Transient VRM Response From a Large Circular Loop Over a Conductive and Magnetically Viscous Half-Space

1Appendix¶

We wish to solve expression (39) for $R_B=1$ to obtain expression (42). This is equivalent to solving an expression of the form:

where

Changing the variable $u=t^{-3/2}$, and with some algebra, we can rewrite Eq. (1) as

Solutions to an expression of the form $x e^x = C$ are defined as branches of the Lambert W function $W[n,C]$, where $n$ are integer values Corless et al., 1996. Therefore, the solutions $u_n$ to Eq. (3) are

We can use Eqs. (41) and (2) to show $A = \frac{2}{3} t_\beta^{3/2}$. By replacing $u_n = t_n^{-3/2}$:

Real-valued solutions $W[n,x]$ only exist for $n=-1,0$ Corless et al. (1996). Additionally, for $t_n$ to occur after the primary field has been removed ($t_n \geq 0$), $W [n,x]$ requires $-1/e \leq x \leq 0$. Thus, by Eq. (5):

Recall that our choice in after-effect function (11) is only valid for $\tau_1 \ll t \ll \tau_2$. Therefore, the condition defined in expression (6) is reasonable under the assumption that $t_\beta \ll \tau_2$. We evaluated Eq. (5) for $n=0$ and noticed the solutions were $t_0 \not \ll \tau_2$. This violates our conditions for the after-effect function and is therefore not a valid solution. On the other hand, solutions of Eq. (5) for $n=-1$ did not violate conditions for the after-effect function. The solutions obtained using $W[-1,x]$ consistently showed $t_\alpha \leq t_\beta$. As a result, the time $t_\alpha$ which solves $R_B=1$ in expression (39) is given by: