9.1. Transformer#
Magnetic field flux, assuming a uniform field or in terms of the average field:
\[\phi = b \, A\]Magnetic field flux linked to \(N\) windings:
\[\psi = N \, \phi\]Relationship between the voltage at the inductor terminals and the linked flux, applying Faraday’s law to irrotational parts:
\[v = \dot{\psi}\]
9.1.1. Ideal Transformer#
In the absence of stray fluxes and reluctance in the air gap, the loop law in the air gap implies:
\[0 = m_1 + m_2 = N_1 \, i_1 + N_2 \, i_2\]
The magnetic field flux can be written in terms of the flux linked to the windings:
\[\phi = \frac{\psi_1}{N_1} = \frac{\psi_2}{N_2}\]
The time derivative of this relation, with a constant number of windings over time, implies:
\[\frac{v_2}{N_2} = \frac{v_1}{N_1} \ .\]
9.1.2. Transformer with Stray Fluxes#
\[\begin{split}\begin{cases}
\phi_1 - \phi_{1,d} = \phi \\
\phi_2 - \phi_{2,d} = \phi \\
m_1 = \theta_{1,d} \phi_{1,d} \\
m_2 = \theta_{2,d} \phi_{2,d} \\
m_1 + m_2 = 0
\end{cases}\end{split}\]
\[\rightarrow \qquad 0 = m_1 + m_2 = N_1 \, i_1 + N_2 \, i_2\]
\[\begin{split}\begin{aligned}
0 & = \phi_2 - \phi_1 - \phi_{2,d} + \phi_{1,d} \\
& = \phi_2 - \phi_1 - \frac{m_2}{\theta_{2,d}} + \frac{m_1}{\theta_{1,d}} \\
\end{aligned}\end{split}\]
\[\rightarrow \qquad \frac{\psi_2}{N_2} - \frac{m_2}{\theta_{2,d}} = \frac{\psi_1}{N_1} - \frac{m_1}{\theta_{1,d}} \ .\]
\[\rightarrow \qquad \frac{1}{N_2} \left( v_2 - \frac{N_2^2}{\theta_{2,d}} \dfrac{d i_2}{d t} \right) =
\frac{1}{N_1} \left( v_1 - \frac{N_1^2}{\theta_{1,d}} \dfrac{d i_1}{d t} \right) \ .\]
9.1.3. Transformer with Stray Fluxes and Reluctance \(\theta_{Fe}\) in the Air Gap#
\[\begin{split}\begin{cases}
\phi_{1} - \phi_{1,d} = \phi \\
\phi_{2} - \phi_{2,d} = \phi \\
m_{1} = \theta_{1,d} \phi_{1,d} \\
m_{2} = \theta_{2,d} \phi_{2,d} \\
m_1 + m_2 = \theta_{Fe} \, \phi
\end{cases}\end{split}\]
todo complete and verify the calculations; draw the equivalent circuit