## 2 TFL Method  
The proposed TFL (Total Flux Leakage) method employs a solenoid similar to the EM (electromagnetic) sensor with a primary and secondary coil in order to estimate the sectional loss in the external tendon. The EM sensor was introduced to measure the residual stress in the tendon for NDE. The EM sensor consists of the primary coil to generate a magnetic field on the external tendon and the secondary coil installed at the center to detect the generated magnetic field. The governing equation of the EM sensor is well established [25, 26]. Once a current is passed through the primary coil installed at the surface of HDPE duct of the external tendon, a magnetic field is generated inside and outside of the duct as shown in Figure 1.  
If a ferromagnetic material like the tendon is placed inside the duct along the longitudinal direction of the solenoid as shown in Figure 2, and if the tendon is intact, the magnetic flux developed by the current in the primary coil will be distributed along the longitudinal direction of the tendon. From the perspective of magnetic material, the grout filling the external tendon has the same relative permeability with the air (\(\mu_{r}=1\)) and has thus no effect on the magnetic flux developed by the current in the primary coil. If, additionally, a secondary coil is installed at the center of the solenoid as shown in Figure 3a, any change in the magnetic flux can be detected by the change in the induced voltage in the secondary coil. Unlike the undamaged tendon case (Figure 3a), when the tendon suffers partial damage like a sectional loss due to the rupture of some strands, a part of the magnetic flux around the damaged area  
Figure 1: Magnetic flux distribution around electromagnetic solenoid. Figure 2: Magnetic flux of solenoid installed on surface of tendon of the prestressed concrete (PSC) structure. is absorbed by the neighboring and undamaged strands and the remaining part of the magnetic flux is leaked (Figure 3b). In such a case, the secondary coil around the damaged area detects the remaining magnetic flux that has not been leaked. The electromotive force of the secondary coil induced by the current in the primary coil of the solenoid is expressed in Equation (1). The induced electromotive force, \(V_{ind}\), is simply the product of the number \(N_{2 nd}\) of turns in the secondary coil and \(\frac{d\Phi}{dt}\), the first time derivative of the magnetic flux. Note that \(V_{ind}\) acts in the direction opposite the change of the magnetic flux. The magnetic flux \(\Phi\) is represented as the integration of the magnetic flux density, \(B\), over the total internal cross-sectional area, \(A\), of the solenoid. \[V_{ind}(t)=-N_{2 nd}\frac{d\Phi}{dt}=-N_{2 nd}\frac{d}{dt}\int_{A}\overset{ \rightarrow}{B}\cdot\overset{\rightarrow}{dA} \tag{1}\]  
where the total internal cross-sectional area of the solenoid \(A=A_{tendon}+A_{air}+A_{duct}+A_{concrete}\); \(A_{air}\) is the cross-sectional area of air in the cross-section of the solenoid; \(A_{duct}\) is the cross-sectional area of the HDPE duct in the cross-section of the solenoid; \(A_{concrete}\) is the cross-sectional area of concrete in the cross-section of the solenoid; and, \(A_{tendon}\) is the cross-sectional area of the tendon in the cross-section of the solenoid. When a time-variant AC flows in the primary coil of the solenoid, the magnetic field changes together and results in the change of the magnetic flux around the solenoid. The magnitude of the induced electromagnetic field is a function of \(A=A_{tendon}+(A_{air}+A_{duct}+A_{concrete})=A_{\mu}+A_{\mu_{0}}\). Since \(A_{air}\), \(A_{duct}\) and \(A_{concrete}\) in \(A\) can be assumed as constants \(A_{\mu_{0}}\), the induced electromotive force a function of \(A_{tendon}=A_{\mu}\). Compared to the permeability \(\mu\) of the ferromagnetic tendon, the permeability of the air, concrete, and HDPE duct is very small and practically identical to the permeability in vacuum, \(\mu_{0}\).