Parallel transformers - checking my models

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fahraynk
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Circuit schematic: left, a voltage source Vo*cos(wt) in series with R1 and inductor L1, magnetically coupled to inductor L2 in series with R2; right, a magnetic reluctance circuit with Rc1, Rc2, Rc3 and magnetomotive sources N1I1 and N2I2.

So on the left is a circuit which shows a voltage source and 2 coupled inductors in a parallel magnetic circuit shown on the right (transformer with a parallel core).
Can someone tell me if I am modelling it correctly with the following equations?
Rc1,Rc2,Rc3 are the reluctance of each branch of the transformer core
M = mutual inductance between the coils.
φ1 and φ2 are the flux on left loop and right loop of the magnetic circuit.
φ10 and φ20 are the total flux from L1 and L2 alone
$$M=k\sqrt{L1*L2}\\k=\frac{\phi_{12}}{\phi{1_0}}=\frac{\phi_{21}}{\phi{2_0}}$$

Here are the equations I came up with :

$$
MI_2'+L_1I_1'+R_1I_1+V_0cos(wt)=0\\
MI_1'+L_2I_2'+R_2I_2=0\\
\phi_1R_{c1}+R_{c2}(\phi_1-\phi_2)=N1I1\\
\phi_2R_{c3}+R_{c2}(\phi2-\phi1)+N1I1-N2I2=0\\$$
so the total flux through path 2 (φ2)- the total flux produced by the 2nd inductor (φ20)= φ12

$$L_2I_2=N2\phi2_0\\k=\frac{\phi_{12}}{\phi1_0} \hspace{10 mm}\phi_{12}=\phi_2-L_2\frac{I_2}{N2}\\
k=\frac{(N2\phi_2-L_2I_2)N1}{N2L_1I_1}$$

Is this enough equations to solve... or am I missing an equation? If so... what equation am I missing? I think there needs to be something relating I2 to I1 without the derivatives... ?
 
Attachments
  • Circuit schematic: on the left a sinusoidal voltage source in series with resistor R1 and inductor L1; beside it, inductor L2 in parallel with resistor R2. On the right, a magnetic circuit with reluctances Rc1, Rc2 and Rc3 drawn as zigzag symbols.
    upload_2017-1-23_22-35-38.png
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Discussion

Tried to pay a tutor to answer but they said I don't have to pay because their answer might be wrong -_-
 
This man isn't serious, he got his answer, never tried to pay the tutor...
 
valentin bogatu said:
This man isn't serious, he got his answer, never tried to pay the tutor...
I tried to pay for an answer, meaning If I got an answer for the current I1, I2. But after like 48 hours I went through like 5 people and did not get the answer? I pretty much got my own work shown back to me with no current.