Application of Kirchhoff's laws in circuits with inductors

Question

As we know, the Kirchhoff circuit laws are applicable for conservative electric fields. Now it is applicable for circuits where inductors are present but the field there is not conservative.

So how are the Kirchhoff's laws applicable there?

Answer 1

Typically, with inductors, we use the complex impedance, $Z=i\omega L$ with current frequency $\omega$ and inductance $L$, for the voltage: $$ V_{ind}=IZ,\quad V_{rms}=I_{rms}|Z| $$ where the left equation is the voltage of the inductor and the right equation the root-mean-square.

Surely, however, what goes on inside the inductor doesn't matter, it only is only the voltage drop across the inductor that actually matters with Kirchoff's voltage law. Kirchoff's current law, on the other hand, states that the current across a complete circuit is zero because otherwise a charge builds up, so again the non-conservative field is irrelevant.

Professor Lewin (MIT) explains that if we have a magnetic potential driving a circuit, rather than a constant voltage source, the Kirchoff voltage law does not come up with the same answer. This is based on a different definition of Kirchoff's law than what is normally taught. What Lewin means, of course, is that $$ \oint\mathbf E\cdot d\mathbf l=0\tag{1} $$ He writes this on the board integrating from points $D$ to $A$ on a circuit board going to the left & going to the right, the two directions are not equal. However, most textbooks use the formulation,

sum of voltage drops around a loop is equal to the sum of emf around a loop

which is more equivalent to $$ \oint\mathbf E\cdot d\mathbf l=-\frac{d\phi}{dt}\tag{2} $$ which is really Faraday's law that always works.

With definition (1) & using Professor Lewin's example, we have that the net voltage is 1V (as he described) being driven by an emf of 1 V; all is well with Kirchoff's laws indeed.

Answer 2

Kirchhoff's laws follow from the conservation of charge and energy in a circuit. Neither is violated in non-conservative fields. A circuit description of such a field would merely act like an active source in a circuit, no different from a battery, which also supplies net energy. For the purposes of circuit analysis the source of that net energy is usually irrelevant.

Answer 3

According to Stoke's theorem, $$\oint_C\vec{E}\cdot d\vec{l}=-\int_S\partial_t\vec{B}\cdot d\vec{S}=-\frac{d\Phi}{dt}.$$ If a circuit is of laboratory size and $\partial_t \vec{B}$ not too large, then the integral of $\vec{E}$ around the circuit $C$ is approximately zero. This is Kirchhoff's circuit law (KVL). Kirchhoff's current law (KCL) is simpler: it is merely charge conservation applied at a node and then time differentiated.

Answer 4

KVL is essentially a statement of energy conservation, while KCL follows from conservation of charge. KVL, as the name implies, depends on a definition of voltage (scalar potential). This definition is only possible when the electric fields are conservative such that $\nabla \times E = 0$, so that we can define $E = -\nabla V$.

In an inductor, this is clearly not the case because we have time-varying magnetic fields (e.g., even for a DC circuit there's a transient during turn-on). In general, for such situations we must turn to Maxwell's equations for a solution.

However, since most of the circuit does not violate this condition, we can treat the remainder of the circuit using KVL, and remedy the inductor by introducing the notion of a back EMF. The back EMF comes from an independent application of Maxwell's equations to the inductor.

There are situations where non-conservative fields cannot be treated in a lumped fashion. This includes for example RF transmission lines in a circuit. In such situations, KVL breaks down and we have to resort to Maxwell's equations for solutions.