Are there 2 correct answers to this transformer homework problem?

[alan123hk]

Below is my corrected equations$N_1I_1+N_2I_2 = \chi_1~ \phi_1 + \chi_2~ \phi_2 ~~~~~,~~~~~N_1I_1 = \chi_1~ \phi_1 + \chi_3~ \phi_3 ~~~~~,~~~~~\phi_3 =\phi_1-\phi_2$

$\phi_1 = \frac {V_1} {N_1J\omega} ~~~~,~~~~ \phi_2 = \frac { \chi_3 \left(\frac {V_1} {N_1 } \right) } {J\omega ( \chi_2+ \chi_3) + \frac {(\omega N_2)^2} {R} } ~~~~,~~~~ \phi_3 = \left[ \frac {1}{J\omega} - \frac { \chi_3 } {J\omega ( \chi_2+ \chi_3) + \frac {(\omega N_2)^2} {R} } \right]\frac {V_1 } {N_1 }$

$I_1 = \left[ \frac {\chi_1+\chi_3}{N_1J\omega} - \frac { \frac {(\chi_3)^2} {N_1} } {J\omega ( \chi_2 + \chi_3) + \frac {(\omega N_2)^2} {R} } \right]\frac {V_1 } {N_1 }~~~~,~~~~I_2= \frac { j\omega\chi_3 \left(\frac {N_2} {N_1 } \right) V_1} { J\omega ( \chi_2+ \chi_3) {R}+ (\omega N_2)^2}$where
$\chi_3 = \chi_{3c} + \chi_{3g}$
$\chi_{3c}= \text {reluctance of the magnetic core segment}$
$\chi_{3g} = \text {reluctance of the air gap segment}$

 

[alan123hk]

For the expression equation of $I_1$, after some algebraic manipulation, the following results can be obtained.

 

[Charles Link]

I originally just solved for $i_2$ in terms of $i_1$. See post 25. I now have also solved for $\phi_1$ in terms of $i_1$, and then for $i_1$ in terms of $v_1$. It took a little work, but I am now in agreement with @alan123hk 's $i_1$.

When I did this additional calculation, I made an algebraic error, that caused my $i_ 1$ result to be in error. I finally located the error. (I had incorrectly copied onto my next page an $X_1$ that was an $X_2$. It caused my final result for $i_1$ in terms of $v_1$ to contain numerous errors. ). One factor that may still differ is the sign of $v_ 1$ or $i_1$ or $v_2$ or $i_2$ relative to the other variables. The reader is free to call either direction of the windings as positive, but other than a matter of sign of this kind, we are now in complete agreement. :)

Meanwhile, good job, @alan123hk :), and also for post 31 above.

additional note, so the reader may follow more easily: For the last ten posts we have been working the problem shown in the "link" of post 23, which is closely related to the OP's original problem. Note also in looking at the result of post 25 for $i_2$ in terms of $i_1$, and also post 31 where for large $X_3$,(which makes it into an ordinary transformer), $i_2=\frac{N_2 v_1}{N_1 R}$. We do see the behavior of $i_2$ as $N_2$ decreases depends on whether $i_1$ is specified , or if $v_1$ is specified, as pointed out in post 17. See also post 24, conclusion (1) .

 

[pkc111]

Thank you so much for your replies everybody that helps a lot!