= Coupled RL induction model
{title2=$\mathbf L\dot{\mathbf i}+\mathbf R\mathbf i=\mathbf v$}
Take two <electrical circuits> containing <coupled inductors>. Let their constant <self-inductances> be $L_1,L_2>0$, <mutual inductance> $M$, and positive <electrical resistances> $R_1,R_2$. Choose current directions so that the off-diagonal entries of the <inductance matrix> are $+M$. Assume a linear, quasistatic model, with no capacitive effects or core <hysteresis>. The <magnetic flux linkages> are $\lambda_1=L_1i_1+Mi_2$ and $\lambda_2=Mi_1+L_2i_2$. Combining their time derivatives with resistive <electric potential differences> gives
$$
L_1\dot i_1+M\dot i_2+R_1i_1=V(t),\qquad
M\dot i_1+L_2\dot i_2+R_2i_2=0.
$$
The secondary is a closed, undriven <electrical circuit>. Require $M^2<L_1L_2$, making $\mathbf L$ a <positive-definite matrix>. The chosen current convention fixes the signs; reversing the secondary's reference direction changes the sign assigned to $M$ and $i_2$ without changing the physical circuit.
There is an essential switching assumption: \b[the primary loop remains closed when $V$ becomes zero]. An ideal <voltage source> then imposes zero <electric potential difference>, allowing the current to decay. Literally opening an inductive circuit requires additional physics to determine the interruption, including possible <capacitance> and an <electric arc>. This closed-return model explains transient induction while keeping that distinction from Faraday's historical making and breaking of contact explicit.
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