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I've just read a book on basic circuit theory and I'd like to understand the pros and cons of each analysis method.

Time Domain Usually ends up in differential equations for reactive circuits. Gives the transient and steady-state response.

Phasor Domain Phasor domain allows us to find the sinusodial steady state response of a AC circuit. Does not find transient response Does it also work on DC circuits? (Or is that just the Laplace transform then)

Laplace Domain Equivalent to solving differentials. Gives us transient and steady-state response.

Sorry if this a stupid question. I would just like to have it clear in my mind what each circuit tool does.

jaurunjljgrtutkwcy
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    Laplace is an easy way to solve differential equations. Turns lots of nth order differentials into algebra equations. But for t vs s domains, [this site](https://lpsa.swarthmore.edu/Transient/TransInputs/TransStep.html) reads pretty well. Study some of what others have written and try out a simple circuit in all ways about which you wonder. That's the better way to clarify. And practice. Phasors are, mostly from my experience anyway, just dealing with a single source frequency. Time domain equations can get [very complicated looking](https://electronics.stackexchange.com/a/373393/38098). – jonk Jun 01 '21 at 00:39

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Time Domain solutions with differential equations give you a complete solution for an electrical circuit starting with some initial conditions usually. Time Domain solutions with a numerical tool like SPICE or EMTP produce the same results using numerical integration - and are preferable for problems larger than the typical simple circuits of textbooks.

Phasor Domain solutions are useful if you care about a particular steady-state of a circuit driven by AC sources. A good example of this are short-circuit programs used by utility protection engineers like CAPE, ASPEN, or DigSILENT. These are phasor based programs and typically are setup to provide subtransient (first cycle or so after a fault occurs), transient (up to around 20-30 cycles), and synchronous short-circuit voltages & current phasors. The bulk of protective relays used in the utility system are phasor based or RMS based elements. Here is an SE question I answered and shows use of phasors for a directional element.

Laplace Domain, as @jonk mentions in his comment, is an easy way to solve differential circuit problems (but not too large as it gets unwieldy) with the result being closed-form solutions for quantities of interest. Here are a few SE posts I have answered that can help you in this regard: LC circuit, LC circuit with driving functions, and a neat RC circuit.

relayman357
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  • Short note: SPICE is solving algebraic-differential equations DAE. While you can tell almost everything from eigenvalues in ODE circuit formulation (if it exists), you have literally nothing in the case of DAEs. So transient analysis never ever can give you a complete description as you said because there are also unstable solutions or solution even doesn't exist (but SPICE will get you some because of some cheating). – struct Jun 04 '21 at 12:57
  • Good comment @struct, I agree. – relayman357 Jun 07 '21 at 19:05