Nodal Analysis of an electrical circuit

Question

I have several doubts about solving circuits.

1. Can any circuit be solved using Nodal Analysis?
2. If some circuit can be solved using Nodal Analysis, can it be solved using Mesh Analysis too?
3. Why do we need these techniques to solve circuits?

Answer 1

Yes, we can solve any question using nodal analysis, and it is also possible to solve that question using mesh analysis.

Answer 2

It is prudent to remain somewhat dubious about methods in solving circuits at higher frequencies. Solving simple and complex circuits take into account the "Lumped Matter Discipline" (LMD) which defines the properties of some element (e.g. resistor, capacitor, etc.) as the voltage across the terminals, $V(t)$, and current through the element $I(t)$.

Why do we have the LMD? It was proposed as a way to make our lives much simpler in that every time we need to solve a simple or complex circuit, we do not have to use Maxwell's equations, which can be quite tedious.

Lumped Matter Discipline Assumptions

• The rate of magnetic flux linked with any closed link outside an element must be zero for all time: $\frac{\partial \phi_{B}}{\partial t} = 0$
• There is no time varying charge within the element for all time: $\frac{\partial q}{\partial t} = 0$, where $q$ is the total charge within the element.
• Use LMD to operate within small signal timescales relative to the propogation delay of EM waves across lumped elements.

As engineers are beginning to find out, we are pushing #3 to its limit because we are approaching higher GHz frequencies. This, in turn, affects the other assumptions of the LMD. It is for this reason that in circuit simulation, such as LTSpice (which uses nodal analysis), you have to be very careful about the analyses of higher-frequency circuits.

It is worth noting that electronic components are by no means "ideal". A capacitor, for example, would have a marginal Equivalent Series Reistance (ESR) that affects its performance depending on the frequency taken into account. If you were to pick and choose a manufacture's capacitor to use in your intended product, simulating it would require that you include a resistor to give a more accurate representation in real-life. Furthermore, you would also have to include some inductance and even capacitance.