1 Introduction

The cables of cable-stayed suspension bridges are normally made of galvanized steel wires. Figure 1 shows the suspension bridge and the steel stable. The steel cable is normally composed of hundreds of galvanized steel wires with a diameter of about 7 mm. The diameter of the cable is over 100 mm and is covered by high-density polyethylene with a thickness of about 10 mm. The galvanized layer of the steel wire is used to protect the steel from corrosion with a sacrificial zinc alloy coating. When the zinc alloy coating is corroded, white rust is produced (Zn(OH)2, ZnO). After the zinc alloy coating layer is damaged, the steel is corroded and red rust (Fe2O3∙H2O), yellow rust (FeO(OH) ∙H2O), brown rust (Fe2O3) or black rust (Fe3O4) was produced.

Fig. 1
figure 1

a Cable-stayed suspension bridge. b Corrosion section of the steel cable made of galvanized steel wire

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Some researchers adopted the accelerated corrosion test method to study the environmental factors that influenced the corrosion speed of galvanized steel wires [1,2,3]. The water and the high temperature inside the cable can increase the speed of the corrosion of galvanized steel wires. It was found that the side wires and the bottom wires in the cable were normally significantly corroded due to the continuous wet environment, but the upper and center wires were rarely corroded due to the relatively dry environment. It was predicted that the zinc layer would be consumed within 10 years when wires were kept in wet or soaked environments, 34 years with a relative humidity of 100%, and 211 years with a relative humidity below 60%. After serving a long time, it is necessary to evaluate the corrosion of the cables of cable-stayed suspension bridges.

For the galvanized steel wire, the zinc layer, the steel, and their corrosion products have different electrical conductivity and permeability. They are shown in Table 1. Due to the different electromagnetic responses of zinc, steel, and their corrosion products, it is possible to judge the corrosion level through the electromagnetic method.

Table 1 Permeabilities and conductivities of zinc, steel, and their corrosion products
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Microwave radar systems [4], and X-ray methods [5] have been used to evaluate the corrosion of the steel. For microwave radar systems, environmental noise may have a big influence on detection accuracy. For the X-ray method, the transmitter and the receiver are often on the opposite side, and the equipment is heavy and expensive, so it is not convenient to do field experiments using an X-ray system.

The magnetic flux leakage (MFL) method was an economical and effective method for the broken wire detection of steel cables [5,6,7,8,9,10,11]. However, the system is normally quite heavy and the detection speed is low, and the MFL method is not so effective in evaluating the corrosion of steel wires.

Eddy current method had been developed to evaluate the corrosion of steel sheets [12,13,14], or the broken wires in steel cables [15, 16]. The early-stage corrosion of steel wire produces a very small signal, which is difficult to be evaluated by a normal eddy current system.

In the paper, we improved the eddy current testing probe and developed a compact eddy current system to evaluate the corrosion of the galvanized steel wires with different corrosion levels.

2 Principle and method

Figure 2 shows an equivalent model of the eddy current testing. AC magnetic field is produced by the excitation coil when an AC current flows in it, then eddy current is induced in the conductive material and the detection coil is used to detect the eddy current.

Fig. 2
figure 2

Equivalent model of eddy current testing

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The AC current ia flowing in the excitation coil with the amplitude of Ia and the frequency of f or ω = 2πf can be expressed as:

$${\mathrm{i}}_{\mathrm{a}}={\mathrm{I}}_{\mathrm{a}}\mathrm{cos}\left(\mathrm{\omega t}\right)=\frac{{\mathrm{I}}_{\mathrm{a}}}{2}\left({\mathrm{e}}^{\mathrm{j\omega t}}+{\mathrm{e}}^{-\mathrm{j\omega t}}\right)$$
(1)

LS and RS are the equivalent inductance and equivalent resistance for the eddy current induced in the conductive material. ϕ is the flux through LS, which is produced by the excitation coil L. If the mutual inductance between L and LS is M, the induced voltage VS and the eddy current iS flowing in the conductive sample can be expressed as:

