Kirchhoff's Voltage Law for AC circuits

Question

I'm having some difficulty with understanding why KVL (Kirchhoff voltage law) is valid when considering an AC circuit. I mean, the system is by definition not in steady state, right? That in turn means that the change in current causes a change in the magnetic field, which is manifested within the electric potential. I realize the potential is still defined and thus a line integral over a crossed curve of ALL of it's component should be zero.

However, what I often see (for, say, RLC circuits with an AC emf) is total disregard to the time varying components of the electric potential. Only the emf of the different components are considered and their sum is said to be zero. How can that be? What about the "second" part of the potentials?

Answer 1

Strictly speaking, Kirchoff's circuit laws are not valid in AC circuits. However they are often good enough for engineering work.

That in turn means that the change in current causes a change in the magnetic field, which is manifested within the electric potential. I realize the potential is still defined and thus a line integral over a crossed curve of ALL of it's component should be zero.

The first thing you should have been taught about Kirchoff's laws is that they are valid for lumped circuits. This means the laws are an approximation that is valid when the circuit can be approximated by an idealized lumped circuit model.

The lumped circuit approximation requires that the circuit be small relative to the wavelength associated with the highest frequency signals to be modeled in the circuit. In engineering we usually take this to mean that the largest dimension of the circuit is less than 1/10 of the shortest important wavelength.

The lumped approximation allows us to neglect the magnetic flux through the surface enclosed by the circuit (so that KVL is valid). And it allows us to neglect charge accumulating in the wires connecting the circuit elements (so that KCL is valid).

Of course inside individual components of the circuit (inductors and capacitors) there can be significant magnetic flux or accumulated charge. But we model that by the consitutive relations for those elements. What we neglect is the magnetic flux and charge storage on the wires between the elements.

Answer 2

This is incorrect:

I realize the potential is still defined and thus a line integral over a crossed curve of ALL of it's component should be zero.

In a dynamic setting, the circulation of the electric field is not zero; instead, it obeys the Faraday induction law $$ \oint_{\partial S}\mathbf E \cdot\mathrm d\mathbf l = -\frac{\mathrm d}{\mathrm dt}\iint_S\mathbf B\cdot\mathrm d\mathbf S. $$ For normal circuits, the magnetic flux on the right-hand side can usually be calculated via the self and mutual inductances of any loops present, which can then be related to the rate of change of the current through the loop itself or to any other loops it is coupled to. If the time dependence is purely harmonic at some frequency $\omega$, this allows us to assign a given complex impedance $Z$ to those elements and to pretend that they're 'voltage drops' that can be included on the left-hand side, leaving a zero on the right.

There is no "disregard" of time-dependent aspects of the potential, though the notation sometimes skips aspects that are considered obvious if you know the dynamics well enough. Whenever you see a phasor analysis, that's likely to be what's going on.

Answer 3

I'm having some difficulty with understanding why KVL (Kirchhoff voltage law) is valid when considering an AC circuit. I mean, the system is by definition not in steady state, right? That in turn means that the change in current causes a change in the magnetic field, which is manifested within the electric potential. I realize the potential is still defined and thus a line integral over a crossed curve of ALL of it's component should be zero.

That is correct. However, it is important to realize that the line integral mentioned above is line integral of the conservative component of total electric field $\mathbf E_{C}$, not integral of total electric field $\mathbf E = \mathbf E_{C} + \mathbf E_i$, where $\mathbf E_i$ is non-conservative part of the field, often due to induced electric field. So, generally

$$ \oint_{any~loop} \mathbf E_{C} \cdot d\mathbf s = 0. $$

If charge distribution in the system is not moving fast, the field $\mathbf E_C$ can be visualized as the Coulomb integral of charge density over all space.

This is relevant because electric potential is related only to this conservative component of the electric field. It is not possible to define electric potential using total electric field $\mathbf E$, because in general, any line integral of $\mathbf E$ depends not merely on endpoints, but also on the path.

However, what I often see (for, say, RLC circuits with an AC emf) is total disregard to the time varying components of the electric potential. Only the emf of the different components are considered and their sum is said to be zero. How can that be? What about the "second" part of the potentials?

Not sure what you mean here. If the AC circuit is powered by AC source, then potential at any point will indeed oscillate in time, with same frequency.

Maybe you mean that when solving equations for simple RLC circuits, the time-dependent factor $e^{i\omega t}$ is dropped. This is done because those equations are linear and this factor is common to all terms in the equation, so it can be dropped in order to simplify the equations. Differential equations then become algebraic equations where time has no role.