Step 1: Why a phase angle appears.
Inductors and capacitors store and return energy instead of dissipating it, so in an AC circuit containing them the current \(i\) leads or lags the applied voltage \(v\) by a phase angle \(\phi\). Only the resistance dissipates energy.
Step 2: Meaning of power factor.
Power factor is the number that tells us what fraction of the apparent (voltage \(\times\) current) power is genuinely turned into useful work or heat. Writing it from the impedance triangle,
\[\cos\phi = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}} = \frac{R}{Z}\]
A high power factor (close to 1) means the circuit is mostly resistive and efficient; a low power factor means most of the current is being pushed back and forth uselessly.
Step 3: Wattless current by energy view.
Resolve the current into a part in step with the voltage, \(I\cos\phi\), and a part quarter-cycle out of step, \(I\sin\phi\). Over one full cycle the out-of-step part delivers positive energy in one quarter and takes exactly that much back in the next quarter, so its net energy transfer is zero. Because it carries current but transfers no watt, \(I\sin\phi\) is named the wattless (idle) current.
Step 4: Limiting case.
In an ideal choke coil (\(R = 0\)) the phase angle is \(90^\circ\); then \(\cos\phi = 0\) and the entire current is wattless, which is why an ideal inductor controls current without heating up.
\[\boxed{\cos\phi = R/Z;\ \ I_{wattless} = I\sin\phi}\]