$${\mathrm{\varnothing }=\mathrm{Mi}}_{\mathrm{a}}={\mathrm{MI}}_{\mathrm{a}}\mathrm{cos}\left(\mathrm{\omega t}\right)=\frac{{\mathrm{MI}}_{\mathrm{a}}}{2}\left({\mathrm{e}}^{\mathrm{j\omega t}}+{\mathrm{e}}^{-\mathrm{j\omega t}}\right)$$
(2)
$${\mathrm{V}}_{\mathrm{S}}=-\frac{\mathrm{d\varnothing }}{\mathrm{dt}}=-\frac{{\mathrm{MI}}_{\mathrm{a}}\mathrm{j\omega }}{2}\left({\mathrm{e}}^{\mathrm{j\omega t}}-{\mathrm{e}}^{-\mathrm{j\omega t}}\right)$$
(3)
$${\mathrm{i}}_{\mathrm{S}}=\frac{{\mathrm{V}}_{\mathrm{S}}}{\mathrm{j\omega }{\mathrm{L}}_{\mathrm{S}}+{\mathrm{R}}_{\mathrm{S}}}=-\frac{{\mathrm{MI}}_{\mathrm{a}}\mathrm{j\omega }}{2\left(\mathrm{j\omega }{\mathrm{L}}_{\mathrm{S}}+{\mathrm{R}}_{\mathrm{S}}\right)}\left({\mathrm{e}}^{\mathrm{j\omega t}}-{\mathrm{e}}^{-\mathrm{j\omega t}}\right)$$
(4)

The magnetic flux through the pickup coil includes two parts: the background magnetic flux produced by the excitation coil k1ia and the magnetic flux produced by the eddy current in the sample k2iS. k1 is the coupling coefficient between the excitation coil and the detection coil; k2 is the coupling coefficient between the equivalent inductance of the eddy current and the detection coil.

$${\mathrm{\varnothing }}_{\mathrm{p}}={\mathrm{k}}_{1}{\mathrm{i}}_{\mathrm{a}}+{\mathrm{k}}_{2}{\mathrm{i}}_{\mathrm{S}}={\mathrm{k}}_{1}\frac{{\mathrm{I}}_{\mathrm{a}}}{2}\left({\mathrm{e}}^{\mathrm{j\omega t}}+{\mathrm{e}}^{-\mathrm{j\omega t}}\right)+\frac{-{\mathrm{k}}_{2}{\mathrm{MI}}_{\mathrm{a}}\mathrm{j\omega }}{2\left(\mathrm{j\omega }{\mathrm{L}}_{\mathrm{S}}+{\mathrm{R}}_{\mathrm{S}}\right)}\left({\mathrm{e}}^{\mathrm{j\omega t}}-{\mathrm{e}}^{-\mathrm{j\omega t}}\right)$$
(5)
$${\mathrm{\varnothing }}_{\mathrm{p}}=\left({\mathrm{k}}_{1}+\frac{{\mathrm{k}}_{2}\mathrm{M}{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}\right){\mathrm{I}}_{\mathrm{a}}\mathrm{cos}\left(\mathrm{\omega t}\right)+\frac{{\mathrm{k}}_{2}\mathrm{M\omega }{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}{\mathrm{I}}_{\mathrm{a}}\mathrm{sin}\left(\mathrm{\omega t}\right)$$
(6)

The induced voltage of the pickup coil can be expressed:

$$\mathrm{V}=-\frac{\mathrm{d}{\mathrm{\varnothing }}_{\mathrm{P}}}{\mathrm{dt}}=\left({\mathrm{k}}_{1}+\frac{{\mathrm{k}}_{2}\mathrm{M}{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}\right){\mathrm{I}}_{\mathrm{a}}\mathrm{\omega sin}\left(\mathrm{\omega t}\right)-\frac{{\mathrm{k}}_{2}\mathrm{M\omega }{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}{\mathrm{I}}_{\mathrm{a}}\mathrm{\omega cos}\left(\mathrm{\omega t}\right)$$
(7)

Using a lock-in amplifier, we can get the X signal: the same phase signal with the excitation current, and the Y signal: 90-degree phase difference signal with the excitation current.

$$\mathrm{X}=-\frac{{\mathrm{k}}_{2}\mathrm{M\omega }{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}\frac{{\mathrm{I}}_{\mathrm{a}}\upomega }{2}$$
(8)
$$\mathrm{Y}=\left({\mathrm{k}}_{1}+\frac{{\mathrm{k}}_{2}\mathrm{M}{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}\right)\frac{{\mathrm{I}}_{\mathrm{a}}\upomega }{2}$$
(9)

