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## Main Question or Discussion Point

So as a learning project I am trying to solve a transformer with losses but I am stuck.

1 coil L1 with N1 turns is excited by a sinusoidal voltage V1 with a resistor R1 in series. Wound around a common core with a second coil L2 with N2 turns and a resistor R2 also in series.

$$V_1+R_1I_1+L_1\frac{dI_1}{dt}-M\frac{dI_2}{dt}=0\\

R_2I_2+L_2\frac{dI_2}{dt}-M\frac{dI_1}{dt}=0\\

M\frac{dI_1}{dt} = N_2\frac{d\phi_{12}}{dt}\\

M\frac{dI_2}{dt}=N_1\frac{d\phi_{21}}{dt}\\

V_1+R_1I_1+N_1\frac{d\phi_{11}}{dt}-N_2\frac{d\phi_{21}}{dt}=0\\

N_2\frac{d\phi_{22}}{dt}+R_2I_2-N_2\frac{d\phi_{12}}{dt}=0

$$

So I have the last 2 equations with 8 unknowns (phi self and mutual and I1 I2 and their derivatives)

Now :

$$\phi_{11} = \phi_{12}\\

\phi_{21}=\phi_{22}$$

I think this has to be the case, because all of the flux from the first coil that hits the second coil must also go through the first coil, since all the flux is confined to the core. So I will call them $$\phi_1\textrm{ and }\phi_2$$

$$

V_1+R_1I_1+N_1\frac{d\phi_{1}}{dt}-N_2\frac{d\phi_{2}}{dt}=0\\

N_2\frac{d\phi_{2}}{dt}+R_2I_2-N_2\frac{d\phi_{1}}{dt}=0

$$

4 unknowns. Now I set up a magnetic circuit

$$\textrm{Median length of core }= l\\

\textrm{Permeability of core }= \mu\\

\textrm{cross section of core }= A_x\\

\textrm{Reluctance of core }(R_c)=\frac{l}{\mu A_x}\\$$

$$N_1I_1=R_c\phi_1\\

N_2I_2=R_c\phi_2\\

\phi_1=\frac{N_1I_1}{R_c}\\

\phi_2=\frac{N_2I_2}{R_c}$$

I think I can just break the above 2 equations for the magnetic flux into 2 by superposition... Which gives 2 equations 4 unknowns:

$$

V_1+R_1I_1+N_1\frac{d}{dt}\frac{N_1I_1}{R_c}-N_2\frac{d}{dt}\frac{N_2I_2}{R_c}=0\\

N_2\frac{d}{dt}\frac{N_2I_2}{R_c}+R_2I_2-N_2\frac{d}{dt}\frac{N_1I_1}{R_c}=0

$$

or

$$

V_1+R_1I_1+\frac{N_1^2}{R_c}\frac{dI_1}{dt}-\frac{N_2^2}{R_c}\frac{dI_2}{dt}=0\\

\frac{N_2^2}{R_c}\frac{dI_2}{dt}+R_2I_2-\frac{N_2N_1}{R_c}\frac{dI_1}{dt}=0

$$

I think I am missing a step... because I think I1, I2 are dependent on each other somehow. What am I missing. Since this is not an ideal transformer, I don't think N1I1=N2I2.. so can someone tell me how I can relate them or what I did wrong please?

1 coil L1 with N1 turns is excited by a sinusoidal voltage V1 with a resistor R1 in series. Wound around a common core with a second coil L2 with N2 turns and a resistor R2 also in series.

$$V_1+R_1I_1+L_1\frac{dI_1}{dt}-M\frac{dI_2}{dt}=0\\

R_2I_2+L_2\frac{dI_2}{dt}-M\frac{dI_1}{dt}=0\\

M\frac{dI_1}{dt} = N_2\frac{d\phi_{12}}{dt}\\

M\frac{dI_2}{dt}=N_1\frac{d\phi_{21}}{dt}\\

V_1+R_1I_1+N_1\frac{d\phi_{11}}{dt}-N_2\frac{d\phi_{21}}{dt}=0\\

N_2\frac{d\phi_{22}}{dt}+R_2I_2-N_2\frac{d\phi_{12}}{dt}=0

$$

So I have the last 2 equations with 8 unknowns (phi self and mutual and I1 I2 and their derivatives)

Now :

$$\phi_{11} = \phi_{12}\\

\phi_{21}=\phi_{22}$$

I think this has to be the case, because all of the flux from the first coil that hits the second coil must also go through the first coil, since all the flux is confined to the core. So I will call them $$\phi_1\textrm{ and }\phi_2$$

$$

V_1+R_1I_1+N_1\frac{d\phi_{1}}{dt}-N_2\frac{d\phi_{2}}{dt}=0\\

N_2\frac{d\phi_{2}}{dt}+R_2I_2-N_2\frac{d\phi_{1}}{dt}=0

$$

4 unknowns. Now I set up a magnetic circuit

$$\textrm{Median length of core }= l\\

\textrm{Permeability of core }= \mu\\

\textrm{cross section of core }= A_x\\

\textrm{Reluctance of core }(R_c)=\frac{l}{\mu A_x}\\$$

$$N_1I_1=R_c\phi_1\\

N_2I_2=R_c\phi_2\\

\phi_1=\frac{N_1I_1}{R_c}\\

\phi_2=\frac{N_2I_2}{R_c}$$

I think I can just break the above 2 equations for the magnetic flux into 2 by superposition... Which gives 2 equations 4 unknowns:

$$

V_1+R_1I_1+N_1\frac{d}{dt}\frac{N_1I_1}{R_c}-N_2\frac{d}{dt}\frac{N_2I_2}{R_c}=0\\

N_2\frac{d}{dt}\frac{N_2I_2}{R_c}+R_2I_2-N_2\frac{d}{dt}\frac{N_1I_1}{R_c}=0

$$

or

$$

V_1+R_1I_1+\frac{N_1^2}{R_c}\frac{dI_1}{dt}-\frac{N_2^2}{R_c}\frac{dI_2}{dt}=0\\

\frac{N_2^2}{R_c}\frac{dI_2}{dt}+R_2I_2-\frac{N_2N_1}{R_c}\frac{dI_1}{dt}=0

$$

I think I am missing a step... because I think I1, I2 are dependent on each other somehow. What am I missing. Since this is not an ideal transformer, I don't think N1I1=N2I2.. so can someone tell me how I can relate them or what I did wrong please?