Basis of Kirchhoffs Laws

Hi everybuddy out there!

I have a topic for discussion and I would like you to submit your comments on it which is
" Did you ever think about the basis of Kirchhoff's Laws?. Can you think of any situation where you can't use these laws?.Try to give a mathematical argument in support of your points"
Thanks!

 

as most of us are aware kirchoff's laws consists of two parts:-
1. current law
2. voltage law
→the current law is based on the law of conservation if charge and that in a given interval of time charges entering a source is equal to the charges exiting it.
→the voltage law however is based on the principle of conservation of energy.
the sum of all potential differences in a closed loop is zero provided the field is conservative, as in the case of normal resistive circuits.
however in circuits having induced electric fields(which are non-conservative in nature) the sume of all potential differences is non-zero.
→mathematically,
for static charges
-E dr = dV
ie -∫Edr = V

if integral "over the loop" (often denoted by ∫sign with a loop in the centre)
of -Edr=0, then the field is conservative.
≠ 0, then the field is non conservative. in case of induced fileds this value is equal to the value of the induced potential difference(emf)

 

In addition to what has been said above, Kirchoff's First Law (Kirchoff's Current Law (KCL)) is also characteristic of the Conservation of Energy. Haditya, I wasn't sure if you classed Law of Conservation as the same thing as Conservation of Energy.

By definition, the Conservation of Energy states that:

ΔEin = ΔEstored + ΔEout

This has to be absolutely statisfied. We simply know this definition as energy cannot be created or destroyed, it can only be converted from one form to another.

Consideration of current (which is a form of energy) at a node in a circuit, shows that what goes into the node, must come out of it. No current can be destroyed at the node and no current can be created - this would violate the absolute definition of the Conservation of Energy.

Therefore at any node:

Iin = Iout

or

Iin - Iout = 0

Which gives rise to the full definition of KCL:

The algebraic sum of the currents meeting at a junction or node is zero

KCL is true regardless of the complexity or scale of the system analysed.

Kirchoff's Second Law (Kirchoff's Voltage Law (KVL)) is slightly more complicated in a strict mathemtical context because of the nature of components in an electrical circuit.

By definition KVL states:

The algebraic sum of the voltage drops and emfs around any closed electrical circuit is zero.

Mathematically stated: ∑ E - IR = 0

As haditya has correctly stated above, this is very simple for basic electrical circuits comprising of resistors which behave in a passive, linear fashion. The whole constituent of the ΔEstored expression in the Conservation of Energy is attributed to the heat loss in the resistor and such circuits behave in phase - all voltage drops are proportional to the applied emf and current through the resistors. This behaviour is true and predictable for DC and AC networks.

Sadly such simplicity is not applicable to other passive, linear components such as inductors and capactors because of two factors. Firstly, inductors and capacitors have the property of energy storage, in addition to the small energy loss as a result of the "resistor" in such circuits. Remember all real (particularly)inductive and capacitive components have some form of 'resistance', hence the ΔEstored component is slightly more involved. Secondly, we know that voltage across inductors and capacitors are not in phase with the applied emf, therefore the relative voltage across either of the devices does not necessarily have any direct proportion to the applied emf (thinkl about the calculation for the hypothenuse on a right angle triangle) . We know that these two ideas are connected and number one can be used to explain number two.

Also consideration must be given to DC and AC networks. We know that for DC, an inductor becomes a short circuit (with ideally no voltage drop across it) and a capacitor becomes an open circuit. From this we can see that KVL is not applicable to inductors and capacitors under DC conditions. Under AC conditions, we consider reactance (like the 'effective resistance' for inductors and capacitors) and can calculate the voltage dropped across these components from Ohm's Law. Reactances and voltage drops are phasors because of the phase angle shift due to the nature of inductors and capacitors. I we analse the circuit as a phasor diagram and can assume that the circuit is conservative from the expression for KVL above.

If there are any queries or factual errors in my post please feel free to post back and we'll try to clarify them, afterall I was posting this in the early hours of the morning!

 

im doing BCS, my name is Muhammad ALim Khan.
watch the attachment...!

 

hi Alim,

Where is the attachment?.I can't see here.Are you doing BCS from VU?.If you do than it is gud coz me 2

 

{
>The algebraic sum of the voltage drops and emfs around any closed electrical circuit >is zero.

>Mathematically stated: ∑ E - IR = 0


>Secondly, we know that voltage across inductors and capacitors are not in phase with >the applied emf, therefore the relative voltage across either of the devices does not >necessarily have any direct proportion to the applied emf (thinkl about the calculation >for the hypothenuse on a right angle triangle) . We know that these two ideas are >connected and number one can be used to explain number two.}

My comment:

If we really follow the definition 'voltage' (or 'potential difference'), i.e. via the line integral of E-field, we should find that V = 0 for an inductor, right? Although most of the texbooks use the term 'voltage' for inductors, we should really understand what we mean here is the emf associated with the inductor through Gauss' and Lenz's laws, rather than if there is any true 'voltage' there!