$2-4$ Figure $2-4$P shows an excitation system for a synchronous generator. The generator field winding is excited by a main exciter that in turn is excited by a pilot exciter. The pilot exciter, the main exciter, and the generator field winding circuit, respectively, are identified by the subscripts $1,2,$ and $\backslash ($ F $\backslash )$; the resistance, inductance, voltage, and current, respectively, are denoted by $\backslash ($ r$,$ L$,$ v $\backslash ),$ and $\backslash ($ i $\backslash )$; and the speed voltage of the pilot exciter is $\backslash ($ k$\_\{1\}$ i$\_\{\backslash$mathrm$\{$f$\}1\}\backslash )$ and that of the main exciter $\backslash ($ k$\_\{\backslash$mathrm$\{$e$\}\}$ i$\_\{\backslash$mathrm$\{$f$\}2\}\backslash ).$ Find the transfer function of the excitation system in terms of time constants and gains with $\backslash ($ v$\_\{\backslash$mathrm$\{$f$\}1\}\backslash )$ of the pilot exciter as the input and $\backslash ($ i$\_\{\backslash$mathrm$\{$F$\}\}\backslash )$ of the generator as the output.

Fig. $2-4$P A rotating excitation system.