Magnetic circuit: What is the cross section area I need to use? I am struggling with this magnetic circuit.
Magnetic circuit of a coil with moving part
What is the equivalent cross section area I need to use to find the magnetic flux?
Should I use $A_c$ for the moving part or $A_g$? Or should I use $A_g$ for the stationary core? $A_g=A_c(1-\frac{x}{X_0})$
Which solution is correct?
\begin{align}
\phi=\frac{NI}{R_c+2R_g+R_p}=\frac{NI}{\frac{l_c}{\mu {\color{Red}{A_c}}}+2\frac{g}{\mu_0 A_g}+\frac{l_p}{\mu {\color{Red}{A_g}}}} \\\\
\phi=\frac{NI}{R_c+2R_g+R_p}=\frac{NI}{\frac{l_c}{\mu {\color{Red}{A_c}}}+2\frac{g}{\mu_0 A_g}+\frac{l_p}{\mu {\color{Red}{A_c}}}} \\\\
\phi=\frac{NI}{R_c+2R_g+R_p}=\frac{NI}{\frac{l_c}{\mu {\color{Red} {A_g}}}+2\frac{g}{\mu_0 A_g}+\frac{l_p}{\mu {\color{Red} {A_g}}}}
\end{align}
I have used FEMM to understand how flux lines behave. It is not clear.
Magnetic flux lines and flux density for two positions of the plunger
 A: The reluctance of the core has nothing to do with $A_g$, so
$$R_c = \frac{l_c}{\mu A_c}. $$
The two overlap of the core and the plunger is $A_g$, so you would write
$$R_g = \frac{g}{\mu_0 A_g}$$
The right value would probably be some average of $A_c$ and $A_g$.
Note that the reluctances of the plunger and the gap are in parallel, so you would write
$$ R_p = \left[\left(\frac{l_p}{\mu A_p}\right)^{-1}+ \left(\frac{l_p}{\mu_0 (A_c - A_g)} \right)^{-1} \right]^{-1} $$
where $A_p$ is the area of the plunger. If $\mu \gg \mu_0$ and the plunger isn't too far out, this can be simplified to
$$ R_p = \frac{l_p}{\mu A_p}. $$
The truth is the correct areas to use aren't always obvious: simple models like this are useful for quick hand calculations but don't always predict the actual results very accurately. For example, in this case the reluctance $\frac{l_p}{\mu A_p}$ of the plunger may not be accurate if the plunger is very thin: then the correct area would be some average of $A_p$ and $A_g$. Note that this analysis neglects also neglects fringe fields and uses a rather simple equation for the core reluctance, so don't expect it to match up exactly with your simulation results.