Using data procession method, it is easy to remove the background signal k1iaω/2. Then, we have:

$${Y}^{\prime}=\frac{{\mathrm{k}}_{2}\mathrm{M}{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{S}}^{2}+{\upomega }^{2}{\mathrm{L}}_{\mathrm{S}}^{2}}\frac{{\mathrm{I}}_{\mathrm{a}}\upomega }{2}$$
(10)
$$\frac{{Y}^{\prime}}{X}=\frac{\omega {L}_{S}}{{R}_{S}}$$
(11)

LS and RS are mainly determined by the electrical conductivity and the permeability of the material. If corrosion happens with the steel wire, the conductivity and the permeability change, which causes the change of the slope of the X–Y graph plotted using the X, Y output signals of the eddy current testing. This method was once used to evaluate the corrosion of the steel rebar in concrete [17], With the similar principle, we improved the probe and developed a compact hand-held system to evaluate the corrosion of galvanized steel wires. Figure 3 shows the block diagram of the system. In our experiments, the excitation coil was made of ϕ0.3 mm copper wire, and the diameter of the excitation coil was 3 cm with 100 turns; the detection coil was made of ϕ0.1 mm copper wire, and the diameter of the detection coil was 1 cm with 60 turns. A compensation coil was put above the detection coil. The compensation coil was 30 turns with a diameter of 1 cm. By adjusting the resistor R2, the influence of the background signal produced by the excitation was reduced.

Fig. 3
figure 3

Principle of the electromagnetic method to evaluate the corrosion of the cable made of galvanized steel wires

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The amplifier was made using an operation amplifier AD797. Figure 4(a) shows it. The gain of the amplifier was about 30 times. The signal generator CG402R2 produced an 80 kHz sinusoidal signal, which was sent to the excitation coil, the compensation coil, and the reference port of the lock-in amplifier. Figure 4(b) shows it. Figure 4(c) shows the circuit of the 90-degree phase shifter. The output of the phase shifter was sent to the reference port of one of the lock-in amplifiers. Figure 4(d) shows the lock-in amplifier composed by the analog multiplier AD633. Two AD633 were used, one was used to get the X signal and another was used to get the Y signal of the eddy current testing. After the output of the multipliers, low pass filters with the bandwidth of about 100 Hz were used.

Fig. 4
figure 4

a Amplifier. b Sinusoidal wave signal generator. c 90-degree Phase shifter. d Lock-in amplifier

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For different corrosion levels of the galvanized steel wires, different corrosion products with different electrical conductivities and permeabilities were produced. In the beginning, the Zinc coating was corroded and the white rust of ZnO was produced; then steel was corroded and the brown rust of Fe(OH)3 was produced. The large area of the black rust of Fe(OH)3 normally happened for severe and longtime corrosion. For different corrosion levels, the surface conductivity and permeability of the steel wire were different. Using the X, Y signals from the lock-in amplifiers, we plotted the X–Y graph of the signal. From the slope of the X–Y graph, it was possible to evaluate the corrosion level of steel wires.

We developed a compact eddy current testing system. Figure 5 shows it. The excitation coil, the detection coil, the amplifier, the lock-in amplifier, and the AD converter all were put in a small box of about 85 mm × 170 mm × 60 mm. Only one USB cable was used to connect with a computer for the data transfer and the power supplied to the system. The electric power required by the system was less than 1 W.

Fig. 5
figure 5

Compact system for the corrosion evaluation of galvanized steel wires

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Labview program was developed for data acquisition and data procession. Figure 6 shows the interface of the Labview program. When there was no sample, the output signal could be reset to zero to remove the background signal, and there was phase adjustment in the program for the calibration to make sure the same results were obtained with different systems.

Fig. 6
figure 6

Interface of the Labview program for the data acquisition and procession of the system

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3 The specimen of galvanized steel wires with different corrosion levels

For the galvanized steel wires, the corrosion products were different at different corrosion levels. From the color and the corroded area of the sample, corrosion levels were classified into 4 stages: no corrosion, slight corrosion, middle corrosion and severe corrosion. The samples of the steel wires were cut from the cable of a real suspension brides in service for over 30 years. Normally, the wires of the outer layer were corroded severely, and the wires of the inner layer were corroded slightly or not corroded. Figure 7 shows the samples of galvanized steel wires we used in our experiments. The diameter of the galvanized steel wire was 7 mm and the thickness of the zinc coating was about 50 μm. For the sample (a), the galvanized steel wire was not corroded. For sample (b), the area of brown or black rust was smaller than the area of white rust. Only the Zn coating was corroded and the corrosion product of ZnO was mainly produced. There was no corrosion with the steel. For the sample (c), the area of the brown or black rust was bigger than the area of the white rust. Some Zn coating was left and some parts of the steel were corroded and the corrosion products of ZnO, Fe(OH)3 were produced. For the ample (d), the zinc coating was completely destroyed and there was only brown or black rust. Fe(OH)3.

