# One Hundred and Fifty Years of Skin Effect
## 4 Self-Inductance of a Pair of Conductors  
Self-induction of a pair of conductors is characterized by the quantity self-inductance, \(L\). \(L\) determines the voltage induced in the conductors and is part of conductor impedance. There is no universal method for the calculation of \(L\), because its value, among other things, depends on the current density in the conductors. The simple formulae given in the literature have been derived on the assumption of constant current density. This is a rather simplifying assumption because, in such a case, the induction phenomenon does not occur. In the calculation of \(L\), it is necessary to respect Formula (10), according to which the maximum distance of the conductors and the frequency of the current flowing through the conductors are limited. With a conductor distance of several metres, the frequency should be less than \(10^{7}\) Hz. In ref. [13], the values are given for the energy in the magnetic and electric fields of coaxial conductors. From a comparison of these values, it results that neglecting the displacement current in the insulation between the conductors is acceptable for supply voltage frequencies not exceeding \(10^{7}\) Hz if there is vacuum between the conductors. For a different insulation, the upper frequency limit would be lower. It is therefore necessary to assume the fulfilment of the condition \(f<10^{7}\) Hz; similar to ref. [35], p. 132, frequencies of up to 1 MHz are assumed so that quasi-stationary approximation may be assumed. Formulae for the calculation of the self-inductance of a pair of long conductors are given in the literature, sometimes also with a derivation, e.g., [33; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47]. Papers regarding the calculation of current density in a pair of long conductors often contain the method for calculating the self-inductance of a segment of the two conductors or the equivalent impedance of this segment [1; 2; 4; 14; 15; 18; 20; 22; 24; 34]. Figure 6 gives the equivalent schematic of the segment of conductors between the planes \(z=z_{1}\) and \(z=z_{2}\), where \(z_{2}>z_{1}\). Figure 6: Equivalent schematic of a segment of two conductors between the planes \(z=z_{1}\) and \(z=z_{2}\). \(U_{L}(t)\) is the induced voltage, \(U_{R}(t)\) is the voltage across the segment of the two conductors. \(V(z_{1},t)\) and \(V(z_{2},t)\) are the voltages across the conductors for \(z=z_{1}\) and \(z=z_{2}\), respectively. The inductor is characterized by the self-inductance coefficient \(L\), which is used to calculate the voltage induced on the inductor,  
\[U_{L}(t)=L\frac{\mathrm{d}I(t)}{\mathrm{d}t}, \tag{11}\]  
where \(I(t)\) is the current in the loop segment or in the circuit in Figure 6. The result of solving the basic task is the vector of current density in the loop conductors, and \(I(t)\) is the flux of this vector through the conductor cross section. Solving the basic task thus also yields \(I(t)\). If the voltage \(U(t)=V(z_{1},t)-V(z_{2},t)\) is given, then [PERSON]'s voltage law holds for the loop and for the circuit in Figure 6,  
\[U_{L}(t)=U(t)-U_{R}(t),\text{ where }U_{R}(t)=RI(t), \tag{12}\]  
with \(R\) being the given resistance of the loop segment. This means that determining \(L\) is redundant, and in the equivalent diagram (Figure 6) the inductor characterized by \(L\) is replaced by the voltage source \(U_{L}(t)\). Unlike the calculation using \(L\), the calculation of \(U_{L}(t)\) using (12) is accurate, in particular in cases when the proximity effect manifests itself or when \(U_{L}(t)\) is not sinusoidal. In the literature, the calculation of \(L\) is, in most cases, part of the calculation of the impedance of the circuit in Figure 6,  
\[Z=\Re(Z)+\mathrm{j}\omega L, \tag{13}\]  
where \(\omega\) is the angular frequency of all the voltages and currents in the circuit. The application of (13) is conditional on the steady state of the circuit and on the distance of the conductors at which the proximity effect can be neglected.