I understand the Laplace transform ideas above (used to TA the class where it's taught) and I agree with your that they are more definitional than explanatory.
To provide actual explanatory power, you'd have to explain what the reactive portion ("imaginary part" in the phasor notation) of the impedance is actually doing in real terms.
What it's doing is to cause the current to lead or lag the voltage in the AC signal. In a purely resistive circuit, current and voltage vary exactly the same way at every instant.
You can actually go a long way with this just by understanding that, e.g., for an inductor,
V = L dI/dt
so that small current changes will imply very large voltage changes. If you're in to intuition, this means "voltage leads current" (i.e., the sinusoid of voltage is 90 degrees ahead of the sinusoid of current).
The above equation is also why lights in your house dim when you turn on a motor (= inductor). There is a significant current change, and the voltage changes even more, so the lights dim.
Or, you can be even more intuitive, and note that it's obvious that when you put a coil of wire (= motor = inductor) across the two sides of your house's electrical wiring, the coil of wire will pull the + and - conductors together in voltage at first. Until the magnetic field sets up to provide reactance, the coil of wire is basically a short circuit.
And finally, a capacitor is just the dual of the inductor.
"The above equation is also why lights in your house dim when you turn on a motor (= inductor). There is a significant current change, and the voltage changes even more, so the lights dim."
Uh, no. Most of the voltage drop in the source occurs in the resistance of the wiring, and you didn't even figure resistance into the equation. Intuitively, the surge current more closely matches that of a capacitor charging through resistances. It isn't even a matter of the motor inductance or the frequency of the power. In fact the early surge would be seen even if we used D.C. power and a D.C. motor. The time constant matching the light dimming involves the mass of the rotor being brought up to speed. The load is actually more resistive at startup than when an A.C. motor is up to speed and coasting.
A pure inductor starts at zero current the instant voltage is applied and it ramps up at a rate determined by the inductance, at D.C. or very low frequencies rising to where it becomes limited by the series resistance. It's pure capacitors that draw a maximum current at the instant voltage is applied to the circuit.
I believe that many people get lost in the math and lose a feel for what is happening. What you said is completely backwards from reality. Mechanical analogies are far easier for people to get a feel for than Laplace Transforms. Compare the behavior of an RLC circuit to a machanical system with a shock absorber, spring, and a mass.
People may get confused over a critically damped circuit, but the idea of a car oscillating up and down after a bump with no shock absorber is easy to grasp.
There is no way a motor is well-modeled by a capacitor, either analytically or intuitively. It's a coil of wire, practically a poster child for an inductor.
This does have lumped resistance elements, but by the time you put the two inductors and two resistors in to the differential equation with forcing
sin (2*pi*60*t) u(t),
you're not going to be in a good place as far as intuition goes.
In fact, overall, I'm not seeing much intuitive sense in your summary. Once you really delve in to it, the subject of starting electric motors (AC and DC) is very complex.
To provide actual explanatory power, you'd have to explain what the reactive portion ("imaginary part" in the phasor notation) of the impedance is actually doing in real terms.
What it's doing is to cause the current to lead or lag the voltage in the AC signal. In a purely resistive circuit, current and voltage vary exactly the same way at every instant.
You can actually go a long way with this just by understanding that, e.g., for an inductor,
V = L dI/dt
so that small current changes will imply very large voltage changes. If you're in to intuition, this means "voltage leads current" (i.e., the sinusoid of voltage is 90 degrees ahead of the sinusoid of current).
The above equation is also why lights in your house dim when you turn on a motor (= inductor). There is a significant current change, and the voltage changes even more, so the lights dim.
Or, you can be even more intuitive, and note that it's obvious that when you put a coil of wire (= motor = inductor) across the two sides of your house's electrical wiring, the coil of wire will pull the + and - conductors together in voltage at first. Until the magnetic field sets up to provide reactance, the coil of wire is basically a short circuit.
And finally, a capacitor is just the dual of the inductor.