Fig. 7
figure 7

Samples of galvanized steel wires with different corrosion levels

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4 Experiments and results

The galvanized steel wires were covered by a 1 cm thick acrylic plate. The acrylic plate was non-conductive material, so it had no influence to the signal of the eddy current testing. The probe was put above the samples (> 10 cm) and the signal was reset to zero to remove the background signal using the labview program. Then, the probe was moved toward the surfaces of the samples and the labview program plotted an X–Y graph using the X and Y signal of the eddy current testing system. Figure 8 shows the method of the measurement.

Fig. 8
figure 8

Method of measurement. The probe was moved from above toward the surface of the samples

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The penetration depth of eddy current can be estimated by δ = (πfμσ)−1/2, where, δ was the penetration depth, f was the excitation frequency, μ was the permeability of the material, and σ was the electrical conductivity of the material. For lower excitation frequency, the penetration was bigger. For example, the eddy current penetration of steel was about 0.2 mm at 10 kHz. The inside steel had a big contribution to the signal and the surface property of steel wire had less contribution to the signal, so it was difficult to evaluate the surface corrosion of the steel wire using the excitation frequency below 10 kHz. If the excitation frequency was too high, such as 200 kHz, the stability of the system became worse. According to our experiment, the optimal excitation frequency was about 80 kHz. The eddy current penetration depth was about 0.07 mm at 80 kHz. Therefore, an 80 kHz excitation frequency was used in our evaluation system.

Figure 9 shows the scanning results of the galvanized steel wires with different corrosion levels. The differences of the slopes were clear for the samples with different corrosion levels.

Fig. 9
figure 9

Scanning results of the galvanized steel wires with different corrosion levels. The X–Y graph was plotted using the X, Y output signals of the lock-in amplifiers

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Then we put each corroded steel wire in a bunch of uncorroded steel wires and did the measurements. Figure 10 shows the experimental layouts of the steel wires when the samples of (a), (b), (c,) and (d) were put in the first layer of a bunch of steel wires. The samples were covered by a 10 mm thick polyethylene arc shape covering.

Fig. 10
figure 10

Experimental layouts of the steel wires when the samples of (a), (b), (c), and (d) were put in the first layer

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Figure 11 shows the scanning results. The signals were different for the samples with different corrosion levels.

Fig. 11
figure 11

Scanning results of the steel wires with the samples of (a), (b), (c), and (d) put in the first layer

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Figure 12 shows the experimental layouts of the steel wires when the samples of (a), (b), (b), and (d) were put in the second layer of a bunch of steel wires. The samples were covered by a 10 mm thick polyethylene arc shape covering.

Fig. 12
figure 12

Experimental layouts of the steel wires when the samples of (a), (b), (c), and (d) were put in the second layer

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Figure 13 shows the scanning results. The differences in the signals for the samples with and without corrosion were clear, but the signal differences for different corrosion levels were not clear.

Fig. 13
figure 13

Scanning results of the steel wires with the samples of (a), (b), (c), and (d) put in the second layer

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Figure 14 shows the experimental layouts of the steel wires when the samples of (a), (b), (c), and (d) were put in the third layer. The samples were covered by a 10 mm thick polyethylene arc shape covering.

Fig. 14
figure 14

Experimental layouts of the steel wires when the samples of (a), (b), (c), and (d) were put in the third layer

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Figure 15 shows the scanning results. It was difficult to evaluate the corrosion levels from the signals for the steel wires in the third layer.

Fig. 15
figure 15

Scanning results of the steel wires with the samples of (a), (b), (c), and (d) put in the third layer

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5 Conclusions

We developed a compact electromagnetic system to evaluate the corrosion of galvanized steel wires. Using the frequency of about 80 kHz and the X, Y output signals of the lock-in amplifier, the corrosion of the steel wire can be evaluated using the slope of the X–Y graph. Some experiments were done in the laboratory. When the corroded galvanized steel wire was in the first layer or second layer of a bunch of steel wires, it was possible to evaluate the corrosion using the electromagnetic method. We will use this system to evaluate the corrosion of steel cables of real suspension bridges in the future.