Kirchhoff's Voltage Law:

Kirchhoff's circuit laws are the two equalities which deal by the conservation of charge and energy in the electrical circuits, and were first introduced in the year 1845 via Gustav Kirchhoff and they are broadly employed in electrical engineering.

They are as well termed as Kirchhoff's rules or simply Kirchhoff's laws. Both the circuit rules can be directly derived from the Maxwell's equations.

Kirchhoff's Voltage Law might be stated therefore: The sum of all the voltages around a loop is equivalent to zero.

v₁ + v₂ + v₃ - v₄ = 0

This law is as well termed as Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule and the principle of conservation of energy means that the directed sum of the electrical potential differences (or voltage) around any closed circuit is zero. 

Or more simply; the sum of the emf in any closed loop is equal to the sum of the potential drops in that loop.

Or the arithmetical sum of the products of the resistances of the conductors and the currents in them in a closed loop is equivalent to the net emf available in that loop. 

Likewise to Kirchhoff's Current Law, it can be defined as:

_k=1Σⁿ V_k = 0

Here, 'n' is the total number of voltages measured. The voltages might as well be complex.

The law is mainly based on the conservation of energy given or taken by potential field (not comprising energy taken through dissipation). Given a voltage potential, a charge that consists completed a closed loop does not lose or gain energy as it has gone back to the initial potential level.

You will find out that this law holds true even if the resistance (that causes the dissipation of energy) is present in a circuit. The validity of this law in this situation can be understood when one realizes that a charge actually does not go back to its beginning point, due to the dissipation of energy. A charge will merely terminate at the negative terminal, rather than positive terminal. This signifies all the energy given via the potential difference has been completely consumed by resistance that in turn loses the energy as heat dissipation.

We summarize: Kirchhoff's voltage law consists of nothing to do with gain or loss of energy through electronic components (that is, resistors, capacitors and so on). This is a law referring to the potential field produced through voltage sources. In this potential field, in spite of what electronic components are present, the gain or loss in energy provided by the potential field should be zero if a charge completes a closed loop.

Electric field and electric potential:

We can suppose that the Kirchhoff's voltage law as an effect of the principle of conservation of energy. Or else, it would be possible to make a perpetual motion machine which passed a current in a circle around the circuit.

Assuming that the electric potential is stated as a line integral over an electric field, Kirchhoff's voltage law can be deduced equally as:

E.dI = 0

Limitations:

In the presence of a changing magnetic field, the electric field is not conservative and it can't thus state a pure scalar potential-the line integral of the electric field around the circuit is not zero. This is due to the reason as energy is being transferred from the magnetic field to the current (or vice-versa). In order to 'fix' Kirchhoff's voltage law for circuits having inductors, an effective potential drop, or electromotive force (emf), is related by each and every inductance of the circuit, exactly equivalent to the amount through which the line integral of the electric field is not zero by Faraday's law of induction.

Kirchhoff's Current Law:

The current entering any junction is equivalent to the current leaving that junction.

i₁ + i₄ = i₂ + i₃

This law is as well termed as Kirchhoff's point rule, Kirchhoff's junction rule (or nodal rule) and as well Kirchhoff's first rule.

The principle of conservation of electric charge means that: At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equivalent to the sum of currents flowing out of that node.  

Or the arithmetical sum of currents in a network of conductors meeting at a point is zero. (Supposing that current entering the junction is taken as positive and current leaving the junction is taken as negative). 

If you will remind that current is a signed (negative or positive) quantity reflecting direction towards or away from a node, then it will be observed that this principle can be defined as:

_k=1Σⁿ I_k = 0

Here, 'n' is the total number of branches having currents flowing towards or away from the node.

The law is mainly based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amperes) and the time (that is measured in seconds).

Changing charge density:

The limit concerning capacitor plates signifies that Kirchhoff's current law is just valid if the charge density remains constant in the point that it is applied to. This is generally not a problem because of the strength of the electrostatic forces: the charge build-up would cause repulsive forces to scatter the charges.

Though, a charge build-up can take place in a capacitor, where the charge is usually spread over broad parallel plates, having a physical break in the circuit which prevents the positive and negative charge accumulations over the two plates from coming altogether and cancelling. In this situation, the sum of the currents flowing into one plate of the capacitor is not zero, however instead equivalent to the rate of charge accumulation. Though, when the displacement current dD/dt is comprised, Kirchhoff's current law once again holds. (This is in actuality only needed when one wants to apply the current law to a point on a capacitor plate. In circuit analyses, though, the capacitor as a whole is in general treated as a unit, in which case the ordinary current law holds since precisely the current that enters the capacitor on the one side leaves it on the other side.)

More precisely, Kirchhoff's current law can be found by taking the divergence of Ampere's law by Maxwell's correction and combining by Gauss's law, resulting:

∇.J = -∇.∂D/∂t = - ∂ρ/∂t

This is simply the charge conservation equation (that is, in integral form, it states that the current flowing out of a closed surface is equivalent to the rate of loss of charge in the enclosed volume (that is, Divergence theorem). Kirchhoff's current law is equal to the statement that the divergence of the current is zero, true for time-invariant ρ, or for all time true when the displacement current is comprised by J.

