follows: d air,opt (I 1,50Hz ) = e I 1,50Hz 390 − 1.