Is my resulting transformer correct?

[LoveLaika]

I worked out equations for an ideal buck-boost converter, but I think I'm not sure if I wrote it correctly. So, I wanted to ask here.

In my attached image, I have two equations: one for V_L and one for I_C. Drawing out 'rough' circuits for them yield the two halves, and seeing the relation of the dependent sources,  you can replace the set of dependent sources with an equivalent transformer (on paper). I was wondering if I did it correctly though?

The equation for V_L yields the circuit on the left using KVL. The equation for I_C yields the circuit on the right using KCL. Using the top node as the node to implement KCL (and assuming that the capacitor current is 0), the current going into the node should equal the current leaving the node. However, the signs are tripping me up.  If you look on the set of equations at the top right to describe the right circuit, since I_R is negative, then V would be negative (oops, that resistor should just be V, not V/R. My bad). Given the way the circuit is drawn, did I replace the sources correctly with the transformer's dot notation in the right locations?

Also, if I draw out an equivalent Thevenin circuit for the secondary load R, do I have to do anything special with regards to the tranformer's polarity?

[ledtester]

Also, if I draw out an equivalent Thevenin circuit for the secondary load R, do I have to do anything special with regards to the tranformer's polarity?

The two sides of your transformer do not have any reference to a common voltage so I don't think the phasing dots matter.

In LTspice, for instance, every part of your circuit needs to have a reference to ground and so you don't have this situation.

[ledtester]

you can replace the set of dependent sources with an equivalent transformer (on paper). I was wondering if I did it correctly though?

Here's a video on deriving the network equations for an ideal transformer:

[LoveLaika]

Thanks for the reply, as well as the link to the video. Sorry for not being able to reply sooner. I see that I forgot to put down a common voltage, but to be clear, the bottom node on both sides of the circuit could be considered the reference/common voltage linking the two circuits.