Maximum Power Transfer Theorem:

Maximum power transfer theorem defines that, to get maximum external power from a source having a finite internal resistance, the resistance of the load should be equivalent to the resistance of the source as observing from the output terminals. Moritz von Jacobi introduced the maximum power (or transfer) theorem in the year 1840 that is as well termed to as 'Jacobi's law'. 

It should be noted that this theorem outcomes in maximum power transfer, and not maximum efficiency. When the resistance of the load is made bigger than the resistance of the source, then efficiency is higher, as a higher percentage of the source power is transferred to the load, however the magnitude of the load power is lower as the net circuit resistance goes up.

Whenever you make the load resistance smaller than the source resistance, then most of the power ends up being dissipated in the source, and however the net power dissipated is higher, due to a lower net resistance, it turns out that the amount dissipated in the load is decreased.  

You can expand the theorem to AC circuits that comprise reactance and state that maximum power transfer takes place if the load impedance is equivalent to the complex conjugate of the source impedance.

Maximizing power transfer versus power efficiency:

This theorem was initially misunderstood (notably through Joule) to imply that a system comprising of an electric motor driven through a battery couldn't be more than 50% efficient as, if the impedances were matched, the power lost as heat in the battery would for all time be equivalent to the power delivered to the motor. In the year 1880 this supposition was revealed to be false by either Edison or his colleague Francis Robbins Upton, who recognized that maximum efficiency was not similar as maximum power transfer. To accomplish the maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. By employing this new understanding, they acquired an efficiency of around 90%, and proved that the electric motor was a practical option to the heat engine.

The condition of maximum power transfer doesn't outcome in maximum efficiency. When we state the efficiency 'η' as the ratio of power dissipated through the load to power developed by the source, then it is straightforward to compute:

η = R_load/(R_load + R_source) = 1/[1 + (R_source/R_load)]

Impedance matching in reactive circuits:

The theorem as well applies where the source and/or load are not totally resistive. This raises a refinement of the maximum power theorem that states that any reactive components of source and load must be of equivalent magnitude however opposite phase. This signifies that the source and load impedances must be complex conjugates of one other. In case of purely resistive circuits, the two concepts are similar. Though, physically realizable sources and loads are not generally completely resistive; having a few inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, however exist.

If the source is completely inductive (capacitive), then a fully capacitive (inductive) load, in the absence of resistive losses, would get 100% of the energy from the source however send it back after a quarter cycle. The resultant circuit is nothing other than a resonant LC circuit in which the energy carries on to oscillate to and fro. This is termed as reactive power.

The power factor correction (that is, where an inductive reactance is employed to balance out a capacitive one), is necessarily the similar idea as complex conjugate impedance matching however it is done for completely different reasons.

For a fixed reactive source, the maximum power theorem maximizes the real power (P) delivered to the load via complex conjugate matching the load to the source.

For a fixed reactive load, the power factor correction minimizes the apparent power (S) (and redundant current) conducted through the transmission lines, as maintaining the similar amount of real power transfer. This is completed by adding a reactance to the load to balance out the load's own reactance, modifying the reactive load impedance to the resistive load impedance.

Miller Theorem:

Miller theorem signifies to the method of making equivalent circuits. The general circuit theorem states that a floating impedance element supplied by two connected in series voltage sources might be divided into two grounded elements having corresponding impedances. There is as well a dual Miller theorem with regards to the impedance supplied through two connected in parallel current sources. The two versions are mainly based on the two Kirchhoff's circuit laws.

Miller theorems are not merely pure mathematical expressions. These arrangements describe significant circuit phenomena about modifying impedance (that is, Miller effect, virtual ground, bootstrapping, negative impedance and so on) and assist designing and understanding different popular circuits (that is, feedback amplifiers, resistive and time-dependent converters, negative impedance converters and so on). They are helpful in the area of circuit analysis particularly for analyzing circuits having feedback and certain transistor amplifiers at high frequencies.

There is a close relationship between the Miller theorem and Miller effect: the theorem might be considered as a generalization of the effect and the effect might be thought as a special case of the theorem.

Miller theorem (for voltages):

Miller theorem sets up that in a linear circuit, if there exists a branch having impedance 'Z', joining two nodes having nodal voltages V₁ and V₂, we can substitute this branch through two branches joining the corresponding nodes to ground by impedances correspondingly Z/(1  - K) and KZ/(K  -  1), here K = V₂/V₁. Miller theorem might be confirmed by employing the equivalent two-port network method to substitute the two-port to its equivalent and by applying the source absorption theorem. This version of Miller theorem is mainly based on the Kirchhoff's voltage law; for that reason, it is named as well Miller theorem for voltages.

Miller theorem means that an impedance element is supplied through two arbitrary (not essentially dependent) voltage sources that are joined in series via the common ground. In practice, one of them acts as a main (or independent) voltage source having voltage V₁ and the other - as an additional (that is, linearly dependent) voltage source having voltage V₂ = KV₁. The idea of Miller theorem (modifying circuit impedances seen from the sides of the input and output sources) is revealed beneath through comparing the two conditions - without and with joined an additional voltage source V₂.

When V₂ was zero (that is, there was not a second voltage source or the right end of the element having impedance Z was just grounded), the input current flowing via the element would be find out, according to the Ohm's law, merely through V₁:

I_ino = V₁/Z

and the input impedance of the circuit would be:

Z_ino = V₁/I_ino = Z

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