Basically there are few different things here that appear lumped together in your mind and which may be better seen separately.

First you have the classical methods for solving circuits, such as the mesh analyses, node-voltage analyses, full equation system, reduced equation system and maybe some more I forgot.

First, the full equation system has no excuse to be used, so don't use it. The reduced equation system should only be used for very simple circuits where the preparation to use one of the two other systems would take more work than solving the circuit using reduced system. After some practice, you should be able to determine how much work you'll need to solve the circuit using each system (so try using all 3 systems on some problems, just so you can compare how much work each requires).

Next, you should have good theoretical knowledge of how each system works. The circuit may be such that some system cannot be easily used with it. You should also note the special cases where the mesh analyses and nodal analyses are especially useful. For example, in mesh analyses, if you have lots of constant current sources, you get lots of trivial equations for mesh currents. Same thing with nodal analyses and voltage sources. So basically, you should calculate how many equations you'll get with each and pick one with least equations, but do take into account special cases of current source and voltage sources when calculating the number of equations!

After that, you have the theorems which you use. I'd say that you should first see if any theorems can be applied to a circuit and only after applying them try to solve it using the systems. For example, you should use Norton/Thevenin when you have a big circuit where one for few elements change. For example, you have a circuit with a rheostat and you need to calculate say current through it on various settings. In this case, just replace the rest of the circuit with Thevenin's generator, since it doesn't change. In case of superposition theorem, it's useful in cases where you have sources that turn on and off and you should calculate their effects on some part of the circuit. Same thing goes for other theorems, like bisection (where you have symmetrical circuit, so you only need to solve one half of it) or compensation (which is useful when you have sources whose values change).

So for theorems, the general idea is to find use cases where each of them will actually allow the circuit to be simplified. So when you have a problem, ask yourself not "How am I going to solve this using method X?" but "Why am I going to solve this using method X?". This should work even on problems in textbooks where they are divided by areas. So as I said before, try solving one problem using several different methods. See which ones can be applied to the problem, which ones can't be applied to the problem, ask yourself why for each and then take a look and see which method is the most optimal (in sense that the least number of equations needs to be solved or that you get a significant number of simple equations) and try to understand why the most optimal method is the most optimal method. This way, you'll see when it's counterproductive to apply some theorem, when you gain nothing by applying some theorem and when applying a theorem actually helps. Same story goes for reduced system, nodal and mesh analyses too.

That real exponential term represents the transient response of the system. Generally, when doing steady-state sinusoidal analysis, you can simply ignore the transient response altogether, since the real part of the exponent is usually a negative value times *t* (time), which goes to zero as *t* → ∞. If not, it means the system is unstable to begin with.

## Best Answer

Thevenin's theorem is extremely important in practice. Every time an engineer talks about input or output impedance (or resistance, if you are concerned only in DC) it is applying that theorem implicitly.

Moreover, the theorem is applied whenever an engineer tries to estimate the input or output impedance of a complex circuit, where many components are connected together. Usually the application of the theorem is done visually, without doing explicit calculations if the circuit is relatively simple. But in some cases some back-of-the-envelope calculations may be needed.

Think of this common example: any time you speak about the internal resistance of a battery cell you are implicitly applying Thevenin's theorem. There is no actual resistor in a cell, besides the tiny resistance of the connection wires. The internal resistance is Thevenin's way to model the resistance of the connection wires

together with all the resistive phenomena that take place in the chemistry of the battery.For the same reasons its dual twin, Norton's theorem, is used (a bit less commonly).

And just to tell you something that is seldom written in basic circuit analysis books, although those theorems are strictly valid only if the circuit being substituted is linear, they are indeed applied also to non-linear circuits, if appropriate conditions are met.

Take as an example the output of a logic gate. You will see that sometimes, when discussing fan-out issues or when analog stuff is connected to that output, we talk about the gate output impedance, which in itself should be taboo since gates are strongly non-linear devices! But if you consider the two separate situations of when the output is high and when the output is low, you can talk about the output impedance, which of course could be different in the two situations!

What an engineer seldom uses in practice, especially nowadays in the era of SPICE simulators, are all those methods for calculating all the electrical quantities in a circuit, like nodal or mesh analysis and the like. Although an engineer should know about them for theoretical reasons, spending too much time in becoming the master of nodal analysis resolution is not time well spent. Much better to learn how to apply Thevenin's and Norton theorems (and a couple of other tricks) at a glance even in complicate circuits.

If you really want to understand how a real design engineer thinks, try the excellent book of Horowitz and Hill: The Art of Electronics. A bit on the expensive side, but it's a 1200 page monster covering most aspects of electronic design and it is written in a very easy-to-read style.