$$ V_i + \sum_{k=0}^n V_k = 0 $$
Which is Kirchhoff's voltage law as seen in circuit theory.
This is not the usual form of KVL that we use in circuit theory. For example,
- K. S. Suresh Kumar in Electric Circuits and Networks states the law as "The algebraic sum of voltages in any closed path in a circuit is zero."
- Dorf and Svoboda, Introduction to Electric Circuits, gives, "The algebraic sum of the voltages around any closed loop in a circuit is identically zero for all time."
- C. L. Wadhwa, Network Analysis, has "The sum of voltage rises and drops in a closed loop at any instant of time are equal"
These are all essentially just the $\sum_k V_k$ term from your statement of the law.
By including the $V_i$ term to account for changing magnetic flux encircled by the circuit, you have indeed formed a version of the law that accounts for changing magnetic flux encircled by the circuit. But that doesn't mean that the most common forms of KVL account for it.
The definition of KVL given above is necessary to analyze circuits including such as elements as transformers
This isn't really true. Usually we simply model transformers or inductors as devices that directly affect only the voltages between the nodes they're directly connected to. For example we model an ideal inductor connected between nodes a and b with the relation
$$ V_{ab} = L\frac{{\rm d}I}{{\rm d}t}.$$
With this relation the inductor voltage is simply one of the $V_k$ in our KVL equation, and we don't need to introduce $V_i$.
What we can't model without the $V_i$ term, and don't normally model under KVL, is the effect of flux through the loops enclosed by the circuit elements and the wires that connect them.
For example, in this simple circuit
the usual KVL can't account for EMF produced by changing magnetic flux in the loop formed by the switch, cell, and bulb and the wires connecting them.
(image source here)
Yes this issue often causes confusion. The basic thing you need to think about here is the electric field. In the first instance, remove any conducting wire and just suppose there is a region of space where there is a magnetic field, uniform spatially (i.e. uniform direction and size), but changing with time. Let's say it is increasing. In this scenario there is also an electric field in that region of space. The electric field in this situation runs in circular loops around the magnetic field lines.
Ok so far so good: we have a changing magnetic field, and, in that same region of space, also an electric field running in circular loops.
Now suppose you put a conducting wire in a loop in that same region, following the direction of the electric field lines, but for the moment do not close the circuit. That is, you have a loop of wire but with a gap in it so that it does not close. What will happen?
The electrons in the wire will be pushed by the electric field, and they will move, such that they start to pile up on one side of the gap. This imbalance in the charge distribution in the wire causes a counter-balancing electric field. The electrons keep moving until this counter-balancing electric field (caused by the electrons) is equal and opposite to the one caused by the changing magnetic field. Thus when the system settles down the net electric field inside the conducting wire is zero (I am assuming the rate of change of the $B$ field here is constant).
At this point there is a build-up of negative electric charge on one side of the gap in the wire loop, and a corresponding positive charge on the other side of the gap. Also there is potential difference across that gap: it is equal to the e.m.f. which you can calculate using Faraday's law. So if you were to now connect a resistor or a light bulb or something like that across the gap, then a current would flow.
When a resistor is connected across the gap, there is a potential difference across the resistor and an electric field inside the resistor. There is no electric field inside the conducting wire (if we assume zero resistance of the wire), and the electric field just outside it is also affected by the distribution of charge in the wire.
On electric potential difference
In the above I did not mention the concept of potential difference until near the end. This is because in electromagnetism it is best to regard the fields and the charges as the main idea, and then concepts such as potential difference come in as useful tools to help in calculating and in getting insight.
Electric potential comes into its own, as a concept, in static conditions, because then we can find a function $V(x,y,z)$ such that the electric field can be written as
$$
{\bf E} = - {\bf \nabla} V.
$$
(This is a standard shorthand notation for a gradient. In terms of components it means:
$$
E_x = - \frac{\partial V}{\partial x},\\
E_y = - \frac{\partial V}{\partial y},\\
E_z = - \frac{\partial V}{\partial z}.)
$$
In non-static cases, as for example in the presence of a changing magnetic field, things are not so simple, because now the electric field is such that it can point around a loop, and this means there is no function $V$, having a single value at each location, such that $\bf E$ is its gradient. However we can still investigate quantities such as
$$
-\int_{P_1}^{P_2} {\bf E} \cdot d{\bf l}
$$
where $P_1$ and $P_2$ are two points in space (or possibly the same point) and the integral is taken along a path running between those points. In a static problem, this integral would give the potential difference between $P_2$ and $P_1$. In a non-static problem, we may choose to give another name to the result of this integral. It is often called 'electromotive force' or 'emf' (this is arguably not a very clear name but is adopted for historical reasons). But this is just a name. You could call it a potential difference if you like, as long as you realise that you are really talking about the integral of electric field along a given path. What you know is that if a charge were to move along that path, then the net energy given to the charge by the electric field is minus the amount given by this integral. This can be deduced because the force on such a charge would be
$$
{\bf f} = q ( {\bf E} + {\bf v} \times {\bf B} )
$$
and therefore the work done when the charge moves through a displacement $d{\bf r}$ is
$$
{\bf f} \cdot d{\bf r} = q {\bf E} \cdot d{\bf r}.
$$
Here the magnetic field term does not contribute because for a charge moving along a path described by $\bf r$ we have that $\bf v$ and $d{\bf r}$ are parallel so the magnetic force is perpendicular to $d{\bf r}$.
The main point of this final section of my answer is to say that 'potential difference' and 'emf' are different words for essentially the same thing, namely the integral of electric field along a path. The reason to have two terms is that the first one (potential difference) draws attention to a useful property of static fields, and the second one draws attention to the fact that the situation under consideration is not static so we have to proceed a little more carefully in our reasoning. In particular, for a non-static case we should not assume that there is any function $V(x,y,z)$ (having a single value at each point $(x,y,z)$) which gives the electric field as its gradient.
Best Answer
Strictly speaking, Kirchoff's circuit laws are not valid in AC circuits. However they are often good enough for engineering work.
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.