As you know, impedance is defined as a complex number.
Ideal capacitors: $$ \frac {1} {j \omega C} \hspace{0.5 pc} \mathrm{or} \hspace{0.5 pc} \frac {1} {sC} $$
Ideal inductors: $$ j \omega L \hspace{0.5 pc} \mathrm{or} \hspace{0.5 pc} sL $$
I know that the reason why they 'invented' the concept of impedance is because it makes it easy to work with circuits in the frequency domain (or complex frequency domain).
However, since in real-life circuits both voltages and currents are real numbers, I'm wondering if there is any actual physical meaning behind the imaginary component of impedance.
Answer
The physical 'meaning' of the imaginary part of the impedance is that it represents the energy storage part of the circuit element.
To see this, let the sinusoidal current $i = I\cos(\omega t)$ be the current through a series RL circuit.
The voltage across the combination is
$$v = Ri + L\frac{di}{dt} = RI\cos(\omega t) - \omega LI\sin(\omega t)$$
The instantaneous power is the product of the voltage and current
$$p(t) = v \cdot i = RI^2\cos^2(\omega t) - \omega LI^2\sin(\omega t)\cos(\omega t) $$
Using the well known trigonometric formulas, the power is
$$p(t) = \frac{RI^2}{2}[1 + \cos(2\omega t)] - \frac{\omega LI^2}{2}\sin(2\omega t) $$
Note that the first term on the RHS is never less than zero - power is always delivered to the resistor.
However, the power for the second term has zero average value and alternates symmetrically positive and negative - the inductor stores energy half the time and releases the energy the other half.
But note that $\omega L$ is the imaginary part of the impedance of the series RL circuit:
$$Z = R + j\omega L$$
Indeed, via the complex power S, we see that the imaginary part of the impedance is related the reactive power Q
$$S = P + jQ = \tilde I^2Z = \frac{I^2}{2}Z = \frac{RI^2}{2} + j\frac{\omega L I^2}{2} $$
Thus, as promised, the imaginary part of the impedance is the energy storage part while the real part of the impedance is the dissipative part.
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