# Transient Current Density in a Pair of Long Parallel Conductors
## 3 Results and Discussion  
### Example 1  
The cross sections of the conductors forming the long loop are illustrated in Figure 1. The temperature of the conductors is 300 K, both conductors are of copper and their resistivity \(\varrho=1.725\times 10^{-8}\Omega\cdot\)m [13]. At time \(t=0\), the conductors are connected to the sinusoidal voltage source \(U(t)=\tilde{U}\sin(\omega t+\alpha)\), \(\tilde{U}=1\) V\(\cdot\)m\({}^{-1}\), the frequency \(f=\omega/(2\pi)\) and the period \(T=1/f\). This example is for \(f=1\) kHz, the same as in [2], Variant 4; however, in Figure 6 in [2], the results of the calculation of current density in conductors are only given for the steady state, i.e., for \(t\to\infty\). Figure 2 illustrates the dependences of current density on \(x\) and \(t\) for a constant \(y\) in the conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\). The calculation of transient current density provides a large amount of data that can be displayed in a similar way to Figure 2. For example, the maximum value of \(|J_{i}|\) is equal to 6.095 MA\(\cdot\)m\({}^{-2}\) in the partial conductor that is down on the left in the conductor \(\mathcal{A}_{2}\), in the interval \(t\in[0,T]\) at time \(t=3.3\times 10^{-4}\) s, for \(\alpha=0,f=1\) kHz. The current densities \(J_{i}(t),\forall i\), are non-periodic oscillating functions that with increasing \(t\) converge to periodic functions. For \(t\to\infty\), it can be assumed that for the sinusoidal voltage source \(\tilde{U}\sin(\omega t+\alpha)\) and the cosinusoidal voltage source \(\tilde{U}\cos(\omega t+\alpha)\) the current density in the \(i\)th partial conductor is  
\[J_{i}(t)=\hat{J}_{i}\sin(\omega t+\varepsilon_{i}),\forall i \tag{8}\]  
and  
\[J_{i}(t)=\hat{J}_{i}\cos(\omega t+\varepsilon_{i}),\forall i \tag{9}\]  
respectively. Substituting the assumed solutions to (8) and (9) into (1) and (4) yields the first and second equation systems, respectively. The two systems form a system of \(2N\) equations for \(N\) unknown amplitudes \(\hat{J}_{i}\) and \(N\) unknown initial phases \(\varepsilon_{i}\). Probably the simplest way of finding a solution to the two systems is to multiply the first system by the imaginary unit j and add the second system of equations. This after adjustment yields a system of \(N\) equations for the complex current densities \(\hat{J}_{i}\exp(\mathrm{i}\varepsilon_{i})\exp(\mathrm{j}\omega t),\forall i\). Solving this system gives current density phasors in the partial conductors, as stated in [1]. With increasing \(t\), the transient current converges to the steady sinusoidal current  
Figure 2: Example 1, \(\alpha=0,f=1\) kHz: The dependences of current density on \(x\) and \(t\) for a constant \(y\) in the conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\). The curve marked by the character \(o\) represents the current density at time \(t=o\times T/4\); \(j=0\) for \(t=0\). The axis \(x\) at the point marked by the arrow ends in the value \(x=16\) mm and continues with the value \(x=56\) mm. Between these values is the gap between the conductors, and the current density is zero. \[I_{\sin}(t)=\hat{I}\sin(\omega t+\beta).\]  
Figure 3 illustrates the current \(I(t)\) in three time intervals. Theoretically speaking, the steady state is achieved after an infinitely long time. In practice, the steady state is reached at time \(t\geq t_{\sin}\), where \(t_{\sin}\) is determined using the chosen value of the deviation \(\Delta_{I}\). \(t_{\sin}\) is such a least time that for \(t\geq t_{\sin}\) it holds  
\[\left|1-\left|I_{\rm Im}(t)\right|/\hat{I}\right|\leq\Delta_{I}, \tag{10}\]  
where \(I_{\rm Im}(t)\) is the value of the current at the local maximum or minimum. In Example 1 (\(\alpha=0,f=1\) kHz), it holds \(t_{\sin}=29.49\,T\) for \(\Delta_{I}=0.0032\). For \(\Delta_{I}=0.003\) in Example 1, it holds \(t_{\sin}=0.03-0.04\) s for \(f\in[50,10^{6}]\) Hz. The time \(t_{\sin}\) depends substantially on the initial phase \(\alpha\) of the voltage source at time \(t=0\).